Hagen–Poiseuille equation

9663:. The equation states that flow rate is proportional to the radius to the fourth power, meaning that a small increase in the internal diameter of the cannula yields a significant increase in flow rate of IV fluids. The radius of IV cannulas is typically measured in "gauge", which is inversely proportional to the radius. Peripheral IV cannulas are typically available as (from large to small) 14G, 16G, 18G, 20G, 22G, 26G. As an example, assuming cannula lengths are equal, the flow of a 14G cannula is 1.73 times that of a 16G cannula, and 4.16 times that of a 20G cannula. It also states that flow is inversely proportional to length, meaning that longer lines have lower flow rates. This is important to remember as in an emergency, many clinicians favor shorter, larger catheters compared to longer, narrower catheters. While of less clinical importance, an increased change in pressure ( 6577: 7380: 6025: 5604: 2276: 8886: 6814: 6572:{\displaystyle {\begin{aligned}u(y,z)&={\frac {G}{2\mu }}y(h-y)-{\frac {4Gh^{2}}{\mu \pi ^{3}}}\sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{3}}}{\frac {\sinh(\beta _{n}z)+\sinh}{\sinh(\beta _{n}l)}}\sin(\beta _{n}y),\quad \beta _{n}={\frac {(2n-1)\pi }{h}},\\Q&={\frac {Gh^{3}l}{12\mu }}-{\frac {16Gh^{4}}{\pi ^{5}\mu }}\sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{5}}}{\frac {\cosh(\beta _{n}l)-1}{\sinh(\beta _{n}l)}}.\end{aligned}}} 3541: 5184: 1141: 4258: 2365: 7666: 5656: 4345: 3095: 4797: 8466: 7375:{\displaystyle {\begin{aligned}u(y,z)&={\frac {G}{2\mu }}(y+z)(\pi -y)-{\frac {G}{\pi \mu }}\sum _{n=1}^{\infty }{\frac {1}{\beta _{n}^{3}\sinh(2\pi \beta _{n})}}\left\{\sinh\sin-\sinh\sin\right\},\quad \beta _{n}=n+{\tfrac {1}{2}},\\Q&={\frac {G\pi ^{4}}{12\mu }}-{\frac {G}{2\pi \mu }}\sum _{n=1}^{\infty }{\frac {1}{\beta _{n}^{5}}}\left.\end{aligned}}} 3279: 3294: 5599:{\displaystyle {\begin{aligned}F_{1}(kr)&={\frac {\mathrm {ber} (kr)\mathrm {ber} (kR)+\mathrm {bei} (kr)\mathrm {bei} (kR)}{\mathrm {ber} ^{2}(kR)+\mathrm {bei} ^{2}(kR)}},\\F_{2}(kr)&={\frac {\mathrm {ber} (kr)\mathrm {bei} (kR)-\mathrm {bei} (kr)\mathrm {ber} (kR)}{\mathrm {ber} ^{2}(kR)+\mathrm {bei} ^{2}(kR)}},\end{aligned}}} 8831: 950: 3951: 7399: 1159:

case of turbulent flow, even though the flow profile in turbulent flow is strictly speaking not actually parabolic. In both cases, laminar or turbulent, the pressure drop is related to the stress at the wall, which determines the so-called friction factor. The wall stress can be determined phenomenologically by the

3865: 2293:. Laminar flow in a round pipe prescribes that there are a bunch of circular layers (lamina) of liquid, each having a velocity determined only by their radial distance from the center of the tube. Also assume the center is moving fastest while the liquid touching the walls of the tube is stationary (due to the 2898: 1633: 3680: 1263:

is the mean flow velocity, which is half the maximal flow velocity in the case of laminar flow. It proves more useful to define the Reynolds number in terms of the mean flow velocity because this quantity remains well defined even in the case of turbulent flow, whereas the maximal flow velocity may

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and the axial velocity are not constant along the tube; but the mass flow rate is constant along the tube length. The volumetric flow rate is usually expressed at the outlet pressure. As fluid is compressed or expanded, work is done and the fluid is heated or cooled. This means that the flow rate

1158:

Normally, Hagen–Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic. However, the result for the pressure drop can be extended to turbulent flow by inferring an effective turbulent viscosity in the

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flow is recovered. More explicit solutions with cross-sections such as snail-shaped sections, sections having the shape of a notch circle following a semicircle, annular sections between homofocal ellipses, annular sections between non-concentric circles are also available, as reviewed by

3536:{\displaystyle 0=-\Delta p2\pi r\,\mathrm {d} r+2\pi \mu \,\mathrm {d} r\,\Delta x{\frac {\mathrm {d} v}{\mathrm {d} r}}+2\pi r\mu \,\mathrm {d} r\,\Delta x{\frac {\mathrm {d} ^{2}v}{\mathrm {d} r^{2}}}+2\pi \mu (\mathrm {d} r)^{2}\,\Delta x{\frac {\mathrm {d} ^{2}v}{\mathrm {d} r^{2}}}.} 1285:

in the laminar flow at very low velocities in cylindrical tube. The theoretical derivation of a slightly different form of the law was made independently by Wiedman in 1856 and Neumann and E. Hagenbach in 1858 (1859, 1860). Hagenbach was the first who called this law Poiseuille's law.

5173: 2169: 8598: 2654: 1136:{\displaystyle {\begin{aligned}\Delta p={\frac {1}{2}}\rho {\overline {v}}_{\text{max}}^{2}&={\frac {1}{2}}\rho \left({\frac {Q_{\text{max}}}{\pi R^{2}}}\right)^{2}\\\Rightarrow \quad Q_{\max }{}&=\pi R^{2}{\sqrt {\frac {2\Delta p}{\rho }}},\end{aligned}}} 4253:{\displaystyle u(r,t)={\frac {G}{4\mu }}\left(R^{2}-r^{2}\right)-{\frac {2GR^{2}}{\mu }}\sum _{n=1}^{\infty }{\frac {1}{\lambda _{n}^{3}}}{\frac {J_{0}(\lambda _{n}r/R)}{J_{1}(\lambda _{n})}}e^{-\lambda _{n}^{2}\nu t/R^{2}},\quad J_{0}\left(\lambda _{n}\right)=0} 7846: 7661:{\displaystyle {\begin{aligned}u(y,z)&={\frac {G}{2\mu \left({\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}\right)}}\left(1-{\frac {y^{2}}{a^{2}}}-{\frac {z^{2}}{b^{2}}}\right),\\Q&={\frac {\pi Ga^{3}b^{3}}{4\mu \left(a^{2}+b^{2}\right)}}.\end{aligned}}} 8471:

Here we assumed the local pressure gradient is not too great to have any compressibility effects. Though locally we ignored the effects of pressure variation due to density variation, over long distances these effects are taken into account. Since

9670:) — such as by pressurizing the bag of fluid, squeezing the bag, or hanging the bag higher (relative to the level of the cannula) — can be used to speed up flow rate. It is also useful to understand that viscous fluids will flow slower (e.g. in 2498:, the force on the slower liquid is equal and opposite (no negative sign) to the force on the faster liquid. This equation assumes that the area of contact is so large that we can ignore any effects from the edges and that the fluids behave as 735:

the viscosity of the fluid; other types of pressure drops may still occur in a fluid (see a demonstration here). For example, the pressure needed to drive a viscous fluid up against gravity would contain both that as needed in Poiseuille's law

8035: 3737: 1242: 3090:{\displaystyle 0=-\Delta p2\pi r\,\mathrm {d} r-2\pi r\mu \,\Delta x\left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right|_{r}+2\pi (r+\mathrm {d} r)\mu \,\Delta x\,\left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right\vert _{r+\mathrm {d} r}.} 4913: 2887: 1526: 3560: 4792:{\displaystyle {\begin{aligned}u(r)&={\frac {G}{4\mu }}\left(R_{1}^{2}-r^{2}\right)+{\frac {G}{4\mu }}\left(R_{2}^{2}-R_{1}^{2}\right){\frac {\ln r/R_{1}}{\ln R_{2}/R_{1}}},\\Q&={\frac {G\pi }{8\mu }}\left.\end{aligned}}} 8148:

case, where the temperature of the fluid is permitted to equilibrate with its surroundings, an approximate relation for the pressure drop can be derived. Using ideal gas equation of state for constant temperature process (i.e.,

872: 6614: 5771: 2489: 8461:{\displaystyle -{\frac {\mathrm {d} p}{\mathrm {d} x}}={\frac {8\mu Q}{\pi R^{4}}}={\frac {8\mu Q_{2}p_{2}}{\pi pR^{4}}}\quad \Rightarrow \quad -p{\frac {\mathrm {d} p}{\mathrm {d} x}}={\frac {8\mu Q_{2}p_{2}}{\pi R^{4}}}.} 5949: 1730: 3274:{\displaystyle \left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right|_{r+\mathrm {d} r}=\left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right|_{r}+\left.{\frac {\mathrm {d} ^{2}v}{\mathrm {d} r^{2}}}\right|_{r}\,\mathrm {d} r.} 9635:, so its velocity does not depend on the distance to the walls of the conductor. The resistance is due to the interaction between the flowing electrons and the atoms of the conductor. Therefore, Poiseuille's law and the 8587: 2699: 9405: 9528: 4940: 8950:

is still conceptually useful for understanding circuits. This analogy is also used to study the frequency response of fluid-mechanical networks using circuit tools, in which case the fluid network is termed a

7932: 1987: 8112: 7404: 6819: 6619: 6030: 5189: 4413: 2554:. We don't know the exact form for the velocity of the liquid within the tube yet, but we do know (from our assumption above) that it is dependent on the radius. Therefore, the velocity gradient is the 1828: 9236: 2039: 2253: 2372:

direction. The liquid on top is moving faster and will be pulled in the negative direction by the bottom liquid while the bottom liquid will be pulled in the positive direction by the top liquid.

8826:{\displaystyle Q_{2}={\frac {\pi R^{4}}{16\mu L}}\left({\frac {p_{1}^{2}-p_{2}^{2}}{p_{2}}}\right)={\frac {\pi R^{4}\left(p_{1}-p_{2}\right)}{8\mu L}}{\frac {\left(p_{1}+p_{2}\right)}{2p_{2}}}.} 955: 932:

The equation fails in the limit of low viscosity, wide and/or short pipe. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as the

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is applied between two ends of a long pipe, the flow will not immediately obtain Poiseuille profile, rather it develops through time and reaches the Poiseuille profile at steady state. The

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Next let's find the force of drag from the slower lamina. We need to calculate the same values that we did for the force from the faster lamina. In this case, the area of contact is at

3940: 2562:. So, considering that this force will be positive with respect to the movement of the liquid (but the derivative of the velocity is negative), the final form of the equation becomes 9606: 10248:

Berker, R. (1963). "Intégration des équations du mouvement d'un fluide visqueux incompressible" [Integration of the equations of motion of a viscous incompressible fluid].

5838: 2693:. Also, we need to remember that this force opposes the direction of movement of the liquid and will therefore be negative (and that the derivative of the velocity is negative). 8213: 2286:

A cross section of the tube shows the lamina moving at different speeds. Those closest to the edge of the tube are moving slowly while those near the center are moving quickly.

3860:{\displaystyle {\frac {\partial u}{\partial t}}={\frac {G}{\rho }}+\nu \left({\frac {\partial ^{2}u}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial u}{\partial r}}\right)} 7946: 1324:

through a pipe of uniform (circular) cross-section is known as Hagen–Poiseuille flow. The equations governing the Hagen–Poiseuille flow can be derived directly from the

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The negative sign is in there because we are concerned with the faster moving liquid (top in figure), which is being slowed by the slower liquid (bottom in figure). By

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derived the velocity profile and volume flow rate in 1868 for rectangular channel and tubes of equilateral triangular cross-section and for elliptical cross-section.

940:

for the Hagen–Poiseuille law to be valid. If the pipe is too short, the Hagen–Poiseuille equation may result in unphysically high flow rates; the flow is bounded by

8247:

can be obtained. Over a short section of the pipe, the gas flowing through the pipe can be assumed to be incompressible so that Poiseuille law can be used locally,

1518: 1628:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial u}{\partial r}}\right)={\frac {1}{\mu }}{\frac {\mathrm {d} p}{\mathrm {d} x}}} 4836: 3675:{\displaystyle {\frac {1}{\mu }}{\frac {\Delta p}{\Delta x}}={\frac {\mathrm {d} ^{2}v}{\mathrm {d} r^{2}}}+{\frac {1}{r}}{\frac {\mathrm {d} v}{\mathrm {d} r}}} 2833: 9879: 4830:

Flow through pipes with an oscillating pressure gradient finds applications in blood flow through large arteries. The imposed pressure gradient is given by

6780:{\displaystyle {\begin{aligned}u(y,z)&=-{\frac {G}{4\mu h}}(y-h)\left(y^{2}-3z^{2}\right),\\Q&={\frac {Gh^{4}}{60{\sqrt {3}}\mu }}.\end{aligned}}} 781: 10087:

Lambossy, P. (1952). "Oscillations forcees d'un liquide incompressibile et visqueux dans un tube rigide et horizontal. Calcul de la force frottement".

5705: 2425: 5849: 1667: 2804:{\displaystyle F_{\text{viscosity, slow}}=2\pi (r+\mathrm {d} r)\mu \,\Delta x\left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right|_{r+\mathrm {d} r}} 8486: 9316: 8907: 7694: 9445: 763:

pressure drop along the direction of flow, which is proportional to length traveled (as per Poiseuille's law). Both effects contribute to the

701:, and published by Hagen in 1839 and then by Poiseuille in 1840–41 and 1846. The theoretical justification of the Poiseuille law was given by 5168:{\displaystyle u(r,t)={\frac {G}{4\mu }}\left(R^{2}-r^{2}\right)+{\frac {\cos \omega t}{\rho \omega }}+{\frac {\sin \omega t}{\rho \omega }}} 3284:

The expression is valid for all laminae. Grouping like terms and dropping the vertical bar since all derivatives are assumed to be at radius

629: 9780:

Stokes, G. G. (1845). "On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids".

1650:, implying that both terms must be the same constant. Evaluating this constant is straightforward. If we take the length of the pipe to be 1167:, given a relationship for the friction factor in terms of the Reynolds number. In the case of laminar flow, for a circular cross section: 9961: 3685:

The above equation is the same as the one obtained from the Navier–Stokes equations and the derivation from here on follows as before.

759:), and its pressure will be lower than in a larger diameter (due to Bernoulli's equation). However, the viscosity of blood will cause 7857: 10155:

Uchida, S. (1956). "The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a circular pipe".

1918: 8046: 2164:{\displaystyle {u}_{\mathrm {avg} }={\frac {1}{\pi R^{2}}}\int _{0}^{R}2\pi ru\mathrm {d} r={\tfrac {1}{2}}{u}_{\mathrm {max} }.} 1745: 1325: 9169: 10281: 10013: 9839: 2196: 10488: 2321:. This force is in the direction of the motion of the liquid. The negative sign comes from the conventional way we define 10001: 2649:{\displaystyle F_{\text{viscosity, fast}}=-2\pi r\mu \,\Delta x\,\left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right|_{r}} 10483: 10108:"Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known" 9015: 2819:

To find the solution for the flow of a laminar layer through a tube, we need to make one last assumption. There is no

10327: 9944: 8933: 7841:{\displaystyle {\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=-{\frac {G}{\mu }}.} 4326: 4298: 622: 8915: 10493: 9730: 9639:

are useful only within certain limits when applied to electricity. Both Ohm's law and Poiseuille's law illustrate

720:

through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no

694: 682:

flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in

9918: 36: 724:

of fluid in the pipe. For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but

10198:

Boussinesq, Joseph (1868). "Mémoire sur l'influence des Frottements dans les Mouvements Réguliers des Fluids".

9564:

of the resistor, which is different from the electrical formula. The electrical relation for the resistance is

8911: 702: 698: 595: 3876: 2381: 133: 9570: 5695:

is applied in the direction of flow. The flow is essentially unidirectional because of infinite length. The

744:, such that any point in the flow would have a pressure greater than zero (otherwise no flow would happen). 5663:

Plane Poiseuille flow is flow created between two infinitely long parallel plates, separated by a distance

615: 336: 222: 5786: 5696: 3728: 2827:, there is no net force. If there is no net force then we can add all of the forces together to get zero 2264: 1317: 1160: 933: 291: 200: 83: 2540:. The area of contact between the lamina and the faster one is simply the surface area of the cylinder: 9118:

is their total charge. This is the charge that flows through the cross section per unit time, i.e. the

8180: 6582:

The velocity and the volume flow rate of tube with equilateral triangular cross-section of side length

5959: 2824: 2495: 1642:

is the dynamic viscosity of the fluid. In the above equation, the left-hand side is only a function of

207: 10468: 10463: 10458: 10267: 8896: 8030:{\displaystyle {\frac {\partial ^{2}U}{\partial y^{2}}}+{\frac {\partial ^{2}U}{\partial z^{2}}}=0} 2030: 941: 502: 497: 286: 279: 112: 9623:

of the resistor. The reason why Poiseuille's law leads to a different formula for the resistance

8900: 1237:{\displaystyle \Lambda ={\frac {64}{\mathrm {Re} }},\quad \mathrm {Re} ={\frac {\rho vd}{\mu }},} 741: 565: 560: 229: 2376:

When two layers of liquid in contact with each other move at different speeds, there will be a

117: 9411: 9119: 8152: 3547: 1266: 540: 158: 10271: 4908:{\displaystyle {\frac {\partial p}{\partial x}}=-G-\alpha \cos \omega t-\beta \sin \omega t} 2307:

force pushing the liquid through the tube is the change in pressure multiplied by the area:

1264:

not be, or in any case, it may be difficult to infer. In this form the law approximates the

10395: 10164: 10053: 9739: 8126: 2882:{\displaystyle 0=F_{\text{pressure}}+F_{\text{viscosity, fast}}+F_{\text{viscosity, slow}}} 1496: 910: 756: 378: 195: 175: 163: 107: 8: 10343:

Fütterer, C.; et al. (2004). "Injection and flow control system for microchannels".

9652: 9640: 3101: 1450: 752: 580: 428: 321: 27: 10407: 10399: 10168: 10057: 9804: 9751: 9743: 2299:

To figure out the motion of the liquid, all forces acting on each lamina must be known:

10425: 10297: 10180: 10132: 10107: 10069: 9910: 9898: 1855: 1146:

because it is impossible to have negative (absolute) pressure (not to be confused with

686: 600: 234: 190: 185: 2349:

effects will pull from the faster lamina immediately closer to the center of the tube.

10429: 10360: 10323: 10277: 10184: 10137: 10073: 10009: 9940: 9902: 9845: 9835: 9703: 9671: 9656: 9636: 8952: 8947: 5777: 2355:

effects will drag from the slower lamina immediately closer to the walls of the tube.

2294: 1445:). Here however, this can be proved via mass conservation, and the above assumptions. 900: 728:, leading to larger pressure drops than calculated by the Hagen–Poiseuille equation. 690: 217: 168: 9984: 9914: 10421: 10403: 10352: 10230: 10172: 10127: 10123: 10119: 10061: 9894: 9747: 7938: 5618: 3100:

First, to get everything happening at the same point, use the first two terms of a

867:{\displaystyle \Delta p={\frac {8\mu LQ}{\pi R^{4}}}={\frac {8\pi \mu LQ}{A^{2}}},} 748: 555: 530: 443: 418: 413: 368: 9256:

is the total charge in the volume of the tube. The volume of the tube is equal to

5843:

Therefore, the velocity distribution and the volume flow rate per unit length are

4825: 9829: 9728:

Sutera, Salvatore P.; Skalak, Richard (1993). "The History of Poiseuille's Law".

8476:

is independent of pressure, the above equation can be integrated over the length

7703: 7699: 5963: 5766:{\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} y^{2}}}=-{\frac {G}{\mu }}} 2499: 2484:{\displaystyle F_{\text{viscosity, top}}=-\mu A{\frac {\Delta v_{x}}{\Delta y}}.} 2267:, an alternative method of deriving the Hagen–Poiseuille equation is as follows. 1252: 937: 713: 675: 545: 469: 433: 383: 314: 303: 248: 150: 10383: 10223:

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science

9688: 9646: 9082: 5944:{\displaystyle u(y)={\frac {G}{2\mu }}y(h-y),\quad Q={\frac {Gh^{3}}{12\mu }}.} 1725:{\displaystyle -{\frac {\mathrm {d} p}{\mathrm {d} x}}={\frac {\Delta p}{L}}=G} 1305: 1147: 709: 671: 643: 550: 408: 373: 274: 180: 10234: 8836:

This equation can be seen as Poiseuille's law with an extra correction factor

7385:

The velocity distribution for tubes of elliptical cross-section with semiaxes

5994:

be the constant pressure gradient acting in direction parallel to the motion.

2558:

at the intersection of these two laminae. That intersection is at a radius of

10477: 10263: 9849: 9632: 9542:

of the resistor, which is true. However, it also follows that the resistance

6790:

The velocity and the volume flow rate in the right-angled isosceles triangle

4402:, the velocity distribution and the volume flux through the annular pipe are 936:. The ratio of length to radius of a pipe should be greater than 1/48 of the 667: 590: 423: 10364: 10141: 9683: 8956: 8582:{\displaystyle p_{1}^{2}-p_{2}^{2}={\frac {16\mu LQ_{2}p_{2}}{\pi R^{4}}}.} 5954: 2820: 2290: 1321: 1294: 1290: 721: 717: 679: 663: 575: 570: 535: 267: 10273:

The Navier–Stokes equations: a classification of flows and exact solutions

10044:

Sexl, T. (1930). "Über den von EG Richardson entdeckten 'Annulareffekt'".

9400:{\displaystyle V={\frac {8\mu LI}{n^{2}\pi r^{4}\left(q^{*}\right)^{2}}}.} 2174:

The easily measurable quantity in experiments is the volumetric flow rate

10433: 9906: 5997:

The velocity and the volume flow rate in a rectangular channel of height

2377: 2275: 585: 488: 16:

Law describing the pressure drop in an incompressible and Newtonian fluid

9767:

Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen

9554:

is inversely proportional to the second power of the cross section area

9523:{\displaystyle R={\frac {8\mu L}{n^{2}\pi r^{4}\left(q^{*}\right)^{2}}}} 10315: 10176: 10065: 9831:

Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices

8145: 2664: 2555: 1654:

and denote the pressure difference between the two ends of the pipe by

1298: 1164: 725: 507: 403: 7937:

then it is easy to see that the problem reduces to that integrating a

10356: 8141: 2352: 2346: 1372:

The radial and azimuthal components of the fluid velocity are zero (

479: 474: 308: 10218: 8885: 8120: 2029:. The average velocity can be obtained by integrating over the pipe 9659:

that may be achieved using various sizes of peripheral and central

9627:

is the difference between the fluid flow and the electric current.

2522: 2304: 1909: 1906: 1479: 1453:

are identically satisfied. The radial momentum equation reduces to

458: 363: 343: 329: 8946:

Electricity was originally understood to be a kind of fluid. This

7711: 10420:. Vol. 31, no. 2 (published Mar 1976). pp. 273–5. 9660: 9628: 9298: 8872:

expressing the average pressure relative to the outlet pressure.

212: 9266:, so the number of charged particles in this volume is equal to 1661:(high pressure minus low pressure), then the constant is simply 9995: 9993: 7927:{\displaystyle U=u+{\frac {G}{4\mu }}\left(y^{2}+z^{2}\right),} 4826:

Poiseuille flow in a pipe with an oscillating pressure gradient

2556:

change of the velocity with respect to the change in the radius

2511: 2392:, the velocity gradient perpendicular to the direction of flow 353: 9074:

be the concentration of free charged particles (in m) and let

8592:

Hence the volumetric flow rate at the pipe outlet is given by

2364: 1326:

Navier–Stokes momentum equations in 3D cylindrical coordinates

751:, its speed will be greater than in a larger diameter (due to 9693: 2510:

Assume that we are figuring out the force on the lamina with

2419:, and a proportionality constant (viscosity) and is given by 1982:{\displaystyle u={\frac {G}{4\mu }}\left(R^{2}-r^{2}\right).} 257: 9990: 9647:

Medical applications – intravenous access and fluid delivery

9546:

is inversely proportional to the fourth power of the radius

8107:{\displaystyle U={\frac {G}{4\mu }}\left(y^{2}+z^{2}\right)} 2259:

Elaborate derivation starting directly from first principles

2190:. Rearrangement of this gives the Hagen–Poiseuille equation 1449:

Then the angular equation in the momentum equations and the

10342: 9698: 9651:

The Hagen–Poiseuille equation is useful in determining the

8177:

is constant) and the conservation of mass flow rate (i.e.,

8140:

depends on the heat transfer to and from the fluid. For an

3209: 3168: 3116: 3037: 2959: 2754: 2610: 2518: 2385: 1823:{\displaystyle u=-{\frac {Gr^{2}}{4\mu }}+c_{1}\ln r+c_{2}} 683: 10304:. Vol. 1, Incompressible Flow. Van Nostrand Reinhold. 7681:, Poiseuille flow for circular pipe is recovered and when 1153: 5655: 393: 9231:{\displaystyle \Delta F={\frac {8\mu LI}{nr^{2}q^{*}}}.} 5955:

Poiseuille flow through some non-circular cross-sections

4339: 3690: 929:

The equation does not hold close to the pipe entrance.

8969:. Since the net force acting on the fluid is equal to 7177: 5966:

derived the same for isosceles triangles in 1914. Let

4344: 2248:{\displaystyle \Delta p={\frac {8\mu QL}{\pi R^{4}}}.} 2127: 1316:

The Hagen–Poiseuille equation can be derived from the

1308:

by L. R. Wilberforce, based on Hagenbach's work.

708:

The assumptions of the equation are that the fluid is

9573: 9448: 9319: 9172: 9018: 8601: 8489: 8256: 8183: 8155: 8049: 7949: 7860: 7747: 7402: 6817: 6617: 6028: 5852: 5789: 5708: 5187: 4943: 4839: 4411: 3954: 3879: 3740: 3563: 3297: 3113: 2901: 2836: 2702: 2571: 2428: 2199: 2042: 1921: 1748: 1670: 1529: 1499: 1176: 953: 784: 39: 10219:"Notes on the motion of viscous liquids in channels" 9619:

is inversely proportional to the cross section area

1992:

The maximum velocity occurs at the pipe centerline (

747:

Another example is when blood flows into a narrower

689:, or the flow through a drinking straw or through a 9937:

Life in Moving Fluids: The Physical Biology of Flow

9782:

Transactions of the Cambridge Philosophical Society

1646:and the right-hand side term is only a function of 10459:Poiseuille's law for power-law non-Newtonian fluid 9600: 9522: 9399: 9230: 9058: 8825: 8581: 8460: 8207: 8169: 8106: 8029: 7926: 7840: 7660: 7374: 6779: 6571: 5943: 5832: 5765: 5598: 5167: 4907: 4791: 4252: 3934: 3859: 3674: 3535: 3273: 3089: 2881: 2803: 2648: 2483: 2247: 2163: 1981: 1822: 1724: 1627: 1512: 1236: 1135: 866: 731:Poiseuille's equation describes the pressure drop 74: 9059:{\displaystyle \Delta F={\frac {8\mu LQ}{r^{2}}}} 8121:Poiseuille's equation for an ideal isothermal gas 4934:is the frequency. The velocity field is given by 2667:indicates that it should be taken at a radius of 693:. It was experimentally derived independently by 10475: 10157:Zeitschrift für angewandte Mathematik und Physik 9962:"Energetic analysis of the Hagen–Poiseuille law" 7738:on the walls. The governing equation reduces to 1079: 887:is the pressure difference between the two ends, 7712:Poiseuille flow through arbitrary cross-section 4374:applied pressure gradient between the two ends 2517:. From the equation above, we need to know the 1304:Poiseuille's law was later in 1891 extended to 10416:Pfitzner, J (1976). "Poiseuille and his law". 10314: 9009:, then from Poiseuille's law, it follows that 8875: 3546:Finally, put this expression in the form of a 2263:Although more lengthy than directly using the 2270: 623: 10441: 10381: 9727: 7851:If we introduce a new dependent variable as 4308:are the positive roots of this function and 1343:by making the following set of assumptions: 10464:Poiseuille's law in a slightly tapered tube 10386:(1993). "The history of Poiseuille's law". 10296: 10262: 9959: 8914:. Unsourced material may be challenged and 75:{\displaystyle J=-D{\frac {d\varphi }{dx}}} 10197: 2814: 2525:. Think of the lamina as a ring of radius 630: 616: 10322:. Pergamon Press. p. 55, problem 6. 10131: 10105: 10028: 9999: 9107:is the number of particles in the volume 8934:Learn how and when to remove this message 7716:The flow through arbitrary cross-section 3485: 3407: 3398: 3352: 3343: 3322: 3259: 3034: 3027: 2950: 2926: 2745: 2607: 2600: 2368:Two fluids moving past each other in the 1520:. The axial momentum equation reduces to 10415: 10216: 10086: 9877: 9834:. New York: Cambridge University Press. 9615:is the resistivity; i.e. the resistance 5654: 5650: 4343: 3935:{\displaystyle u(r,0)=0,\quad u(R,t)=0.} 2363: 2274: 944:, under less restrictive conditions, by 9985:Determinants of blood vessel resistance 8125:For a compressible fluid in a tube the 1154:Relation to the Darcy–Weisbach equation 10476: 10247: 10154: 9779: 9723: 9721: 9719: 9601:{\displaystyle R={\frac {\rho L}{S}},} 3945:The velocity distribution is given by 3870:with initial and boundary conditions, 1485:is a function of the axial coordinate 10442:Bennett, C. O.; Myers, J. E. (1962). 9934: 9862: 9827: 9764: 9410:This is exactly Ohm's law, where the 4822:, the original problem is recovered. 4340:Poiseuille flow in an annular section 2659:where the vertical bar and subscript 2282:A tube showing the imaginary lamina. 1905:. Thus we have finally the following 775:In standard fluid-kinetics notation: 10469:Hagen–Poiseuille equation calculator 10043: 9856: 9821: 9799: 9797: 9795: 8912:adding citations to reliable sources 8879: 4336:, Poiseuille solution is recovered. 3691:Startup of Poiseuille flow in a pipe 925:is the cross-sectional area of pipe. 10408:10.1146/annurev.fl.25.010193.000245 10002:"An Introduction to Fluid Dynamics" 9752:10.1146/annurev.fl.25.010193.000245 9716: 9709: 9081:be the charge of each particle (in 5833:{\displaystyle u(0)=0,\quad u(h)=0} 1875:(radius of the pipe), which yields 13: 10426:10.1111/j.1365-2044.1976.tb11804.x 9899:10.1111/j.1365-2044.1976.tb11804.x 9173: 9019: 8955:. Poiseuille's law corresponds to 8395: 8385: 8274: 8264: 8005: 7991: 7968: 7954: 7803: 7789: 7766: 7752: 7269: 6926: 6459: 6147: 5730: 5714: 5667:with a constant pressure gradient 5564: 5561: 5558: 5531: 5528: 5525: 5505: 5502: 5499: 5482: 5479: 5476: 5456: 5453: 5450: 5433: 5430: 5427: 5362: 5359: 5356: 5329: 5326: 5323: 5303: 5300: 5297: 5280: 5277: 5274: 5254: 5251: 5248: 5231: 5228: 5225: 4851: 4843: 4370:is the outer cylinder radii, with 4348:Poiseuille flow in annular section 4068: 3843: 3835: 3803: 3789: 3752: 3744: 3662: 3652: 3618: 3602: 3585: 3577: 3513: 3497: 3486: 3468: 3435: 3419: 3408: 3400: 3373: 3363: 3353: 3345: 3324: 3307: 3261: 3232: 3216: 3184: 3174: 3154: 3132: 3122: 3075: 3053: 3043: 3028: 3014: 2975: 2965: 2951: 2928: 2911: 2792: 2770: 2760: 2746: 2732: 2626: 2616: 2601: 2469: 2454: 2200: 2152: 2149: 2146: 2116: 2057: 2054: 2051: 1704: 1688: 1678: 1615: 1605: 1574: 1566: 1546: 1542: 1289:The law is also very important in 1206: 1203: 1192: 1189: 1177: 1113: 958: 785: 14: 10505: 10452: 10444:Momentum, Heat, and Mass Transfer 9792: 8208:{\displaystyle {\dot {m}}=\rho Q} 4327:Bessel function of the first kind 4299:Bessel function of the first kind 3550:, dropping the term quadratic in 10388:Annual Review of Fluid Mechanics 10000:Batchelor, George Keith (2000). 9731:Annual Review of Fluid Mechanics 8884: 4361:is the inner cylinder radii and 2674: 2505: 10336: 10308: 10290: 10256: 10241: 10210: 10191: 10148: 10099: 10080: 10037: 10022: 9978: 9534:It follows that the resistance 8374: 8370: 7156: 6312: 5904: 5811: 4215: 3907: 1858:at the pipe wall requires that 1201: 1073: 10276:. Cambridge University Press. 10252:. Vol. 3. pp. 1–384. 10124:10.1113/jphysiol.1955.sp005276 10029:Rosenhead, Louis, ed. (1963). 9953: 9928: 9871: 9773: 9758: 9538:is proportional to the length 8371: 7422: 7410: 7357: 7338: 7326: 7307: 7145: 7142: 7130: 7117: 7108: 7105: 7093: 7080: 7068: 7065: 7053: 7040: 7031: 7028: 7007: 6994: 6977: 6958: 6889: 6877: 6874: 6862: 6837: 6825: 6680: 6668: 6637: 6625: 6556: 6540: 6523: 6507: 6486: 6470: 6344: 6329: 6306: 6290: 6278: 6262: 6251: 6248: 6236: 6223: 6211: 6195: 6174: 6158: 6088: 6076: 6048: 6036: 5898: 5886: 5862: 5856: 5821: 5815: 5799: 5793: 5583: 5574: 5550: 5541: 5518: 5509: 5495: 5486: 5469: 5460: 5446: 5437: 5413: 5404: 5381: 5372: 5348: 5339: 5316: 5307: 5293: 5284: 5267: 5258: 5244: 5235: 5211: 5202: 5136: 5133: 5114: 5092: 5060: 5057: 5038: 5016: 4959: 4947: 4425: 4419: 4160: 4147: 4132: 4108: 3970: 3958: 3923: 3911: 3895: 3883: 3476: 3464: 3021: 3004: 2823:of liquid in the pipe, and by 2739: 2722: 1739:is positive. The solution is 1417:The flow is fully developed ( 1070: 699:Gotthilf Heinrich Ludwig Hagen 1: 10375: 9809:hyperphysics.phy-astr.gsu.edu 9070:For electrical circuits, let 7731:satisfies the condition that 1311: 1150:) in an incompressible flow. 695:Jean Léonard Marie Poiseuille 9439:is described by the formula 9279:, and their total charge is 2521:of contact and the velocity 2496:Newton's third law of motion 2380:between them. This force is 2359: 986: 7: 10489:Equations of fluid dynamics 9769:. Basel: Birkhäuser Verlag. 9677: 8876:Electrical circuits analogy 8215:is constant), the relation 2289:Assume the liquid exhibits 1392:The flow is axisymmetric ( 770: 10: 10510: 10318:; Lifshitz, E. M. (1987). 10006:Cambridge University Press 9960:tec-science (2020-04-02). 3104:of the velocity gradient: 2271:Liquid flow through a pipe 1259:is the fluid density, and 10484:Eponymous laws of physics 10235:10.1080/14786440708635179 10106:Womersley, J. R. (1955). 9865:Theoretical Microfluidics 8959:for electrical circuits, 8040:satisfying the condition 1272:energy (head) loss factor 648:Hagen–Poiseuille equation 10300:; Davies, H. J. (1971). 9880:"Poiseuille and his law" 134:Clausius–Duhem (entropy) 84:Fick's laws of diffusion 10494:Mathematics in medicine 10031:Laminar Boundary Layers 9939:. PWS Kent Publishers. 9657:intravenous (IV) fluids 9655:and hence flow rate of 8170:{\displaystyle p/\rho } 5697:Navier–Stokes equations 3729:Navier–Stokes equations 3102:Taylor series expansion 2815:Putting it all together 2265:Navier–Stokes equations 1489:only. For brevity, use 1318:Navier–Stokes equations 1280:Darcy (friction) factor 1161:Darcy–Weisbach equation 934:Darcy–Weisbach equation 292:Navier–Stokes equations 230:Material failure theory 10046:Zeitschrift für Physik 9935:Vogel, Steven (1981). 9765:Szabó, István (1979). 9602: 9550:, i.e. the resistance 9524: 9401: 9232: 9060: 8827: 8583: 8462: 8209: 8171: 8108: 8031: 7928: 7842: 7662: 7376: 7273: 6930: 6781: 6573: 6463: 6151: 5945: 5834: 5767: 5660: 5600: 5169: 4909: 4793: 4349: 4254: 4072: 3936: 3861: 3676: 3537: 3275: 3091: 2883: 2805: 2650: 2485: 2373: 2287: 2249: 2165: 1983: 1837:needs to be finite at 1824: 1726: 1629: 1514: 1238: 1137: 893:is the length of pipe, 868: 76: 10302:Modern Fluid Dynamics 10217:Proudman, J. (1914). 10112:Journal of Physiology 9878:Pfitzner, J. (1976). 9828:Kirby, B. J. (2010). 9603: 9525: 9402: 9233: 9061: 8828: 8584: 8463: 8210: 8172: 8109: 8032: 7929: 7843: 7663: 7377: 7253: 6910: 6782: 6574: 6443: 6131: 5946: 5835: 5768: 5659:Plane Poiseuille flow 5658: 5651:Plane Poiseuille flow 5601: 5170: 4910: 4794: 4347: 4255: 4052: 3937: 3862: 3677: 3548:differential equation 3538: 3276: 3092: 2884: 2806: 2651: 2486: 2367: 2278: 2250: 2166: 1984: 1825: 1727: 1630: 1515: 1513:{\displaystyle u_{x}} 1347:The flow is steady ( 1267:Darcy friction factor 1239: 1138: 942:Bernoulli's principle 869: 287:Bernoulli's principle 280:Archimedes' principle 77: 9571: 9446: 9317: 9170: 9016: 8908:improve this section 8599: 8487: 8254: 8181: 8153: 8127:volumetric flow rate 8047: 7947: 7858: 7745: 7400: 6815: 6615: 6026: 5850: 5787: 5706: 5185: 4941: 4837: 4409: 3952: 3877: 3738: 3561: 3295: 3111: 2899: 2834: 2700: 2569: 2426: 2197: 2040: 1919: 1746: 1668: 1527: 1497: 1276:friction loss factor 1174: 951: 911:volumetric flow rate 782: 757:volumetric flow rate 742:Bernoulli's equation 652:Hagen–Poiseuille law 650:, also known as the 379:Cohesion (chemistry) 201:Infinitesimal strain 37: 10400:1993AnRFM..25....1S 10298:Curle, Samuel Newby 10250:Handbuch der Physik 10200:J. Math. Pures Appl 10169:1956ZaMP....7..403U 10058:1930ZPhy...61..349S 9744:1993AnRFM..25....1S 9653:vascular resistance 9641:transport phenomena 9085:). (For electrons, 8684: 8666: 8522: 8504: 7293: 6951: 4730: 4712: 4686: 4668: 4543: 4525: 4469: 4188: 4092: 2102: 1451:continuity equation 1001: 919:is the pipe radius, 660:Poiseuille equation 297:Poiseuille equation 28:Continuum mechanics 22:Part of a series on 10177:10.1007/BF01606327 10066:10.1007/BF01340631 10033:. Clarendon Press. 9863:Bruus, H. (2007). 9598: 9520: 9397: 9310:, it follows then 9228: 9056: 8823: 8670: 8652: 8579: 8508: 8490: 8458: 8205: 8167: 8104: 8027: 7924: 7838: 7658: 7656: 7372: 7370: 7279: 7186: 6937: 6777: 6775: 6569: 6567: 5941: 5830: 5763: 5661: 5596: 5594: 5165: 4930:are constants and 4905: 4789: 4787: 4716: 4698: 4672: 4654: 4529: 4511: 4455: 4350: 4301:of order zero and 4250: 4174: 4078: 3932: 3857: 3699:pressure gradient 3672: 3533: 3271: 3087: 2879: 2825:Newton's first law 2801: 2646: 2481: 2374: 2288: 2245: 2161: 2136: 2088: 1979: 1856:boundary condition 1820: 1735:defined such that 1722: 1625: 1510: 1234: 1133: 1131: 980: 864: 740:that as needed in 503:Magnetorheological 498:Electrorheological 235:Fracture mechanics 72: 10283:978-0-521-68162-9 10264:Drazin, Philip G. 10015:978-0-521-66396-0 9841:978-0-521-11903-0 9704:Hydraulic circuit 9672:blood transfusion 9637:hydraulic analogy 9593: 9518: 9392: 9223: 9054: 8953:hydraulic circuit 8948:hydraulic analogy 8944: 8943: 8936: 8818: 8766: 8696: 8643: 8574: 8453: 8403: 8368: 8315: 8282: 8193: 8069: 8019: 7982: 7886: 7833: 7817: 7780: 7649: 7554: 7527: 7492: 7484: 7464: 7294: 7251: 7230: 7185: 6981: 6908: 6860: 6768: 6762: 6666: 6560: 6496: 6441: 6401: 6354: 6282: 6184: 6129: 6071: 5960:Joseph Boussinesq 5936: 5881: 5778:no-slip condition 5761: 5745: 5587: 5385: 5163: 5087: 4978: 4858: 4775: 4647: 4610: 4504: 4448: 4329:of order one. As 4164: 4093: 4050: 3989: 3850: 3830: 3817: 3772: 3759: 3670: 3646: 3633: 3592: 3572: 3528: 3450: 3381: 3247: 3192: 3140: 3061: 2983: 2876: 2863: 2850: 2778: 2710: 2634: 2579: 2476: 2436: 2295:no-slip condition 2240: 2135: 2086: 1941: 1783: 1714: 1696: 1623: 1599: 1581: 1553: 1538: 1297:, both fields of 1229: 1196: 1124: 1123: 1054: 1036: 1017: 994: 989: 975: 901:dynamic viscosity 859: 825: 691:hypodermic needle 640: 639: 515: 514: 449: 448: 218:Contact mechanics 141: 140: 70: 10501: 10447: 10437: 10411: 10369: 10368: 10357:10.1039/B316729A 10340: 10334: 10333: 10312: 10306: 10305: 10294: 10288: 10287: 10260: 10254: 10253: 10245: 10239: 10238: 10214: 10208: 10207: 10195: 10189: 10188: 10152: 10146: 10145: 10135: 10103: 10097: 10096: 10089:Helv. Phys. Acta 10084: 10078: 10077: 10052:(5–6): 349–362. 10041: 10035: 10034: 10026: 10020: 10019: 9997: 9988: 9982: 9976: 9975: 9973: 9972: 9957: 9951: 9950: 9932: 9926: 9925: 9923: 9917:. Archived from 9884: 9875: 9869: 9868: 9860: 9854: 9853: 9825: 9819: 9818: 9816: 9815: 9801: 9790: 9789: 9777: 9771: 9770: 9762: 9756: 9755: 9725: 9710:Cited references 9669: 9626: 9622: 9618: 9614: 9607: 9605: 9604: 9599: 9594: 9589: 9581: 9563: 9553: 9549: 9545: 9541: 9537: 9529: 9527: 9526: 9521: 9519: 9517: 9516: 9515: 9510: 9506: 9505: 9491: 9490: 9478: 9477: 9467: 9456: 9438: 9437: 9435: 9434: 9429: 9426: 9406: 9404: 9403: 9398: 9393: 9391: 9390: 9389: 9384: 9380: 9379: 9365: 9364: 9352: 9351: 9341: 9327: 9309: 9296: 9278: 9265: 9255: 9251: 9237: 9235: 9234: 9229: 9224: 9222: 9221: 9220: 9211: 9210: 9197: 9183: 9162: 9161: 9159: 9158: 9152: 9149: 9136:. Consequently, 9135: 9124: 9117: 9110: 9106: 9102: 9101: 9099: 9080: 9073: 9065: 9063: 9062: 9057: 9055: 9053: 9052: 9043: 9029: 9008: 8993: 8983: 8968: 8939: 8932: 8928: 8925: 8919: 8888: 8880: 8871: 8870: 8868: 8867: 8858: 8855: 8832: 8830: 8829: 8824: 8819: 8817: 8816: 8815: 8802: 8798: 8797: 8796: 8784: 8783: 8769: 8767: 8765: 8754: 8753: 8749: 8748: 8747: 8735: 8734: 8720: 8719: 8706: 8701: 8697: 8695: 8694: 8685: 8683: 8678: 8665: 8660: 8650: 8644: 8642: 8631: 8630: 8629: 8616: 8611: 8610: 8588: 8586: 8585: 8580: 8575: 8573: 8572: 8571: 8558: 8557: 8556: 8547: 8546: 8527: 8521: 8516: 8503: 8498: 8479: 8475: 8467: 8465: 8464: 8459: 8454: 8452: 8451: 8450: 8437: 8436: 8435: 8426: 8425: 8409: 8404: 8402: 8398: 8392: 8388: 8382: 8369: 8367: 8366: 8365: 8349: 8348: 8347: 8338: 8337: 8321: 8316: 8314: 8313: 8312: 8299: 8288: 8283: 8281: 8277: 8271: 8267: 8261: 8246: 8214: 8212: 8211: 8206: 8195: 8194: 8186: 8176: 8174: 8173: 8168: 8163: 8138: 8113: 8111: 8110: 8105: 8103: 8099: 8098: 8097: 8085: 8084: 8070: 8068: 8057: 8036: 8034: 8033: 8028: 8020: 8018: 8017: 8016: 8003: 7999: 7998: 7988: 7983: 7981: 7980: 7979: 7966: 7962: 7961: 7951: 7939:Laplace equation 7933: 7931: 7930: 7925: 7920: 7916: 7915: 7914: 7902: 7901: 7887: 7885: 7874: 7847: 7845: 7844: 7839: 7834: 7826: 7818: 7816: 7815: 7814: 7801: 7797: 7796: 7786: 7781: 7779: 7778: 7777: 7764: 7760: 7759: 7749: 7737: 7730: 7707: 7690:plane Poiseuille 7687: 7680: 7667: 7665: 7664: 7659: 7657: 7650: 7648: 7647: 7643: 7642: 7641: 7629: 7628: 7607: 7606: 7605: 7596: 7595: 7579: 7560: 7556: 7555: 7553: 7552: 7543: 7542: 7533: 7528: 7526: 7525: 7516: 7515: 7506: 7493: 7491: 7490: 7486: 7485: 7483: 7482: 7470: 7465: 7463: 7462: 7450: 7433: 7392: 7388: 7381: 7379: 7378: 7373: 7371: 7364: 7360: 7356: 7355: 7325: 7324: 7295: 7292: 7287: 7275: 7272: 7267: 7252: 7250: 7236: 7231: 7229: 7221: 7220: 7219: 7206: 7187: 7178: 7166: 7165: 7152: 7148: 7129: 7128: 7092: 7091: 7052: 7051: 7006: 7005: 6982: 6980: 6976: 6975: 6950: 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5619:Kelvin functions 5616: 5612: 5605: 5603: 5602: 5597: 5595: 5588: 5586: 5573: 5572: 5567: 5540: 5539: 5534: 5521: 5508: 5485: 5459: 5436: 5424: 5403: 5402: 5386: 5384: 5371: 5370: 5365: 5338: 5337: 5332: 5319: 5306: 5283: 5257: 5234: 5222: 5201: 5200: 5174: 5172: 5171: 5166: 5164: 5162: 5154: 5140: 5126: 5125: 5107: 5106: 5088: 5086: 5078: 5064: 5050: 5049: 5031: 5030: 5012: 5008: 5007: 5006: 4994: 4993: 4979: 4977: 4966: 4933: 4929: 4925: 4921: 4914: 4912: 4911: 4906: 4859: 4857: 4849: 4841: 4821: 4811: 4798: 4796: 4795: 4790: 4788: 4781: 4777: 4776: 4774: 4773: 4772: 4763: 4758: 4757: 4741: 4740: 4735: 4731: 4729: 4724: 4711: 4706: 4691: 4685: 4680: 4667: 4662: 4648: 4646: 4638: 4630: 4611: 4609: 4608: 4607: 4598: 4593: 4592: 4576: 4575: 4574: 4565: 4550: 4548: 4544: 4542: 4537: 4524: 4519: 4505: 4503: 4492: 4487: 4483: 4482: 4481: 4468: 4463: 4449: 4447: 4436: 4401: 4400: 4398: 4397: 4391: 4388: 4369: 4360: 4335: 4324: 4307: 4296: 4294: 4292: 4291: 4286: 4283: 4259: 4257: 4256: 4251: 4243: 4239: 4238: 4225: 4224: 4211: 4210: 4209: 4208: 4199: 4187: 4182: 4165: 4163: 4159: 4158: 4146: 4145: 4135: 4128: 4120: 4119: 4107: 4106: 4096: 4094: 4091: 4086: 4074: 4071: 4066: 4051: 4046: 4045: 4044: 4028: 4023: 4019: 4018: 4017: 4005: 4004: 3990: 3988: 3977: 3941: 3939: 3938: 3933: 3866: 3864: 3863: 3858: 3856: 3852: 3851: 3849: 3841: 3833: 3831: 3823: 3818: 3816: 3815: 3814: 3801: 3797: 3796: 3786: 3773: 3765: 3760: 3758: 3750: 3742: 3726: 3725: 3723: 3722: 3716: 3713: 3681: 3679: 3678: 3673: 3671: 3669: 3665: 3659: 3655: 3649: 3647: 3639: 3634: 3632: 3631: 3630: 3621: 3615: 3611: 3610: 3605: 3598: 3593: 3591: 3583: 3575: 3573: 3565: 3553: 3542: 3540: 3539: 3534: 3529: 3527: 3526: 3525: 3516: 3510: 3506: 3505: 3500: 3493: 3484: 3483: 3471: 3451: 3449: 3448: 3447: 3438: 3432: 3428: 3427: 3422: 3415: 3403: 3382: 3380: 3376: 3370: 3366: 3360: 3348: 3327: 3287: 3280: 3278: 3277: 3272: 3264: 3258: 3257: 3252: 3248: 3246: 3245: 3244: 3235: 3229: 3225: 3224: 3219: 3212: 3203: 3202: 3197: 3193: 3191: 3187: 3181: 3177: 3171: 3162: 3161: 3157: 3145: 3141: 3139: 3135: 3129: 3125: 3119: 3096: 3094: 3093: 3088: 3083: 3082: 3078: 3066: 3062: 3060: 3056: 3050: 3046: 3040: 3017: 2994: 2993: 2988: 2984: 2982: 2978: 2972: 2968: 2962: 2931: 2888: 2886: 2885: 2880: 2878: 2877: 2874: 2865: 2864: 2861: 2852: 2851: 2848: 2810: 2808: 2807: 2802: 2800: 2799: 2795: 2783: 2779: 2777: 2773: 2767: 2763: 2757: 2735: 2712: 2711: 2708: 2692: 2688: 2670: 2662: 2655: 2653: 2652: 2647: 2645: 2644: 2639: 2635: 2633: 2629: 2623: 2619: 2613: 2581: 2580: 2577: 2561: 2553: 2539: 2532: 2528: 2516: 2500:Newtonian fluids 2490: 2488: 2487: 2482: 2477: 2475: 2467: 2466: 2465: 2452: 2438: 2437: 2434: 2418: 2417: 2415: 2414: 2408: 2405: 2391: 2371: 2342: 2320: 2254: 2252: 2251: 2246: 2241: 2239: 2238: 2237: 2224: 2210: 2189: 2170: 2168: 2167: 2162: 2157: 2156: 2155: 2143: 2137: 2128: 2119: 2101: 2096: 2087: 2085: 2084: 2083: 2067: 2062: 2061: 2060: 2048: 2028: 2027: 2025: 2024: 2018: 2015: 1998: 1988: 1986: 1985: 1980: 1975: 1971: 1970: 1969: 1957: 1956: 1942: 1940: 1929: 1904: 1903: 1901: 1900: 1894: 1891: 1874: 1864: 1853: 1843: 1836: 1829: 1827: 1826: 1821: 1819: 1818: 1797: 1796: 1784: 1782: 1774: 1773: 1772: 1759: 1738: 1731: 1729: 1728: 1723: 1715: 1710: 1702: 1697: 1695: 1691: 1685: 1681: 1675: 1660: 1653: 1649: 1645: 1641: 1634: 1632: 1631: 1626: 1624: 1622: 1618: 1612: 1608: 1602: 1600: 1592: 1587: 1583: 1582: 1580: 1572: 1564: 1554: 1552: 1541: 1539: 1531: 1519: 1517: 1516: 1511: 1509: 1508: 1492: 1488: 1484: 1477: 1475: 1473: 1472: 1466: 1463: 1444: 1442: 1440: 1439: 1433: 1430: 1413: 1411: 1409: 1408: 1402: 1399: 1388: 1368: 1366: 1364: 1363: 1357: 1354: 1342: 1284: 1262: 1258: 1250: 1243: 1241: 1240: 1235: 1230: 1225: 1214: 1209: 1197: 1195: 1184: 1163:in the field of 1142: 1140: 1139: 1134: 1132: 1125: 1119: 1108: 1107: 1105: 1104: 1085: 1083: 1082: 1065: 1064: 1059: 1055: 1053: 1052: 1051: 1038: 1037: 1034: 1028: 1018: 1010: 1000: 995: 992: 990: 982: 976: 968: 924: 918: 908: 898: 892: 886: 873: 871: 870: 865: 860: 858: 857: 848: 831: 826: 824: 823: 822: 809: 795: 632: 625: 618: 464: 463: 429:Gay-Lussac's law 419:Combined gas law 369:Capillary action 254: 253: 97: 96: 81: 79: 78: 73: 71: 69: 61: 53: 19: 18: 10509: 10508: 10504: 10503: 10502: 10500: 10499: 10498: 10474: 10473: 10455: 10382:Sutera, S. P.; 10378: 10373: 10372: 10341: 10337: 10330: 10320:Fluid Mechanics 10313: 10309: 10295: 10291: 10284: 10261: 10257: 10246: 10242: 10215: 10211: 10196: 10192: 10153: 10149: 10104: 10100: 10085: 10081: 10042: 10038: 10027: 10023: 10016: 9998: 9991: 9983: 9979: 9970: 9968: 9958: 9954: 9947: 9933: 9929: 9921: 9882: 9876: 9872: 9861: 9857: 9842: 9826: 9822: 9813: 9811: 9803: 9802: 9793: 9778: 9774: 9763: 9759: 9726: 9717: 9712: 9680: 9664: 9649: 9624: 9620: 9616: 9612: 9582: 9580: 9572: 9569: 9568: 9555: 9551: 9547: 9543: 9539: 9535: 9511: 9501: 9497: 9493: 9492: 9486: 9482: 9473: 9469: 9468: 9457: 9455: 9447: 9444: 9443: 9430: 9427: 9422: 9421: 9419: 9414: 9385: 9375: 9371: 9367: 9366: 9360: 9356: 9347: 9343: 9342: 9328: 9326: 9318: 9315: 9314: 9301: 9280: 9267: 9257: 9253: 9242: 9216: 9212: 9206: 9202: 9198: 9184: 9182: 9171: 9168: 9167: 9153: 9150: 9145: 9144: 9142: 9137: 9126: 9122: 9112: 9108: 9104: 9097: 9095: 9086: 9075: 9071: 9048: 9044: 9030: 9028: 9017: 9014: 9013: 8995: 8985: 8970: 8960: 8940: 8929: 8923: 8920: 8905: 8889: 8878: 8866: 8859: 8856: 8854: 8847: 8841: 8840: 8838: 8837: 8811: 8807: 8803: 8792: 8788: 8779: 8775: 8774: 8770: 8768: 8755: 8743: 8739: 8730: 8726: 8725: 8721: 8715: 8711: 8707: 8705: 8690: 8686: 8679: 8674: 8661: 8656: 8651: 8649: 8645: 8632: 8625: 8621: 8617: 8615: 8606: 8602: 8600: 8597: 8596: 8567: 8563: 8559: 8552: 8548: 8542: 8538: 8528: 8526: 8517: 8512: 8499: 8494: 8488: 8485: 8484: 8477: 8473: 8446: 8442: 8438: 8431: 8427: 8421: 8417: 8410: 8408: 8394: 8393: 8384: 8383: 8381: 8361: 8357: 8350: 8343: 8339: 8333: 8329: 8322: 8320: 8308: 8304: 8300: 8289: 8287: 8273: 8272: 8263: 8262: 8260: 8255: 8252: 8251: 8245: 8239: 8232: 8226: 8216: 8185: 8184: 8182: 8179: 8178: 8159: 8154: 8151: 8150: 8129: 8123: 8093: 8089: 8080: 8076: 8075: 8071: 8061: 8056: 8048: 8045: 8044: 8012: 8008: 8004: 7994: 7990: 7989: 7987: 7975: 7971: 7967: 7957: 7953: 7952: 7950: 7948: 7945: 7944: 7910: 7906: 7897: 7893: 7892: 7888: 7878: 7873: 7859: 7856: 7855: 7825: 7810: 7806: 7802: 7792: 7788: 7787: 7785: 7773: 7769: 7765: 7755: 7751: 7750: 7748: 7746: 7743: 7742: 7732: 7717: 7714: 7697: 7682: 7672: 7655: 7654: 7637: 7633: 7624: 7620: 7619: 7615: 7608: 7601: 7597: 7591: 7587: 7580: 7578: 7571: 7565: 7564: 7548: 7544: 7538: 7534: 7532: 7521: 7517: 7511: 7507: 7505: 7498: 7494: 7478: 7474: 7469: 7458: 7454: 7449: 7448: 7444: 7437: 7432: 7425: 7403: 7401: 7398: 7397: 7390: 7386: 7369: 7368: 7351: 7347: 7320: 7316: 7300: 7296: 7288: 7283: 7274: 7268: 7257: 7240: 7235: 7222: 7215: 7211: 7207: 7205: 7198: 7192: 7191: 7176: 7161: 7157: 7124: 7120: 7087: 7083: 7047: 7043: 7001: 6997: 6987: 6983: 6971: 6967: 6946: 6941: 6936: 6931: 6925: 6914: 6900: 6895: 6852: 6847: 6840: 6818: 6816: 6813: 6812: 6798: 6791: 6774: 6773: 6757: 6753: 6746: 6742: 6738: 6736: 6729: 6723: 6722: 6708: 6704: 6692: 6688: 6687: 6683: 6655: 6650: 6640: 6618: 6616: 6613: 6612: 6599: 6597: 6596: 6593: 6587: 6586: 6584: 6583: 6566: 6565: 6547: 6543: 6533: 6514: 6510: 6500: 6498: 6489: 6485: 6469: 6464: 6458: 6447: 6431: 6427: 6426: 6419: 6415: 6408: 6406: 6393: 6383: 6379: 6375: 6373: 6366: 6360: 6359: 6328: 6326: 6317: 6313: 6297: 6293: 6269: 6265: 6255: 6230: 6226: 6202: 6198: 6188: 6186: 6177: 6173: 6157: 6152: 6146: 6135: 6122: 6118: 6114: 6107: 6103: 6096: 6094: 6063: 6058: 6051: 6029: 6027: 6024: 6023: 6009: 5998: 5984: 5981: 5975: 5974: 5972: 5967: 5964:Joseph Proudman 5957: 5928: 5921: 5917: 5913: 5911: 5873: 5868: 5851: 5848: 5847: 5788: 5785: 5784: 5753: 5738: 5734: 5729: 5728: 5718: 5713: 5712: 5711: 5709: 5707: 5704: 5703: 5685: 5682: 5676: 5675: 5673: 5668: 5664: 5653: 5638: 5635: 5630: 5629: 5627: 5622: 5614: 5610: 5593: 5592: 5568: 5557: 5556: 5535: 5524: 5523: 5522: 5498: 5475: 5449: 5426: 5425: 5423: 5416: 5398: 5394: 5391: 5390: 5366: 5355: 5354: 5333: 5322: 5321: 5320: 5296: 5273: 5247: 5224: 5223: 5221: 5214: 5196: 5192: 5188: 5186: 5183: 5182: 5155: 5141: 5139: 5121: 5117: 5102: 5098: 5079: 5065: 5063: 5045: 5041: 5026: 5022: 5002: 4998: 4989: 4985: 4984: 4980: 4970: 4965: 4942: 4939: 4938: 4931: 4927: 4923: 4919: 4850: 4842: 4840: 4838: 4835: 4834: 4828: 4819: 4813: 4809: 4803: 4786: 4785: 4768: 4764: 4759: 4753: 4749: 4742: 4736: 4725: 4720: 4707: 4702: 4697: 4693: 4692: 4690: 4681: 4676: 4663: 4658: 4653: 4649: 4639: 4631: 4629: 4622: 4616: 4615: 4603: 4599: 4594: 4588: 4584: 4577: 4570: 4566: 4561: 4551: 4549: 4538: 4533: 4520: 4515: 4510: 4506: 4496: 4491: 4477: 4473: 4464: 4459: 4454: 4450: 4440: 4435: 4428: 4412: 4410: 4407: 4406: 4392: 4389: 4383: 4382: 4380: 4375: 4368: 4362: 4359: 4353: 4342: 4330: 4321: 4315: 4309: 4306: 4302: 4287: 4284: 4280: 4275: 4274: 4272: 4270: 4264: 4234: 4230: 4226: 4220: 4216: 4204: 4200: 4195: 4183: 4178: 4170: 4166: 4154: 4150: 4141: 4137: 4136: 4124: 4115: 4111: 4102: 4098: 4097: 4095: 4087: 4082: 4073: 4067: 4056: 4040: 4036: 4029: 4027: 4013: 4009: 4000: 3996: 3995: 3991: 3981: 3976: 3953: 3950: 3949: 3878: 3875: 3874: 3842: 3834: 3832: 3822: 3810: 3806: 3802: 3792: 3788: 3787: 3785: 3784: 3780: 3764: 3751: 3743: 3741: 3739: 3736: 3735: 3717: 3714: 3708: 3707: 3705: 3700: 3693: 3688: 3687: 3661: 3660: 3651: 3650: 3648: 3638: 3626: 3622: 3617: 3616: 3606: 3601: 3600: 3599: 3597: 3584: 3576: 3574: 3564: 3562: 3559: 3558: 3551: 3521: 3517: 3512: 3511: 3501: 3496: 3495: 3494: 3492: 3479: 3475: 3467: 3443: 3439: 3434: 3433: 3423: 3418: 3417: 3416: 3414: 3399: 3372: 3371: 3362: 3361: 3359: 3344: 3323: 3296: 3293: 3292: 3285: 3260: 3253: 3240: 3236: 3231: 3230: 3220: 3215: 3214: 3213: 3211: 3208: 3207: 3198: 3183: 3182: 3173: 3172: 3170: 3167: 3166: 3153: 3146: 3131: 3130: 3121: 3120: 3118: 3115: 3114: 3112: 3109: 3108: 3074: 3067: 3052: 3051: 3042: 3041: 3039: 3036: 3035: 3013: 2989: 2974: 2973: 2964: 2963: 2961: 2958: 2957: 2927: 2900: 2897: 2896: 2875:viscosity, slow 2873: 2869: 2862:viscosity, fast 2860: 2856: 2847: 2843: 2835: 2832: 2831: 2817: 2791: 2784: 2769: 2768: 2759: 2758: 2756: 2753: 2752: 2731: 2709:viscosity, slow 2707: 2703: 2701: 2698: 2697: 2690: 2680: 2677: 2668: 2660: 2640: 2625: 2624: 2615: 2614: 2612: 2609: 2608: 2578:viscosity, fast 2576: 2572: 2570: 2567: 2566: 2559: 2541: 2534: 2530: 2526: 2514: 2508: 2468: 2461: 2457: 2453: 2451: 2433: 2429: 2427: 2424: 2423: 2409: 2406: 2403: 2397: 2396: 2394: 2393: 2389: 2369: 2362: 2340: 2333: 2322: 2308: 2273: 2260: 2233: 2229: 2225: 2211: 2209: 2198: 2195: 2194: 2188: 2175: 2145: 2144: 2139: 2138: 2126: 2115: 2097: 2092: 2079: 2075: 2071: 2066: 2050: 2049: 2044: 2043: 2041: 2038: 2037: 2019: 2016: 2011: 2010: 2008: 2006: 2000: 1993: 1965: 1961: 1952: 1948: 1947: 1943: 1933: 1928: 1920: 1917: 1916: 1895: 1892: 1887: 1886: 1884: 1882: 1876: 1866: 1859: 1851: 1845: 1838: 1834: 1814: 1810: 1792: 1788: 1775: 1768: 1764: 1760: 1758: 1747: 1744: 1743: 1736: 1703: 1701: 1687: 1686: 1677: 1676: 1674: 1669: 1666: 1665: 1655: 1651: 1647: 1643: 1639: 1614: 1613: 1604: 1603: 1601: 1591: 1573: 1565: 1563: 1559: 1555: 1545: 1540: 1530: 1528: 1525: 1524: 1504: 1500: 1498: 1495: 1494: 1490: 1486: 1482: 1467: 1464: 1458: 1457: 1455: 1454: 1434: 1431: 1428: 1422: 1421: 1419: 1418: 1403: 1400: 1397: 1396: 1394: 1393: 1385: 1378: 1373: 1358: 1355: 1352: 1351: 1349: 1348: 1328: 1314: 1282: 1260: 1256: 1253:Reynolds number 1248: 1215: 1213: 1202: 1188: 1183: 1175: 1172: 1171: 1156: 1130: 1129: 1109: 1106: 1100: 1096: 1086: 1084: 1078: 1074: 1067: 1066: 1060: 1047: 1043: 1039: 1033: 1029: 1027: 1023: 1022: 1009: 1002: 996: 991: 981: 967: 954: 952: 949: 948: 938:Reynolds number 922: 916: 906: 896: 890: 881: 853: 849: 832: 830: 818: 814: 810: 796: 794: 783: 780: 779: 773: 767:pressure drop. 718:flow is laminar 666:that gives the 636: 607: 606: 605: 525: 517: 516: 470:Viscoelasticity 461: 451: 450: 438: 388: 384:Surface tension 348: 251: 249:Fluid mechanics 241: 240: 239: 153: 151:Solid mechanics 143: 142: 94: 86: 62: 54: 52: 38: 35: 34: 17: 12: 11: 5: 10507: 10497: 10496: 10491: 10486: 10472: 10471: 10466: 10461: 10454: 10453:External links 10451: 10450: 10449: 10446:. McGraw-Hill. 10439: 10413: 10377: 10374: 10371: 10370: 10351:(4): 351–356. 10335: 10328: 10307: 10289: 10282: 10255: 10240: 10229:(163): 30–36. 10209: 10190: 10163:(5): 403–422. 10147: 10118:(3): 553–563. 10098: 10079: 10036: 10021: 10014: 9989: 9977: 9952: 9945: 9927: 9924:on 2017-08-10. 9893:(2): 273–275. 9870: 9855: 9840: 9820: 9791: 9772: 9757: 9714: 9713: 9711: 9708: 9707: 9706: 9701: 9696: 9691: 9686: 9679: 9676: 9648: 9645: 9609: 9608: 9597: 9592: 9588: 9585: 9579: 9576: 9532: 9531: 9514: 9509: 9504: 9500: 9496: 9489: 9485: 9481: 9476: 9472: 9466: 9463: 9460: 9454: 9451: 9408: 9407: 9396: 9388: 9383: 9378: 9374: 9370: 9363: 9359: 9355: 9350: 9346: 9340: 9337: 9334: 9331: 9325: 9322: 9239: 9238: 9227: 9219: 9215: 9209: 9205: 9201: 9196: 9193: 9190: 9187: 9181: 9178: 9175: 9068: 9067: 9051: 9047: 9042: 9039: 9036: 9033: 9027: 9024: 9021: 8942: 8941: 8924:September 2016 8892: 8890: 8883: 8877: 8874: 8864: 8852: 8845: 8834: 8833: 8822: 8814: 8810: 8806: 8801: 8795: 8791: 8787: 8782: 8778: 8773: 8764: 8761: 8758: 8752: 8746: 8742: 8738: 8733: 8729: 8724: 8718: 8714: 8710: 8704: 8700: 8693: 8689: 8682: 8677: 8673: 8669: 8664: 8659: 8655: 8648: 8641: 8638: 8635: 8628: 8624: 8620: 8614: 8609: 8605: 8590: 8589: 8578: 8570: 8566: 8562: 8555: 8551: 8545: 8541: 8537: 8534: 8531: 8525: 8520: 8515: 8511: 8507: 8502: 8497: 8493: 8469: 8468: 8457: 8449: 8445: 8441: 8434: 8430: 8424: 8420: 8416: 8413: 8407: 8401: 8397: 8391: 8387: 8380: 8377: 8373: 8364: 8360: 8356: 8353: 8346: 8342: 8336: 8332: 8328: 8325: 8319: 8311: 8307: 8303: 8298: 8295: 8292: 8286: 8280: 8276: 8270: 8266: 8259: 8243: 8237: 8230: 8224: 8204: 8201: 8198: 8192: 8189: 8166: 8162: 8158: 8122: 8119: 8115: 8114: 8102: 8096: 8092: 8088: 8083: 8079: 8074: 8067: 8064: 8060: 8055: 8052: 8038: 8037: 8026: 8023: 8015: 8011: 8007: 8002: 7997: 7993: 7986: 7978: 7974: 7970: 7965: 7960: 7956: 7935: 7934: 7923: 7919: 7913: 7909: 7905: 7900: 7896: 7891: 7884: 7881: 7877: 7872: 7869: 7866: 7863: 7849: 7848: 7837: 7832: 7829: 7824: 7821: 7813: 7809: 7805: 7800: 7795: 7791: 7784: 7776: 7772: 7768: 7763: 7758: 7754: 7713: 7710: 7669: 7668: 7653: 7646: 7640: 7636: 7632: 7627: 7623: 7618: 7614: 7611: 7604: 7600: 7594: 7590: 7586: 7583: 7577: 7574: 7572: 7570: 7567: 7566: 7563: 7559: 7551: 7547: 7541: 7537: 7531: 7524: 7520: 7514: 7510: 7504: 7501: 7497: 7489: 7481: 7477: 7473: 7468: 7461: 7457: 7453: 7447: 7443: 7440: 7436: 7431: 7428: 7426: 7424: 7421: 7418: 7415: 7412: 7409: 7406: 7405: 7383: 7382: 7367: 7363: 7359: 7354: 7350: 7346: 7343: 7340: 7337: 7334: 7331: 7328: 7323: 7319: 7315: 7312: 7309: 7306: 7303: 7299: 7291: 7286: 7282: 7278: 7271: 7266: 7263: 7260: 7256: 7249: 7246: 7243: 7239: 7234: 7228: 7225: 7218: 7214: 7210: 7204: 7201: 7199: 7197: 7194: 7193: 7190: 7184: 7181: 7175: 7172: 7169: 7164: 7160: 7155: 7151: 7147: 7144: 7141: 7138: 7135: 7132: 7127: 7123: 7119: 7116: 7113: 7110: 7107: 7104: 7101: 7098: 7095: 7090: 7086: 7082: 7079: 7076: 7073: 7070: 7067: 7064: 7061: 7058: 7055: 7050: 7046: 7042: 7039: 7036: 7033: 7030: 7027: 7024: 7021: 7018: 7015: 7012: 7009: 7004: 7000: 6996: 6993: 6990: 6986: 6979: 6974: 6970: 6966: 6963: 6960: 6957: 6954: 6949: 6944: 6940: 6935: 6928: 6923: 6920: 6917: 6913: 6906: 6903: 6899: 6894: 6891: 6888: 6885: 6882: 6879: 6876: 6873: 6870: 6867: 6864: 6858: 6855: 6851: 6846: 6843: 6841: 6839: 6836: 6833: 6830: 6827: 6824: 6821: 6820: 6788: 6787: 6772: 6766: 6761: 6756: 6749: 6745: 6741: 6735: 6732: 6730: 6728: 6725: 6724: 6721: 6717: 6711: 6707: 6703: 6700: 6695: 6691: 6686: 6682: 6679: 6676: 6673: 6670: 6664: 6661: 6658: 6654: 6649: 6646: 6643: 6641: 6639: 6636: 6633: 6630: 6627: 6624: 6621: 6620: 6580: 6579: 6564: 6558: 6555: 6550: 6546: 6542: 6539: 6536: 6531: 6528: 6525: 6522: 6517: 6513: 6509: 6506: 6503: 6492: 6488: 6484: 6481: 6478: 6475: 6472: 6468: 6461: 6456: 6453: 6450: 6446: 6439: 6434: 6430: 6422: 6418: 6414: 6411: 6405: 6399: 6396: 6391: 6386: 6382: 6378: 6372: 6369: 6367: 6365: 6362: 6361: 6358: 6353: 6349: 6346: 6343: 6340: 6337: 6334: 6331: 6325: 6320: 6316: 6311: 6308: 6305: 6300: 6296: 6292: 6289: 6286: 6280: 6277: 6272: 6268: 6264: 6261: 6258: 6253: 6250: 6247: 6244: 6241: 6238: 6233: 6229: 6225: 6222: 6219: 6216: 6213: 6210: 6205: 6201: 6197: 6194: 6191: 6180: 6176: 6172: 6169: 6166: 6163: 6160: 6156: 6149: 6144: 6141: 6138: 6134: 6125: 6121: 6117: 6110: 6106: 6102: 6099: 6093: 6090: 6087: 6084: 6081: 6078: 6075: 6069: 6066: 6062: 6057: 6054: 6052: 6050: 6047: 6044: 6041: 6038: 6035: 6032: 6031: 5956: 5953: 5952: 5951: 5940: 5934: 5931: 5924: 5920: 5916: 5910: 5907: 5903: 5900: 5897: 5894: 5891: 5888: 5885: 5879: 5876: 5872: 5867: 5864: 5861: 5858: 5855: 5841: 5840: 5829: 5826: 5823: 5820: 5817: 5814: 5810: 5807: 5804: 5801: 5798: 5795: 5792: 5780:on both walls 5774: 5773: 5760: 5757: 5752: 5749: 5741: 5737: 5732: 5726: 5721: 5716: 5652: 5649: 5607: 5606: 5591: 5585: 5582: 5579: 5576: 5571: 5566: 5563: 5560: 5555: 5552: 5549: 5546: 5543: 5538: 5533: 5530: 5527: 5520: 5517: 5514: 5511: 5507: 5504: 5501: 5497: 5494: 5491: 5488: 5484: 5481: 5478: 5474: 5471: 5468: 5465: 5462: 5458: 5455: 5452: 5448: 5445: 5442: 5439: 5435: 5432: 5429: 5422: 5419: 5417: 5415: 5412: 5409: 5406: 5401: 5397: 5393: 5392: 5389: 5383: 5380: 5377: 5374: 5369: 5364: 5361: 5358: 5353: 5350: 5347: 5344: 5341: 5336: 5331: 5328: 5325: 5318: 5315: 5312: 5309: 5305: 5302: 5299: 5295: 5292: 5289: 5286: 5282: 5279: 5276: 5272: 5269: 5266: 5263: 5260: 5256: 5253: 5250: 5246: 5243: 5240: 5237: 5233: 5230: 5227: 5220: 5217: 5215: 5213: 5210: 5207: 5204: 5199: 5195: 5191: 5190: 5176: 5175: 5161: 5158: 5153: 5150: 5147: 5144: 5138: 5135: 5132: 5129: 5124: 5120: 5116: 5113: 5110: 5105: 5101: 5097: 5094: 5091: 5085: 5082: 5077: 5074: 5071: 5068: 5062: 5059: 5056: 5053: 5048: 5044: 5040: 5037: 5034: 5029: 5025: 5021: 5018: 5015: 5011: 5005: 5001: 4997: 4992: 4988: 4983: 4976: 4973: 4969: 4964: 4961: 4958: 4955: 4952: 4949: 4946: 4916: 4915: 4904: 4901: 4898: 4895: 4892: 4889: 4886: 4883: 4880: 4877: 4874: 4871: 4868: 4865: 4862: 4856: 4853: 4848: 4845: 4827: 4824: 4817: 4807: 4800: 4799: 4784: 4780: 4771: 4767: 4762: 4756: 4752: 4748: 4745: 4739: 4734: 4728: 4723: 4719: 4715: 4710: 4705: 4701: 4696: 4689: 4684: 4679: 4675: 4671: 4666: 4661: 4657: 4652: 4645: 4642: 4637: 4634: 4628: 4625: 4623: 4621: 4618: 4617: 4614: 4606: 4602: 4597: 4591: 4587: 4583: 4580: 4573: 4569: 4564: 4560: 4557: 4554: 4547: 4541: 4536: 4532: 4528: 4523: 4518: 4514: 4509: 4502: 4499: 4495: 4490: 4486: 4480: 4476: 4472: 4467: 4462: 4458: 4453: 4446: 4443: 4439: 4434: 4431: 4429: 4427: 4424: 4421: 4418: 4415: 4414: 4366: 4357: 4341: 4338: 4319: 4313: 4304: 4278: 4268: 4261: 4260: 4249: 4246: 4242: 4237: 4233: 4229: 4223: 4219: 4214: 4207: 4203: 4198: 4194: 4191: 4186: 4181: 4177: 4173: 4169: 4162: 4157: 4153: 4149: 4144: 4140: 4134: 4131: 4127: 4123: 4118: 4114: 4110: 4105: 4101: 4090: 4085: 4081: 4077: 4070: 4065: 4062: 4059: 4055: 4049: 4043: 4039: 4035: 4032: 4026: 4022: 4016: 4012: 4008: 4003: 3999: 3994: 3987: 3984: 3980: 3975: 3972: 3969: 3966: 3963: 3960: 3957: 3943: 3942: 3931: 3928: 3925: 3922: 3919: 3916: 3913: 3910: 3906: 3903: 3900: 3897: 3894: 3891: 3888: 3885: 3882: 3868: 3867: 3855: 3848: 3845: 3840: 3837: 3829: 3826: 3821: 3813: 3809: 3805: 3800: 3795: 3791: 3783: 3779: 3776: 3771: 3768: 3763: 3757: 3754: 3749: 3746: 3692: 3689: 3683: 3682: 3668: 3664: 3658: 3654: 3645: 3642: 3637: 3629: 3625: 3620: 3614: 3609: 3604: 3596: 3590: 3587: 3582: 3579: 3571: 3568: 3544: 3543: 3532: 3524: 3520: 3515: 3509: 3504: 3499: 3491: 3488: 3482: 3478: 3474: 3470: 3466: 3463: 3460: 3457: 3454: 3446: 3442: 3437: 3431: 3426: 3421: 3413: 3410: 3406: 3402: 3397: 3394: 3391: 3388: 3385: 3379: 3375: 3369: 3365: 3358: 3355: 3351: 3347: 3342: 3339: 3336: 3333: 3330: 3326: 3321: 3318: 3315: 3312: 3309: 3306: 3303: 3300: 3282: 3281: 3270: 3267: 3263: 3256: 3251: 3243: 3239: 3234: 3228: 3223: 3218: 3210: 3206: 3201: 3196: 3190: 3186: 3180: 3176: 3169: 3165: 3160: 3156: 3152: 3149: 3144: 3138: 3134: 3128: 3124: 3117: 3098: 3097: 3086: 3081: 3077: 3073: 3070: 3065: 3059: 3055: 3049: 3045: 3038: 3033: 3030: 3026: 3023: 3020: 3016: 3012: 3009: 3006: 3003: 3000: 2997: 2992: 2987: 2981: 2977: 2971: 2967: 2960: 2956: 2953: 2949: 2946: 2943: 2940: 2937: 2934: 2930: 2925: 2922: 2919: 2916: 2913: 2910: 2907: 2904: 2890: 2889: 2872: 2868: 2859: 2855: 2846: 2842: 2839: 2816: 2813: 2812: 2811: 2798: 2794: 2790: 2787: 2782: 2776: 2772: 2766: 2762: 2755: 2751: 2748: 2744: 2741: 2738: 2734: 2730: 2727: 2724: 2721: 2718: 2715: 2706: 2676: 2673: 2663:following the 2657: 2656: 2643: 2638: 2632: 2628: 2622: 2618: 2611: 2606: 2603: 2599: 2596: 2593: 2590: 2587: 2584: 2575: 2507: 2504: 2492: 2491: 2480: 2474: 2471: 2464: 2460: 2456: 2450: 2447: 2444: 2441: 2435:viscosity, top 2432: 2401: 2361: 2358: 2357: 2356: 2350: 2344: 2338: 2331: 2272: 2269: 2261: 2258: 2257: 2256: 2255: 2244: 2236: 2232: 2228: 2223: 2220: 2217: 2214: 2208: 2205: 2202: 2186: 2172: 2171: 2160: 2154: 2151: 2148: 2142: 2134: 2131: 2125: 2122: 2118: 2114: 2111: 2108: 2105: 2100: 2095: 2091: 2082: 2078: 2074: 2070: 2065: 2059: 2056: 2053: 2047: 2004: 1990: 1989: 1978: 1974: 1968: 1964: 1960: 1955: 1951: 1946: 1939: 1936: 1932: 1927: 1924: 1880: 1854:. The no slip 1849: 1831: 1830: 1817: 1813: 1809: 1806: 1803: 1800: 1795: 1791: 1787: 1781: 1778: 1771: 1767: 1763: 1757: 1754: 1751: 1733: 1732: 1721: 1718: 1713: 1709: 1706: 1700: 1694: 1690: 1684: 1680: 1673: 1636: 1635: 1621: 1617: 1611: 1607: 1598: 1595: 1590: 1586: 1579: 1576: 1571: 1568: 1562: 1558: 1551: 1548: 1544: 1537: 1534: 1507: 1503: 1447: 1446: 1426: 1415: 1390: 1383: 1376: 1370: 1313: 1310: 1306:turbulent flow 1245: 1244: 1233: 1228: 1224: 1221: 1218: 1212: 1208: 1205: 1200: 1194: 1191: 1187: 1182: 1179: 1155: 1152: 1148:gauge pressure 1144: 1143: 1128: 1122: 1118: 1115: 1112: 1103: 1099: 1095: 1092: 1089: 1087: 1081: 1077: 1072: 1069: 1068: 1063: 1058: 1050: 1046: 1042: 1032: 1026: 1021: 1016: 1013: 1008: 1005: 1003: 999: 988: 985: 979: 974: 971: 966: 963: 960: 957: 956: 927: 926: 920: 914: 904: 894: 888: 875: 874: 863: 856: 852: 847: 844: 841: 838: 835: 829: 821: 817: 813: 808: 805: 802: 799: 793: 790: 787: 772: 769: 710:incompressible 672:incompressible 656:Poiseuille law 644:fluid dynamics 638: 637: 635: 634: 627: 620: 612: 609: 608: 604: 603: 598: 593: 588: 583: 578: 573: 568: 563: 558: 553: 548: 543: 538: 533: 527: 526: 523: 522: 519: 518: 513: 512: 511: 510: 505: 500: 492: 491: 485: 484: 483: 482: 477: 472: 462: 457: 456: 453: 452: 447: 446: 440: 439: 437: 436: 431: 426: 421: 416: 411: 406: 400: 397: 396: 390: 389: 387: 386: 381: 376: 374:Chromatography 371: 366: 360: 357: 356: 350: 349: 347: 346: 327: 326: 325: 306: 294: 289: 277: 264: 261: 260: 252: 247: 246: 243: 242: 238: 237: 232: 227: 226: 225: 215: 210: 205: 204: 203: 198: 188: 183: 178: 173: 172: 171: 161: 155: 154: 149: 148: 145: 144: 139: 138: 137: 136: 128: 127: 123: 122: 121: 120: 115: 110: 102: 101: 95: 92: 91: 88: 87: 82: 68: 65: 60: 57: 51: 48: 45: 42: 31: 30: 24: 23: 15: 9: 6: 4: 3: 2: 10506: 10495: 10492: 10490: 10487: 10485: 10482: 10481: 10479: 10470: 10467: 10465: 10462: 10460: 10457: 10456: 10445: 10440: 10435: 10431: 10427: 10423: 10419: 10414: 10409: 10405: 10401: 10397: 10393: 10389: 10385: 10380: 10379: 10366: 10362: 10358: 10354: 10350: 10346: 10345:Lab on a Chip 10339: 10331: 10329:0-08-033933-6 10325: 10321: 10317: 10316:Landau, L. D. 10311: 10303: 10299: 10293: 10285: 10279: 10275: 10274: 10269: 10268:Riley, Norman 10265: 10259: 10251: 10244: 10236: 10232: 10228: 10224: 10220: 10213: 10206:(2): 377–424. 10205: 10201: 10194: 10186: 10182: 10178: 10174: 10170: 10166: 10162: 10158: 10151: 10143: 10139: 10134: 10129: 10125: 10121: 10117: 10113: 10109: 10102: 10094: 10090: 10083: 10075: 10071: 10067: 10063: 10059: 10055: 10051: 10047: 10040: 10032: 10025: 10017: 10011: 10007: 10003: 9996: 9994: 9986: 9981: 9967: 9963: 9956: 9948: 9946:0-87150-749-8 9942: 9938: 9931: 9920: 9916: 9912: 9908: 9904: 9900: 9896: 9892: 9888: 9881: 9874: 9866: 9859: 9851: 9847: 9843: 9837: 9833: 9832: 9824: 9810: 9806: 9800: 9798: 9796: 9787: 9783: 9776: 9768: 9761: 9753: 9749: 9745: 9741: 9737: 9733: 9732: 9724: 9722: 9720: 9715: 9705: 9702: 9700: 9697: 9695: 9692: 9690: 9687: 9685: 9682: 9681: 9675: 9673: 9668: 9662: 9658: 9654: 9644: 9642: 9638: 9634: 9630: 9595: 9590: 9586: 9583: 9577: 9574: 9567: 9566: 9565: 9562: 9558: 9512: 9507: 9502: 9498: 9494: 9487: 9483: 9479: 9474: 9470: 9464: 9461: 9458: 9452: 9449: 9442: 9441: 9440: 9433: 9425: 9417: 9413: 9394: 9386: 9381: 9376: 9372: 9368: 9361: 9357: 9353: 9348: 9344: 9338: 9335: 9332: 9329: 9323: 9320: 9313: 9312: 9311: 9308: 9304: 9300: 9294: 9291: 9287: 9283: 9277: 9274: 9270: 9264: 9261: 9250: 9246: 9225: 9217: 9213: 9207: 9203: 9199: 9194: 9191: 9188: 9185: 9179: 9176: 9166: 9165: 9164: 9156: 9148: 9140: 9133: 9129: 9125:. Therefore, 9121: 9115: 9093: 9089: 9084: 9078: 9049: 9045: 9040: 9037: 9034: 9031: 9025: 9022: 9012: 9011: 9010: 9007: 9003: 8999: 8992: 8988: 8982: 8978: 8974: 8967: 8963: 8958: 8954: 8949: 8938: 8935: 8927: 8917: 8913: 8909: 8903: 8902: 8898: 8893:This section 8891: 8887: 8882: 8881: 8873: 8863: 8851: 8844: 8820: 8812: 8808: 8804: 8799: 8793: 8789: 8785: 8780: 8776: 8771: 8762: 8759: 8756: 8750: 8744: 8740: 8736: 8731: 8727: 8722: 8716: 8712: 8708: 8702: 8698: 8691: 8687: 8680: 8675: 8671: 8667: 8662: 8657: 8653: 8646: 8639: 8636: 8633: 8626: 8622: 8618: 8612: 8607: 8603: 8595: 8594: 8593: 8576: 8568: 8564: 8560: 8553: 8549: 8543: 8539: 8535: 8532: 8529: 8523: 8518: 8513: 8509: 8505: 8500: 8495: 8491: 8483: 8482: 8481: 8455: 8447: 8443: 8439: 8432: 8428: 8422: 8418: 8414: 8411: 8405: 8399: 8389: 8378: 8375: 8362: 8358: 8354: 8351: 8344: 8340: 8334: 8330: 8326: 8323: 8317: 8309: 8305: 8301: 8296: 8293: 8290: 8284: 8278: 8268: 8257: 8250: 8249: 8248: 8242: 8236: 8229: 8223: 8219: 8202: 8199: 8196: 8190: 8187: 8164: 8160: 8156: 8147: 8143: 8136: 8132: 8128: 8118: 8117:on the wall. 8100: 8094: 8090: 8086: 8081: 8077: 8072: 8065: 8062: 8058: 8053: 8050: 8043: 8042: 8041: 8024: 8021: 8013: 8009: 8000: 7995: 7984: 7976: 7972: 7963: 7958: 7943: 7942: 7941: 7940: 7921: 7917: 7911: 7907: 7903: 7898: 7894: 7889: 7882: 7879: 7875: 7870: 7867: 7864: 7861: 7854: 7853: 7852: 7835: 7830: 7827: 7822: 7819: 7811: 7807: 7798: 7793: 7782: 7774: 7770: 7761: 7756: 7741: 7740: 7739: 7735: 7728: 7724: 7720: 7709: 7705: 7701: 7696: 7691: 7685: 7679: 7675: 7651: 7644: 7638: 7634: 7630: 7625: 7621: 7616: 7612: 7609: 7602: 7598: 7592: 7588: 7584: 7581: 7575: 7573: 7568: 7561: 7557: 7549: 7545: 7539: 7535: 7529: 7522: 7518: 7512: 7508: 7502: 7499: 7495: 7487: 7479: 7475: 7471: 7466: 7459: 7455: 7451: 7445: 7441: 7438: 7434: 7429: 7427: 7419: 7416: 7413: 7407: 7396: 7395: 7394: 7365: 7361: 7352: 7348: 7344: 7341: 7335: 7332: 7329: 7321: 7317: 7313: 7310: 7304: 7301: 7297: 7289: 7284: 7280: 7276: 7264: 7261: 7258: 7254: 7247: 7244: 7241: 7237: 7232: 7226: 7223: 7216: 7212: 7208: 7202: 7200: 7195: 7188: 7182: 7179: 7173: 7170: 7167: 7162: 7158: 7153: 7149: 7139: 7136: 7133: 7125: 7121: 7114: 7111: 7102: 7099: 7096: 7088: 7084: 7077: 7074: 7071: 7062: 7059: 7056: 7048: 7044: 7037: 7034: 7025: 7022: 7019: 7016: 7013: 7010: 7002: 6998: 6991: 6988: 6984: 6972: 6968: 6964: 6961: 6955: 6952: 6947: 6942: 6938: 6933: 6921: 6918: 6915: 6911: 6904: 6901: 6897: 6892: 6886: 6883: 6880: 6871: 6868: 6865: 6856: 6853: 6849: 6844: 6842: 6834: 6831: 6828: 6822: 6811: 6810: 6809: 6805: 6801: 6794: 6770: 6764: 6759: 6754: 6747: 6743: 6739: 6733: 6731: 6726: 6719: 6715: 6709: 6705: 6701: 6698: 6693: 6689: 6684: 6677: 6674: 6671: 6662: 6659: 6656: 6652: 6647: 6644: 6642: 6634: 6631: 6628: 6622: 6611: 6610: 6609: 6591: 6562: 6553: 6548: 6544: 6537: 6534: 6529: 6526: 6520: 6515: 6511: 6504: 6501: 6490: 6482: 6479: 6476: 6473: 6466: 6454: 6451: 6448: 6444: 6437: 6432: 6428: 6420: 6416: 6412: 6409: 6403: 6397: 6394: 6389: 6384: 6380: 6376: 6370: 6368: 6363: 6356: 6351: 6347: 6341: 6338: 6335: 6332: 6323: 6318: 6314: 6309: 6303: 6298: 6294: 6287: 6284: 6275: 6270: 6266: 6259: 6256: 6245: 6242: 6239: 6231: 6227: 6220: 6217: 6214: 6208: 6203: 6199: 6192: 6189: 6178: 6170: 6167: 6164: 6161: 6154: 6142: 6139: 6136: 6132: 6123: 6119: 6115: 6108: 6104: 6100: 6097: 6091: 6085: 6082: 6079: 6073: 6067: 6064: 6060: 6055: 6053: 6045: 6042: 6039: 6033: 6022: 6021: 6020: 6017: 6013: 6006: 6002: 5995: 5988: 5979: 5970: 5965: 5961: 5938: 5932: 5929: 5922: 5918: 5914: 5908: 5905: 5901: 5895: 5892: 5889: 5883: 5877: 5874: 5870: 5865: 5859: 5853: 5846: 5845: 5844: 5827: 5824: 5818: 5812: 5808: 5805: 5802: 5796: 5790: 5783: 5782: 5781: 5779: 5758: 5755: 5750: 5747: 5739: 5735: 5724: 5719: 5702: 5701: 5700: 5698: 5689: 5680: 5671: 5657: 5648: 5641: 5633: 5625: 5620: 5589: 5580: 5577: 5569: 5553: 5547: 5544: 5536: 5515: 5512: 5492: 5489: 5472: 5466: 5463: 5443: 5440: 5420: 5418: 5410: 5407: 5399: 5395: 5387: 5378: 5375: 5367: 5351: 5345: 5342: 5334: 5313: 5310: 5290: 5287: 5270: 5264: 5261: 5241: 5238: 5218: 5216: 5208: 5205: 5197: 5193: 5181: 5180: 5179: 5159: 5156: 5151: 5148: 5145: 5142: 5130: 5127: 5122: 5118: 5111: 5108: 5103: 5099: 5095: 5089: 5083: 5080: 5075: 5072: 5069: 5066: 5054: 5051: 5046: 5042: 5035: 5032: 5027: 5023: 5019: 5013: 5009: 5003: 4999: 4995: 4990: 4986: 4981: 4974: 4971: 4967: 4962: 4956: 4953: 4950: 4944: 4937: 4936: 4935: 4902: 4899: 4896: 4893: 4890: 4887: 4884: 4881: 4878: 4875: 4872: 4869: 4866: 4863: 4860: 4854: 4846: 4833: 4832: 4831: 4823: 4816: 4806: 4782: 4778: 4769: 4765: 4760: 4754: 4750: 4746: 4743: 4737: 4732: 4726: 4721: 4717: 4713: 4708: 4703: 4699: 4694: 4687: 4682: 4677: 4673: 4669: 4664: 4659: 4655: 4650: 4643: 4640: 4635: 4632: 4626: 4624: 4619: 4612: 4604: 4600: 4595: 4589: 4585: 4581: 4578: 4571: 4567: 4562: 4558: 4555: 4552: 4545: 4539: 4534: 4530: 4526: 4521: 4516: 4512: 4507: 4500: 4497: 4493: 4488: 4484: 4478: 4474: 4470: 4465: 4460: 4456: 4451: 4444: 4441: 4437: 4432: 4430: 4422: 4416: 4405: 4404: 4403: 4396: 4387: 4378: 4373: 4365: 4356: 4346: 4337: 4333: 4328: 4322: 4312: 4300: 4290: 4282: 4267: 4247: 4244: 4240: 4235: 4231: 4227: 4221: 4217: 4212: 4205: 4201: 4196: 4192: 4189: 4184: 4179: 4175: 4171: 4167: 4155: 4151: 4142: 4138: 4129: 4125: 4121: 4116: 4112: 4103: 4099: 4088: 4083: 4079: 4075: 4063: 4060: 4057: 4053: 4047: 4041: 4037: 4033: 4030: 4024: 4020: 4014: 4010: 4006: 4001: 3997: 3992: 3985: 3982: 3978: 3973: 3967: 3964: 3961: 3955: 3948: 3947: 3946: 3929: 3926: 3920: 3917: 3914: 3908: 3904: 3901: 3898: 3892: 3889: 3886: 3880: 3873: 3872: 3871: 3853: 3846: 3838: 3827: 3824: 3819: 3811: 3807: 3798: 3793: 3781: 3777: 3774: 3769: 3766: 3761: 3755: 3747: 3734: 3733: 3732: 3730: 3721: 3712: 3703: 3698: 3686: 3666: 3656: 3643: 3640: 3635: 3627: 3623: 3612: 3607: 3594: 3588: 3580: 3569: 3566: 3557: 3556: 3555: 3549: 3530: 3522: 3518: 3507: 3502: 3489: 3480: 3472: 3461: 3458: 3455: 3452: 3444: 3440: 3429: 3424: 3411: 3404: 3395: 3392: 3389: 3386: 3383: 3377: 3367: 3356: 3349: 3340: 3337: 3334: 3331: 3328: 3319: 3316: 3313: 3310: 3304: 3301: 3298: 3291: 3290: 3289: 3268: 3265: 3254: 3249: 3241: 3237: 3226: 3221: 3204: 3199: 3194: 3188: 3178: 3163: 3158: 3150: 3147: 3142: 3136: 3126: 3107: 3106: 3105: 3103: 3084: 3079: 3071: 3068: 3063: 3057: 3047: 3031: 3024: 3018: 3010: 3007: 3001: 2998: 2995: 2990: 2985: 2979: 2969: 2954: 2947: 2944: 2941: 2938: 2935: 2932: 2923: 2920: 2917: 2914: 2908: 2905: 2902: 2895: 2894: 2893: 2870: 2866: 2857: 2853: 2844: 2840: 2837: 2830: 2829: 2828: 2826: 2822: 2796: 2788: 2785: 2780: 2774: 2764: 2749: 2742: 2736: 2728: 2725: 2719: 2716: 2713: 2704: 2696: 2695: 2694: 2687: 2683: 2675:Slower lamina 2672: 2666: 2641: 2636: 2630: 2620: 2604: 2597: 2594: 2591: 2588: 2585: 2582: 2573: 2565: 2564: 2563: 2557: 2552: 2548: 2544: 2538: 2533:, and length 2524: 2520: 2513: 2506:Faster lamina 2503: 2501: 2497: 2478: 2472: 2462: 2458: 2448: 2445: 2442: 2439: 2430: 2422: 2421: 2420: 2413: 2404: 2387: 2383: 2379: 2366: 2354: 2351: 2348: 2345: 2337: 2330: 2326: 2319: 2315: 2311: 2306: 2302: 2301: 2300: 2296: 2292: 2285: 2281: 2277: 2268: 2266: 2242: 2234: 2230: 2226: 2221: 2218: 2215: 2212: 2206: 2203: 2193: 2192: 2191: 2185: 2182: 2178: 2158: 2140: 2132: 2129: 2123: 2120: 2112: 2109: 2106: 2103: 2098: 2093: 2089: 2080: 2076: 2072: 2068: 2063: 2045: 2036: 2035: 2034: 2032: 2031:cross section 2023: 2014: 2003: 1996: 1976: 1972: 1966: 1962: 1958: 1953: 1949: 1944: 1937: 1934: 1930: 1925: 1922: 1915: 1914: 1913: 1911: 1908: 1899: 1890: 1879: 1873: 1869: 1862: 1857: 1848: 1841: 1815: 1811: 1807: 1804: 1801: 1798: 1793: 1789: 1785: 1779: 1776: 1769: 1765: 1761: 1755: 1752: 1749: 1742: 1741: 1740: 1719: 1716: 1711: 1707: 1698: 1692: 1682: 1671: 1664: 1663: 1662: 1659: 1619: 1609: 1596: 1593: 1588: 1584: 1577: 1569: 1560: 1556: 1549: 1535: 1532: 1523: 1522: 1521: 1505: 1501: 1481: 1471: 1462: 1452: 1438: 1429: 1416: 1407: 1391: 1386: 1379: 1371: 1362: 1346: 1345: 1344: 1340: 1336: 1332: 1327: 1323: 1319: 1309: 1307: 1302: 1300: 1296: 1292: 1287: 1281: 1277: 1273: 1269: 1268: 1254: 1231: 1226: 1222: 1219: 1216: 1210: 1198: 1185: 1180: 1170: 1169: 1168: 1166: 1162: 1151: 1149: 1126: 1120: 1116: 1110: 1101: 1097: 1093: 1090: 1088: 1075: 1061: 1056: 1048: 1044: 1040: 1030: 1024: 1019: 1014: 1011: 1006: 1004: 997: 983: 977: 972: 969: 964: 961: 947: 946: 945: 943: 939: 935: 930: 921: 915: 912: 905: 902: 895: 889: 885: 880: 879: 878: 861: 854: 850: 845: 842: 839: 836: 833: 827: 819: 815: 811: 806: 803: 800: 797: 791: 788: 778: 777: 776: 768: 766: 762: 758: 754: 750: 745: 743: 739: 734: 729: 727: 723: 719: 715: 711: 706: 704: 703:George Stokes 700: 696: 692: 688: 685: 681: 677: 673: 669: 668:pressure drop 665: 661: 657: 653: 649: 645: 633: 628: 626: 621: 619: 614: 613: 611: 610: 602: 599: 597: 594: 592: 589: 587: 584: 582: 579: 577: 574: 572: 569: 567: 564: 562: 559: 557: 554: 552: 549: 547: 544: 542: 539: 537: 534: 532: 529: 528: 521: 520: 509: 506: 504: 501: 499: 496: 495: 494: 493: 490: 487: 486: 481: 478: 476: 473: 471: 468: 467: 466: 465: 460: 455: 454: 445: 442: 441: 435: 432: 430: 427: 425: 422: 420: 417: 415: 414:Charles's law 412: 410: 407: 405: 402: 401: 399: 398: 395: 392: 391: 385: 382: 380: 377: 375: 372: 370: 367: 365: 362: 361: 359: 358: 355: 352: 351: 345: 342: 338: 335: 331: 328: 323: 322:non-Newtonian 320: 316: 312: 311: 310: 307: 305: 302: 298: 295: 293: 290: 288: 285: 281: 278: 276: 273: 269: 266: 265: 263: 262: 259: 256: 255: 250: 245: 244: 236: 233: 231: 228: 224: 221: 220: 219: 216: 214: 211: 209: 208:Compatibility 206: 202: 199: 197: 196:Finite strain 194: 193: 192: 189: 187: 184: 182: 179: 177: 174: 170: 167: 166: 165: 162: 160: 157: 156: 152: 147: 146: 135: 132: 131: 130: 129: 125: 124: 119: 116: 114: 111: 109: 106: 105: 104: 103: 100:Conservations 99: 98: 90: 89: 85: 66: 63: 58: 55: 49: 46: 43: 40: 33: 32: 29: 26: 25: 21: 20: 10443: 10417: 10391: 10387: 10348: 10344: 10338: 10319: 10310: 10301: 10292: 10272: 10258: 10249: 10243: 10226: 10222: 10212: 10203: 10199: 10193: 10160: 10156: 10150: 10115: 10111: 10101: 10092: 10088: 10082: 10049: 10045: 10039: 10030: 10024: 10005: 9980: 9969:. Retrieved 9965: 9955: 9936: 9930: 9919:the original 9890: 9886: 9873: 9864: 9858: 9830: 9823: 9812:. Retrieved 9808: 9785: 9781: 9775: 9766: 9760: 9735: 9729: 9684:Couette flow 9666: 9650: 9629:Electron gas 9610: 9560: 9556: 9533: 9431: 9423: 9415: 9409: 9306: 9302: 9297:. Since the 9292: 9289: 9285: 9281: 9275: 9272: 9268: 9262: 9259: 9248: 9244: 9240: 9154: 9146: 9138: 9131: 9127: 9113: 9091: 9087: 9076: 9069: 9005: 9001: 8997: 8990: 8986: 8980: 8976: 8972: 8965: 8961: 8945: 8930: 8921: 8906:Please help 8894: 8861: 8849: 8842: 8835: 8591: 8470: 8240: 8234: 8227: 8221: 8217: 8134: 8130: 8124: 8116: 8039: 7936: 7850: 7733: 7726: 7722: 7718: 7715: 7695:Ratip Berker 7689: 7683: 7677: 7673: 7670: 7384: 6803: 6799: 6792: 6789: 6589: 6581: 6015: 6011: 6004: 6000: 5996: 5986: 5977: 5968: 5958: 5842: 5775: 5687: 5678: 5669: 5662: 5639: 5631: 5623: 5608: 5177: 4917: 4829: 4814: 4804: 4801: 4394: 4385: 4376: 4371: 4363: 4354: 4351: 4331: 4317: 4310: 4288: 4276: 4265: 4262: 3944: 3869: 3719: 3710: 3701: 3696: 3694: 3684: 3545: 3283: 3099: 2891: 2821:acceleration 2818: 2685: 2681: 2678: 2658: 2550: 2546: 2542: 2536: 2529:, thickness 2509: 2493: 2411: 2399: 2382:proportional 2375: 2335: 2328: 2324: 2317: 2313: 2309: 2298: 2291:laminar flow 2283: 2279: 2262: 2183: 2180: 2176: 2173: 2021: 2012: 2001: 1994: 1991: 1897: 1888: 1877: 1871: 1867: 1860: 1846: 1839: 1832: 1734: 1657: 1637: 1478:, i.e., the 1469: 1460: 1448: 1436: 1424: 1405: 1381: 1374: 1360: 1338: 1334: 1330: 1322:laminar flow 1315: 1303: 1295:hemodynamics 1291:hemorheology 1288: 1279: 1275: 1271: 1265: 1246: 1157: 1145: 931: 928: 883: 876: 774: 764: 760: 749:constriction 746: 737: 732: 730: 722:acceleration 707: 697:in 1838 and 680:laminar flow 664:physical law 659: 655: 651: 647: 642:In nonideal 641: 489:Smart fluids 434:Graham's law 340: 333: 318: 304:Pascal's law 300: 296: 283: 271: 126:Inequalities 10418:Anaesthesia 9966:tec-science 9887:Anaesthesia 9689:Darcy's law 7698: [ 7671:Here, when 2689:instead of 2388:of contact 2378:shear force 1493:instead of 508:Ferrofluids 409:Boyle's law 181:Hooke's law 159:Deformation 10478:Categories 10384:Skalak, R. 10376:References 10095:: 371–386. 9971:2020-05-07 9814:2019-12-15 9805:"Pressure" 9788:: 287–341. 9412:resistance 8146:isothermal 6008:and width 5699:reduce to 3731:reduce to 2665:derivative 1312:Derivation 1299:physiology 1165:hydraulics 761:additional 753:continuity 561:Gay-Lussac 524:Scientists 424:Fick's law 404:Atmosphere 223:frictional 176:Plasticity 164:Elasticity 10185:123217023 10074:119771908 9850:665837940 9584:ρ 9503:∗ 9480:π 9462:μ 9377:∗ 9354:π 9333:μ 9218:∗ 9189:μ 9174:Δ 9100:10 C 9035:μ 9020:Δ 8957:Ohm's law 8895:does not 8760:μ 8737:− 8709:π 8668:− 8637:μ 8619:π 8561:π 8533:μ 8506:− 8440:π 8415:μ 8376:− 8372:⇒ 8352:π 8327:μ 8302:π 8294:μ 8258:− 8200:ρ 8191:˙ 8165:ρ 8142:ideal gas 8066:μ 8006:∂ 7992:∂ 7969:∂ 7955:∂ 7883:μ 7831:μ 7823:− 7804:∂ 7790:∂ 7767:∂ 7753:∂ 7613:μ 7582:π 7530:− 7503:− 7442:μ 7349:β 7345:π 7336:⁡ 7318:β 7314:π 7305:⁡ 7281:β 7270:∞ 7255:∑ 7248:μ 7245:π 7233:− 7227:μ 7213:π 7159:β 7137:− 7122:β 7115:⁡ 7085:β 7078:⁡ 7072:− 7045:β 7038:⁡ 7017:− 7014:π 6999:β 6992:⁡ 6969:β 6965:π 6956:⁡ 6939:β 6927:∞ 6912:∑ 6905:μ 6902:π 6893:− 6884:− 6881:π 6857:μ 6765:μ 6699:− 6675:− 6660:μ 6648:− 6545:β 6538:⁡ 6527:− 6512:β 6505:⁡ 6480:− 6460:∞ 6445:∑ 6438:μ 6429:π 6404:− 6398:μ 6348:π 6339:− 6315:β 6295:β 6288:⁡ 6267:β 6260:⁡ 6243:− 6228:β 6221:⁡ 6200:β 6193:⁡ 6168:− 6148:∞ 6133:∑ 6120:π 6116:μ 6092:− 6083:− 6068:μ 5933:μ 5893:− 5878:μ 5759:μ 5751:− 5473:− 5160:ω 5157:ρ 5149:ω 5146:⁡ 5128:− 5112:α 5109:− 5096:β 5084:ω 5081:ρ 5073:ω 5070:⁡ 5052:− 5036:β 5020:α 4996:− 4975:μ 4900:ω 4897:⁡ 4891:β 4888:− 4882:ω 4879:⁡ 4873:α 4870:− 4864:− 4852:∂ 4844:∂ 4747:⁡ 4714:− 4688:− 4670:− 4644:μ 4636:π 4582:⁡ 4556:⁡ 4527:− 4501:μ 4471:− 4445:μ 4232:λ 4190:ν 4176:λ 4172:− 4152:λ 4113:λ 4080:λ 4069:∞ 4054:∑ 4048:μ 4025:− 4007:− 3986:μ 3844:∂ 3836:∂ 3804:∂ 3790:∂ 3778:ν 3770:ρ 3753:∂ 3745:∂ 3586:Δ 3578:Δ 3570:μ 3487:Δ 3462:μ 3459:π 3409:Δ 3396:μ 3390:π 3354:Δ 3341:μ 3338:π 3317:π 3308:Δ 3305:− 3029:Δ 3025:μ 3002:π 2952:Δ 2948:μ 2942:π 2936:− 2921:π 2912:Δ 2909:− 2747:Δ 2743:μ 2720:π 2602:Δ 2598:μ 2592:π 2586:− 2470:Δ 2455:Δ 2446:μ 2443:− 2360:Viscosity 2353:Viscosity 2347:Viscosity 2227:π 2216:μ 2201:Δ 2107:π 2090:∫ 2073:π 1959:− 1938:μ 1912:profile: 1907:parabolic 1802:⁡ 1780:μ 1756:− 1705:Δ 1672:− 1597:μ 1575:∂ 1567:∂ 1547:∂ 1543:∂ 1227:μ 1217:ρ 1178:Λ 1121:ρ 1114:Δ 1094:π 1071:⇒ 1041:π 1020:ρ 987:¯ 978:ρ 959:Δ 840:μ 837:π 812:π 801:μ 786:Δ 726:turbulent 714:Newtonian 705:in 1845. 678:fluid in 676:Newtonian 601:Truesdell 531:Bernoulli 480:Rheometer 475:Rheometry 315:Newtonian 309:Viscosity 59:φ 47:− 10394:: 1–19. 10365:15269803 10270:(2006). 10142:14368548 9915:40607063 9738:: 1–19. 9678:See also 9661:cannulas 9633:inviscid 9252:, where 9103:.) Then 9083:coulombs 8984:, where 8480:to give 5617:are the 4372:constant 3697:constant 2849:pressure 2523:gradient 2305:pressure 1910:velocity 1480:pressure 771:Equation 459:Rheology 364:Adhesion 344:Pressure 330:Buoyancy 275:Dynamics 113:Momentum 10396:Bibcode 10165:Bibcode 10133:1365740 10054:Bibcode 9740:Bibcode 9436:⁠ 9420:⁠ 9299:voltage 9160:⁠ 9143:⁠ 9120:current 8994:, i.e. 8916:removed 8901:sources 8869:⁠ 8839:⁠ 8144:in the 6605:⁠ 6598:√ 6585:⁠ 5991:⁠ 5973:⁠ 5692:⁠ 5674:⁠ 5644:⁠ 5628:⁠ 4399:⁠ 4381:⁠ 4325:is the 4297:is the 4293:⁠ 4273:⁠ 3724:⁠ 3706:⁠ 3695:When a 2416:⁠ 2395:⁠ 2384:to the 2026:⁠ 2009:⁠ 1902:⁠ 1885:⁠ 1474:⁠ 1456:⁠ 1441:⁠ 1420:⁠ 1410:⁠ 1395:⁠ 1365:⁠ 1350:⁠ 1251:is the 909:is the 899:is the 687:alveoli 662:, is a 546:Charles 354:Liquids 268:Statics 213:Bending 10434:779509 10432: 10363: 10326: 10280: 10183: 10140: 10130: 10072: 10012: 9943: 9913: 9907:779509 9905: 9848: 9838: 9611:where 9163:, and 9111:, and 5609:where 5178:where 4918:where 4263:where 2512:radius 2341:< 0 1833:Since 1638:where 1320:. The 1270:, the 1247:where 877:where 765:actual 733:due to 716:; the 670:in an 646:, the 596:Stokes 591:Pascal 581:Navier 576:Newton 566:Graham 541:Cauchy 444:Plasma 339: 337:Mixing 332: 317: 299: 282: 270: 258:Fluids 191:Strain 186:Stress 169:linear 118:Energy 10181:S2CID 10070:S2CID 9922:(PDF) 9911:S2CID 9883:(PDF) 9694:Pulse 7706:] 5776:with 4802:When 571:Hooke 551:Euler 536:Boyle 394:Gases 10430:PMID 10361:PMID 10324:ISBN 10278:ISBN 10138:PMID 10010:ISBN 9941:ISBN 9903:PMID 9846:OCLC 9836:ISBN 9699:Wave 9241:But 9090:* = 8899:any 8897:cite 7389:and 7302:coth 7075:sinh 6989:sinh 6953:sinh 6808:are 6608:are 6535:sinh 6502:cosh 6257:sinh 6218:sinh 6190:sinh 6019:are 6010:0 ≤ 5999:0 ≤ 5621:and 5613:and 4926:and 2545:= 2π 2519:area 2386:area 2303:The 1398:∂... 1353:∂... 1293:and 738:plus 712:and 684:lung 674:and 586:Noll 556:Fick 108:Mass 93:Laws 10422:doi 10404:doi 10353:doi 10231:doi 10173:doi 10128:PMC 10120:doi 10116:127 10062:doi 9895:doi 9748:doi 9674:). 9631:is 9559:= π 9132:nQq 9114:nQq 9096:1.6 9000:= π 8989:= π 8910:by 7736:= 0 7686:→ ∞ 7393:is 7333:csc 7112:sin 7035:sin 6806:= 0 6795:= π 6285:sin 5971:= − 5672:= − 5615:bei 5611:ber 5143:sin 5067:cos 4894:sin 4876:cos 4820:= 0 4810:= R 4379:= − 4352:If 4334:→ ∞ 3704:= − 2892:or 2684:+ d 2339:top 2332:end 2312:= − 2297:). 2187:avg 2179:= π 2005:max 1999:), 1997:= 0 1865:at 1863:= 0 1852:= 0 1842:= 0 1476:= 0 1443:= 0 1412:= 0 1387:= 0 1367:= 0 1278:or 1080:max 1035:max 993:max 755:of 658:or 10480:: 10428:. 10402:. 10392:25 10390:. 10359:. 10347:. 10266:; 10227:28 10225:. 10221:. 10204:13 10202:. 10179:. 10171:. 10159:. 10136:. 10126:. 10114:. 10110:. 10093:25 10091:. 10068:. 10060:. 10050:61 10048:. 10008:. 10004:. 9992:^ 9964:. 9909:. 9901:. 9891:31 9889:. 9885:. 9844:. 9807:. 9794:^ 9784:. 9746:. 9736:25 9734:. 9718:^ 9643:. 9418:= 9307:EL 9305:= 9293:Lq 9284:= 9249:Eq 9247:= 9155:nq 9141:= 9130:= 9105:nQ 9094:= 8975:= 8966:IR 8964:= 8848:+ 8634:16 8530:16 8233:= 8220:= 8218:Qp 7708:. 7704:de 7702:; 7700:tr 7688:, 7676:= 7224:12 6802:± 6797:, 6755:60 6410:16 6395:12 6014:≤ 6003:≤ 5930:12 5647:. 5632:ρω 5626:= 4922:, 4812:, 4744:ln 4579:ln 4553:ln 3930:0. 3554:. 3552:dr 3288:, 2671:. 2531:dr 2502:. 2334:− 2327:= 2284:b) 2280:a) 2033:, 2013:GR 2007:= 1889:GR 1883:= 1870:= 1844:, 1799:ln 1414:). 1389:). 1380:= 1369:). 1301:. 1274:, 1255:, 1249:Re 1186:64 654:, 10448:. 10438:. 10436:. 10424:: 10412:. 10410:. 10406:: 10398:: 10367:. 10355:: 10349:4 10332:. 10286:. 10237:. 10233:: 10187:. 10175:: 10167:: 10161:7 10144:. 10122:: 10076:. 10064:: 10056:: 10018:. 9987:. 9974:. 9949:. 9897:: 9867:. 9852:. 9817:. 9786:8 9754:. 9750:: 9742:: 9667:p 9665:∆ 9625:R 9621:S 9617:R 9613:ρ 9596:, 9591:S 9587:L 9578:= 9575:R 9561:r 9557:S 9552:R 9548:r 9544:R 9540:L 9536:R 9530:. 9513:2 9508:) 9499:q 9495:( 9488:4 9484:r 9475:2 9471:n 9465:L 9459:8 9453:= 9450:R 9432:I 9428:/ 9424:V 9416:R 9395:. 9387:2 9382:) 9373:q 9369:( 9362:4 9358:r 9349:2 9345:n 9339:I 9336:L 9330:8 9324:= 9321:V 9303:V 9295:* 9290:r 9288:π 9286:n 9282:q 9276:L 9273:r 9271:π 9269:n 9263:L 9260:r 9258:π 9254:q 9245:F 9243:Δ 9226:. 9214:q 9208:2 9204:r 9200:n 9195:I 9192:L 9186:8 9180:= 9177:F 9157:* 9151:/ 9147:I 9139:Q 9134:* 9128:I 9123:I 9116:* 9109:Q 9098:× 9092:e 9088:q 9079:* 9077:q 9072:n 9066:. 9050:2 9046:r 9041:Q 9038:L 9032:8 9026:= 9023:F 9006:P 9004:Δ 9002:r 8998:F 8996:Δ 8991:r 8987:S 8981:p 8979:Δ 8977:S 8973:F 8971:Δ 8962:V 8937:) 8931:( 8926:) 8922:( 8918:. 8904:. 8865:2 8862:p 8860:2 8857:/ 8853:2 8850:p 8846:1 8843:p 8821:. 8813:2 8809:p 8805:2 8800:) 8794:2 8790:p 8786:+ 8781:1 8777:p 8772:( 8763:L 8757:8 8751:) 8745:2 8741:p 8732:1 8728:p 8723:( 8717:4 8713:R 8703:= 8699:) 8692:2 8688:p 8681:2 8676:2 8672:p 8663:2 8658:1 8654:p 8647:( 8640:L 8627:4 8623:R 8613:= 8608:2 8604:Q 8577:. 8569:4 8565:R 8554:2 8550:p 8544:2 8540:Q 8536:L 8524:= 8519:2 8514:2 8510:p 8501:2 8496:1 8492:p 8478:L 8474:μ 8456:. 8448:4 8444:R 8433:2 8429:p 8423:2 8419:Q 8412:8 8406:= 8400:x 8396:d 8390:p 8386:d 8379:p 8363:4 8359:R 8355:p 8345:2 8341:p 8335:2 8331:Q 8324:8 8318:= 8310:4 8306:R 8297:Q 8291:8 8285:= 8279:x 8275:d 8269:p 8265:d 8244:2 8241:p 8238:2 8235:Q 8231:1 8228:p 8225:1 8222:Q 8203:Q 8197:= 8188:m 8161:/ 8157:p 8137:) 8135:x 8133:( 8131:Q 8101:) 8095:2 8091:z 8087:+ 8082:2 8078:y 8073:( 8063:4 8059:G 8054:= 8051:U 8025:0 8022:= 8014:2 8010:z 8001:U 7996:2 7985:+ 7977:2 7973:y 7964:U 7959:2 7922:, 7918:) 7912:2 7908:z 7904:+ 7899:2 7895:y 7890:( 7880:4 7876:G 7871:+ 7868:u 7865:= 7862:U 7836:. 7828:G 7820:= 7812:2 7808:z 7799:u 7794:2 7783:+ 7775:2 7771:y 7762:u 7757:2 7734:u 7729:) 7727:z 7725:, 7723:y 7721:( 7719:u 7684:a 7678:b 7674:a 7652:. 7645:) 7639:2 7635:b 7631:+ 7626:2 7622:a 7617:( 7610:4 7603:3 7599:b 7593:3 7589:a 7585:G 7576:= 7569:Q 7562:, 7558:) 7550:2 7546:b 7540:2 7536:z 7523:2 7519:a 7513:2 7509:y 7500:1 7496:( 7488:) 7480:2 7476:b 7472:1 7467:+ 7460:2 7456:a 7452:1 7446:( 7439:2 7435:G 7430:= 7423:) 7420:z 7417:, 7414:y 7411:( 7408:u 7391:b 7387:a 7366:. 7362:] 7358:) 7353:n 7342:2 7339:( 7330:+ 7327:) 7322:n 7311:2 7308:( 7298:[ 7290:5 7285:n 7277:1 7265:1 7262:= 7259:n 7242:2 7238:G 7217:4 7209:G 7203:= 7196:Q 7189:, 7183:2 7180:1 7174:+ 7171:n 7168:= 7163:n 7154:, 7150:} 7146:] 7143:) 7140:z 7134:y 7131:( 7126:n 7118:[ 7109:] 7106:) 7103:z 7100:+ 7097:y 7094:( 7089:n 7081:[ 7069:] 7066:) 7063:z 7060:+ 7057:y 7054:( 7049:n 7041:[ 7032:] 7029:) 7026:z 7023:+ 7020:y 7011:2 7008:( 7003:n 6995:[ 6985:{ 6978:) 6973:n 6962:2 6959:( 6948:3 6943:n 6934:1 6922:1 6919:= 6916:n 6898:G 6890:) 6887:y 6878:( 6875:) 6872:z 6869:+ 6866:y 6863:( 6854:2 6850:G 6845:= 6838:) 6835:z 6832:, 6829:y 6826:( 6823:u 6804:z 6800:y 6793:y 6771:. 6760:3 6748:4 6744:h 6740:G 6734:= 6727:Q 6720:, 6716:) 6710:2 6706:z 6702:3 6694:2 6690:y 6685:( 6681:) 6678:h 6672:y 6669:( 6663:h 6657:4 6653:G 6645:= 6638:) 6635:z 6632:, 6629:y 6626:( 6623:u 6600:3 6594:/ 6590:h 6588:2 6563:. 6557:) 6554:l 6549:n 6541:( 6530:1 6524:) 6521:l 6516:n 6508:( 6491:5 6487:) 6483:1 6477:n 6474:2 6471:( 6467:1 6455:1 6452:= 6449:n 6433:5 6421:4 6417:h 6413:G 6390:l 6385:3 6381:h 6377:G 6371:= 6364:Q 6357:, 6352:h 6345:) 6342:1 6336:n 6333:2 6330:( 6324:= 6319:n 6310:, 6307:) 6304:y 6299:n 6291:( 6279:) 6276:l 6271:n 6263:( 6252:] 6249:) 6246:z 6240:l 6237:( 6232:n 6224:[ 6215:+ 6212:) 6209:z 6204:n 6196:( 6179:3 6175:) 6171:1 6165:n 6162:2 6159:( 6155:1 6143:1 6140:= 6137:n 6124:3 6109:2 6105:h 6101:G 6098:4 6089:) 6086:y 6080:h 6077:( 6074:y 6065:2 6061:G 6056:= 6049:) 6046:z 6043:, 6040:y 6037:( 6034:u 6016:l 6012:z 6005:h 6001:y 5987:x 5985:d 5982:/ 5978:p 5976:d 5969:G 5939:. 5923:3 5919:h 5915:G 5909:= 5906:Q 5902:, 5899:) 5896:y 5890:h 5887:( 5884:y 5875:2 5871:G 5866:= 5863:) 5860:y 5857:( 5854:u 5828:0 5825:= 5822:) 5819:h 5816:( 5813:u 5809:, 5806:0 5803:= 5800:) 5797:0 5794:( 5791:u 5756:G 5748:= 5740:2 5736:y 5731:d 5725:u 5720:2 5715:d 5688:x 5686:d 5683:/ 5679:p 5677:d 5670:G 5665:h 5640:μ 5636:/ 5624:k 5590:, 5584:) 5581:R 5578:k 5575:( 5570:2 5565:i 5562:e 5559:b 5554:+ 5551:) 5548:R 5545:k 5542:( 5537:2 5532:r 5529:e 5526:b 5519:) 5516:R 5513:k 5510:( 5506:r 5503:e 5500:b 5496:) 5493:r 5490:k 5487:( 5483:i 5480:e 5477:b 5470:) 5467:R 5464:k 5461:( 5457:i 5454:e 5451:b 5447:) 5444:r 5441:k 5438:( 5434:r 5431:e 5428:b 5421:= 5414:) 5411:r 5408:k 5405:( 5400:2 5396:F 5388:, 5382:) 5379:R 5376:k 5373:( 5368:2 5363:i 5360:e 5357:b 5352:+ 5349:) 5346:R 5343:k 5340:( 5335:2 5330:r 5327:e 5324:b 5317:) 5314:R 5311:k 5308:( 5304:i 5301:e 5298:b 5294:) 5291:r 5288:k 5285:( 5281:i 5278:e 5275:b 5271:+ 5268:) 5265:R 5262:k 5259:( 5255:r 5252:e 5249:b 5245:) 5242:r 5239:k 5236:( 5232:r 5229:e 5226:b 5219:= 5212:) 5209:r 5206:k 5203:( 5198:1 5194:F 5152:t 5137:] 5134:) 5131:1 5123:1 5119:F 5115:( 5104:2 5100:F 5093:[ 5090:+ 5076:t 5061:] 5058:) 5055:1 5047:1 5043:F 5039:( 5033:+ 5028:2 5024:F 5017:[ 5014:+ 5010:) 5004:2 5000:r 4991:2 4987:R 4982:( 4972:4 4968:G 4963:= 4960:) 4957:t 4954:, 4951:r 4948:( 4945:u 4932:ω 4928:β 4924:α 4920:G 4903:t 4885:t 4867:G 4861:= 4855:x 4847:p 4818:1 4815:R 4808:2 4805:R 4783:. 4779:] 4770:1 4766:R 4761:/ 4755:2 4751:R 4738:2 4733:) 4727:2 4722:1 4718:R 4709:2 4704:2 4700:R 4695:( 4683:4 4678:1 4674:R 4665:4 4660:2 4656:R 4651:[ 4641:8 4633:G 4627:= 4620:Q 4613:, 4605:1 4601:R 4596:/ 4590:2 4586:R 4572:1 4568:R 4563:/ 4559:r 4546:) 4540:2 4535:1 4531:R 4522:2 4517:2 4513:R 4508:( 4498:4 4494:G 4489:+ 4485:) 4479:2 4475:r 4466:2 4461:1 4457:R 4452:( 4442:4 4438:G 4433:= 4426:) 4423:r 4420:( 4417:u 4395:x 4393:d 4390:/ 4386:p 4384:d 4377:G 4367:2 4364:R 4358:1 4355:R 4332:t 4323:) 4320:n 4318:λ 4316:( 4314:1 4311:J 4305:n 4303:λ 4295:) 4289:R 4285:/ 4281:r 4279:n 4277:λ 4271:( 4269:0 4266:J 4248:0 4245:= 4241:) 4236:n 4228:( 4222:0 4218:J 4213:, 4206:2 4202:R 4197:/ 4193:t 4185:2 4180:n 4168:e 4161:) 4156:n 4148:( 4143:1 4139:J 4133:) 4130:R 4126:/ 4122:r 4117:n 4109:( 4104:0 4100:J 4089:3 4084:n 4076:1 4064:1 4061:= 4058:n 4042:2 4038:R 4034:G 4031:2 4021:) 4015:2 4011:r 4002:2 3998:R 3993:( 3983:4 3979:G 3974:= 3971:) 3968:t 3965:, 3962:r 3959:( 3956:u 3927:= 3924:) 3921:t 3918:, 3915:R 3912:( 3909:u 3905:, 3902:0 3899:= 3896:) 3893:0 3890:, 3887:r 3884:( 3881:u 3854:) 3847:r 3839:u 3828:r 3825:1 3820:+ 3812:2 3808:r 3799:u 3794:2 3782:( 3775:+ 3767:G 3762:= 3756:t 3748:u 3720:x 3718:d 3715:/ 3711:p 3709:d 3702:G 3667:r 3663:d 3657:v 3653:d 3644:r 3641:1 3636:+ 3628:2 3624:r 3619:d 3613:v 3608:2 3603:d 3595:= 3589:x 3581:p 3567:1 3531:. 3523:2 3519:r 3514:d 3508:v 3503:2 3498:d 3490:x 3481:2 3477:) 3473:r 3469:d 3465:( 3456:2 3453:+ 3445:2 3441:r 3436:d 3430:v 3425:2 3420:d 3412:x 3405:r 3401:d 3393:r 3387:2 3384:+ 3378:r 3374:d 3368:v 3364:d 3357:x 3350:r 3346:d 3335:2 3332:+ 3329:r 3325:d 3320:r 3314:2 3311:p 3302:= 3299:0 3286:r 3269:. 3266:r 3262:d 3255:r 3250:| 3242:2 3238:r 3233:d 3227:v 3222:2 3217:d 3205:+ 3200:r 3195:| 3189:r 3185:d 3179:v 3175:d 3164:= 3159:r 3155:d 3151:+ 3148:r 3143:| 3137:r 3133:d 3127:v 3123:d 3085:. 3080:r 3076:d 3072:+ 3069:r 3064:| 3058:r 3054:d 3048:v 3044:d 3032:x 3022:) 3019:r 3015:d 3011:+ 3008:r 3005:( 2999:2 2996:+ 2991:r 2986:| 2980:r 2976:d 2970:v 2966:d 2955:x 2945:r 2939:2 2933:r 2929:d 2924:r 2918:2 2915:p 2906:= 2903:0 2871:F 2867:+ 2858:F 2854:+ 2845:F 2841:= 2838:0 2797:r 2793:d 2789:+ 2786:r 2781:| 2775:r 2771:d 2765:v 2761:d 2750:x 2740:) 2737:r 2733:d 2729:+ 2726:r 2723:( 2717:2 2714:= 2705:F 2691:r 2686:r 2682:r 2669:r 2661:r 2642:r 2637:| 2631:r 2627:d 2621:v 2617:d 2605:x 2595:r 2589:2 2583:= 2574:F 2560:r 2551:x 2549:Δ 2547:r 2543:A 2537:x 2535:Δ 2527:r 2515:r 2479:. 2473:y 2463:x 2459:v 2449:A 2440:= 2431:F 2412:y 2410:Δ 2407:/ 2402:x 2400:v 2398:Δ 2390:A 2370:x 2343:. 2336:p 2329:p 2325:p 2323:Δ 2318:p 2316:Δ 2314:A 2310:F 2243:. 2235:4 2231:R 2222:L 2219:Q 2213:8 2207:= 2204:p 2184:u 2181:R 2177:Q 2159:. 2153:x 2150:a 2147:m 2141:u 2133:2 2130:1 2124:= 2121:r 2117:d 2113:u 2110:r 2104:2 2099:R 2094:0 2081:2 2077:R 2069:1 2064:= 2058:g 2055:v 2052:a 2046:u 2022:μ 2020:4 2017:/ 2002:u 1995:r 1977:. 1973:) 1967:2 1963:r 1954:2 1950:R 1945:( 1935:4 1931:G 1926:= 1923:u 1898:μ 1896:4 1893:/ 1881:2 1878:c 1872:R 1868:r 1861:u 1850:1 1847:c 1840:r 1835:u 1816:2 1812:c 1808:+ 1805:r 1794:1 1790:c 1786:+ 1777:4 1770:2 1766:r 1762:G 1753:= 1750:u 1737:G 1720:G 1717:= 1712:L 1708:p 1699:= 1693:x 1689:d 1683:p 1679:d 1658:p 1656:Δ 1652:L 1648:x 1644:r 1640:μ 1620:x 1616:d 1610:p 1606:d 1594:1 1589:= 1585:) 1578:r 1570:u 1561:r 1557:( 1550:r 1536:r 1533:1 1506:x 1502:u 1491:u 1487:x 1483:p 1470:r 1468:∂ 1465:/ 1461:p 1459:∂ 1437:x 1435:∂ 1432:/ 1427:x 1425:u 1423:∂ 1406:θ 1404:∂ 1401:/ 1384:θ 1382:u 1377:r 1375:u 1361:t 1359:∂ 1356:/ 1341:) 1339:x 1337:, 1335:θ 1333:, 1331:r 1329:( 1283:Λ 1261:v 1257:ρ 1232:, 1223:d 1220:v 1211:= 1207:e 1204:R 1199:, 1193:e 1190:R 1181:= 1127:, 1117:p 1111:2 1102:2 1098:R 1091:= 1076:Q 1062:2 1057:) 1049:2 1045:R 1031:Q 1025:( 1015:2 1012:1 1007:= 998:2 984:v 973:2 970:1 965:= 962:p 923:A 917:R 913:, 907:Q 903:, 897:μ 891:L 884:p 882:Δ 862:, 855:2 851:A 846:Q 843:L 834:8 828:= 820:4 816:R 807:Q 804:L 798:8 792:= 789:p 631:e 624:t 617:v 341:· 334:· 324:) 319:· 313:( 301:· 284:· 272:· 67:x 64:d 56:d 50:D 44:= 41:J

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Knowledge

Hagen–Poiseuille equation

Index