12246:
11001:
12241:{\displaystyle {\begin{aligned}r&:&{\frac {\partial u_{r}}{\partial t}}+u_{r}{\frac {\partial u_{r}}{\partial r}}+{\frac {u_{\phi }}{r}}{\frac {\partial u_{r}}{\partial \phi }}+u_{z}{\frac {\partial u_{r}}{\partial z}}-{\frac {u_{\phi }^{2}}{r}}&=-{\frac {1}{\rho }}{\frac {\partial P}{\partial r}}+{\frac {1}{r\rho }}{\frac {\partial \left(r\tau _{rr}\right)}{\partial r}}+{\frac {1}{r\rho }}{\frac {\partial \tau _{\phi r}}{\partial \phi }}+{\frac {1}{\rho }}{\frac {\partial \tau _{zr}}{\partial z}}-{\frac {\tau _{\phi \phi }}{r\rho }}+f_{r}\\\phi &:&{\frac {\partial u_{\phi }}{\partial t}}+u_{r}{\frac {\partial u_{\phi }}{\partial r}}+{\frac {u_{\phi }}{r}}{\frac {\partial u_{\phi }}{\partial \phi }}+u_{z}{\frac {\partial u_{\phi }}{\partial z}}+{\frac {u_{r}u_{\phi }}{r}}&=-{\frac {1}{r\rho }}{\frac {\partial P}{\partial \phi }}+{\frac {1}{r\rho }}{\frac {\partial \tau _{\phi \phi }}{\partial \phi }}+{\frac {1}{r^{2}\rho }}{\frac {\partial \left(r^{2}\tau _{r\phi }\right)}{\partial r}}+{\frac {1}{\rho }}{\frac {\partial \tau _{z\phi }}{\partial z}}+f_{\phi }\\z&:&{\frac {\partial u_{z}}{\partial t}}+u_{r}{\frac {\partial u_{z}}{\partial r}}+{\frac {u_{\phi }}{r}}{\frac {\partial u_{z}}{\partial \phi }}+u_{z}{\frac {\partial u_{z}}{\partial z}}&=-{\frac {1}{\rho }}{\frac {\partial P}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial \tau _{zz}}{\partial z}}+{\frac {1}{r\rho }}{\frac {\partial \tau _{\phi z}}{\partial \phi }}+{\frac {1}{r\rho }}{\frac {\partial \left(r\tau _{rz}\right)}{\partial r}}+f_{z}\end{aligned}}}
10909:
9996:
10904:{\displaystyle {\begin{aligned}x&:&{\frac {\partial u_{x}}{\partial t}}+u_{x}{\frac {\partial u_{x}}{\partial x}}+u_{y}{\frac {\partial u_{x}}{\partial y}}+u_{z}{\frac {\partial u_{x}}{\partial z}}&={\frac {1}{\rho }}\left({\frac {\partial \sigma _{xx}}{\partial x}}+{\frac {\partial \sigma _{yx}}{\partial y}}+{\frac {\partial \sigma _{zx}}{\partial z}}\right)+f_{x}\\y&:&{\frac {\partial u_{y}}{\partial t}}+u_{x}{\frac {\partial u_{y}}{\partial x}}+u_{y}{\frac {\partial u_{y}}{\partial y}}+u_{z}{\frac {\partial u_{y}}{\partial z}}&={\frac {1}{\rho }}\left({\frac {\partial \sigma _{xy}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}+{\frac {\partial \sigma _{zy}}{\partial z}}\right)+f_{y}\\z&:&{\frac {\partial u_{z}}{\partial t}}+u_{x}{\frac {\partial u_{z}}{\partial x}}+u_{y}{\frac {\partial u_{z}}{\partial y}}+u_{z}{\frac {\partial u_{z}}{\partial z}}&={\frac {1}{\rho }}\left({\frac {\partial \sigma _{xz}}{\partial x}}+{\frac {\partial \sigma _{yz}}{\partial y}}+{\frac {\partial \sigma _{zz}}{\partial z}}\right)+f_{z}\end{aligned}}}
4064:
3477:
12852:
4059:{\displaystyle {\begin{aligned}F_{p}^{x}&={\frac {\partial \sigma _{xx}}{\partial x}}\,dx\,dy\,dz+{\frac {\partial \sigma _{yx}}{\partial y}}\,dy\,dx\,dz+{\frac {\partial \sigma _{zx}}{\partial z}}\,dz\,dx\,dy\\F_{p}^{y}&={\frac {\partial \sigma _{xy}}{\partial x}}\,dx\,dy\,dz+{\frac {\partial \sigma _{yy}}{\partial y}}\,dy\,dx\,dz+{\frac {\partial \sigma _{zy}}{\partial z}}\,dz\,dx\,dy\\F_{p}^{z}&={\frac {\partial \sigma _{xz}}{\partial x}}\,dx\,dy\,dz+{\frac {\partial \sigma _{yz}}{\partial y}}\,dy\,dx\,dz+{\frac {\partial \sigma _{zz}}{\partial z}}\,dz\,dx\,dy{\vphantom {\begin{matrix}\\\\\end{matrix}}}\end{aligned}}}
12311:
1061:
8579:
5464:
705:
12847:{\displaystyle {\begin{aligned}{\mathbf {j} }&={\begin{pmatrix}\rho u_{1}\\\rho u_{2}\\\rho u_{3}\end{pmatrix}}\\{\mathbf {s} }&={\begin{pmatrix}\rho g_{1}\\\rho g_{2}\\\rho g_{3}\end{pmatrix}}\\{\mathbf {F} }&={\begin{pmatrix}\rho u_{1}^{2}+\sigma _{11}&\rho u_{1}u_{2}+\sigma _{12}&\rho u_{1}u_{3}+\sigma _{13}\\\rho u_{2}u_{1}+\sigma _{12}&\rho u_{2}^{2}+\sigma _{22}&\rho u_{2}u_{3}+\sigma _{23}\\\rho u_{3}u_{1}+\sigma _{13}&\rho u_{3}u_{2}+\sigma _{23}&\rho u_{3}^{2}+\sigma _{33}\end{pmatrix}}\end{aligned}}}
5785:
8259:
5021:
1056:{\displaystyle \nabla \cdot {\boldsymbol {\sigma }}={\begin{bmatrix}{\dfrac {\partial \sigma _{xx}}{\partial x}}+{\dfrac {\partial \sigma _{yx}}{\partial y}}+{\dfrac {\partial \sigma _{zx}}{\partial z}}\\{\dfrac {\partial \sigma _{xy}}{\partial x}}+{\dfrac {\partial \sigma _{yy}}{\partial y}}+{\dfrac {\partial \sigma _{zy}}{\partial z}}\\{\dfrac {\partial \sigma _{xz}}{\partial x}}+{\dfrac {\partial \sigma _{yz}}{\partial y}}+{\dfrac {\partial \sigma _{zz}}{\partial z}}\\\end{bmatrix}}}
8892:
9204:
3386:
8574:{\displaystyle {\begin{aligned}\rho ^{*}&\equiv {\frac {\rho }{\rho _{0}}}&u^{*}&\equiv {\frac {u}{u_{0}}}&r^{*}&\equiv {\frac {r}{r_{0}}}&t^{*}&\equiv {\frac {u_{0}}{r_{0}}}t\\\nabla ^{*}&\equiv r_{0}\nabla &\mathbf {f} ^{*}&\equiv {\frac {\mathbf {f} }{f_{0}}}&p^{*}&\equiv {\frac {p}{p_{0}}}&{\boldsymbol {\tau }}^{*}&\equiv {\frac {\boldsymbol {\tau }}{\tau _{0}}}\end{aligned}}}
5459:{\displaystyle {\begin{aligned}\int _{\Omega }\rho {\frac {Du_{i}}{Dt}}\,dV&=\int _{\Omega }\nabla _{j}\sigma _{i}^{j}\,dV+\int _{\Omega }\rho f_{i}\,dV\\\int _{\Omega }\left(\rho {\frac {Du_{i}}{Dt}}-\nabla _{j}\sigma _{i}^{j}-\rho f_{i}\right)\,dV&=0\\\rho {\frac {Du_{i}}{Dt}}-\nabla _{j}\sigma _{i}^{j}-\rho f_{i}&=0\\{\frac {Du_{i}}{Dt}}-{\frac {\nabla _{j}\sigma _{i}^{j}}{\rho }}-f_{i}&=0\end{aligned}}}
8588:
8901:
6774:
6945:
1620:
1612:
3038:
9706:
9856:
7083:
6506:
6618:
2912:
7225:
6791:
5686:
6609:
8887:{\displaystyle {\frac {\rho _{0}u_{0}^{2}}{r_{0}}}{\frac {\partial \rho ^{*}\mathbf {u} ^{*}}{\partial t^{*}}}+{\frac {\nabla ^{*}}{r_{0}}}\cdot \left(\rho _{0}u_{0}^{2}\rho ^{*}\mathbf {u} ^{*}\otimes \mathbf {u} ^{*}+p_{0}p^{*}\right)=-{\frac {\tau _{0}}{r_{0}}}\nabla ^{*}\cdot {\boldsymbol {\tau }}^{*}+f_{0}\mathbf {f} ^{*}}
9199:{\displaystyle {\frac {\partial \mathbf {\rho } ^{*}u^{*}}{\partial t^{*}}}+\nabla ^{*}\cdot \left(\rho ^{*}\mathbf {u} ^{*}\otimes \mathbf {u} ^{*}+{\frac {p_{0}}{\rho _{0}u_{0}^{2}}}p^{*}\right)=-{\frac {\tau _{0}}{\rho _{0}u_{0}^{2}}}\nabla ^{*}\cdot {\boldsymbol {\tau }}^{*}+{\frac {f_{0}r_{0}}{u_{0}^{2}}}\mathbf {f} ^{*}}
9556:
3381:{\displaystyle F_{p}^{x}=\left(\sigma _{xx}+{\frac {\partial \sigma _{xx}}{\partial x}}dx\right)dy\,dz-\sigma _{xx}dy\,dz+\left(\sigma _{yx}+{\frac {\partial \sigma _{yx}}{\partial y}}dy\right)dx\,dz-\sigma _{yx}dx\,dz+\left(\sigma _{zx}+{\frac {\partial \sigma _{zx}}{\partial z}}dz\right)dx\,dy-\sigma _{zx}dx\,dy}
6182:
6311:
6954:
6378:
5792:
A significant feature of the NavierβStokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent acceleration, the convective acceleration of the flow
8031:
are yet unknown, so this equation alone cannot be used to solve problems. Besides the equations of motionβNewton's second lawβa force model is needed relating the stresses to the flow motion. For this reason, assumptions based on natural observations are often applied to specify the stresses in
7388:
9976:
7102:
5576:
9729:
6515:
4759:
2661:
6769:{\displaystyle \nabla \cdot \left({\frac {1}{2}}u^{2}-{\frac {\boldsymbol {\sigma }}{\rho }}\right)-\mathbf {f} ={\frac {1}{\rho ^{2}}}{\boldsymbol {\sigma }}\cdot \nabla \rho +\mathbf {u} \times (\nabla \times \mathbf {u} )-{\frac {\partial \mathbf {u} }{\partial t}}}
6940:{\displaystyle \nabla \cdot \left({\frac {1}{2}}u^{2}+\phi -{\frac {\boldsymbol {\sigma }}{\rho }}\right)={\frac {1}{\rho ^{2}}}{\boldsymbol {\sigma }}\cdot \nabla \rho +\mathbf {u} \times (\nabla \times \mathbf {u} )-{\frac {\partial \mathbf {u} }{\partial t}}}
9538:
5472:
represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main step (not done above) in deriving this equation is establishing that the
9701:{\displaystyle {\frac {\partial \mathbf {j} }{\partial t}}+\nabla \cdot \left({\frac {1}{\rho }}\mathbf {j} \otimes \mathbf {j} +\mathrm {Eu} \,p\right)=-{\frac {C_{\mathrm {f} }}{2}}\nabla \cdot {\boldsymbol {\tau }}+{\frac {1}{\mathrm {Fr} }}\mathbf {g} }
4937:
123:
6086:
6210:
7800:
8202:
7524:
8014:
5566:
1236:
7276:
13471:
13110:
13028:
6367:
4460:
4629:
4145:
9886:
2506:
The second element requiring explanation is the approximation of the values of stress acting on the walls opposite the walls covering the axes. Let us focus on the right wall where the stress is an approximation of stress
350:
10994:
1377:
653:
7903:
9851:{\displaystyle {\frac {\partial \mathbf {j} }{\partial t}}+\nabla \cdot \left({\frac {1}{\rho }}\mathbf {j} \otimes \mathbf {j} +\mathrm {Eu} \,p\right)=-{\frac {C_{\mathrm {f} }}{2}}\nabla \cdot {\boldsymbol {\tau }}}
7605:
7078:{\displaystyle \nabla \cdot \left({\frac {1}{2}}u^{2}+\phi -{\frac {\boldsymbol {\sigma }}{\rho }}\right)={\frac {1}{\rho ^{2}}}{\boldsymbol {\sigma }}\cdot \nabla \rho +\mathbf {u} \times (\nabla \times \mathbf {u} )}
9453:
2633:
6501:{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+{\frac {1}{2}}\nabla \left(u^{2}\right)+(\nabla \times \mathbf {u} )\times \mathbf {u} ={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} }
2125:
6064:
4624:
4316:
4866:
2907:{\displaystyle {\frac {\sigma _{xx}(x+dx)-(\sigma _{xx}(x)+dx{\frac {\partial \sigma _{xx}(x)}{\partial x}})}{dx}}={\frac {\sigma _{xx}(x+dx)-\sigma _{xx}(x)}{dx}}-{\frac {\partial \sigma _{xx}(x)}{\partial x}}}
2403:
9364:
11006:
10001:
7220:{\displaystyle \mathbf {u} \cdot \nabla \cdot \left({\frac {1}{2}}u^{2}+\phi -{\frac {\boldsymbol {\sigma }}{\rho }}\right)={\frac {1}{\rho ^{2}}}\mathbf {u} \cdot ({\boldsymbol {\sigma }}\cdot \nabla \rho )}
3482:
572:
9284:
5681:{\displaystyle {\begin{aligned}{\mathbf {j} }&=\rho \mathbf {u} \\{\mathbf {F} }&=\rho \mathbf {u} \otimes \mathbf {u} -{\boldsymbol {\sigma }}\\{\mathbf {s} }&=\rho \mathbf {f} \end{aligned}}}
9474:
1679:" (purple square from the upper graph). In the bottom graph, the yellow line is completely covered by the blue one, thus not visible. In the bottom figure, two equivalent derivative forms have been used:
4237:
8264:
1139:
6604:{\displaystyle \nabla \cdot \left({\frac {\boldsymbol {\sigma }}{\rho }}\right)={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}-{\frac {1}{\rho ^{2}}}{\boldsymbol {\sigma }}\cdot \nabla \rho }
1996:
1764:
7263:
4871:
57:
5788:
An example of convective acceleration. The flow is steady (time-independent), but the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow).
4530:
1599:
12316:
9479:
7728:
7433:
5581:
5026:
1840:
8228:
rather than the body force term. This may even include antisymmetric stresses (inputs of angular momentum), in contrast to the usually symmetrical internal contributions to the stress tensor.
7653:
8109:
7446:
2501:
2284:
273:
7962:
3020:
1180:
8226:
481:
2049:
4365:
3470:
2449:
457:
698:
387:
4802:
1457:
7721:
6205:
in his famous classical book
Hydrodynamics (1895), used this identity to change the convective term of the flow velocity in rotational form, i.e. without a tensor derivative:
1922:
1883:
184:
4370:
2203:
2170:
1532:
1417:
5004:
4169:
2945:
2535:
299:
232:
150:
5510:
3420:
1145:
Commonly used SI units are given in parentheses although the equations are general in nature and other units can be entered into them or units can be removed at all by
6177:{\displaystyle \mathbf {v} \times \left(\nabla \times \mathbf {a} \right)=\nabla _{a}\left(\mathbf {v} \cdot \mathbf {a} \right)-\mathbf {v} \cdot \nabla \mathbf {a} }
4073:
1503:
1480:
6306:{\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} =\nabla \left({\frac {\|\mathbf {u} \|^{2}}{2}}\right)+\left(\nabla \times \mathbf {u} \right)\times \mathbf {u} }
1677:
1283:
409:
13417:
13056:
9458:
2974:
2564:
1650:
7864:
7537:
2656:
51:
9377:
8254:
need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
206:
1278:
which can be written as follows: "The change in system momentum is proportional to the resulting force acting on this system". It is expressed by the formula:
12948:
7659:
the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant
6318:
5920:
4539:
5800:
effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in
4249:
13182:
7383:{\displaystyle \mathbf {u} \cdot \nabla \left({\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\right)+{\frac {p}{\rho ^{2}}}\mathbf {u} \cdot \nabla \rho =0}
9971:{\displaystyle {\frac {D\mathbf {u} }{Dt}}+\mathrm {Eu} \,{\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}={\frac {1}{\mathrm {Fr} }}\mathbf {f} }
9297:
8054:
acceleration, but may include others, such as electromagnetic forces. In non-inertial coordinate frames, other "inertial accelerations" associated with
304:
10921:
9217:
579:
4174:
2569:
1928:
1615:
The X component of the forces acting on walls of a cubic fluid element (green for top-bottom walls; red for left-right; black for front-back).
2139:
It requires some explanation why stress applied to the walls covering the coordinate axes takes a minus sign (e.g. for the left wall we have
2054:
4754:{\displaystyle {\frac {d\mathbf {u} }{dt}}\rho \,dx\,dy\,dz=(\nabla \cdot {\boldsymbol {\sigma }})dx\,dy\,dz+\mathbf {f} \rho \,dx\,dy\,dz}
4474:
1543:
7397:
4807:
2293:
7614:
1652:(blue line) using a finite difference (yellow line). In the bottom graph we see "infinitely many times enlarged neighborhood of point
490:
1161:
1157:
2004:
component of these forces was marked in the figure with a cubic element (in the form of a product of stress and surface area e.g.
13505:
13495:
1070:
13137:
8018:
The effect of the pressure gradient on the flow is to accelerate the flow in the direction from high pressure to low pressure.
7089:
13147:
7232:
8090:. Because pressure from such gravitation arises only as a gradient, we may include it in the pressure term as a body force
1769:
3472:(they have not been shown in the figure, but these would be vectors parallel to the Y and Z axes, respectively) we get:
1682:
9533:{\displaystyle {\begin{aligned}\mathbf {j} &=\rho \mathbf {u} \\\mathbf {g} &=\rho \mathbf {f} \end{aligned}}}
1168:, then we should calculate and write equations in covariant ("row vectors") or contravariant ("column vectors") form.
13341:
13266:
13223:
13195:
13117:
12919:
12256:
7951:
5773:
4966:
13332:
4149:
There are mass forces acting on the inside of the control volume. We can write them using the acceleration field
2454:
2208:
239:
6949:
In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:
5772:
In the
Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the
9865:. The limit of high Froude numbers (low external field) is thus notable for such equations and is studied with
4932:{\displaystyle {\frac {D\mathbf {u} }{Dt}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} }
2979:
118:{\displaystyle {\frac {D\mathbf {u} }{Dt}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} }
8209:
7439:
the mass conservation for a steady incompressible flow states that the density along a streamline is constant
464:
24:
12945:(Report). AIR FORCE WRIGHT AERONAUTICAL LABORATORIES. p. 13 (Below the main equation, authors describe
7795:{\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} =\nabla \left({\frac {\|\mathbf {u} \|^{2}}{2}}\right).}
13490:
12271:
7930:, can be derived by beginning with the Cauchy momentum equation and specifying the stress tensor through a
2007:
12261:
7943:
4323:
3425:
2976:
has been shown in the figure below the cube. We proceed with similar reasoning for stress approximations
2408:
1153:
422:
666:
355:
10918:
Below, we write the main equation in pressure-tau form assuming that the stress tensor is symmetrical (
7927:
5493:
5009:
4766:
1426:
660:
13408:
7849:, which usually describes viscous forces; for incompressible flow, this is only a shear effect. Thus,
7690:
8197:{\displaystyle -\nabla p+\mathbf {f} =-\nabla p+\nabla \chi =-\nabla \left(p-\chi \right)=-\nabla h.}
7519:{\displaystyle \mathbf {u} \cdot \nabla \left({\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\right)=0}
7271:
is the identity tensor), and the Euler momentum equation in the steady incompressible case becomes:
1891:
1852:
159:
13240:
7856:
6074:
2175:
2142:
1508:
1384:
13164:
8009:{\displaystyle \nabla \cdot {\boldsymbol {\sigma }}=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}.}
6078:
5561:{\displaystyle {\frac {\partial \mathbf {j} }{\partial t}}+\nabla \cdot \mathbf {F} =\mathbf {s} }
4152:
2920:
2510:
1231:{\displaystyle {\frac {\partial \mathbf {j} }{\partial t}}+\nabla \cdot \mathbf {F} =\mathbf {s} }
282:
215:
133:
12911:
1165:
9289:
4950:
3393:
2133:
Explanation of the value of forces (approximations and minus signs) acting on the cube walls.
1485:
1462:
1156:) above for clarity, but the equation is written using physical components (which are neither
13466:{\displaystyle \nabla \cdot \mathbb {\sigma } =\nabla \cdot (p\mathbf {I} +\mathbf {\tau } )}
13105:{\displaystyle \nabla \cdot \mathbb {\sigma } =\nabla \cdot (p\mathbf {I} +\mathbf {\tau } )}
7942:, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the
7931:
7842:
that occur in almost all situations. The anisotropic part of the stress tensor gives rise to
7094:
1146:
13365:
12879:
8104:. The pressure and force terms on the right-hand side of the NavierβStokes equation become
7835:
2287:
1655:
484:
394:
13286:
2950:
2540:
1626:
8:
9866:
5701:
5013:
416:
276:
39:
35:
13369:
13023:{\displaystyle \sigma _{ki,k}=\partial \sigma _{ki}/\partial x_{k}=\nabla \cdot \sigma }
12939:
2638:
13389:
13323:
12862:
8076:
7670:
6779:
6362:{\displaystyle \mathbf {l} =\left(\nabla \times \mathbf {u} \right)\times \mathbf {u} }
5796:
Regardless of what kind of continuum is being dealt with, convective acceleration is a
191:
13045:
4455:{\displaystyle {\frac {d{\vec {p}}}{dt}}={\frac {d\mathbf {u} }{dt}}\rho \,dx\,dy\,dz}
13500:
13381:
13337:
13283:
13262:
13219:
13191:
13143:
13113:
12915:
12266:
5848:
5474:
1275:
1172:
13393:
5784:
1064:
13373:
13254:
9862:
9369:
8055:
2451:(we use similar reasoning for stresses acting on the bottom and back walls, i.e.:
13319:
8583:
Substitution of these inverted relations in the Euler momentum equations yields:
7914:
4140:{\displaystyle {\vec {F}}_{p}=(\nabla \cdot {\boldsymbol {\sigma }})\,dx\,dy\,dz}
13044:
Papanastasiou, Tasos C.; Georgiou, Georgios C.; Alexandrou, Andreas N. (2000).
8061:
Often, these forces may be represented as the gradient of some scalar quantity
7939:
5731:
4958:
345:{\displaystyle \partial _{t}\mathbf {u} +\mathbf {u} \cdot \nabla \mathbf {u} }
10989:{\displaystyle \sigma _{ij}=\sigma _{ji}\Longrightarrow \tau _{ij}=\tau _{ji}}
9991:
For asymmetric stress tensors, equations in general take the following forms:
9715:(corresponding to negligible external field) are named free Cauchy equations:
7441:. This leads to a considerable simplification of the Euler momentum equation:
1372:{\displaystyle {\vec {p}}(t+\Delta t)-{\vec {p}}(t)=\Delta t{\vec {\bar {F}}}}
648:{\displaystyle \mathbf {f} ={\begin{bmatrix}f_{x}\\f_{y}\\f_{z}\end{bmatrix}}}
13484:
13385:
13327:
9466:
9209:
7947:
7839:
5801:
1886:
153:
7898:{\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} +{\boldsymbol {\tau }}}
7087:
And by projecting the momentum equation on the flow direction, i.e. along a
5793:
field is a spatial effect, one example being fluid speeding up in a nozzle.
7600:{\displaystyle b_{l}\equiv {\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\,,}
5899:, which is the component-wise derivative of the velocity vector defined by
2000:
Surface forces act on walls of the cubic fluid element. For each wall, the
9448:{\displaystyle C_{\mathrm {f} }={\frac {2\tau _{0}}{\rho _{0}u_{0}^{2}}},}
6195:
is used, which means the subscripted gradient operates only on the factor
2628:{\displaystyle \sigma _{xx}+{\frac {\partial \sigma _{xx}}{\partial x}}dx}
12940:
Behavior of a
Vorticity-Influenced Asymmetric Stress Tensor in Fluid Flow
6370:
6202:
5804:(also called Stokes flow). Convective acceleration is represented by the
8206:
It is also possible to include external influences into the stress term
7093:, the cross product disappears due to a vector calculus identity of the
2120:{\textstyle \mathrm {Pa\cdot m\cdot m={\frac {N}{m^{2}}}\cdot m^{2}=N} }
8047:
7529:
6059:{\displaystyle \left_{i}=\sum _{m}v_{m}\partial _{m}v_{i}=\left_{i}\,.}
5717:
4619:{\displaystyle {\frac {d{\vec {p}}}{dt}}={\vec {F}}_{p}+{\vec {F}}_{m}}
2205:. The minus sign is due to the fact that a vector normal to this wall
1847:
1535:
1263:
656:
13356:
Dahler, J. S.; Scriven, L. E. (1961). "Angular
Momentum of Continua".
12290:
In 3D for example, with respect to some coordinate system, the vector
8236:
In order to make the equations dimensionless, a characteristic length
4311:{\displaystyle {\vec {p}}=\mathbf {u} m=\mathbf {u} \rho \,dx\,dy\,dz}
13377:
13291:
9462:
9368:
and the coefficient of skin-friction or the one usually referred as '
7935:
7674:
5877:
5805:
5797:
5697:
1247:
5498:
The Cauchy momentum equation can also be put in the following form:
4861:{\textstyle {\frac {d\mathbf {u} }{dt}}={\frac {D\mathbf {u} }{Dt}}}
1171:
After an appropriate change of variables, it can also be written in
7834:
is the pressure gradient and arises from the isotropic part of the
7824:
5892:. This contrasts with the expression in terms of tensor derivative
2398:{\displaystyle {\vec {s}}={\vec {n}}\cdot {\boldsymbol {\sigma }}=}
32:
13336:, Reading, Massachusetts: Addison-Wesley, Vol. 1, Β§9β4 and Β§12β1,
12308:
have 9 (3Γ3), so the explicit forms written as matrices would be:
9359:{\displaystyle \mathrm {Eu} ={\frac {p_{0}}{\rho _{0}u_{0}^{2}}},}
7926:
All non-relativistic momentum conservation equations, such as the
7229:
If the stress tensor is isotropic, then only the pressure enters:
12882:), since dyads of vectors with themselves are always symmetrical.
8051:
8032:
terms of the other flow variables, such as velocity and density.
7809:
The effect of stress in the continuum flow is represented by the
412:
9542:
the equations are finally expressed (now omitting the indexes):
567:{\displaystyle \mathrm {Pa=N/m^{2}=kg\cdot m^{-1}\cdot s^{-2}} }
13043:
7392:
In the steady incompressible case the mass equation is simply:
5760:
28:
13281:
9279:{\displaystyle \mathrm {Fr} ={\frac {u_{0}^{2}}{f_{0}r_{0}}},}
2917:
A more intuitive representation of the value of approximation
2172:). For simplicity, let us focus on the left wall with tension
1619:
1611:
8021:
As written in the Cauchy momentum equation, the stress terms
7669:
The Lamb form is also useful in irrotational flow, where the
4232:{\displaystyle {\vec {F}}_{m}=\mathbf {f} \rho \,dx\,dy\,dz}
5477:
of the stress tensor is one of the forces that constitutes
2914:
goes to zero as well, by definition of partial derivative.
209:
7950:, the NavierβStokes equations can further simplify to the
7827:
of surface forces, analogous to stresses in a solid. Here
5860:
1134:{\displaystyle \mathrm {Pa/m=kg\cdot m^{-2}\cdot s^{-2}} }
655:
is a vector containing all of the accelerations caused by
13300:
5700:
at the point considered in the continuum (for which the
1991:{\displaystyle {\vec {F}}={\vec {F}}_{p}+{\vec {F}}_{m}}
12878:'s symmetry (which will be present for the most common
7687:
is equal to zero. In that case, the convection term in
4068:
We can then write it in the symbolic operational form:
12500:
12419:
12338:
7957:
The divergence of the stress tensor can be written as
7609:
in fact, the above equation can be simply written as:
7258:{\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} }
4810:
4040:
3033:
components) acting on each of the cube walls, we get:
2057:
1685:
728:
596:
156:
vector field, which depends on time and space, (unit:
13420:
13059:
12951:
12314:
11004:
10924:
9999:
9889:
9732:
9559:
9477:
9380:
9300:
9220:
8904:
8591:
8262:
8212:
8112:
7965:
7867:
7731:
7693:
7617:
7540:
7449:
7400:
7279:
7235:
7105:
6957:
6794:
6621:
6518:
6381:
6321:
6213:
6089:
5923:
5579:
5513:
5024:
4969:
4874:
4769:
4632:
4542:
4477:
4373:
4326:
4252:
4177:
4155:
4076:
3480:
3428:
3396:
3041:
2982:
2953:
2923:
2664:
2641:
2572:
2543:
2513:
2457:
2411:
2296:
2211:
2178:
2145:
2010:
1931:
1894:
1855:
1772:
1658:
1629:
1546:
1511:
1488:
1465:
1429:
1387:
1286:
1183:
1073:
1014:
979:
944:
908:
873:
838:
802:
767:
732:
708:
669:
582:
493:
467:
425:
397:
358:
307:
285:
242:
218:
194:
162:
136:
60:
13318:
13139:
Introduction to
Chemical Engineering Fluid Mechanics
13053:. CRC Press. pp. 66, 68, 143, 182 (Authors use
5710:
is the flux associated to the momentum density, and
4525:{\displaystyle {\frac {d{\vec {p}}}{dt}}={\vec {F}}}
1594:{\displaystyle {\frac {d{\vec {p}}}{dt}}={\vec {F}}}
1256:
is the flux associated to the momentum density, and
7428:{\displaystyle \mathbf {u} \cdot \nabla \rho =0\,,}
5753:as the flow speed and the body acceleration, while
2286:is a negative unit vector. Then, we calculated the
1835:{\displaystyle \Delta f=f(x_{1}+\Delta x)-f(x_{1})}
54:) form the Cauchy momentum equation is written as:
13465:
13104:
13022:
12846:
12240:
10988:
10903:
9970:
9850:
9700:
9532:
9447:
9358:
9278:
9198:
8886:
8573:
8220:
8196:
8008:
7897:
7794:
7715:
7647:
7599:
7518:
7427:
7382:
7257:
7219:
7077:
6939:
6768:
6603:
6500:
6361:
6305:
6176:
6058:
5680:
5560:
5458:
4998:
4931:
4860:
4796:
4753:
4618:
4524:
4454:
4359:
4310:
4231:
4163:
4139:
4058:
3464:
3414:
3380:
3014:
2968:
2939:
2906:
2650:
2627:
2558:
2529:
2495:
2443:
2397:
2278:
2197:
2164:
2119:
2043:
1990:
1916:
1877:
1834:
1759:{\textstyle f'(x_{1})={\frac {df(x_{1})}{dx_{1}}}}
1758:
1671:
1644:
1623:In the top graph we see approximation of function
1593:
1526:
1497:
1474:
1451:
1411:
1371:
1230:
1133:
1055:
692:
647:
566:
475:
451:
403:
381:
344:
293:
267:
226:
200:
178:
144:
117:
13349:
4464:
1164:("row") ). However, if we chose a non-orthogonal
415:at a given point of the continuum (for which the
13482:
13142:. Cambridge University Press. pp. 133β136.
8050:per unit mass. Typically, these consist of only
7648:{\displaystyle \mathbf {u} \cdot \nabla b_{l}=0}
3422:and performing similar reasoning for components
1603:Let us analyse each side of the equation above.
13162:
13039:
13037:
9981:
12938:Berdahl, C. I.; Strang, W. Z. (October 1986).
9872:Finally in convective form the equations are:
2537:from the left wall at points with coordinates
13355:
12937:
10913:
7934:. By expressing the shear tensor in terms of
7532:for an inviscid liquid flow is now apparent:
5857:. Both interpretations give the same result.
1152:Note that only we use column vectors (in the
13218:(second ed.). CRC Press. pp. 6β7.
13034:
7770:
7761:
6252:
6243:
13261:. Cambridge University Press. p. 160.
9986:
8896:and by dividing for the first coefficient:
8083:direction, for example, is the gradient of
2496:{\displaystyle -\sigma _{yx},-\sigma _{zx}}
2279:{\displaystyle {\vec {n}}==-{\vec {e}}_{x}}
1276:generalized momentum conservation principle
268:{\displaystyle {\frac {D\mathbf {u} }{Dt}}}
5779:
4320:Because we assume that tested mass (cube)
1269:
13306:
13253:
13209:
13207:
13190:. New York: McGraw-Hill. pp. 61β64.
13156:
13131:
13129:
12933:
12931:
12901:
12899:
9923:
9803:
9630:
7593:
7421:
6052:
5256:
5151:
5118:
5069:
4787:
4780:
4773:
4744:
4737:
4730:
4712:
4705:
4672:
4665:
4658:
4445:
4438:
4431:
4350:
4343:
4336:
4301:
4294:
4287:
4222:
4215:
4208:
4130:
4123:
4116:
4029:
4022:
4015:
3975:
3968:
3961:
3921:
3914:
3907:
3844:
3837:
3830:
3790:
3783:
3776:
3736:
3729:
3722:
3659:
3652:
3645:
3605:
3598:
3591:
3551:
3544:
3537:
3371:
3342:
3264:
3235:
3157:
3128:
3015:{\displaystyle \sigma _{yx},\sigma _{zx}}
2405:, thus the X component of this vector is
2034:
2027:
13180:
8231:
6373:. The Cauchy momentum equation becomes:
5783:
2635:. This approximation suffices since, as
1618:
1610:
13213:
13174:
12905:
9941:
9844:
9671:
9130:
8849:
8552:
8533:
8221:{\displaystyle {\boldsymbol {\sigma }}}
8214:
7999:
7973:
7891:
7869:
7237:
7201:
7156:
7034:
7000:
6871:
6837:
6700:
6658:
6588:
6563:
6531:
6486:
5861:Advection operator vs tensor derivative
5644:
4917:
4692:
4246:Let us calculate momentum of the cube:
4109:
2328:
716:
476:{\displaystyle {\boldsymbol {\sigma }}}
469:
103:
13483:
13406:
13204:
13126:
12928:
12896:
4961:in the continuum being modeled gives:
4944:
13282:
9711:Cauchy equations in the Froude limit
7859:, and the stress tensor is equal to:
7664:
6186:where the Feynman subscript notation
5818:, which may be interpreted either as
2044:{\displaystyle -\sigma _{xx}\,dy\,dz}
13238:
13135:
12854:Note, however, that if symmetrical,
12296:has 3 components, while the tensors
5487:
31:that describes the non-relativistic
13243:(6th ed.). Dover Publications.
4360:{\displaystyle m=\rho \,dx\,dy\,dz}
4171:(e.g. gravitational acceleration):
3465:{\displaystyle F_{p}^{y},F_{p}^{z}}
2444:{\displaystyle s_{x}=-\sigma _{xx}}
452:{\displaystyle \mathrm {kg/m^{3}} }
13:
13435:
13421:
13074:
13060:
13011:
12995:
12974:
12212:
12181:
12151:
12133:
12103:
12085:
12060:
12052:
12020:
12005:
11980:
11965:
11933:
11918:
11893:
11878:
11837:
11819:
11794:
11756:
11719:
11701:
11671:
11663:
11594:
11579:
11554:
11539:
11507:
11492:
11467:
11452:
11383:
11365:
11340:
11322:
11292:
11261:
11231:
11223:
11166:
11151:
11126:
11111:
11079:
11064:
11039:
11024:
10870:
10852:
10837:
10819:
10804:
10786:
10752:
10737:
10712:
10697:
10672:
10657:
10632:
10617:
10571:
10553:
10538:
10520:
10505:
10487:
10453:
10438:
10413:
10398:
10373:
10358:
10333:
10318:
10272:
10254:
10239:
10221:
10206:
10188:
10154:
10139:
10114:
10099:
10074:
10059:
10034:
10019:
9957:
9954:
9934:
9919:
9916:
9837:
9826:
9799:
9796:
9758:
9746:
9736:
9687:
9684:
9664:
9653:
9626:
9623:
9585:
9573:
9563:
9387:
9305:
9302:
9225:
9222:
9116:
8955:
8935:
8908:
8835:
8685:
8663:
8636:
8450:
8424:
8185:
8157:
8145:
8136:
8116:
8035:
7992:
7983:
7966:
7751:
7740:
7626:
7458:
7409:
7368:
7288:
7208:
7114:
7061:
7041:
6958:
6928:
6918:
6898:
6878:
6795:
6757:
6747:
6727:
6707:
6622:
6595:
6556:
6519:
6479:
6444:
6417:
6395:
6385:
6335:
6279:
6233:
6222:
6166:
6123:
6103:
6030:
5991:
5943:
5539:
5527:
5517:
5399:
5311:
5211:
5167:
5133:
5094:
5088:
5034:
4910:
4685:
4102:
4006:
3988:
3952:
3934:
3898:
3880:
3821:
3803:
3767:
3749:
3713:
3695:
3636:
3618:
3582:
3564:
3528:
3510:
3316:
3298:
3209:
3191:
3102:
3084:
2895:
2868:
2766:
2739:
2610:
2592:
2113:
2101:
2086:
2082:
2074:
2068:
2062:
2059:
1801:
1773:
1512:
1489:
1466:
1344:
1308:
1209:
1197:
1187:
1118:
1102:
1095:
1092:
1086:
1078:
1075:
1035:
1017:
1000:
982:
965:
947:
929:
911:
894:
876:
859:
841:
823:
805:
788:
770:
753:
735:
709:
693:{\displaystyle \mathrm {m/s^{2}} }
680:
671:
551:
535:
528:
525:
513:
504:
498:
495:
439:
430:
427:
382:{\displaystyle \mathrm {m/s^{2}} }
369:
360:
334:
309:
220:
172:
164:
96:
14:
13517:
13259:An Introduction to Fluid Dynamics
4797:{\displaystyle \rho \,dx\,dy\,dz}
1452:{\displaystyle {\vec {\bar {F}}}}
13448:
13087:
12483:
12402:
12321:
12257:Euler equations (fluid dynamics)
9964:
9897:
9788:
9780:
9740:
9723:nondimensional conservative form
9694:
9615:
9607:
9567:
9550:nondimensional conservative form
9522:
9507:
9498:
9483:
9372:' in the field of aerodynamics:
9186:
8999:
8984:
8874:
8766:
8751:
8651:
8477:
8457:
8126:
7883:
7765:
7744:
7733:
7716:{\displaystyle D\mathbf {u} /Dt}
7698:
7619:
7528:The convenience of defining the
7451:
7402:
7361:
7281:
7251:
7190:
7107:
7068:
7051:
6922:
6905:
6888:
6778:In fact, in case of an external
6751:
6734:
6717:
6675:
6494:
6462:
6451:
6389:
6355:
6342:
6323:
6299:
6286:
6247:
6226:
6215:
6170:
6159:
6146:
6138:
6110:
6091:
6037:
6023:
5947:
5931:
5670:
5654:
5636:
5628:
5612:
5602:
5586:
5554:
5546:
5521:
4925:
4882:
4843:
4818:
4723:
4640:
4413:
4280:
4269:
4201:
4157:
1224:
1216:
1191:
584:
338:
327:
319:
287:
250:
138:
111:
68:
45:
13400:
13333:The Feynman Lectures on Physics
13312:
13275:
13247:
9457:by passing respectively to the
5749:have same number of dimensions
4941:which finishes the derivation.
13506:Partial differential equations
13496:Eponymous equations of physics
13460:
13441:
13407:Powell, Adam (12 April 2010).
13232:
13181:Anderson, John D. Jr. (1995).
13099:
13080:
12284:
10954:
9880:nondimensional convective form
8245:and a characteristic velocity
8075:in which case they are called
7214:
7197:
7072:
7058:
6909:
6895:
6738:
6724:
6455:
6441:
6033:
6019:
4696:
4682:
4604:
4582:
4555:
4516:
4490:
4465:Left and Right side comparison
4386:
4259:
4185:
4113:
4099:
4084:
2890:
2884:
2848:
2842:
2823:
2808:
2775:
2761:
2755:
2724:
2718:
2702:
2696:
2681:
2392:
2335:
2318:
2303:
2264:
2248:
2227:
2218:
1976:
1954:
1938:
1917:{\displaystyle {\vec {F}}_{p}}
1902:
1878:{\displaystyle {\vec {F}}_{m}}
1863:
1829:
1816:
1807:
1785:
1735:
1722:
1707:
1694:
1639:
1633:
1585:
1559:
1518:
1443:
1438:
1406:
1400:
1394:
1363:
1358:
1338:
1332:
1326:
1314:
1299:
1293:
179:{\displaystyle \mathrm {m/s} }
1:
13409:"The Navier-Stokes Equations"
13165:"A Primer on Tensor Calculus"
12889:
12872:'s symmetry is equivalent to
9719:Free Cauchy momentum equation
6613:the Cauchy equation becomes:
5886:acting on the velocity field
2198:{\displaystyle -\sigma _{xx}}
2165:{\displaystyle -\sigma _{xx}}
1606:
1527:{\displaystyle \Delta t\to 0}
1412:{\displaystyle {\vec {p}}(t)}
1250:at a given space-time point,
1166:curvilinear coordinate system
25:partial differential equation
13184:Computational Fluid Dynamics
9982:3D explicit convective forms
7917:in the space considered and
7838:. This part is given by the
6782:, by defining its potential
6068:
5865:The convective acceleration
4999:{\displaystyle ma_{i}=F_{i}}
4241:
4164:{\displaystyle \mathbf {f} }
4038:
2940:{\displaystyle \sigma _{xx}}
2530:{\displaystyle \sigma _{xx}}
2129:
294:{\displaystyle \mathbf {u} }
227:{\displaystyle \mathrm {s} }
145:{\displaystyle \mathbf {u} }
7:
12250:
7804:
1154:Cartesian coordinate system
10:
13522:
10914:Cylindrical 3D coordinates
5494:Conservation law (physics)
5491:
5010:Reynolds transport theorem
661:gravitational acceleration
13414:. p. 2 (Author uses
13216:Analytical fluid dynamics
13136:Deen, William M. (2016).
12908:Elementary Fluid Dynamics
5876:can be thought of as the
3415:{\displaystyle F_{p}^{x}}
1846:We split the forces into
1505:and passing to the limit
1067:of stress tensor. (unit:
13163:David A. Clarke (2011).
12277:
12272:ChapmanβEnskog expansion
9987:Cartesian 3D coordinates
9876:Cauchy momentum equation
9546:Cauchy momentum equation
7857:deviatoric stress tensor
7673:of the velocity (called
6075:vector calculus identity
5502:Cauchy momentum equation
5016:notation, one can write
4367:is constant in time, so
1498:{\displaystyle \Delta t}
1475:{\displaystyle \Delta t}
21:Cauchy momentum equation
13287:"Convective Derivative"
12912:Oxford University Press
12906:Acheson, D. J. (1990).
12262:NavierβStokes equations
7944:NavierβStokes equations
6079:cross product of a curl
5851:of the velocity vector
5780:Convective acceleration
1766:], and the designation
1459:is force averaged over
1270:Differential derivation
13467:
13170:. p. 11 (pdf 15).
13106:
13024:
12848:
12242:
10990:
10905:
9972:
9863:conservation equations
9861:and can be eventually
9852:
9702:
9534:
9459:conservative variables
9449:
9360:
9280:
9200:
8888:
8575:
8222:
8198:
8010:
7928:NavierβStokes equation
7899:
7796:
7717:
7649:
7601:
7520:
7429:
7384:
7259:
7221:
7079:
6941:
6770:
6605:
6502:
6363:
6307:
6178:
6060:
5789:
5682:
5562:
5460:
5000:
4933:
4862:
4798:
4755:
4620:
4526:
4456:
4361:
4312:
4233:
4165:
4141:
4060:
3466:
3416:
3382:
3016:
2970:
2941:
2908:
2652:
2629:
2560:
2531:
2497:
2445:
2399:
2280:
2199:
2166:
2121:
2045:
1992:
1918:
1879:
1843:
1836:
1760:
1673:
1646:
1616:
1595:
1528:
1499:
1476:
1453:
1413:
1373:
1274:Let us start with the
1232:
1135:
1057:
694:
649:
568:
477:
453:
405:
383:
346:
295:
269:
228:
202:
180:
146:
119:
13468:
13239:Lamb, Horace (1932).
13107:
13025:
12880:Cauchy stress tensors
12849:
12243:
10991:
10906:
9973:
9853:
9703:
9535:
9450:
9361:
9281:
9201:
8889:
8576:
8232:Nondimensionalisation
8223:
8199:
8011:
7932:constitutive relation
7900:
7797:
7718:
7650:
7602:
7521:
7430:
7385:
7260:
7222:
7095:triple scalar product
7080:
6942:
6771:
6606:
6503:
6364:
6308:
6179:
6061:
5787:
5683:
5563:
5461:
5001:
4934:
4863:
4799:
4763:Divide both sides by
4756:
4621:
4527:
4457:
4362:
4313:
4234:
4166:
4142:
4061:
3467:
3417:
3383:
3029:Adding forces (their
3017:
2971:
2942:
2909:
2653:
2630:
2561:
2532:
2498:
2446:
2400:
2281:
2200:
2167:
2122:
2046:
1993:
1919:
1880:
1837:
1761:
1674:
1672:{\displaystyle x_{1}}
1647:
1622:
1614:
1596:
1529:
1500:
1477:
1454:
1414:
1374:
1233:
1147:nondimensionalization
1136:
1058:
695:
650:
569:
478:
454:
406:
404:{\displaystyle \rho }
384:
347:
296:
270:
229:
203:
181:
147:
120:
13418:
13214:Emanuel, G. (2001).
13057:
12949:
12860:will only contain 6
12312:
11002:
10922:
9997:
9887:
9730:
9557:
9475:
9378:
9298:
9218:
8902:
8589:
8260:
8210:
8110:
8056:rotating coordinates
7963:
7865:
7836:Cauchy stress tensor
7729:
7691:
7615:
7538:
7447:
7398:
7277:
7233:
7103:
6955:
6792:
6619:
6516:
6510:Using the identity:
6379:
6319:
6211:
6087:
5921:
5716:contains all of the
5577:
5571:simply by defining:
5511:
5022:
4967:
4872:
4808:
4767:
4630:
4540:
4475:
4371:
4324:
4250:
4175:
4153:
4074:
4048:
3478:
3426:
3394:
3039:
2980:
2969:{\displaystyle x+dx}
2951:
2921:
2662:
2639:
2570:
2559:{\displaystyle x+dx}
2541:
2511:
2455:
2409:
2294:
2209:
2176:
2143:
2055:
2008:
1929:
1892:
1853:
1770:
1683:
1656:
1645:{\displaystyle f(x)}
1627:
1544:
1509:
1486:
1482:. After dividing by
1463:
1427:
1419:is momentum at time
1385:
1284:
1262:contains all of the
1181:
1071:
706:
667:
580:
491:
465:
423:
395:
356:
305:
283:
240:
216:
192:
160:
134:
58:
13491:Continuum mechanics
13370:1961Natur.192...36D
13324:Leighton, Robert B.
13320:Feynman, Richard P.
12818:
12669:
12520:
11194:
9867:perturbation theory
9438:
9349:
9248:
9181:
9111:
9049:
8738:
8619:
8077:conservative forces
5702:continuity equation
5505:(conservation form)
5422:
5334:
5234:
5117:
5014:material derivative
5008:Then, based on the
4957:th component) to a
4951:Newton's second law
4945:Integral derivation
4049:
4039:
3869:
3684:
3499:
3461:
3443:
3411:
3056:
2566:and it is equal to
417:continuity equation
277:material derivative
13463:
13284:Weisstein, Eric W.
13102:
13047:Viscous Fluid Flow
13020:
12863:degrees of freedom
12844:
12842:
12834:
12804:
12655:
12506:
12471:
12390:
12238:
12236:
11180:
10986:
10901:
10899:
9968:
9848:
9698:
9530:
9528:
9445:
9424:
9356:
9335:
9276:
9234:
9196:
9167:
9097:
9035:
8884:
8724:
8605:
8571:
8569:
8218:
8194:
8006:
7923:the shear tensor.
7895:
7792:
7713:
7665:Irrotational flows
7645:
7597:
7516:
7425:
7380:
7255:
7217:
7075:
6937:
6780:conservative field
6766:
6601:
6498:
6359:
6303:
6174:
6056:
5979:
5878:advection operator
5790:
5678:
5676:
5558:
5456:
5454:
5408:
5320:
5220:
5103:
4996:
4929:
4858:
4794:
4751:
4616:
4522:
4452:
4357:
4308:
4229:
4161:
4137:
4056:
4054:
4047:
3855:
3670:
3485:
3462:
3447:
3429:
3412:
3397:
3378:
3042:
3012:
2966:
2937:
2904:
2651:{\displaystyle dx}
2648:
2625:
2556:
2527:
2493:
2441:
2395:
2276:
2195:
2162:
2117:
2041:
1988:
1914:
1875:
1844:
1832:
1756:
1669:
1642:
1617:
1591:
1524:
1495:
1472:
1449:
1409:
1369:
1228:
1131:
1053:
1047:
1043:
1008:
973:
937:
902:
867:
831:
796:
761:
690:
659:(sometimes simply
645:
639:
564:
473:
449:
401:
379:
342:
291:
265:
224:
198:
176:
142:
115:
50:In convective (or
13149:978-1-107-12377-9
12267:Burnett equations
12219:
12176:
12158:
12128:
12110:
12080:
12067:
12047:
12027:
11987:
11960:
11940:
11900:
11844:
11814:
11801:
11751:
11726:
11696:
11678:
11658:
11633:
11601:
11561:
11534:
11514:
11474:
11418:
11390:
11360:
11347:
11317:
11299:
11256:
11238:
11218:
11198:
11173:
11133:
11106:
11086:
11046:
10877:
10844:
10811:
10776:
10759:
10719:
10679:
10639:
10578:
10545:
10512:
10477:
10460:
10420:
10380:
10340:
10279:
10246:
10213:
10178:
10161:
10121:
10081:
10041:
9961:
9932:
9910:
9835:
9777:
9753:
9691:
9662:
9604:
9580:
9440:
9351:
9271:
9208:Now defining the
9182:
9113:
9051:
8949:
8832:
8704:
8677:
8631:
8565:
8527:
8491:
8414:
8371:
8335:
8299:
8079:. Gravity in the
8040:The vector field
7823:terms; these are
7783:
7591:
7562:
7503:
7474:
7358:
7333:
7304:
7187:
7162:
7133:
7031:
7006:
6977:
6935:
6868:
6843:
6814:
6764:
6697:
6664:
6641:
6585:
6554:
6537:
6477:
6415:
6402:
6315:where the vector
6265:
5970:
5849:tensor derivative
5734:of the velocity.
5720:per unit volume.
5534:
5488:Conservation form
5427:
5390:
5305:
5205:
5067:
4908:
4895:
4856:
4831:
4653:
4607:
4585:
4570:
4558:
4519:
4505:
4493:
4426:
4401:
4389:
4262:
4188:
4087:
4013:
3959:
3905:
3828:
3774:
3720:
3643:
3589:
3535:
3323:
3216:
3109:
3027:
3026:
2902:
2860:
2787:
2773:
2617:
2321:
2306:
2267:
2221:
2095:
1979:
1957:
1941:
1905:
1866:
1754:
1588:
1574:
1562:
1446:
1441:
1397:
1366:
1361:
1329:
1296:
1266:per unit volume.
1204:
1173:conservation form
1042:
1007:
972:
936:
901:
866:
830:
795:
760:
263:
201:{\displaystyle t}
94:
81:
13513:
13475:
13474:
13472:
13470:
13469:
13464:
13459:
13451:
13431:
13413:
13404:
13398:
13397:
13378:10.1038/192036a0
13353:
13347:
13346:
13316:
13310:
13307:Batchelor (1967)
13304:
13298:
13297:
13296:
13279:
13273:
13272:
13257:(1967). "Β§3.5".
13255:Batchelor, G. K.
13251:
13245:
13244:
13236:
13230:
13229:
13211:
13202:
13201:
13189:
13178:
13172:
13171:
13169:
13160:
13154:
13153:
13133:
13124:
13123:
13111:
13109:
13108:
13103:
13098:
13090:
13070:
13052:
13041:
13032:
13031:
13029:
13027:
13026:
13021:
13007:
13006:
12994:
12989:
12988:
12970:
12969:
12944:
12935:
12926:
12925:
12903:
12883:
12877:
12871:
12859:
12853:
12851:
12850:
12845:
12843:
12839:
12838:
12831:
12830:
12817:
12812:
12798:
12797:
12785:
12784:
12775:
12774:
12760:
12759:
12747:
12746:
12737:
12736:
12720:
12719:
12707:
12706:
12697:
12696:
12682:
12681:
12668:
12663:
12649:
12648:
12636:
12635:
12626:
12625:
12609:
12608:
12596:
12595:
12586:
12585:
12571:
12570:
12558:
12557:
12548:
12547:
12533:
12532:
12519:
12514:
12487:
12486:
12476:
12475:
12468:
12467:
12451:
12450:
12434:
12433:
12406:
12405:
12395:
12394:
12387:
12386:
12370:
12369:
12353:
12352:
12325:
12324:
12307:
12301:
12295:
12288:
12247:
12245:
12244:
12239:
12237:
12233:
12232:
12220:
12218:
12210:
12209:
12205:
12204:
12203:
12179:
12177:
12175:
12164:
12159:
12157:
12149:
12148:
12147:
12131:
12129:
12127:
12116:
12111:
12109:
12101:
12100:
12099:
12083:
12081:
12073:
12068:
12066:
12058:
12050:
12048:
12040:
12028:
12026:
12018:
12017:
12016:
12003:
12001:
12000:
11988:
11986:
11978:
11977:
11976:
11963:
11961:
11956:
11955:
11946:
11941:
11939:
11931:
11930:
11929:
11916:
11914:
11913:
11901:
11899:
11891:
11890:
11889:
11876:
11858:
11857:
11845:
11843:
11835:
11834:
11833:
11817:
11815:
11807:
11802:
11800:
11792:
11791:
11787:
11786:
11785:
11773:
11772:
11754:
11752:
11750:
11746:
11745:
11732:
11727:
11725:
11717:
11716:
11715:
11699:
11697:
11695:
11684:
11679:
11677:
11669:
11661:
11659:
11657:
11646:
11634:
11629:
11628:
11627:
11618:
11617:
11607:
11602:
11600:
11592:
11591:
11590:
11577:
11575:
11574:
11562:
11560:
11552:
11551:
11550:
11537:
11535:
11530:
11529:
11520:
11515:
11513:
11505:
11504:
11503:
11490:
11488:
11487:
11475:
11473:
11465:
11464:
11463:
11450:
11432:
11431:
11419:
11417:
11409:
11408:
11396:
11391:
11389:
11381:
11380:
11379:
11363:
11361:
11353:
11348:
11346:
11338:
11337:
11336:
11320:
11318:
11316:
11305:
11300:
11298:
11290:
11289:
11285:
11284:
11283:
11259:
11257:
11255:
11244:
11239:
11237:
11229:
11221:
11219:
11211:
11199:
11193:
11188:
11179:
11174:
11172:
11164:
11163:
11162:
11149:
11147:
11146:
11134:
11132:
11124:
11123:
11122:
11109:
11107:
11102:
11101:
11092:
11087:
11085:
11077:
11076:
11075:
11062:
11060:
11059:
11047:
11045:
11037:
11036:
11035:
11022:
10995:
10993:
10992:
10987:
10985:
10984:
10969:
10968:
10953:
10952:
10937:
10936:
10910:
10908:
10907:
10902:
10900:
10896:
10895:
10883:
10879:
10878:
10876:
10868:
10867:
10866:
10850:
10845:
10843:
10835:
10834:
10833:
10817:
10812:
10810:
10802:
10801:
10800:
10784:
10777:
10769:
10760:
10758:
10750:
10749:
10748:
10735:
10733:
10732:
10720:
10718:
10710:
10709:
10708:
10695:
10693:
10692:
10680:
10678:
10670:
10669:
10668:
10655:
10653:
10652:
10640:
10638:
10630:
10629:
10628:
10615:
10597:
10596:
10584:
10580:
10579:
10577:
10569:
10568:
10567:
10551:
10546:
10544:
10536:
10535:
10534:
10518:
10513:
10511:
10503:
10502:
10501:
10485:
10478:
10470:
10461:
10459:
10451:
10450:
10449:
10436:
10434:
10433:
10421:
10419:
10411:
10410:
10409:
10396:
10394:
10393:
10381:
10379:
10371:
10370:
10369:
10356:
10354:
10353:
10341:
10339:
10331:
10330:
10329:
10316:
10298:
10297:
10285:
10281:
10280:
10278:
10270:
10269:
10268:
10252:
10247:
10245:
10237:
10236:
10235:
10219:
10214:
10212:
10204:
10203:
10202:
10186:
10179:
10171:
10162:
10160:
10152:
10151:
10150:
10137:
10135:
10134:
10122:
10120:
10112:
10111:
10110:
10097:
10095:
10094:
10082:
10080:
10072:
10071:
10070:
10057:
10055:
10054:
10042:
10040:
10032:
10031:
10030:
10017:
9977:
9975:
9974:
9969:
9967:
9962:
9960:
9949:
9944:
9933:
9925:
9922:
9911:
9909:
9901:
9900:
9891:
9857:
9855:
9854:
9849:
9847:
9836:
9831:
9830:
9829:
9819:
9811:
9807:
9802:
9791:
9783:
9778:
9770:
9754:
9752:
9744:
9743:
9734:
9714:
9707:
9705:
9704:
9699:
9697:
9692:
9690:
9679:
9674:
9663:
9658:
9657:
9656:
9646:
9638:
9634:
9629:
9618:
9610:
9605:
9597:
9581:
9579:
9571:
9570:
9561:
9539:
9537:
9536:
9531:
9529:
9525:
9510:
9501:
9486:
9463:momentum density
9454:
9452:
9451:
9446:
9441:
9439:
9437:
9432:
9423:
9422:
9412:
9411:
9410:
9397:
9392:
9391:
9390:
9370:drag coefficient
9365:
9363:
9362:
9357:
9352:
9350:
9348:
9343:
9334:
9333:
9323:
9322:
9313:
9308:
9285:
9283:
9282:
9277:
9272:
9270:
9269:
9268:
9259:
9258:
9247:
9242:
9233:
9228:
9205:
9203:
9202:
9197:
9195:
9194:
9189:
9183:
9180:
9175:
9166:
9165:
9164:
9155:
9154:
9144:
9139:
9138:
9133:
9124:
9123:
9114:
9112:
9110:
9105:
9096:
9095:
9085:
9084:
9075:
9067:
9063:
9062:
9061:
9052:
9050:
9048:
9043:
9034:
9033:
9023:
9022:
9013:
9008:
9007:
9002:
8993:
8992:
8987:
8981:
8980:
8963:
8962:
8950:
8948:
8947:
8946:
8933:
8932:
8931:
8922:
8921:
8916:
8906:
8893:
8891:
8890:
8885:
8883:
8882:
8877:
8871:
8870:
8858:
8857:
8852:
8843:
8842:
8833:
8831:
8830:
8821:
8820:
8811:
8803:
8799:
8798:
8797:
8788:
8787:
8775:
8774:
8769:
8760:
8759:
8754:
8748:
8747:
8737:
8732:
8723:
8722:
8705:
8703:
8702:
8693:
8692:
8683:
8678:
8676:
8675:
8674:
8661:
8660:
8659:
8654:
8648:
8647:
8634:
8632:
8630:
8629:
8620:
8618:
8613:
8604:
8603:
8593:
8580:
8578:
8577:
8572:
8570:
8566:
8564:
8563:
8551:
8542:
8541:
8536:
8528:
8526:
8525:
8513:
8504:
8503:
8492:
8490:
8489:
8480:
8475:
8466:
8465:
8460:
8449:
8448:
8432:
8431:
8415:
8413:
8412:
8403:
8402:
8393:
8384:
8383:
8372:
8370:
8369:
8357:
8348:
8347:
8336:
8334:
8333:
8321:
8312:
8311:
8300:
8298:
8297:
8285:
8276:
8275:
8253:
8244:
8227:
8225:
8224:
8219:
8217:
8203:
8201:
8200:
8195:
8178:
8174:
8129:
8103:
8089:
8082:
8074:
8064:
8045:
8030:
8024:
8015:
8013:
8012:
8007:
8002:
7976:
7922:
7912:
7904:
7902:
7901:
7896:
7894:
7886:
7872:
7854:
7848:
7833:
7822:
7815:
7801:
7799:
7798:
7793:
7788:
7784:
7779:
7778:
7777:
7768:
7759:
7747:
7736:
7722:
7720:
7719:
7714:
7706:
7701:
7686:
7654:
7652:
7651:
7646:
7638:
7637:
7622:
7606:
7604:
7603:
7598:
7592:
7584:
7573:
7572:
7563:
7555:
7550:
7549:
7525:
7523:
7522:
7517:
7509:
7505:
7504:
7496:
7485:
7484:
7475:
7467:
7454:
7434:
7432:
7431:
7426:
7405:
7389:
7387:
7386:
7381:
7364:
7359:
7357:
7356:
7344:
7339:
7335:
7334:
7326:
7315:
7314:
7305:
7297:
7284:
7270:
7264:
7262:
7261:
7256:
7254:
7240:
7226:
7224:
7223:
7218:
7204:
7193:
7188:
7186:
7185:
7173:
7168:
7164:
7163:
7155:
7144:
7143:
7134:
7126:
7110:
7084:
7082:
7081:
7076:
7071:
7054:
7037:
7032:
7030:
7029:
7017:
7012:
7008:
7007:
6999:
6988:
6987:
6978:
6970:
6946:
6944:
6943:
6938:
6936:
6934:
6926:
6925:
6916:
6908:
6891:
6874:
6869:
6867:
6866:
6854:
6849:
6845:
6844:
6836:
6825:
6824:
6815:
6807:
6785:
6775:
6773:
6772:
6767:
6765:
6763:
6755:
6754:
6745:
6737:
6720:
6703:
6698:
6696:
6695:
6683:
6678:
6670:
6666:
6665:
6657:
6652:
6651:
6642:
6634:
6610:
6608:
6607:
6602:
6591:
6586:
6584:
6583:
6571:
6566:
6555:
6547:
6542:
6538:
6530:
6507:
6505:
6504:
6499:
6497:
6489:
6478:
6470:
6465:
6454:
6437:
6433:
6432:
6416:
6408:
6403:
6401:
6393:
6392:
6383:
6368:
6366:
6365:
6360:
6358:
6350:
6346:
6345:
6326:
6312:
6310:
6309:
6304:
6302:
6294:
6290:
6289:
6270:
6266:
6261:
6260:
6259:
6250:
6241:
6229:
6218:
6198:
6194:
6183:
6181:
6180:
6175:
6173:
6162:
6154:
6150:
6149:
6141:
6131:
6130:
6118:
6114:
6113:
6094:
6065:
6063:
6062:
6057:
6051:
6050:
6045:
6041:
6040:
6026:
6009:
6008:
5999:
5998:
5989:
5988:
5978:
5966:
5965:
5960:
5956:
5955:
5951:
5950:
5934:
5916:
5898:
5891:
5885:
5875:
5856:
5846:
5839:
5828:
5817:
5768:
5758:
5752:
5748:
5742:
5729:
5715:
5709:
5698:momentum density
5695:
5687:
5685:
5684:
5679:
5677:
5673:
5658:
5657:
5647:
5639:
5631:
5616:
5615:
5605:
5590:
5589:
5567:
5565:
5564:
5559:
5557:
5549:
5535:
5533:
5525:
5524:
5515:
5483:
5471:
5465:
5463:
5462:
5457:
5455:
5441:
5440:
5428:
5423:
5421:
5416:
5407:
5406:
5396:
5391:
5389:
5381:
5380:
5379:
5366:
5350:
5349:
5333:
5328:
5319:
5318:
5306:
5304:
5296:
5295:
5294:
5281:
5255:
5251:
5250:
5249:
5233:
5228:
5219:
5218:
5206:
5204:
5196:
5195:
5194:
5181:
5171:
5170:
5150:
5149:
5137:
5136:
5116:
5111:
5102:
5101:
5092:
5091:
5068:
5066:
5058:
5057:
5056:
5043:
5038:
5037:
5005:
5003:
5002:
4997:
4995:
4994:
4982:
4981:
4956:
4938:
4936:
4935:
4930:
4928:
4920:
4909:
4901:
4896:
4894:
4886:
4885:
4876:
4867:
4865:
4864:
4859:
4857:
4855:
4847:
4846:
4837:
4832:
4830:
4822:
4821:
4812:
4803:
4801:
4800:
4795:
4760:
4758:
4757:
4752:
4726:
4695:
4654:
4652:
4644:
4643:
4634:
4625:
4623:
4622:
4617:
4615:
4614:
4609:
4608:
4600:
4593:
4592:
4587:
4586:
4578:
4571:
4569:
4561:
4560:
4559:
4551:
4544:
4531:
4529:
4528:
4523:
4521:
4520:
4512:
4506:
4504:
4496:
4495:
4494:
4486:
4479:
4461:
4459:
4458:
4453:
4427:
4425:
4417:
4416:
4407:
4402:
4400:
4392:
4391:
4390:
4382:
4375:
4366:
4364:
4363:
4358:
4317:
4315:
4314:
4309:
4283:
4272:
4264:
4263:
4255:
4238:
4236:
4235:
4230:
4204:
4196:
4195:
4190:
4189:
4181:
4170:
4168:
4167:
4162:
4160:
4146:
4144:
4143:
4138:
4112:
4095:
4094:
4089:
4088:
4080:
4065:
4063:
4062:
4057:
4055:
4051:
4050:
4045:
4042:
4014:
4012:
4004:
4003:
4002:
3986:
3960:
3958:
3950:
3949:
3948:
3932:
3906:
3904:
3896:
3895:
3894:
3878:
3868:
3863:
3829:
3827:
3819:
3818:
3817:
3801:
3775:
3773:
3765:
3764:
3763:
3747:
3721:
3719:
3711:
3710:
3709:
3693:
3683:
3678:
3644:
3642:
3634:
3633:
3632:
3616:
3590:
3588:
3580:
3579:
3578:
3562:
3536:
3534:
3526:
3525:
3524:
3508:
3498:
3493:
3471:
3469:
3468:
3463:
3460:
3455:
3442:
3437:
3421:
3419:
3418:
3413:
3410:
3405:
3387:
3385:
3384:
3379:
3364:
3363:
3335:
3331:
3324:
3322:
3314:
3313:
3312:
3296:
3291:
3290:
3257:
3256:
3228:
3224:
3217:
3215:
3207:
3206:
3205:
3189:
3184:
3183:
3150:
3149:
3121:
3117:
3110:
3108:
3100:
3099:
3098:
3082:
3077:
3076:
3055:
3050:
3021:
3019:
3018:
3013:
3011:
3010:
2995:
2994:
2975:
2973:
2972:
2967:
2946:
2944:
2943:
2938:
2936:
2935:
2913:
2911:
2910:
2905:
2903:
2901:
2893:
2883:
2882:
2866:
2861:
2859:
2851:
2841:
2840:
2807:
2806:
2793:
2788:
2786:
2778:
2774:
2772:
2764:
2754:
2753:
2737:
2717:
2716:
2680:
2679:
2666:
2657:
2655:
2654:
2649:
2634:
2632:
2631:
2626:
2618:
2616:
2608:
2607:
2606:
2590:
2585:
2584:
2565:
2563:
2562:
2557:
2536:
2534:
2533:
2528:
2526:
2525:
2502:
2500:
2499:
2494:
2492:
2491:
2473:
2472:
2450:
2448:
2447:
2442:
2440:
2439:
2421:
2420:
2404:
2402:
2401:
2396:
2391:
2390:
2372:
2371:
2353:
2352:
2331:
2323:
2322:
2314:
2308:
2307:
2299:
2285:
2283:
2282:
2277:
2275:
2274:
2269:
2268:
2260:
2223:
2222:
2214:
2204:
2202:
2201:
2196:
2194:
2193:
2171:
2169:
2168:
2163:
2161:
2160:
2130:
2126:
2124:
2123:
2118:
2116:
2109:
2108:
2096:
2094:
2093:
2081:
2050:
2048:
2047:
2042:
2026:
2025:
1997:
1995:
1994:
1989:
1987:
1986:
1981:
1980:
1972:
1965:
1964:
1959:
1958:
1950:
1943:
1942:
1934:
1923:
1921:
1920:
1915:
1913:
1912:
1907:
1906:
1898:
1884:
1882:
1881:
1876:
1874:
1873:
1868:
1867:
1859:
1841:
1839:
1838:
1833:
1828:
1827:
1797:
1796:
1765:
1763:
1762:
1757:
1755:
1753:
1752:
1751:
1738:
1734:
1733:
1714:
1706:
1705:
1693:
1678:
1676:
1675:
1670:
1668:
1667:
1651:
1649:
1648:
1643:
1600:
1598:
1597:
1592:
1590:
1589:
1581:
1575:
1573:
1565:
1564:
1563:
1555:
1548:
1533:
1531:
1530:
1525:
1504:
1502:
1501:
1496:
1481:
1479:
1478:
1473:
1458:
1456:
1455:
1450:
1448:
1447:
1442:
1434:
1432:
1422:
1418:
1416:
1415:
1410:
1399:
1398:
1390:
1378:
1376:
1375:
1370:
1368:
1367:
1362:
1354:
1352:
1331:
1330:
1322:
1298:
1297:
1289:
1261:
1255:
1248:momentum density
1245:
1237:
1235:
1234:
1229:
1227:
1219:
1205:
1203:
1195:
1194:
1185:
1140:
1138:
1137:
1132:
1130:
1129:
1128:
1113:
1112:
1085:
1062:
1060:
1059:
1054:
1052:
1051:
1044:
1041:
1033:
1032:
1031:
1015:
1009:
1006:
998:
997:
996:
980:
974:
971:
963:
962:
961:
945:
938:
935:
927:
926:
925:
909:
903:
900:
892:
891:
890:
874:
868:
865:
857:
856:
855:
839:
832:
829:
821:
820:
819:
803:
797:
794:
786:
785:
784:
768:
762:
759:
751:
750:
749:
733:
719:
699:
697:
696:
691:
689:
688:
687:
678:
654:
652:
651:
646:
644:
643:
636:
635:
622:
621:
608:
607:
587:
573:
571:
570:
565:
563:
562:
561:
546:
545:
521:
520:
511:
482:
480:
479:
474:
472:
458:
456:
455:
450:
448:
447:
446:
437:
410:
408:
407:
402:
388:
386:
385:
380:
378:
377:
376:
367:
351:
349:
348:
343:
341:
330:
322:
317:
316:
300:
298:
297:
292:
290:
274:
272:
271:
266:
264:
262:
254:
253:
244:
233:
231:
230:
225:
223:
207:
205:
204:
199:
185:
183:
182:
177:
175:
171:
151:
149:
148:
143:
141:
124:
122:
121:
116:
114:
106:
95:
87:
82:
80:
72:
71:
62:
13521:
13520:
13516:
13515:
13514:
13512:
13511:
13510:
13481:
13480:
13479:
13478:
13455:
13447:
13427:
13419:
13416:
13415:
13411:
13405:
13401:
13364:(4797): 36β37.
13354:
13350:
13344:
13317:
13313:
13305:
13301:
13280:
13276:
13269:
13252:
13248:
13241:"Hydrodynamics"
13237:
13233:
13226:
13212:
13205:
13198:
13187:
13179:
13175:
13167:
13161:
13157:
13150:
13134:
13127:
13120:
13094:
13086:
13066:
13058:
13055:
13054:
13050:
13042:
13035:
13002:
12998:
12990:
12981:
12977:
12956:
12952:
12950:
12947:
12946:
12942:
12936:
12929:
12922:
12914:. p. 205.
12904:
12897:
12892:
12887:
12886:
12873:
12867:
12855:
12841:
12840:
12833:
12832:
12826:
12822:
12813:
12808:
12799:
12793:
12789:
12780:
12776:
12770:
12766:
12761:
12755:
12751:
12742:
12738:
12732:
12728:
12722:
12721:
12715:
12711:
12702:
12698:
12692:
12688:
12683:
12677:
12673:
12664:
12659:
12650:
12644:
12640:
12631:
12627:
12621:
12617:
12611:
12610:
12604:
12600:
12591:
12587:
12581:
12577:
12572:
12566:
12562:
12553:
12549:
12543:
12539:
12534:
12528:
12524:
12515:
12510:
12496:
12495:
12488:
12482:
12481:
12478:
12477:
12470:
12469:
12463:
12459:
12453:
12452:
12446:
12442:
12436:
12435:
12429:
12425:
12415:
12414:
12407:
12401:
12400:
12397:
12396:
12389:
12388:
12382:
12378:
12372:
12371:
12365:
12361:
12355:
12354:
12348:
12344:
12334:
12333:
12326:
12320:
12319:
12315:
12313:
12310:
12309:
12303:
12297:
12291:
12289:
12285:
12280:
12253:
12235:
12234:
12228:
12224:
12211:
12196:
12192:
12188:
12184:
12180:
12178:
12168:
12163:
12150:
12140:
12136:
12132:
12130:
12120:
12115:
12102:
12092:
12088:
12084:
12082:
12072:
12059:
12051:
12049:
12039:
12029:
12019:
12012:
12008:
12004:
12002:
11996:
11992:
11979:
11972:
11968:
11964:
11962:
11951:
11947:
11945:
11932:
11925:
11921:
11917:
11915:
11909:
11905:
11892:
11885:
11881:
11877:
11875:
11873:
11866:
11860:
11859:
11853:
11849:
11836:
11826:
11822:
11818:
11816:
11806:
11793:
11778:
11774:
11768:
11764:
11763:
11759:
11755:
11753:
11741:
11737:
11736:
11731:
11718:
11708:
11704:
11700:
11698:
11688:
11683:
11670:
11662:
11660:
11650:
11645:
11635:
11623:
11619:
11613:
11609:
11608:
11606:
11593:
11586:
11582:
11578:
11576:
11570:
11566:
11553:
11546:
11542:
11538:
11536:
11525:
11521:
11519:
11506:
11499:
11495:
11491:
11489:
11483:
11479:
11466:
11459:
11455:
11451:
11449:
11447:
11440:
11434:
11433:
11427:
11423:
11410:
11401:
11397:
11395:
11382:
11372:
11368:
11364:
11362:
11352:
11339:
11329:
11325:
11321:
11319:
11309:
11304:
11291:
11276:
11272:
11268:
11264:
11260:
11258:
11248:
11243:
11230:
11222:
11220:
11210:
11200:
11189:
11184:
11178:
11165:
11158:
11154:
11150:
11148:
11142:
11138:
11125:
11118:
11114:
11110:
11108:
11097:
11093:
11091:
11078:
11071:
11067:
11063:
11061:
11055:
11051:
11038:
11031:
11027:
11023:
11021:
11019:
11012:
11005:
11003:
11000:
10999:
10977:
10973:
10961:
10957:
10945:
10941:
10929:
10925:
10923:
10920:
10919:
10916:
10898:
10897:
10891:
10887:
10869:
10859:
10855:
10851:
10849:
10836:
10826:
10822:
10818:
10816:
10803:
10793:
10789:
10785:
10783:
10782:
10778:
10768:
10761:
10751:
10744:
10740:
10736:
10734:
10728:
10724:
10711:
10704:
10700:
10696:
10694:
10688:
10684:
10671:
10664:
10660:
10656:
10654:
10648:
10644:
10631:
10624:
10620:
10616:
10614:
10612:
10605:
10599:
10598:
10592:
10588:
10570:
10560:
10556:
10552:
10550:
10537:
10527:
10523:
10519:
10517:
10504:
10494:
10490:
10486:
10484:
10483:
10479:
10469:
10462:
10452:
10445:
10441:
10437:
10435:
10429:
10425:
10412:
10405:
10401:
10397:
10395:
10389:
10385:
10372:
10365:
10361:
10357:
10355:
10349:
10345:
10332:
10325:
10321:
10317:
10315:
10313:
10306:
10300:
10299:
10293:
10289:
10271:
10261:
10257:
10253:
10251:
10238:
10228:
10224:
10220:
10218:
10205:
10195:
10191:
10187:
10185:
10184:
10180:
10170:
10163:
10153:
10146:
10142:
10138:
10136:
10130:
10126:
10113:
10106:
10102:
10098:
10096:
10090:
10086:
10073:
10066:
10062:
10058:
10056:
10050:
10046:
10033:
10026:
10022:
10018:
10016:
10014:
10007:
10000:
9998:
9995:
9994:
9989:
9984:
9979:
9963:
9953:
9948:
9940:
9924:
9915:
9902:
9896:
9892:
9890:
9888:
9885:
9884:
9859:
9843:
9825:
9824:
9820:
9818:
9795:
9787:
9779:
9769:
9768:
9764:
9745:
9739:
9735:
9733:
9731:
9728:
9727:
9712:
9709:
9693:
9683:
9678:
9670:
9652:
9651:
9647:
9645:
9622:
9614:
9606:
9596:
9595:
9591:
9572:
9566:
9562:
9560:
9558:
9555:
9554:
9527:
9526:
9521:
9511:
9506:
9503:
9502:
9497:
9487:
9482:
9478:
9476:
9473:
9472:
9433:
9428:
9418:
9414:
9413:
9406:
9402:
9398:
9396:
9386:
9385:
9381:
9379:
9376:
9375:
9344:
9339:
9329:
9325:
9324:
9318:
9314:
9312:
9301:
9299:
9296:
9295:
9264:
9260:
9254:
9250:
9249:
9243:
9238:
9232:
9221:
9219:
9216:
9215:
9190:
9185:
9184:
9176:
9171:
9160:
9156:
9150:
9146:
9145:
9143:
9134:
9129:
9128:
9119:
9115:
9106:
9101:
9091:
9087:
9086:
9080:
9076:
9074:
9057:
9053:
9044:
9039:
9029:
9025:
9024:
9018:
9014:
9012:
9003:
8998:
8997:
8988:
8983:
8982:
8976:
8972:
8971:
8967:
8958:
8954:
8942:
8938:
8934:
8927:
8923:
8917:
8912:
8911:
8907:
8905:
8903:
8900:
8899:
8878:
8873:
8872:
8866:
8862:
8853:
8848:
8847:
8838:
8834:
8826:
8822:
8816:
8812:
8810:
8793:
8789:
8783:
8779:
8770:
8765:
8764:
8755:
8750:
8749:
8743:
8739:
8733:
8728:
8718:
8714:
8713:
8709:
8698:
8694:
8688:
8684:
8682:
8670:
8666:
8662:
8655:
8650:
8649:
8643:
8639:
8635:
8633:
8625:
8621:
8614:
8609:
8599:
8595:
8594:
8592:
8590:
8587:
8586:
8568:
8567:
8559:
8555:
8550:
8543:
8537:
8532:
8531:
8529:
8521:
8517:
8512:
8505:
8499:
8495:
8493:
8485:
8481:
8476:
8474:
8467:
8461:
8456:
8455:
8453:
8444:
8440:
8433:
8427:
8423:
8420:
8419:
8408:
8404:
8398:
8394:
8392:
8385:
8379:
8375:
8373:
8365:
8361:
8356:
8349:
8343:
8339:
8337:
8329:
8325:
8320:
8313:
8307:
8303:
8301:
8293:
8289:
8284:
8277:
8271:
8267:
8263:
8261:
8258:
8257:
8252:
8246:
8243:
8237:
8234:
8213:
8211:
8208:
8207:
8164:
8160:
8125:
8111:
8108:
8107:
8091:
8084:
8080:
8066:
8062:
8041:
8038:
8036:External forces
8026:
8022:
7998:
7972:
7964:
7961:
7960:
7952:Euler equations
7918:
7915:identity matrix
7908:
7890:
7882:
7868:
7866:
7863:
7862:
7850:
7843:
7840:normal stresses
7828:
7817:
7810:
7807:
7773:
7769:
7764:
7760:
7758:
7754:
7743:
7732:
7730:
7727:
7726:
7702:
7697:
7692:
7689:
7688:
7678:
7667:
7633:
7629:
7618:
7616:
7613:
7612:
7583:
7568:
7564:
7554:
7545:
7541:
7539:
7536:
7535:
7495:
7480:
7476:
7466:
7465:
7461:
7450:
7448:
7445:
7444:
7401:
7399:
7396:
7395:
7360:
7352:
7348:
7343:
7325:
7310:
7306:
7296:
7295:
7291:
7280:
7278:
7275:
7274:
7266:
7250:
7236:
7234:
7231:
7230:
7200:
7189:
7181:
7177:
7172:
7154:
7139:
7135:
7125:
7124:
7120:
7106:
7104:
7101:
7100:
7067:
7050:
7033:
7025:
7021:
7016:
6998:
6983:
6979:
6969:
6968:
6964:
6956:
6953:
6952:
6927:
6921:
6917:
6915:
6904:
6887:
6870:
6862:
6858:
6853:
6835:
6820:
6816:
6806:
6805:
6801:
6793:
6790:
6789:
6783:
6756:
6750:
6746:
6744:
6733:
6716:
6699:
6691:
6687:
6682:
6674:
6656:
6647:
6643:
6633:
6632:
6628:
6620:
6617:
6616:
6587:
6579:
6575:
6570:
6562:
6546:
6529:
6525:
6517:
6514:
6513:
6493:
6485:
6469:
6461:
6450:
6428:
6424:
6420:
6407:
6394:
6388:
6384:
6382:
6380:
6377:
6376:
6354:
6341:
6334:
6330:
6322:
6320:
6317:
6316:
6298:
6285:
6278:
6274:
6255:
6251:
6246:
6242:
6240:
6236:
6225:
6214:
6212:
6209:
6208:
6196:
6193:
6187:
6169:
6158:
6145:
6137:
6136:
6132:
6126:
6122:
6109:
6102:
6098:
6090:
6088:
6085:
6084:
6071:
6046:
6036:
6022:
6018:
6014:
6013:
6004:
6000:
5994:
5990:
5984:
5980:
5974:
5961:
5946:
5942:
5938:
5930:
5929:
5925:
5924:
5922:
5919:
5918:
5914:
5910:
5905:
5900:
5893:
5887:
5880:
5866:
5863:
5852:
5841:
5830:
5819:
5809:
5782:
5774:Euler equations
5764:
5754:
5750:
5744:
5738:
5721:
5711:
5705:
5691:
5675:
5674:
5669:
5659:
5653:
5652:
5649:
5648:
5643:
5635:
5627:
5617:
5611:
5610:
5607:
5606:
5601:
5591:
5585:
5584:
5580:
5578:
5575:
5574:
5569:
5553:
5545:
5526:
5520:
5516:
5514:
5512:
5509:
5508:
5496:
5490:
5482:
5478:
5469:
5453:
5452:
5442:
5436:
5432:
5417:
5412:
5402:
5398:
5397:
5395:
5382:
5375:
5371:
5367:
5365:
5362:
5361:
5351:
5345:
5341:
5329:
5324:
5314:
5310:
5297:
5290:
5286:
5282:
5280:
5274:
5273:
5263:
5245:
5241:
5229:
5224:
5214:
5210:
5197:
5190:
5186:
5182:
5180:
5176:
5172:
5166:
5162:
5159:
5158:
5145:
5141:
5132:
5128:
5112:
5107:
5097:
5093:
5087:
5083:
5076:
5059:
5052:
5048:
5044:
5042:
5033:
5029:
5025:
5023:
5020:
5019:
4990:
4986:
4977:
4973:
4968:
4965:
4964:
4954:
4947:
4924:
4916:
4900:
4887:
4881:
4877:
4875:
4873:
4870:
4869:
4848:
4842:
4838:
4836:
4823:
4817:
4813:
4811:
4809:
4806:
4805:
4768:
4765:
4764:
4722:
4691:
4645:
4639:
4635:
4633:
4631:
4628:
4627:
4610:
4599:
4598:
4597:
4588:
4577:
4576:
4575:
4562:
4550:
4549:
4545:
4543:
4541:
4538:
4537:
4511:
4510:
4497:
4485:
4484:
4480:
4478:
4476:
4473:
4472:
4467:
4418:
4412:
4408:
4406:
4393:
4381:
4380:
4376:
4374:
4372:
4369:
4368:
4325:
4322:
4321:
4279:
4268:
4254:
4253:
4251:
4248:
4247:
4244:
4200:
4191:
4180:
4179:
4178:
4176:
4173:
4172:
4156:
4154:
4151:
4150:
4108:
4090:
4079:
4078:
4077:
4075:
4072:
4071:
4053:
4052:
4046:
4043:
4037:
4036:
4005:
3995:
3991:
3987:
3985:
3951:
3941:
3937:
3933:
3931:
3897:
3887:
3883:
3879:
3877:
3870:
3864:
3859:
3852:
3851:
3820:
3810:
3806:
3802:
3800:
3766:
3756:
3752:
3748:
3746:
3712:
3702:
3698:
3694:
3692:
3685:
3679:
3674:
3667:
3666:
3635:
3625:
3621:
3617:
3615:
3581:
3571:
3567:
3563:
3561:
3527:
3517:
3513:
3509:
3507:
3500:
3494:
3489:
3481:
3479:
3476:
3475:
3456:
3451:
3438:
3433:
3427:
3424:
3423:
3406:
3401:
3395:
3392:
3391:
3390:After ordering
3356:
3352:
3315:
3305:
3301:
3297:
3295:
3283:
3279:
3278:
3274:
3249:
3245:
3208:
3198:
3194:
3190:
3188:
3176:
3172:
3171:
3167:
3142:
3138:
3101:
3091:
3087:
3083:
3081:
3069:
3065:
3064:
3060:
3051:
3046:
3040:
3037:
3036:
3003:
2999:
2987:
2983:
2981:
2978:
2977:
2952:
2949:
2948:
2928:
2924:
2922:
2919:
2918:
2894:
2875:
2871:
2867:
2865:
2852:
2833:
2829:
2799:
2795:
2794:
2792:
2779:
2765:
2746:
2742:
2738:
2736:
2709:
2705:
2672:
2668:
2667:
2665:
2663:
2660:
2659:
2640:
2637:
2636:
2609:
2599:
2595:
2591:
2589:
2577:
2573:
2571:
2568:
2567:
2542:
2539:
2538:
2518:
2514:
2512:
2509:
2508:
2484:
2480:
2465:
2461:
2456:
2453:
2452:
2432:
2428:
2416:
2412:
2410:
2407:
2406:
2383:
2379:
2364:
2360:
2345:
2341:
2327:
2313:
2312:
2298:
2297:
2295:
2292:
2291:
2270:
2259:
2258:
2257:
2213:
2212:
2210:
2207:
2206:
2186:
2182:
2177:
2174:
2173:
2153:
2149:
2144:
2141:
2140:
2104:
2100:
2089:
2085:
2080:
2058:
2056:
2053:
2052:
2018:
2014:
2009:
2006:
2005:
1982:
1971:
1970:
1969:
1960:
1949:
1948:
1947:
1933:
1932:
1930:
1927:
1926:
1908:
1897:
1896:
1895:
1893:
1890:
1889:
1869:
1858:
1857:
1856:
1854:
1851:
1850:
1823:
1819:
1792:
1788:
1771:
1768:
1767:
1747:
1743:
1739:
1729:
1725:
1715:
1713:
1701:
1697:
1686:
1684:
1681:
1680:
1663:
1659:
1657:
1654:
1653:
1628:
1625:
1624:
1609:
1580:
1579:
1566:
1554:
1553:
1549:
1547:
1545:
1542:
1541:
1510:
1507:
1506:
1487:
1484:
1483:
1464:
1461:
1460:
1433:
1431:
1430:
1428:
1425:
1424:
1420:
1389:
1388:
1386:
1383:
1382:
1353:
1351:
1350:
1321:
1320:
1288:
1287:
1285:
1282:
1281:
1272:
1257:
1251:
1241:
1223:
1215:
1196:
1190:
1186:
1184:
1182:
1179:
1178:
1160:("column") nor
1121:
1117:
1105:
1101:
1081:
1074:
1072:
1069:
1068:
1046:
1045:
1034:
1024:
1020:
1016:
1013:
999:
989:
985:
981:
978:
964:
954:
950:
946:
943:
940:
939:
928:
918:
914:
910:
907:
893:
883:
879:
875:
872:
858:
848:
844:
840:
837:
834:
833:
822:
812:
808:
804:
801:
787:
777:
773:
769:
766:
752:
742:
738:
734:
731:
724:
723:
715:
707:
704:
703:
683:
679:
674:
670:
668:
665:
664:
638:
637:
631:
627:
624:
623:
617:
613:
610:
609:
603:
599:
592:
591:
583:
581:
578:
577:
554:
550:
538:
534:
516:
512:
507:
494:
492:
489:
488:
468:
466:
463:
462:
442:
438:
433:
426:
424:
421:
420:
419:holds), (unit:
396:
393:
392:
372:
368:
363:
359:
357:
354:
353:
337:
326:
318:
312:
308:
306:
303:
302:
286:
284:
281:
280:
255:
249:
245:
243:
241:
238:
237:
219:
217:
214:
213:
193:
190:
189:
167:
163:
161:
158:
157:
137:
135:
132:
131:
110:
102:
86:
73:
67:
63:
61:
59:
56:
55:
48:
17:
12:
11:
5:
13519:
13509:
13508:
13503:
13498:
13493:
13477:
13476:
13462:
13458:
13454:
13450:
13446:
13443:
13440:
13437:
13434:
13430:
13426:
13423:
13399:
13348:
13342:
13328:Sands, Matthew
13311:
13309:, p. 142.
13299:
13274:
13267:
13246:
13231:
13224:
13203:
13196:
13173:
13155:
13148:
13125:
13118:
13101:
13097:
13093:
13089:
13085:
13082:
13079:
13076:
13073:
13069:
13065:
13062:
13033:
13019:
13016:
13013:
13010:
13005:
13001:
12997:
12993:
12987:
12984:
12980:
12976:
12973:
12968:
12965:
12962:
12959:
12955:
12927:
12920:
12894:
12893:
12891:
12888:
12885:
12884:
12837:
12829:
12825:
12821:
12816:
12811:
12807:
12803:
12800:
12796:
12792:
12788:
12783:
12779:
12773:
12769:
12765:
12762:
12758:
12754:
12750:
12745:
12741:
12735:
12731:
12727:
12724:
12723:
12718:
12714:
12710:
12705:
12701:
12695:
12691:
12687:
12684:
12680:
12676:
12672:
12667:
12662:
12658:
12654:
12651:
12647:
12643:
12639:
12634:
12630:
12624:
12620:
12616:
12613:
12612:
12607:
12603:
12599:
12594:
12590:
12584:
12580:
12576:
12573:
12569:
12565:
12561:
12556:
12552:
12546:
12542:
12538:
12535:
12531:
12527:
12523:
12518:
12513:
12509:
12505:
12502:
12501:
12499:
12494:
12491:
12489:
12485:
12480:
12479:
12474:
12466:
12462:
12458:
12455:
12454:
12449:
12445:
12441:
12438:
12437:
12432:
12428:
12424:
12421:
12420:
12418:
12413:
12410:
12408:
12404:
12399:
12398:
12393:
12385:
12381:
12377:
12374:
12373:
12368:
12364:
12360:
12357:
12356:
12351:
12347:
12343:
12340:
12339:
12337:
12332:
12329:
12327:
12323:
12318:
12317:
12282:
12281:
12279:
12276:
12275:
12274:
12269:
12264:
12259:
12252:
12249:
12231:
12227:
12223:
12217:
12214:
12208:
12202:
12199:
12195:
12191:
12187:
12183:
12174:
12171:
12167:
12162:
12156:
12153:
12146:
12143:
12139:
12135:
12126:
12123:
12119:
12114:
12108:
12105:
12098:
12095:
12091:
12087:
12079:
12076:
12071:
12065:
12062:
12057:
12054:
12046:
12043:
12038:
12035:
12032:
12030:
12025:
12022:
12015:
12011:
12007:
11999:
11995:
11991:
11985:
11982:
11975:
11971:
11967:
11959:
11954:
11950:
11944:
11938:
11935:
11928:
11924:
11920:
11912:
11908:
11904:
11898:
11895:
11888:
11884:
11880:
11874:
11872:
11869:
11867:
11865:
11862:
11861:
11856:
11852:
11848:
11842:
11839:
11832:
11829:
11825:
11821:
11813:
11810:
11805:
11799:
11796:
11790:
11784:
11781:
11777:
11771:
11767:
11762:
11758:
11749:
11744:
11740:
11735:
11730:
11724:
11721:
11714:
11711:
11707:
11703:
11694:
11691:
11687:
11682:
11676:
11673:
11668:
11665:
11656:
11653:
11649:
11644:
11641:
11638:
11636:
11632:
11626:
11622:
11616:
11612:
11605:
11599:
11596:
11589:
11585:
11581:
11573:
11569:
11565:
11559:
11556:
11549:
11545:
11541:
11533:
11528:
11524:
11518:
11512:
11509:
11502:
11498:
11494:
11486:
11482:
11478:
11472:
11469:
11462:
11458:
11454:
11448:
11446:
11443:
11441:
11439:
11436:
11435:
11430:
11426:
11422:
11416:
11413:
11407:
11404:
11400:
11394:
11388:
11385:
11378:
11375:
11371:
11367:
11359:
11356:
11351:
11345:
11342:
11335:
11332:
11328:
11324:
11315:
11312:
11308:
11303:
11297:
11294:
11288:
11282:
11279:
11275:
11271:
11267:
11263:
11254:
11251:
11247:
11242:
11236:
11233:
11228:
11225:
11217:
11214:
11209:
11206:
11203:
11201:
11197:
11192:
11187:
11183:
11177:
11171:
11168:
11161:
11157:
11153:
11145:
11141:
11137:
11131:
11128:
11121:
11117:
11113:
11105:
11100:
11096:
11090:
11084:
11081:
11074:
11070:
11066:
11058:
11054:
11050:
11044:
11041:
11034:
11030:
11026:
11020:
11018:
11015:
11013:
11011:
11008:
11007:
10983:
10980:
10976:
10972:
10967:
10964:
10960:
10956:
10951:
10948:
10944:
10940:
10935:
10932:
10928:
10915:
10912:
10894:
10890:
10886:
10882:
10875:
10872:
10865:
10862:
10858:
10854:
10848:
10842:
10839:
10832:
10829:
10825:
10821:
10815:
10809:
10806:
10799:
10796:
10792:
10788:
10781:
10775:
10772:
10767:
10764:
10762:
10757:
10754:
10747:
10743:
10739:
10731:
10727:
10723:
10717:
10714:
10707:
10703:
10699:
10691:
10687:
10683:
10677:
10674:
10667:
10663:
10659:
10651:
10647:
10643:
10637:
10634:
10627:
10623:
10619:
10613:
10611:
10608:
10606:
10604:
10601:
10600:
10595:
10591:
10587:
10583:
10576:
10573:
10566:
10563:
10559:
10555:
10549:
10543:
10540:
10533:
10530:
10526:
10522:
10516:
10510:
10507:
10500:
10497:
10493:
10489:
10482:
10476:
10473:
10468:
10465:
10463:
10458:
10455:
10448:
10444:
10440:
10432:
10428:
10424:
10418:
10415:
10408:
10404:
10400:
10392:
10388:
10384:
10378:
10375:
10368:
10364:
10360:
10352:
10348:
10344:
10338:
10335:
10328:
10324:
10320:
10314:
10312:
10309:
10307:
10305:
10302:
10301:
10296:
10292:
10288:
10284:
10277:
10274:
10267:
10264:
10260:
10256:
10250:
10244:
10241:
10234:
10231:
10227:
10223:
10217:
10211:
10208:
10201:
10198:
10194:
10190:
10183:
10177:
10174:
10169:
10166:
10164:
10159:
10156:
10149:
10145:
10141:
10133:
10129:
10125:
10119:
10116:
10109:
10105:
10101:
10093:
10089:
10085:
10079:
10076:
10069:
10065:
10061:
10053:
10049:
10045:
10039:
10036:
10029:
10025:
10021:
10015:
10013:
10010:
10008:
10006:
10003:
10002:
9988:
9985:
9983:
9980:
9966:
9959:
9956:
9952:
9947:
9943:
9939:
9936:
9931:
9928:
9921:
9918:
9914:
9908:
9905:
9899:
9895:
9874:
9846:
9842:
9839:
9834:
9828:
9823:
9817:
9814:
9810:
9806:
9801:
9798:
9794:
9790:
9786:
9782:
9776:
9773:
9767:
9763:
9760:
9757:
9751:
9748:
9742:
9738:
9717:
9696:
9689:
9686:
9682:
9677:
9673:
9669:
9666:
9661:
9655:
9650:
9644:
9641:
9637:
9633:
9628:
9625:
9621:
9617:
9613:
9609:
9603:
9600:
9594:
9590:
9587:
9584:
9578:
9575:
9569:
9565:
9544:
9524:
9520:
9517:
9514:
9512:
9509:
9505:
9504:
9500:
9496:
9493:
9490:
9488:
9485:
9481:
9480:
9444:
9436:
9431:
9427:
9421:
9417:
9409:
9405:
9401:
9395:
9389:
9384:
9355:
9347:
9342:
9338:
9332:
9328:
9321:
9317:
9311:
9307:
9304:
9275:
9267:
9263:
9257:
9253:
9246:
9241:
9237:
9231:
9227:
9224:
9193:
9188:
9179:
9174:
9170:
9163:
9159:
9153:
9149:
9142:
9137:
9132:
9127:
9122:
9118:
9109:
9104:
9100:
9094:
9090:
9083:
9079:
9073:
9070:
9066:
9060:
9056:
9047:
9042:
9038:
9032:
9028:
9021:
9017:
9011:
9006:
9001:
8996:
8991:
8986:
8979:
8975:
8970:
8966:
8961:
8957:
8953:
8945:
8941:
8937:
8930:
8926:
8920:
8915:
8910:
8881:
8876:
8869:
8865:
8861:
8856:
8851:
8846:
8841:
8837:
8829:
8825:
8819:
8815:
8809:
8806:
8802:
8796:
8792:
8786:
8782:
8778:
8773:
8768:
8763:
8758:
8753:
8746:
8742:
8736:
8731:
8727:
8721:
8717:
8712:
8708:
8701:
8697:
8691:
8687:
8681:
8673:
8669:
8665:
8658:
8653:
8646:
8642:
8638:
8628:
8624:
8617:
8612:
8608:
8602:
8598:
8562:
8558:
8554:
8549:
8546:
8544:
8540:
8535:
8530:
8524:
8520:
8516:
8511:
8508:
8506:
8502:
8498:
8494:
8488:
8484:
8479:
8473:
8470:
8468:
8464:
8459:
8454:
8452:
8447:
8443:
8439:
8436:
8434:
8430:
8426:
8422:
8421:
8418:
8411:
8407:
8401:
8397:
8391:
8388:
8386:
8382:
8378:
8374:
8368:
8364:
8360:
8355:
8352:
8350:
8346:
8342:
8338:
8332:
8328:
8324:
8319:
8316:
8314:
8310:
8306:
8302:
8296:
8292:
8288:
8283:
8280:
8278:
8274:
8270:
8266:
8265:
8250:
8241:
8233:
8230:
8216:
8193:
8190:
8187:
8184:
8181:
8177:
8173:
8170:
8167:
8163:
8159:
8156:
8153:
8150:
8147:
8144:
8141:
8138:
8135:
8132:
8128:
8124:
8121:
8118:
8115:
8037:
8034:
8005:
8001:
7997:
7994:
7991:
7988:
7985:
7982:
7979:
7975:
7971:
7968:
7946:. By assuming
7893:
7889:
7885:
7881:
7878:
7875:
7871:
7806:
7803:
7791:
7787:
7782:
7776:
7772:
7767:
7763:
7757:
7753:
7750:
7746:
7742:
7739:
7735:
7712:
7709:
7705:
7700:
7696:
7666:
7663:
7644:
7641:
7636:
7632:
7628:
7625:
7621:
7596:
7590:
7587:
7582:
7579:
7576:
7571:
7567:
7561:
7558:
7553:
7548:
7544:
7515:
7512:
7508:
7502:
7499:
7494:
7491:
7488:
7483:
7479:
7473:
7470:
7464:
7460:
7457:
7453:
7424:
7420:
7417:
7414:
7411:
7408:
7404:
7379:
7376:
7373:
7370:
7367:
7363:
7355:
7351:
7347:
7342:
7338:
7332:
7329:
7324:
7321:
7318:
7313:
7309:
7303:
7300:
7294:
7290:
7287:
7283:
7253:
7249:
7246:
7243:
7239:
7216:
7213:
7210:
7207:
7203:
7199:
7196:
7192:
7184:
7180:
7176:
7171:
7167:
7161:
7158:
7153:
7150:
7147:
7142:
7138:
7132:
7129:
7123:
7119:
7116:
7113:
7109:
7074:
7070:
7066:
7063:
7060:
7057:
7053:
7049:
7046:
7043:
7040:
7036:
7028:
7024:
7020:
7015:
7011:
7005:
7002:
6997:
6994:
6991:
6986:
6982:
6976:
6973:
6967:
6963:
6960:
6933:
6930:
6924:
6920:
6914:
6911:
6907:
6903:
6900:
6897:
6894:
6890:
6886:
6883:
6880:
6877:
6873:
6865:
6861:
6857:
6852:
6848:
6842:
6839:
6834:
6831:
6828:
6823:
6819:
6813:
6810:
6804:
6800:
6797:
6762:
6759:
6753:
6749:
6743:
6740:
6736:
6732:
6729:
6726:
6723:
6719:
6715:
6712:
6709:
6706:
6702:
6694:
6690:
6686:
6681:
6677:
6673:
6669:
6663:
6660:
6655:
6650:
6646:
6640:
6637:
6631:
6627:
6624:
6600:
6597:
6594:
6590:
6582:
6578:
6574:
6569:
6565:
6561:
6558:
6553:
6550:
6545:
6541:
6536:
6533:
6528:
6524:
6521:
6496:
6492:
6488:
6484:
6481:
6476:
6473:
6468:
6464:
6460:
6457:
6453:
6449:
6446:
6443:
6440:
6436:
6431:
6427:
6423:
6419:
6414:
6411:
6406:
6400:
6397:
6391:
6387:
6369:is called the
6357:
6353:
6349:
6344:
6340:
6337:
6333:
6329:
6325:
6301:
6297:
6293:
6288:
6284:
6281:
6277:
6273:
6269:
6264:
6258:
6254:
6249:
6245:
6239:
6235:
6232:
6228:
6224:
6221:
6217:
6189:
6172:
6168:
6165:
6161:
6157:
6153:
6148:
6144:
6140:
6135:
6129:
6125:
6121:
6117:
6112:
6108:
6105:
6101:
6097:
6093:
6070:
6067:
6055:
6049:
6044:
6039:
6035:
6032:
6029:
6025:
6021:
6017:
6012:
6007:
6003:
5997:
5993:
5987:
5983:
5977:
5973:
5969:
5964:
5959:
5954:
5949:
5945:
5941:
5937:
5933:
5928:
5912:
5908:
5901:
5862:
5859:
5781:
5778:
5672:
5668:
5665:
5662:
5660:
5656:
5651:
5650:
5646:
5642:
5638:
5634:
5630:
5626:
5623:
5620:
5618:
5614:
5609:
5608:
5604:
5600:
5597:
5594:
5592:
5588:
5583:
5582:
5556:
5552:
5548:
5544:
5541:
5538:
5532:
5529:
5523:
5519:
5500:
5489:
5486:
5480:
5451:
5448:
5445:
5443:
5439:
5435:
5431:
5426:
5420:
5415:
5411:
5405:
5401:
5394:
5388:
5385:
5378:
5374:
5370:
5364:
5363:
5360:
5357:
5354:
5352:
5348:
5344:
5340:
5337:
5332:
5327:
5323:
5317:
5313:
5309:
5303:
5300:
5293:
5289:
5285:
5279:
5276:
5275:
5272:
5269:
5266:
5264:
5262:
5259:
5254:
5248:
5244:
5240:
5237:
5232:
5227:
5223:
5217:
5213:
5209:
5203:
5200:
5193:
5189:
5185:
5179:
5175:
5169:
5165:
5161:
5160:
5157:
5154:
5148:
5144:
5140:
5135:
5131:
5127:
5124:
5121:
5115:
5110:
5106:
5100:
5096:
5090:
5086:
5082:
5079:
5077:
5075:
5072:
5065:
5062:
5055:
5051:
5047:
5041:
5036:
5032:
5028:
5027:
4993:
4989:
4985:
4980:
4976:
4972:
4959:control volume
4946:
4943:
4927:
4923:
4919:
4915:
4912:
4907:
4904:
4899:
4893:
4890:
4884:
4880:
4854:
4851:
4845:
4841:
4835:
4829:
4826:
4820:
4816:
4804:, and because
4793:
4790:
4786:
4783:
4779:
4776:
4772:
4750:
4747:
4743:
4740:
4736:
4733:
4729:
4725:
4721:
4718:
4715:
4711:
4708:
4704:
4701:
4698:
4694:
4690:
4687:
4684:
4681:
4678:
4675:
4671:
4668:
4664:
4661:
4657:
4651:
4648:
4642:
4638:
4613:
4606:
4603:
4596:
4591:
4584:
4581:
4574:
4568:
4565:
4557:
4554:
4548:
4518:
4515:
4509:
4503:
4500:
4492:
4489:
4483:
4466:
4463:
4451:
4448:
4444:
4441:
4437:
4434:
4430:
4424:
4421:
4415:
4411:
4405:
4399:
4396:
4388:
4385:
4379:
4356:
4353:
4349:
4346:
4342:
4339:
4335:
4332:
4329:
4307:
4304:
4300:
4297:
4293:
4290:
4286:
4282:
4278:
4275:
4271:
4267:
4261:
4258:
4243:
4240:
4228:
4225:
4221:
4218:
4214:
4211:
4207:
4203:
4199:
4194:
4187:
4184:
4159:
4136:
4133:
4129:
4126:
4122:
4119:
4115:
4111:
4107:
4104:
4101:
4098:
4093:
4086:
4083:
4044:
4041:
4035:
4032:
4028:
4025:
4021:
4018:
4011:
4008:
4001:
3998:
3994:
3990:
3984:
3981:
3978:
3974:
3971:
3967:
3964:
3957:
3954:
3947:
3944:
3940:
3936:
3930:
3927:
3924:
3920:
3917:
3913:
3910:
3903:
3900:
3893:
3890:
3886:
3882:
3876:
3873:
3871:
3867:
3862:
3858:
3854:
3853:
3850:
3847:
3843:
3840:
3836:
3833:
3826:
3823:
3816:
3813:
3809:
3805:
3799:
3796:
3793:
3789:
3786:
3782:
3779:
3772:
3769:
3762:
3759:
3755:
3751:
3745:
3742:
3739:
3735:
3732:
3728:
3725:
3718:
3715:
3708:
3705:
3701:
3697:
3691:
3688:
3686:
3682:
3677:
3673:
3669:
3668:
3665:
3662:
3658:
3655:
3651:
3648:
3641:
3638:
3631:
3628:
3624:
3620:
3614:
3611:
3608:
3604:
3601:
3597:
3594:
3587:
3584:
3577:
3574:
3570:
3566:
3560:
3557:
3554:
3550:
3547:
3543:
3540:
3533:
3530:
3523:
3520:
3516:
3512:
3506:
3503:
3501:
3497:
3492:
3488:
3484:
3483:
3459:
3454:
3450:
3446:
3441:
3436:
3432:
3409:
3404:
3400:
3377:
3374:
3370:
3367:
3362:
3359:
3355:
3351:
3348:
3345:
3341:
3338:
3334:
3330:
3327:
3321:
3318:
3311:
3308:
3304:
3300:
3294:
3289:
3286:
3282:
3277:
3273:
3270:
3267:
3263:
3260:
3255:
3252:
3248:
3244:
3241:
3238:
3234:
3231:
3227:
3223:
3220:
3214:
3211:
3204:
3201:
3197:
3193:
3187:
3182:
3179:
3175:
3170:
3166:
3163:
3160:
3156:
3153:
3148:
3145:
3141:
3137:
3134:
3131:
3127:
3124:
3120:
3116:
3113:
3107:
3104:
3097:
3094:
3090:
3086:
3080:
3075:
3072:
3068:
3063:
3059:
3054:
3049:
3045:
3025:
3024:
3009:
3006:
3002:
2998:
2993:
2990:
2986:
2965:
2962:
2959:
2956:
2934:
2931:
2927:
2900:
2897:
2892:
2889:
2886:
2881:
2878:
2874:
2870:
2864:
2858:
2855:
2850:
2847:
2844:
2839:
2836:
2832:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2805:
2802:
2798:
2791:
2785:
2782:
2777:
2771:
2768:
2763:
2760:
2757:
2752:
2749:
2745:
2741:
2735:
2732:
2729:
2726:
2723:
2720:
2715:
2712:
2708:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2678:
2675:
2671:
2658:goes to zero,
2647:
2644:
2624:
2621:
2615:
2612:
2605:
2602:
2598:
2594:
2588:
2583:
2580:
2576:
2555:
2552:
2549:
2546:
2524:
2521:
2517:
2490:
2487:
2483:
2479:
2476:
2471:
2468:
2464:
2460:
2438:
2435:
2431:
2427:
2424:
2419:
2415:
2394:
2389:
2386:
2382:
2378:
2375:
2370:
2367:
2363:
2359:
2356:
2351:
2348:
2344:
2340:
2337:
2334:
2330:
2326:
2320:
2317:
2311:
2305:
2302:
2290:by definition
2273:
2266:
2263:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2220:
2217:
2192:
2189:
2185:
2181:
2159:
2156:
2152:
2148:
2135:
2134:
2115:
2112:
2107:
2103:
2099:
2092:
2088:
2084:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2040:
2037:
2033:
2030:
2024:
2021:
2017:
2013:
1985:
1978:
1975:
1968:
1963:
1956:
1953:
1946:
1940:
1937:
1911:
1904:
1901:
1887:surface forces
1872:
1865:
1862:
1831:
1826:
1822:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1795:
1791:
1787:
1784:
1781:
1778:
1775:
1750:
1746:
1742:
1737:
1732:
1728:
1724:
1721:
1718:
1712:
1709:
1704:
1700:
1696:
1692:
1689:
1666:
1662:
1641:
1638:
1635:
1632:
1608:
1605:
1587:
1584:
1578:
1572:
1569:
1561:
1558:
1552:
1523:
1520:
1517:
1514:
1494:
1491:
1471:
1468:
1445:
1440:
1437:
1408:
1405:
1402:
1396:
1393:
1365:
1360:
1357:
1349:
1346:
1343:
1340:
1337:
1334:
1328:
1325:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1295:
1292:
1271:
1268:
1226:
1222:
1218:
1214:
1211:
1208:
1202:
1199:
1193:
1189:
1162:contravariants
1143:
1142:
1127:
1124:
1120:
1116:
1111:
1108:
1104:
1100:
1097:
1094:
1091:
1088:
1084:
1080:
1077:
1050:
1040:
1037:
1030:
1027:
1023:
1019:
1012:
1005:
1002:
995:
992:
988:
984:
977:
970:
967:
960:
957:
953:
949:
942:
941:
934:
931:
924:
921:
917:
913:
906:
899:
896:
889:
886:
882:
878:
871:
864:
861:
854:
851:
847:
843:
836:
835:
828:
825:
818:
815:
811:
807:
800:
793:
790:
783:
780:
776:
772:
765:
758:
755:
748:
745:
741:
737:
730:
729:
727:
722:
718:
714:
711:
701:
686:
682:
677:
673:
642:
634:
630:
626:
625:
620:
616:
612:
611:
606:
602:
598:
597:
595:
590:
586:
575:
560:
557:
553:
549:
544:
541:
537:
533:
530:
527:
524:
519:
515:
510:
506:
503:
500:
497:
471:
460:
445:
441:
436:
432:
429:
400:
390:
375:
371:
366:
362:
340:
336:
333:
329:
325:
321:
315:
311:
289:
261:
258:
252:
248:
235:
222:
197:
187:
174:
170:
166:
140:
113:
109:
105:
101:
98:
93:
90:
85:
79:
76:
70:
66:
47:
44:
15:
9:
6:
4:
3:
2:
13518:
13507:
13504:
13502:
13499:
13497:
13494:
13492:
13489:
13488:
13486:
13456:
13452:
13444:
13438:
13432:
13428:
13424:
13410:
13403:
13395:
13391:
13387:
13383:
13379:
13375:
13371:
13367:
13363:
13359:
13352:
13345:
13343:0-201-02116-1
13339:
13335:
13334:
13329:
13325:
13321:
13315:
13308:
13303:
13294:
13293:
13288:
13285:
13278:
13270:
13268:0-521-66396-2
13264:
13260:
13256:
13250:
13242:
13235:
13227:
13225:0-8493-9114-8
13221:
13217:
13210:
13208:
13199:
13197:0-07-001685-2
13193:
13186:
13185:
13177:
13166:
13159:
13151:
13145:
13141:
13140:
13132:
13130:
13121:
13119:0-8493-1606-5
13115:
13095:
13091:
13083:
13077:
13071:
13067:
13063:
13049:
13048:
13040:
13038:
13017:
13014:
13008:
13003:
12999:
12991:
12985:
12982:
12978:
12971:
12966:
12963:
12960:
12957:
12953:
12941:
12934:
12932:
12923:
12921:0-19-859679-0
12917:
12913:
12909:
12902:
12900:
12895:
12881:
12876:
12870:
12865:
12864:
12858:
12835:
12827:
12823:
12819:
12814:
12809:
12805:
12801:
12794:
12790:
12786:
12781:
12777:
12771:
12767:
12763:
12756:
12752:
12748:
12743:
12739:
12733:
12729:
12725:
12716:
12712:
12708:
12703:
12699:
12693:
12689:
12685:
12678:
12674:
12670:
12665:
12660:
12656:
12652:
12645:
12641:
12637:
12632:
12628:
12622:
12618:
12614:
12605:
12601:
12597:
12592:
12588:
12582:
12578:
12574:
12567:
12563:
12559:
12554:
12550:
12544:
12540:
12536:
12529:
12525:
12521:
12516:
12511:
12507:
12503:
12497:
12492:
12490:
12472:
12464:
12460:
12456:
12447:
12443:
12439:
12430:
12426:
12422:
12416:
12411:
12409:
12391:
12383:
12379:
12375:
12366:
12362:
12358:
12349:
12345:
12341:
12335:
12330:
12328:
12306:
12300:
12294:
12287:
12283:
12273:
12270:
12268:
12265:
12263:
12260:
12258:
12255:
12254:
12248:
12229:
12225:
12221:
12215:
12206:
12200:
12197:
12193:
12189:
12185:
12172:
12169:
12165:
12160:
12154:
12144:
12141:
12137:
12124:
12121:
12117:
12112:
12106:
12096:
12093:
12089:
12077:
12074:
12069:
12063:
12055:
12044:
12041:
12036:
12033:
12031:
12023:
12013:
12009:
11997:
11993:
11989:
11983:
11973:
11969:
11957:
11952:
11948:
11942:
11936:
11926:
11922:
11910:
11906:
11902:
11896:
11886:
11882:
11870:
11868:
11863:
11854:
11850:
11846:
11840:
11830:
11827:
11823:
11811:
11808:
11803:
11797:
11788:
11782:
11779:
11775:
11769:
11765:
11760:
11747:
11742:
11738:
11733:
11728:
11722:
11712:
11709:
11705:
11692:
11689:
11685:
11680:
11674:
11666:
11654:
11651:
11647:
11642:
11639:
11637:
11630:
11624:
11620:
11614:
11610:
11603:
11597:
11587:
11583:
11571:
11567:
11563:
11557:
11547:
11543:
11531:
11526:
11522:
11516:
11510:
11500:
11496:
11484:
11480:
11476:
11470:
11460:
11456:
11444:
11442:
11437:
11428:
11424:
11420:
11414:
11411:
11405:
11402:
11398:
11392:
11386:
11376:
11373:
11369:
11357:
11354:
11349:
11343:
11333:
11330:
11326:
11313:
11310:
11306:
11301:
11295:
11286:
11280:
11277:
11273:
11269:
11265:
11252:
11249:
11245:
11240:
11234:
11226:
11215:
11212:
11207:
11204:
11202:
11195:
11190:
11185:
11181:
11175:
11169:
11159:
11155:
11143:
11139:
11135:
11129:
11119:
11115:
11103:
11098:
11094:
11088:
11082:
11072:
11068:
11056:
11052:
11048:
11042:
11032:
11028:
11016:
11014:
11009:
10997:
10981:
10978:
10974:
10970:
10965:
10962:
10958:
10949:
10946:
10942:
10938:
10933:
10930:
10926:
10911:
10892:
10888:
10884:
10880:
10873:
10863:
10860:
10856:
10846:
10840:
10830:
10827:
10823:
10813:
10807:
10797:
10794:
10790:
10779:
10773:
10770:
10765:
10763:
10755:
10745:
10741:
10729:
10725:
10721:
10715:
10705:
10701:
10689:
10685:
10681:
10675:
10665:
10661:
10649:
10645:
10641:
10635:
10625:
10621:
10609:
10607:
10602:
10593:
10589:
10585:
10581:
10574:
10564:
10561:
10557:
10547:
10541:
10531:
10528:
10524:
10514:
10508:
10498:
10495:
10491:
10480:
10474:
10471:
10466:
10464:
10456:
10446:
10442:
10430:
10426:
10422:
10416:
10406:
10402:
10390:
10386:
10382:
10376:
10366:
10362:
10350:
10346:
10342:
10336:
10326:
10322:
10310:
10308:
10303:
10294:
10290:
10286:
10282:
10275:
10265:
10262:
10258:
10248:
10242:
10232:
10229:
10225:
10215:
10209:
10199:
10196:
10192:
10181:
10175:
10172:
10167:
10165:
10157:
10147:
10143:
10131:
10127:
10123:
10117:
10107:
10103:
10091:
10087:
10083:
10077:
10067:
10063:
10051:
10047:
10043:
10037:
10027:
10023:
10011:
10009:
10004:
9992:
9978:
9950:
9945:
9937:
9929:
9926:
9912:
9906:
9903:
9893:
9881:
9877:
9873:
9870:
9868:
9864:
9858:
9840:
9832:
9821:
9815:
9812:
9808:
9804:
9792:
9784:
9774:
9771:
9765:
9761:
9755:
9749:
9724:
9720:
9716:
9708:
9680:
9675:
9667:
9659:
9648:
9642:
9639:
9635:
9631:
9619:
9611:
9601:
9598:
9592:
9588:
9582:
9576:
9551:
9547:
9543:
9540:
9518:
9515:
9513:
9494:
9491:
9489:
9470:
9468:
9467:force density
9464:
9460:
9455:
9442:
9434:
9429:
9425:
9419:
9415:
9407:
9403:
9399:
9393:
9382:
9373:
9371:
9366:
9353:
9345:
9340:
9336:
9330:
9326:
9319:
9315:
9309:
9293:
9291:
9286:
9273:
9265:
9261:
9255:
9251:
9244:
9239:
9235:
9229:
9213:
9211:
9210:Froude number
9206:
9191:
9177:
9172:
9168:
9161:
9157:
9151:
9147:
9140:
9135:
9125:
9120:
9107:
9102:
9098:
9092:
9088:
9081:
9077:
9071:
9068:
9064:
9058:
9054:
9045:
9040:
9036:
9030:
9026:
9019:
9015:
9009:
9004:
8994:
8989:
8977:
8973:
8968:
8964:
8959:
8951:
8943:
8939:
8928:
8924:
8918:
8913:
8897:
8894:
8879:
8867:
8863:
8859:
8854:
8844:
8839:
8827:
8823:
8817:
8813:
8807:
8804:
8800:
8794:
8790:
8784:
8780:
8776:
8771:
8761:
8756:
8744:
8740:
8734:
8729:
8725:
8719:
8715:
8710:
8706:
8699:
8695:
8689:
8679:
8671:
8667:
8656:
8644:
8640:
8626:
8622:
8615:
8610:
8606:
8600:
8596:
8584:
8581:
8560:
8556:
8547:
8545:
8538:
8522:
8518:
8514:
8509:
8507:
8500:
8496:
8486:
8482:
8471:
8469:
8462:
8445:
8441:
8437:
8435:
8428:
8416:
8409:
8405:
8399:
8395:
8389:
8387:
8380:
8376:
8366:
8362:
8358:
8353:
8351:
8344:
8340:
8330:
8326:
8322:
8317:
8315:
8308:
8304:
8294:
8290:
8286:
8281:
8279:
8272:
8268:
8255:
8249:
8240:
8229:
8204:
8191:
8188:
8182:
8179:
8175:
8171:
8168:
8165:
8161:
8154:
8151:
8148:
8142:
8139:
8133:
8130:
8122:
8119:
8113:
8105:
8102:
8098:
8094:
8088:
8078:
8073:
8069:
8059:
8057:
8053:
8049:
8044:
8033:
8029:
8019:
8016:
8003:
7995:
7989:
7986:
7980:
7977:
7969:
7958:
7955:
7953:
7949:
7948:inviscid flow
7945:
7941:
7937:
7933:
7929:
7924:
7921:
7916:
7911:
7905:
7887:
7879:
7876:
7873:
7860:
7858:
7853:
7847:
7841:
7837:
7832:
7826:
7821:
7814:
7802:
7789:
7785:
7780:
7774:
7755:
7748:
7737:
7724:
7710:
7707:
7703:
7694:
7685:
7681:
7676:
7672:
7662:
7660:
7655:
7642:
7639:
7634:
7630:
7623:
7610:
7607:
7594:
7588:
7585:
7580:
7577:
7574:
7569:
7565:
7559:
7556:
7551:
7546:
7542:
7533:
7531:
7526:
7513:
7510:
7506:
7500:
7497:
7492:
7489:
7486:
7481:
7477:
7471:
7468:
7462:
7455:
7442:
7440:
7435:
7422:
7418:
7415:
7412:
7406:
7393:
7390:
7377:
7374:
7371:
7365:
7353:
7349:
7345:
7340:
7336:
7330:
7327:
7322:
7319:
7316:
7311:
7307:
7301:
7298:
7292:
7285:
7272:
7269:
7247:
7244:
7241:
7227:
7211:
7205:
7194:
7182:
7178:
7174:
7169:
7165:
7159:
7151:
7148:
7145:
7140:
7136:
7130:
7127:
7121:
7117:
7111:
7098:
7096:
7092:
7091:
7085:
7064:
7055:
7047:
7044:
7038:
7026:
7022:
7018:
7013:
7009:
7003:
6995:
6992:
6989:
6984:
6980:
6974:
6971:
6965:
6961:
6950:
6947:
6931:
6912:
6901:
6892:
6884:
6881:
6875:
6863:
6859:
6855:
6850:
6846:
6840:
6832:
6829:
6826:
6821:
6817:
6811:
6808:
6802:
6798:
6787:
6781:
6776:
6760:
6741:
6730:
6721:
6713:
6710:
6704:
6692:
6688:
6684:
6679:
6671:
6667:
6661:
6653:
6648:
6644:
6638:
6635:
6629:
6625:
6614:
6611:
6598:
6592:
6580:
6576:
6572:
6567:
6559:
6551:
6548:
6543:
6539:
6534:
6526:
6522:
6511:
6508:
6490:
6482:
6474:
6471:
6466:
6458:
6447:
6438:
6434:
6429:
6425:
6421:
6412:
6409:
6404:
6398:
6374:
6372:
6351:
6347:
6338:
6331:
6327:
6313:
6295:
6291:
6282:
6275:
6271:
6267:
6262:
6256:
6237:
6230:
6219:
6206:
6204:
6200:
6192:
6184:
6163:
6155:
6151:
6142:
6133:
6127:
6119:
6115:
6106:
6099:
6095:
6082:
6080:
6076:
6066:
6053:
6047:
6042:
6027:
6015:
6010:
6005:
6001:
5995:
5985:
5981:
5975:
5971:
5967:
5962:
5957:
5952:
5939:
5935:
5926:
5915:
5904:
5897:
5890:
5883:
5879:
5874:
5870:
5858:
5855:
5850:
5845:
5837:
5833:
5827:
5823:
5816:
5812:
5807:
5803:
5802:creeping flow
5799:
5794:
5786:
5777:
5775:
5770:
5767:
5762:
5757:
5747:
5741:
5735:
5733:
5728:
5724:
5719:
5714:
5708:
5703:
5699:
5694:
5688:
5666:
5663:
5661:
5640:
5632:
5624:
5621:
5619:
5598:
5595:
5593:
5572:
5568:
5550:
5542:
5536:
5530:
5506:
5503:
5499:
5495:
5485:
5476:
5466:
5449:
5446:
5444:
5437:
5433:
5429:
5424:
5418:
5413:
5409:
5403:
5392:
5386:
5383:
5376:
5372:
5368:
5358:
5355:
5353:
5346:
5342:
5338:
5335:
5330:
5325:
5321:
5315:
5307:
5301:
5298:
5291:
5287:
5283:
5277:
5270:
5267:
5265:
5260:
5257:
5252:
5246:
5242:
5238:
5235:
5230:
5225:
5221:
5215:
5207:
5201:
5198:
5191:
5187:
5183:
5177:
5173:
5163:
5155:
5152:
5146:
5142:
5138:
5129:
5125:
5122:
5119:
5113:
5108:
5104:
5098:
5084:
5080:
5078:
5073:
5070:
5063:
5060:
5053:
5049:
5045:
5039:
5030:
5017:
5015:
5011:
5006:
4991:
4987:
4983:
4978:
4974:
4970:
4962:
4960:
4952:
4942:
4939:
4921:
4913:
4905:
4902:
4897:
4891:
4888:
4878:
4852:
4849:
4839:
4833:
4827:
4824:
4814:
4791:
4788:
4784:
4781:
4777:
4774:
4770:
4761:
4748:
4745:
4741:
4738:
4734:
4731:
4727:
4719:
4716:
4713:
4709:
4706:
4702:
4699:
4688:
4679:
4676:
4673:
4669:
4666:
4662:
4659:
4655:
4649:
4646:
4636:
4611:
4601:
4594:
4589:
4579:
4572:
4566:
4563:
4552:
4546:
4535:
4532:
4513:
4507:
4501:
4498:
4487:
4481:
4470:
4462:
4449:
4446:
4442:
4439:
4435:
4432:
4428:
4422:
4419:
4409:
4403:
4397:
4394:
4383:
4377:
4354:
4351:
4347:
4344:
4340:
4337:
4333:
4330:
4327:
4318:
4305:
4302:
4298:
4295:
4291:
4288:
4284:
4276:
4273:
4265:
4256:
4239:
4226:
4223:
4219:
4216:
4212:
4209:
4205:
4197:
4192:
4182:
4147:
4134:
4131:
4127:
4124:
4120:
4117:
4105:
4096:
4091:
4081:
4069:
4066:
4033:
4030:
4026:
4023:
4019:
4016:
4009:
3999:
3996:
3992:
3982:
3979:
3976:
3972:
3969:
3965:
3962:
3955:
3945:
3942:
3938:
3928:
3925:
3922:
3918:
3915:
3911:
3908:
3901:
3891:
3888:
3884:
3874:
3872:
3865:
3860:
3856:
3848:
3845:
3841:
3838:
3834:
3831:
3824:
3814:
3811:
3807:
3797:
3794:
3791:
3787:
3784:
3780:
3777:
3770:
3760:
3757:
3753:
3743:
3740:
3737:
3733:
3730:
3726:
3723:
3716:
3706:
3703:
3699:
3689:
3687:
3680:
3675:
3671:
3663:
3660:
3656:
3653:
3649:
3646:
3639:
3629:
3626:
3622:
3612:
3609:
3606:
3602:
3599:
3595:
3592:
3585:
3575:
3572:
3568:
3558:
3555:
3552:
3548:
3545:
3541:
3538:
3531:
3521:
3518:
3514:
3504:
3502:
3495:
3490:
3486:
3473:
3457:
3452:
3448:
3444:
3439:
3434:
3430:
3407:
3402:
3398:
3388:
3375:
3372:
3368:
3365:
3360:
3357:
3353:
3349:
3346:
3343:
3339:
3336:
3332:
3328:
3325:
3319:
3309:
3306:
3302:
3292:
3287:
3284:
3280:
3275:
3271:
3268:
3265:
3261:
3258:
3253:
3250:
3246:
3242:
3239:
3236:
3232:
3229:
3225:
3221:
3218:
3212:
3202:
3199:
3195:
3185:
3180:
3177:
3173:
3168:
3164:
3161:
3158:
3154:
3151:
3146:
3143:
3139:
3135:
3132:
3129:
3125:
3122:
3118:
3114:
3111:
3105:
3095:
3092:
3088:
3078:
3073:
3070:
3066:
3061:
3057:
3052:
3047:
3043:
3034:
3032:
3023:
3007:
3004:
3000:
2996:
2991:
2988:
2984:
2963:
2960:
2957:
2954:
2932:
2929:
2925:
2915:
2898:
2887:
2879:
2876:
2872:
2862:
2856:
2853:
2845:
2837:
2834:
2830:
2826:
2820:
2817:
2814:
2811:
2803:
2800:
2796:
2789:
2783:
2780:
2769:
2758:
2750:
2747:
2743:
2733:
2730:
2727:
2721:
2713:
2710:
2706:
2699:
2693:
2690:
2687:
2684:
2676:
2673:
2669:
2645:
2642:
2622:
2619:
2613:
2603:
2600:
2596:
2586:
2581:
2578:
2574:
2553:
2550:
2547:
2544:
2522:
2519:
2515:
2504:
2488:
2485:
2481:
2477:
2474:
2469:
2466:
2462:
2458:
2436:
2433:
2429:
2425:
2422:
2417:
2413:
2387:
2384:
2380:
2376:
2373:
2368:
2365:
2361:
2357:
2354:
2349:
2346:
2342:
2338:
2332:
2324:
2315:
2309:
2300:
2289:
2288:stress vector
2271:
2261:
2254:
2251:
2245:
2242:
2239:
2236:
2233:
2230:
2224:
2215:
2190:
2187:
2183:
2179:
2157:
2154:
2150:
2146:
2137:
2136:
2132:
2131:
2128:
2110:
2105:
2097:
2090:
2077:
2071:
2065:
2038:
2035:
2031:
2028:
2022:
2019:
2015:
2011:
2003:
1998:
1983:
1973:
1966:
1961:
1951:
1944:
1935:
1924:
1909:
1899:
1888:
1870:
1860:
1849:
1824:
1820:
1813:
1810:
1804:
1798:
1793:
1789:
1782:
1779:
1776:
1748:
1744:
1740:
1730:
1726:
1719:
1716:
1710:
1702:
1698:
1690:
1687:
1664:
1660:
1636:
1630:
1621:
1613:
1604:
1601:
1582:
1576:
1570:
1567:
1556:
1550:
1539:
1537:
1521:
1515:
1492:
1469:
1435:
1403:
1391:
1379:
1355:
1347:
1341:
1335:
1323:
1317:
1311:
1305:
1302:
1290:
1279:
1277:
1267:
1265:
1260:
1254:
1249:
1244:
1238:
1220:
1212:
1206:
1200:
1176:
1174:
1169:
1167:
1163:
1159:
1155:
1150:
1148:
1125:
1122:
1114:
1109:
1106:
1098:
1089:
1082:
1066:
1048:
1038:
1028:
1025:
1021:
1010:
1003:
993:
990:
986:
975:
968:
958:
955:
951:
932:
922:
919:
915:
904:
897:
887:
884:
880:
869:
862:
852:
849:
845:
826:
816:
813:
809:
798:
791:
781:
778:
774:
763:
756:
746:
743:
739:
725:
720:
712:
702:
684:
675:
662:
658:
640:
632:
628:
618:
614:
604:
600:
593:
588:
576:
558:
555:
547:
542:
539:
531:
522:
517:
508:
501:
486:
485:stress tensor
461:
443:
434:
418:
414:
398:
391:
373:
364:
331:
323:
313:
278:
259:
256:
246:
236:
211:
195:
188:
168:
155:
154:flow velocity
130:
129:
128:
125:
107:
99:
91:
88:
83:
77:
74:
64:
53:
46:Main equation
43:
41:
37:
34:
30:
27:put forth by
26:
22:
13402:
13361:
13357:
13351:
13331:
13314:
13302:
13290:
13277:
13258:
13249:
13234:
13215:
13183:
13176:
13158:
13138:
13046:
12907:
12874:
12868:
12861:
12856:
12304:
12298:
12292:
12286:
10998:
10917:
9993:
9990:
9883:
9879:
9875:
9871:
9860:
9726:
9722:
9718:
9710:
9553:
9549:
9545:
9541:
9471:
9456:
9374:
9367:
9294:
9290:Euler number
9287:
9214:
9207:
8898:
8895:
8585:
8582:
8256:
8247:
8238:
8235:
8205:
8106:
8100:
8096:
8092:
8086:
8071:
8067:
8060:
8042:
8039:
8027:
8020:
8017:
7959:
7956:
7925:
7919:
7909:
7906:
7861:
7851:
7845:
7830:
7819:
7812:
7808:
7725:
7683:
7679:
7668:
7658:
7656:
7611:
7608:
7534:
7527:
7443:
7438:
7436:
7394:
7391:
7273:
7267:
7228:
7099:
7088:
7086:
6951:
6948:
6788:
6777:
6615:
6612:
6512:
6509:
6375:
6314:
6207:
6201:
6190:
6185:
6083:
6072:
5907:
5902:
5895:
5888:
5881:
5872:
5868:
5864:
5853:
5843:
5835:
5831:
5825:
5821:
5814:
5810:
5795:
5791:
5771:
5765:
5755:
5745:
5739:
5736:
5726:
5722:
5712:
5706:
5692:
5689:
5573:
5570:
5507:
5504:
5501:
5497:
5467:
5018:
5007:
4963:
4948:
4940:
4762:
4536:
4533:
4471:
4468:
4319:
4245:
4148:
4070:
4067:
3474:
3389:
3035:
3030:
3028:
2916:
2505:
2138:
2001:
1999:
1925:
1845:
1602:
1540:
1380:
1280:
1273:
1258:
1252:
1242:
1239:
1177:
1170:
1151:
1144:
126:
49:
23:is a vector
20:
18:
9461:, i.e. the
8058:may arise.
8048:body forces
8046:represents
7723:reduces to
6371:Lamb vector
5718:body forces
2051:with units
1848:body forces
1264:body forces
657:body forces
301:, equal to
13485:Categories
12890:References
7938:and fluid
7530:total head
7090:streamline
5917:, so that
5759:, being a
5492:See also:
5475:derivative
5012:and using
1607:Right side
1536:derivative
1158:covariants
1065:divergence
663:), (unit:
52:Lagrangian
13457:τ
13439:⋅
13436:∇
13429:σ
13425:⋅
13422:∇
13386:0028-0836
13292:MathWorld
13096:τ
13078:⋅
13075:∇
13068:σ
13064:⋅
13061:∇
13018:σ
13015:⋅
13012:∇
12996:∂
12979:σ
12975:∂
12954:σ
12824:σ
12802:ρ
12791:σ
12764:ρ
12753:σ
12726:ρ
12713:σ
12686:ρ
12675:σ
12653:ρ
12642:σ
12615:ρ
12602:σ
12575:ρ
12564:σ
12537:ρ
12526:σ
12504:ρ
12457:ρ
12440:ρ
12423:ρ
12376:ρ
12359:ρ
12342:ρ
12213:∂
12194:τ
12182:∂
12173:ρ
12155:ϕ
12152:∂
12142:ϕ
12138:τ
12134:∂
12125:ρ
12104:∂
12090:τ
12086:∂
12078:ρ
12061:∂
12053:∂
12045:ρ
12037:−
12021:∂
12006:∂
11984:ϕ
11981:∂
11966:∂
11953:ϕ
11934:∂
11919:∂
11894:∂
11879:∂
11855:ϕ
11838:∂
11831:ϕ
11824:τ
11820:∂
11812:ρ
11795:∂
11783:ϕ
11776:τ
11757:∂
11748:ρ
11723:ϕ
11720:∂
11713:ϕ
11710:ϕ
11706:τ
11702:∂
11693:ρ
11675:ϕ
11672:∂
11664:∂
11655:ρ
11643:−
11625:ϕ
11595:∂
11588:ϕ
11580:∂
11558:ϕ
11555:∂
11548:ϕ
11540:∂
11527:ϕ
11508:∂
11501:ϕ
11493:∂
11468:∂
11461:ϕ
11453:∂
11438:ϕ
11415:ρ
11406:ϕ
11403:ϕ
11399:τ
11393:−
11384:∂
11370:τ
11366:∂
11358:ρ
11344:ϕ
11341:∂
11331:ϕ
11327:τ
11323:∂
11314:ρ
11293:∂
11274:τ
11262:∂
11253:ρ
11232:∂
11224:∂
11216:ρ
11208:−
11186:ϕ
11176:−
11167:∂
11152:∂
11130:ϕ
11127:∂
11112:∂
11099:ϕ
11080:∂
11065:∂
11040:∂
11025:∂
10975:τ
10959:τ
10955:⟹
10943:σ
10927:σ
10871:∂
10857:σ
10853:∂
10838:∂
10824:σ
10820:∂
10805:∂
10791:σ
10787:∂
10774:ρ
10753:∂
10738:∂
10713:∂
10698:∂
10673:∂
10658:∂
10633:∂
10618:∂
10572:∂
10558:σ
10554:∂
10539:∂
10525:σ
10521:∂
10506:∂
10492:σ
10488:∂
10475:ρ
10454:∂
10439:∂
10414:∂
10399:∂
10374:∂
10359:∂
10334:∂
10319:∂
10273:∂
10259:σ
10255:∂
10240:∂
10226:σ
10222:∂
10207:∂
10193:σ
10189:∂
10176:ρ
10155:∂
10140:∂
10115:∂
10100:∂
10075:∂
10060:∂
10035:∂
10020:∂
9942:σ
9938:⋅
9935:∇
9930:ρ
9845:τ
9841:⋅
9838:∇
9816:−
9785:⊗
9775:ρ
9762:⋅
9759:∇
9747:∂
9737:∂
9672:τ
9668:⋅
9665:∇
9643:−
9612:⊗
9602:ρ
9589:⋅
9586:∇
9574:∂
9564:∂
9519:ρ
9495:ρ
9416:ρ
9404:τ
9327:ρ
9192:∗
9136:∗
9131:τ
9126:⋅
9121:∗
9117:∇
9089:ρ
9078:τ
9072:−
9059:∗
9027:ρ
9005:∗
8995:⊗
8990:∗
8978:∗
8974:ρ
8965:⋅
8960:∗
8956:∇
8944:∗
8936:∂
8929:∗
8919:∗
8914:ρ
8909:∂
8880:∗
8855:∗
8850:τ
8845:⋅
8840:∗
8836:∇
8814:τ
8808:−
8795:∗
8772:∗
8762:⊗
8757:∗
8745:∗
8741:ρ
8716:ρ
8707:⋅
8690:∗
8686:∇
8672:∗
8664:∂
8657:∗
8645:∗
8641:ρ
8637:∂
8597:ρ
8557:τ
8553:τ
8548:≡
8539:∗
8534:τ
8510:≡
8501:∗
8472:≡
8463:∗
8451:∇
8438:≡
8429:∗
8425:∇
8390:≡
8381:∗
8354:≡
8345:∗
8318:≡
8309:∗
8291:ρ
8287:ρ
8282:≡
8273:∗
8269:ρ
8215:σ
8186:∇
8183:−
8172:χ
8169:−
8158:∇
8155:−
8149:χ
8146:∇
8137:∇
8134:−
8117:∇
8114:−
8000:τ
7996:⋅
7993:∇
7984:∇
7981:−
7974:σ
7970:⋅
7967:∇
7936:viscosity
7892:τ
7877:−
7870:σ
7825:gradients
7771:‖
7762:‖
7752:∇
7741:∇
7738:⋅
7675:vorticity
7657:That is,
7627:∇
7624:⋅
7589:ρ
7578:ϕ
7552:≡
7501:ρ
7490:ϕ
7459:∇
7456:⋅
7437:that is,
7413:ρ
7410:∇
7407:⋅
7372:ρ
7369:∇
7366:⋅
7350:ρ
7331:ρ
7320:ϕ
7289:∇
7286:⋅
7245:−
7238:σ
7212:ρ
7209:∇
7206:⋅
7202:σ
7195:⋅
7179:ρ
7160:ρ
7157:σ
7152:−
7149:ϕ
7118:⋅
7115:∇
7112:⋅
7065:×
7062:∇
7056:×
7045:ρ
7042:∇
7039:⋅
7035:σ
7023:ρ
7004:ρ
7001:σ
6996:−
6993:ϕ
6962:⋅
6959:∇
6929:∂
6919:∂
6913:−
6902:×
6899:∇
6893:×
6882:ρ
6879:∇
6876:⋅
6872:σ
6860:ρ
6841:ρ
6838:σ
6833:−
6830:ϕ
6799:⋅
6796:∇
6758:∂
6748:∂
6742:−
6731:×
6728:∇
6722:×
6711:ρ
6708:∇
6705:⋅
6701:σ
6689:ρ
6672:−
6662:ρ
6659:σ
6654:−
6626:⋅
6623:∇
6599:ρ
6596:∇
6593:⋅
6589:σ
6577:ρ
6568:−
6564:σ
6560:⋅
6557:∇
6552:ρ
6535:ρ
6532:σ
6523:⋅
6520:∇
6487:σ
6483:⋅
6480:∇
6475:ρ
6459:×
6448:×
6445:∇
6418:∇
6396:∂
6386:∂
6352:×
6339:×
6336:∇
6296:×
6283:×
6280:∇
6253:‖
6244:‖
6234:∇
6223:∇
6220:⋅
6167:∇
6164:⋅
6156:−
6143:⋅
6124:∇
6107:×
6104:∇
6096:×
6069:Lamb form
6031:∇
6028:⋅
5992:∂
5972:∑
5944:∇
5936:⋅
5808:quantity
5806:nonlinear
5798:nonlinear
5667:ρ
5645:σ
5641:−
5633:⊗
5625:ρ
5599:ρ
5543:⋅
5540:∇
5528:∂
5518:∂
5430:−
5425:ρ
5410:σ
5400:∇
5393:−
5339:ρ
5336:−
5322:σ
5312:∇
5308:−
5278:ρ
5239:ρ
5236:−
5222:σ
5212:∇
5208:−
5178:ρ
5168:Ω
5164:∫
5139:ρ
5134:Ω
5130:∫
5105:σ
5095:∇
5089:Ω
5085:∫
5040:ρ
5035:Ω
5031:∫
4949:Applying
4918:σ
4914:⋅
4911:∇
4906:ρ
4771:ρ
4728:ρ
4693:σ
4689:⋅
4686:∇
4656:ρ
4605:→
4583:→
4556:→
4517:→
4491:→
4429:ρ
4387:→
4334:ρ
4285:ρ
4260:→
4242:Left side
4206:ρ
4186:→
4110:σ
4106:⋅
4103:∇
4085:→
4007:∂
3993:σ
3989:∂
3953:∂
3939:σ
3935:∂
3899:∂
3885:σ
3881:∂
3822:∂
3808:σ
3804:∂
3768:∂
3754:σ
3750:∂
3714:∂
3700:σ
3696:∂
3637:∂
3623:σ
3619:∂
3583:∂
3569:σ
3565:∂
3529:∂
3515:σ
3511:∂
3354:σ
3350:−
3317:∂
3303:σ
3299:∂
3281:σ
3247:σ
3243:−
3210:∂
3196:σ
3192:∂
3174:σ
3140:σ
3136:−
3103:∂
3089:σ
3085:∂
3067:σ
3001:σ
2985:σ
2947:in point
2926:σ
2896:∂
2873:σ
2869:∂
2863:−
2831:σ
2827:−
2797:σ
2767:∂
2744:σ
2740:∂
2707:σ
2700:−
2670:σ
2611:∂
2597:σ
2593:∂
2575:σ
2516:σ
2482:σ
2478:−
2463:σ
2459:−
2430:σ
2426:−
2381:σ
2377:−
2362:σ
2358:−
2343:σ
2339:−
2329:σ
2325:⋅
2319:→
2304:→
2265:→
2255:−
2231:−
2219:→
2184:σ
2180:−
2151:σ
2147:−
2098:⋅
2072:⋅
2066:⋅
2016:σ
2012:−
1977:→
1955:→
1939:→
1903:→
1864:→
1842:was used.
1811:−
1802:Δ
1774:Δ
1586:→
1560:→
1519:→
1513:Δ
1490:Δ
1467:Δ
1444:→
1439:¯
1395:→
1364:→
1359:¯
1345:Δ
1327:→
1318:−
1309:Δ
1294:→
1213:⋅
1210:∇
1198:∂
1188:∂
1123:−
1115:⋅
1107:−
1099:⋅
1036:∂
1022:σ
1018:∂
1001:∂
987:σ
983:∂
966:∂
952:σ
948:∂
930:∂
916:σ
912:∂
895:∂
881:σ
877:∂
860:∂
846:σ
842:∂
824:∂
810:σ
806:∂
789:∂
775:σ
771:∂
754:∂
740:σ
736:∂
717:σ
713:⋅
710:∇
556:−
548:⋅
540:−
532:⋅
487:, (unit:
470:σ
399:ρ
352:, (unit:
335:∇
332:⋅
310:∂
212:, (unit:
104:σ
100:⋅
97:∇
92:ρ
40:continuum
36:transport
13501:Momentum
13394:11034749
13330:(1963),
12251:See also
9465:and the
7940:velocity
7805:Stresses
5704:holds),
4868:we get:
4469:We have
1691:′
1534:we get (
33:momentum
16:Equation
13366:Bibcode
8065:, with
8052:gravity
7913:is the
7855:is the
7265:(where
6081:holds:
6077:of the
5840:, with
5730:is the
5696:is the
1246:is the
1063:is the
483:is the
413:density
411:is the
275:is the
152:is the
127:where
38:in any
13392:
13384:
13358:Nature
13340:
13265:
13222:
13194:
13146:
13116:
12918:
12866:. And
9713:Fr β β
7907:where
7682:= β Γ
5829:or as
5763:, has
5761:tensor
5690:where
5468:where
1423:, and
1381:where
1240:where
29:Cauchy
13412:(PDF)
13390:S2CID
13188:(PDF)
13168:(PDF)
13051:(PDF)
12943:(PDF)
12278:Notes
5737:Here
4626:then
4534:then
13382:ISSN
13338:ISBN
13263:ISBN
13220:ISBN
13192:ISBN
13144:ISBN
13114:ISBN
12916:ISBN
12302:and
9288:the
8025:and
7844:β β
7818:β β
7816:and
7671:curl
6203:Lamb
6073:The
5871:β
β)
5847:the
5834:β
(β
5824:β
β)
5743:and
5732:dyad
1885:and
210:time
19:The
13374:doi
13362:192
13112:).
10996:):
8087:Οgz
8070:= β
5906:= β
5884:β
β
5813:β
β
2503:).
2127:).
1538:):
279:of
208:is
13487::
13473:).
13388:.
13380:.
13372:.
13360:.
13326:;
13322:;
13289:.
13206:^
13128:^
13036:^
13030:).
12930:^
12910:.
12898:^
12828:33
12795:23
12757:13
12717:23
12679:22
12646:12
12606:13
12568:12
12530:11
9882:)
9869:.
9725:)
9552:)
9469::
9292::
9212::
8099:β
8095:=
7954:.
7677:)
7661:.
7097::
6786::
6199:.
5903:mi
5776:.
5769:.
5725:β
5484:.
3022:.
1175::
1149:.
42:.
13461:)
13453:+
13449:I
13445:p
13442:(
13433:=
13396:.
13376::
13368::
13295:.
13271:.
13228:.
13200:.
13152:.
13122:.
13100:)
13092:+
13088:I
13084:p
13081:(
13072:=
13009:=
13004:k
13000:x
12992:/
12986:i
12983:k
12972:=
12967:k
12964:,
12961:i
12958:k
12924:.
12875:Ο
12869:F
12857:F
12836:)
12820:+
12815:2
12810:3
12806:u
12787:+
12782:2
12778:u
12772:3
12768:u
12749:+
12744:1
12740:u
12734:3
12730:u
12709:+
12704:3
12700:u
12694:2
12690:u
12671:+
12666:2
12661:2
12657:u
12638:+
12633:1
12629:u
12623:2
12619:u
12598:+
12593:3
12589:u
12583:1
12579:u
12560:+
12555:2
12551:u
12545:1
12541:u
12522:+
12517:2
12512:1
12508:u
12498:(
12493:=
12484:F
12473:)
12465:3
12461:g
12448:2
12444:g
12431:1
12427:g
12417:(
12412:=
12403:s
12392:)
12384:3
12380:u
12367:2
12363:u
12350:1
12346:u
12336:(
12331:=
12322:j
12305:F
12299:Ο
12293:j
12230:z
12226:f
12222:+
12216:r
12207:)
12201:z
12198:r
12190:r
12186:(
12170:r
12166:1
12161:+
12145:z
12122:r
12118:1
12113:+
12107:z
12097:z
12094:z
12075:1
12070:+
12064:z
12056:P
12042:1
12034:=
12024:z
12014:z
12010:u
11998:z
11994:u
11990:+
11974:z
11970:u
11958:r
11949:u
11943:+
11937:r
11927:z
11923:u
11911:r
11907:u
11903:+
11897:t
11887:z
11883:u
11871::
11864:z
11851:f
11847:+
11841:z
11828:z
11809:1
11804:+
11798:r
11789:)
11780:r
11770:2
11766:r
11761:(
11743:2
11739:r
11734:1
11729:+
11690:r
11686:1
11681:+
11667:P
11652:r
11648:1
11640:=
11631:r
11621:u
11615:r
11611:u
11604:+
11598:z
11584:u
11572:z
11568:u
11564:+
11544:u
11532:r
11523:u
11517:+
11511:r
11497:u
11485:r
11481:u
11477:+
11471:t
11457:u
11445::
11429:r
11425:f
11421:+
11412:r
11387:z
11377:r
11374:z
11355:1
11350:+
11334:r
11311:r
11307:1
11302:+
11296:r
11287:)
11281:r
11278:r
11270:r
11266:(
11250:r
11246:1
11241:+
11235:r
11227:P
11213:1
11205:=
11196:r
11191:2
11182:u
11170:z
11160:r
11156:u
11144:z
11140:u
11136:+
11120:r
11116:u
11104:r
11095:u
11089:+
11083:r
11073:r
11069:u
11057:r
11053:u
11049:+
11043:t
11033:r
11029:u
11017::
11010:r
10982:i
10979:j
10971:=
10966:j
10963:i
10950:i
10947:j
10939:=
10934:j
10931:i
10893:z
10889:f
10885:+
10881:)
10874:z
10864:z
10861:z
10847:+
10841:y
10831:z
10828:y
10814:+
10808:x
10798:z
10795:x
10780:(
10771:1
10766:=
10756:z
10746:z
10742:u
10730:z
10726:u
10722:+
10716:y
10706:z
10702:u
10690:y
10686:u
10682:+
10676:x
10666:z
10662:u
10650:x
10646:u
10642:+
10636:t
10626:z
10622:u
10610::
10603:z
10594:y
10590:f
10586:+
10582:)
10575:z
10565:y
10562:z
10548:+
10542:y
10532:y
10529:y
10515:+
10509:x
10499:y
10496:x
10481:(
10472:1
10467:=
10457:z
10447:y
10443:u
10431:z
10427:u
10423:+
10417:y
10407:y
10403:u
10391:y
10387:u
10383:+
10377:x
10367:y
10363:u
10351:x
10347:u
10343:+
10337:t
10327:y
10323:u
10311::
10304:y
10295:x
10291:f
10287:+
10283:)
10276:z
10266:x
10263:z
10249:+
10243:y
10233:x
10230:y
10216:+
10210:x
10200:x
10197:x
10182:(
10173:1
10168:=
10158:z
10148:x
10144:u
10132:z
10128:u
10124:+
10118:y
10108:x
10104:u
10092:y
10088:u
10084:+
10078:x
10068:x
10064:u
10052:x
10048:u
10044:+
10038:t
10028:x
10024:u
10012::
10005:x
9965:f
9958:r
9955:F
9951:1
9946:=
9927:1
9920:u
9917:E
9913:+
9907:t
9904:D
9898:u
9894:D
9878:(
9833:2
9827:f
9822:C
9813:=
9809:)
9805:p
9800:u
9797:E
9793:+
9789:j
9781:j
9772:1
9766:(
9756:+
9750:t
9741:j
9721:(
9695:g
9688:r
9685:F
9681:1
9676:+
9660:2
9654:f
9649:C
9640:=
9636:)
9632:p
9627:u
9624:E
9620:+
9616:j
9608:j
9599:1
9593:(
9583:+
9577:t
9568:j
9548:(
9523:f
9516:=
9508:g
9499:u
9492:=
9484:j
9443:,
9435:2
9430:0
9426:u
9420:0
9408:0
9400:2
9394:=
9388:f
9383:C
9354:,
9346:2
9341:0
9337:u
9331:0
9320:0
9316:p
9310:=
9306:u
9303:E
9274:,
9266:0
9262:r
9256:0
9252:f
9245:2
9240:0
9236:u
9230:=
9226:r
9223:F
9187:f
9178:2
9173:0
9169:u
9162:0
9158:r
9152:0
9148:f
9141:+
9108:2
9103:0
9099:u
9093:0
9082:0
9069:=
9065:)
9055:p
9046:2
9041:0
9037:u
9031:0
9020:0
9016:p
9010:+
9000:u
8985:u
8969:(
8952:+
8940:t
8925:u
8875:f
8868:0
8864:f
8860:+
8828:0
8824:r
8818:0
8805:=
8801:)
8791:p
8785:0
8781:p
8777:+
8767:u
8752:u
8735:2
8730:0
8726:u
8720:0
8711:(
8700:0
8696:r
8680:+
8668:t
8652:u
8627:0
8623:r
8616:2
8611:0
8607:u
8601:0
8561:0
8523:0
8519:p
8515:p
8497:p
8487:0
8483:f
8478:f
8458:f
8446:0
8442:r
8417:t
8410:0
8406:r
8400:0
8396:u
8377:t
8367:0
8363:r
8359:r
8341:r
8331:0
8327:u
8323:u
8305:u
8295:0
8251:0
8248:u
8242:0
8239:r
8192:.
8189:h
8180:=
8176:)
8166:p
8162:(
8152:=
8143:+
8140:p
8131:=
8127:f
8123:+
8120:p
8101:Ο
8097:p
8093:h
8085:β
8081:z
8072:Ο
8068:f
8063:Ο
8043:f
8028:Ο
8023:p
8004:.
7990:+
7987:p
7978:=
7920:Ο
7910:I
7888:+
7884:I
7880:p
7874:=
7852:Ο
7846:Ο
7831:p
7829:β
7820:Ο
7813:p
7811:β
7790:.
7786:)
7781:2
7775:2
7766:u
7756:(
7749:=
7745:u
7734:u
7711:t
7708:D
7704:/
7699:u
7695:D
7684:u
7680:Ο
7643:0
7640:=
7635:l
7631:b
7620:u
7595:,
7586:p
7581:+
7575:+
7570:2
7566:u
7560:2
7557:1
7547:l
7543:b
7514:0
7511:=
7507:)
7498:p
7493:+
7487:+
7482:2
7478:u
7472:2
7469:1
7463:(
7452:u
7423:,
7419:0
7416:=
7403:u
7378:0
7375:=
7362:u
7354:2
7346:p
7341:+
7337:)
7328:p
7323:+
7317:+
7312:2
7308:u
7302:2
7299:1
7293:(
7282:u
7268:I
7252:I
7248:p
7242:=
7215:)
7198:(
7191:u
7183:2
7175:1
7170:=
7166:)
7146:+
7141:2
7137:u
7131:2
7128:1
7122:(
7108:u
7073:)
7069:u
7059:(
7052:u
7048:+
7027:2
7019:1
7014:=
7010:)
6990:+
6985:2
6981:u
6975:2
6972:1
6966:(
6932:t
6923:u
6910:)
6906:u
6896:(
6889:u
6885:+
6864:2
6856:1
6851:=
6847:)
6827:+
6822:2
6818:u
6812:2
6809:1
6803:(
6784:Ο
6761:t
6752:u
6739:)
6735:u
6725:(
6718:u
6714:+
6693:2
6685:1
6680:=
6676:f
6668:)
6649:2
6645:u
6639:2
6636:1
6630:(
6581:2
6573:1
6549:1
6544:=
6540:)
6527:(
6495:f
6491:+
6472:1
6467:=
6463:u
6456:)
6452:u
6442:(
6439:+
6435:)
6430:2
6426:u
6422:(
6413:2
6410:1
6405:+
6399:t
6390:u
6356:u
6348:)
6343:u
6332:(
6328:=
6324:l
6300:u
6292:)
6287:u
6276:(
6272:+
6268:)
6263:2
6257:2
6248:u
6238:(
6231:=
6227:u
6216:u
6197:a
6191:a
6188:β
6171:a
6160:v
6152:)
6147:a
6139:v
6134:(
6128:a
6120:=
6116:)
6111:a
6100:(
6092:v
6054:.
6048:i
6043:]
6038:u
6034:)
6024:u
6020:(
6016:[
6011:=
6006:i
6002:v
5996:m
5986:m
5982:v
5976:m
5968:=
5963:i
5958:]
5953:)
5948:u
5940:(
5932:u
5927:[
5913:i
5911:v
5909:m
5896:u
5894:β
5889:u
5882:u
5873:u
5869:u
5867:(
5854:u
5844:u
5842:β
5838:)
5836:u
5832:u
5826:u
5822:u
5820:(
5815:u
5811:u
5766:N
5756:F
5751:N
5746:s
5740:j
5727:u
5723:u
5713:s
5707:F
5693:j
5671:f
5664:=
5655:s
5637:u
5629:u
5622:=
5613:F
5603:u
5596:=
5587:j
5555:s
5551:=
5547:F
5537:+
5531:t
5522:j
5481:i
5479:F
5470:Ξ©
5450:0
5447:=
5438:i
5434:f
5419:j
5414:i
5404:j
5387:t
5384:D
5377:i
5373:u
5369:D
5359:0
5356:=
5347:i
5343:f
5331:j
5326:i
5316:j
5302:t
5299:D
5292:i
5288:u
5284:D
5271:0
5268:=
5261:V
5258:d
5253:)
5247:i
5243:f
5231:j
5226:i
5216:j
5202:t
5199:D
5192:i
5188:u
5184:D
5174:(
5156:V
5153:d
5147:i
5143:f
5126:+
5123:V
5120:d
5114:j
5109:i
5099:j
5081:=
5074:V
5071:d
5064:t
5061:D
5054:i
5050:u
5046:D
4992:i
4988:F
4984:=
4979:i
4975:a
4971:m
4955:i
4953:(
4926:f
4922:+
4903:1
4898:=
4892:t
4889:D
4883:u
4879:D
4853:t
4850:D
4844:u
4840:D
4834:=
4828:t
4825:d
4819:u
4815:d
4792:z
4789:d
4785:y
4782:d
4778:x
4775:d
4749:z
4746:d
4742:y
4739:d
4735:x
4732:d
4724:f
4720:+
4717:z
4714:d
4710:y
4707:d
4703:x
4700:d
4697:)
4683:(
4680:=
4677:z
4674:d
4670:y
4667:d
4663:x
4660:d
4650:t
4647:d
4641:u
4637:d
4612:m
4602:F
4595:+
4590:p
4580:F
4573:=
4567:t
4564:d
4553:p
4547:d
4514:F
4508:=
4502:t
4499:d
4488:p
4482:d
4450:z
4447:d
4443:y
4440:d
4436:x
4433:d
4423:t
4420:d
4414:u
4410:d
4404:=
4398:t
4395:d
4384:p
4378:d
4355:z
4352:d
4348:y
4345:d
4341:x
4338:d
4331:=
4328:m
4306:z
4303:d
4299:y
4296:d
4292:x
4289:d
4281:u
4277:=
4274:m
4270:u
4266:=
4257:p
4227:z
4224:d
4220:y
4217:d
4213:x
4210:d
4202:f
4198:=
4193:m
4183:F
4158:f
4135:z
4132:d
4128:y
4125:d
4121:x
4118:d
4114:)
4100:(
4097:=
4092:p
4082:F
4034:y
4031:d
4027:x
4024:d
4020:z
4017:d
4010:z
4000:z
3997:z
3983:+
3980:z
3977:d
3973:x
3970:d
3966:y
3963:d
3956:y
3946:z
3943:y
3929:+
3926:z
3923:d
3919:y
3916:d
3912:x
3909:d
3902:x
3892:z
3889:x
3875:=
3866:z
3861:p
3857:F
3849:y
3846:d
3842:x
3839:d
3835:z
3832:d
3825:z
3815:y
3812:z
3798:+
3795:z
3792:d
3788:x
3785:d
3781:y
3778:d
3771:y
3761:y
3758:y
3744:+
3741:z
3738:d
3734:y
3731:d
3727:x
3724:d
3717:x
3707:y
3704:x
3690:=
3681:y
3676:p
3672:F
3664:y
3661:d
3657:x
3654:d
3650:z
3647:d
3640:z
3630:x
3627:z
3613:+
3610:z
3607:d
3603:x
3600:d
3596:y
3593:d
3586:y
3576:x
3573:y
3559:+
3556:z
3553:d
3549:y
3546:d
3542:x
3539:d
3532:x
3522:x
3519:x
3505:=
3496:x
3491:p
3487:F
3458:z
3453:p
3449:F
3445:,
3440:y
3435:p
3431:F
3408:x
3403:p
3399:F
3376:y
3373:d
3369:x
3366:d
3361:x
3358:z
3347:y
3344:d
3340:x
3337:d
3333:)
3329:z
3326:d
3320:z
3310:x
3307:z
3293:+
3288:x
3285:z
3276:(
3272:+
3269:z
3266:d
3262:x
3259:d
3254:x
3251:y
3240:z
3237:d
3233:x
3230:d
3226:)
3222:y
3219:d
3213:y
3203:x
3200:y
3186:+
3181:x
3178:y
3169:(
3165:+
3162:z
3159:d
3155:y
3152:d
3147:x
3144:x
3133:z
3130:d
3126:y
3123:d
3119:)
3115:x
3112:d
3106:x
3096:x
3093:x
3079:+
3074:x
3071:x
3062:(
3058:=
3053:x
3048:p
3044:F
3031:X
3008:x
3005:z
2997:,
2992:x
2989:y
2964:x
2961:d
2958:+
2955:x
2933:x
2930:x
2899:x
2891:)
2888:x
2885:(
2880:x
2877:x
2857:x
2854:d
2849:)
2846:x
2843:(
2838:x
2835:x
2824:)
2821:x
2818:d
2815:+
2812:x
2809:(
2804:x
2801:x
2790:=
2784:x
2781:d
2776:)
2770:x
2762:)
2759:x
2756:(
2751:x
2748:x
2734:x
2731:d
2728:+
2725:)
2722:x
2719:(
2714:x
2711:x
2703:(
2697:)
2694:x
2691:d
2688:+
2685:x
2682:(
2677:x
2674:x
2646:x
2643:d
2623:x
2620:d
2614:x
2604:x
2601:x
2587:+
2582:x
2579:x
2554:x
2551:d
2548:+
2545:x
2523:x
2520:x
2489:x
2486:z
2475:,
2470:x
2467:y
2437:x
2434:x
2423:=
2418:x
2414:s
2393:]
2388:z
2385:x
2374:,
2369:y
2366:x
2355:,
2350:x
2347:x
2336:[
2333:=
2316:n
2310:=
2301:s
2272:x
2262:e
2252:=
2249:]
2246:0
2243:,
2240:0
2237:,
2234:1
2228:[
2225:=
2216:n
2191:x
2188:x
2158:x
2155:x
2114:N
2111:=
2106:2
2102:m
2091:2
2087:m
2083:N
2078:=
2075:m
2069:m
2063:a
2060:P
2039:z
2036:d
2032:y
2029:d
2023:x
2020:x
2002:X
1984:m
1974:F
1967:+
1962:p
1952:F
1945:=
1936:F
1910:p
1900:F
1871:m
1861:F
1830:)
1825:1
1821:x
1817:(
1814:f
1808:)
1805:x
1799:+
1794:1
1790:x
1786:(
1783:f
1780:=
1777:f
1749:1
1745:x
1741:d
1736:)
1731:1
1727:x
1723:(
1720:f
1717:d
1711:=
1708:)
1703:1
1699:x
1695:(
1688:f
1665:1
1661:x
1640:)
1637:x
1634:(
1631:f
1583:F
1577:=
1571:t
1568:d
1557:p
1551:d
1522:0
1516:t
1493:t
1470:t
1436:F
1421:t
1407:)
1404:t
1401:(
1392:p
1356:F
1348:t
1342:=
1339:)
1336:t
1333:(
1324:p
1315:)
1312:t
1306:+
1303:t
1300:(
1291:p
1259:s
1253:F
1243:j
1225:s
1221:=
1217:F
1207:+
1201:t
1192:j
1141:)
1126:2
1119:s
1110:2
1103:m
1096:g
1093:k
1090:=
1087:m
1083:/
1079:a
1076:P
1049:]
1039:z
1029:z
1026:z
1011:+
1004:y
994:z
991:y
976:+
969:x
959:z
956:x
933:z
923:y
920:z
905:+
898:y
888:y
885:y
870:+
863:x
853:y
850:x
827:z
817:x
814:z
799:+
792:y
782:x
779:y
764:+
757:x
747:x
744:x
726:[
721:=
700:)
685:2
681:s
676:/
672:m
641:]
633:z
629:f
619:y
615:f
605:x
601:f
594:[
589:=
585:f
574:)
559:2
552:s
543:1
536:m
529:g
526:k
523:=
518:2
514:m
509:/
505:N
502:=
499:a
496:P
459:)
444:3
440:m
435:/
431:g
428:k
389:)
374:2
370:s
365:/
361:m
339:u
328:u
324:+
320:u
314:t
288:u
260:t
257:D
251:u
247:D
234:)
221:s
196:t
186:)
173:s
169:/
165:m
139:u
112:f
108:+
89:1
84:=
78:t
75:D
69:u
65:D
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