1058:
22:
377:
634:, because the density can change as observed from a fixed position as fluid flows through the control volume. This approach maintains generality, and not requiring that the partial time derivative of the density vanish illustrates that compressible fluids can still undergo incompressible flow. What interests us is the change in density of a control volume that moves along with the flow velocity,
855:
828:
1996:, the anelastic constraint extends incompressible flow validity to stratified density and/or temperature as well as pressure. This allows the thermodynamic variables to relax to an 'atmospheric' base state seen in the lower atmosphere when used in the field of meteorology, for example. This condition can also be used for various astrophysical systems.
554:
2062:
perturbations in density and/or temperature. The assumption is that the flow remains within a Mach number limit (normally less than 0.3) for any solution using such a constraint to be valid. Again, in accordance with all incompressible flows the pressure deviation must be small in comparison to the
1053:{\displaystyle {\mathrm {d} \rho \over \mathrm {d} t}={\partial \rho \over \partial t}+{\partial \rho \over \partial x}{\mathrm {d} x \over \mathrm {d} t}+{\partial \rho \over \partial y}{\mathrm {d} y \over \mathrm {d} t}+{\partial \rho \over \partial z}{\mathrm {d} z \over \mathrm {d} t}.}
1865:
690:
1307:
And so beginning with the conservation of mass and the constraint that the density within a moving volume of fluid remains constant, it has been shown that an equivalent condition required for incompressible flow is that the divergence of the flow velocity vanishes.
1549:
1178:
1878:
In fluid dynamics, a flow is considered incompressible if the divergence of the flow velocity is zero. However, related formulations can sometimes be used, depending on the flow system being modelled. Some versions are described below:
450:
2057:
constraint can be derived from the compressible Euler equations using scale analysis of non-dimensional quantities. The restraint, like the previous in this section, allows for the removal of acoustic waves, but also allows for
1380:
1256:
617:
1761:
2112:
2051:
823:{\displaystyle {\partial \rho \over \partial t}+{\nabla \cdot \left(\rho \mathbf {u} \right)}={\partial \rho \over \partial t}+{\nabla \rho \cdot \mathbf {u} }+{\rho \left(\nabla \cdot \mathbf {u} \right)}=0.}
1990:
311:
439:
The negative sign in the above expression ensures that outward flow results in a decrease in the mass with respect to time, using the convention that the surface area vector points outward. Now, using the
1265:, had changed), which we have prohibited. We must then require that the material derivative of the density vanishes, and equivalently (for non-zero density) so must the divergence of the flow velocity:
433:
1715:
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Thus homogeneous materials always undergo flow that is incompressible, but the converse is not true. That is, compressible materials might not experience compression in the flow.
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1558:
of the density is zero. Thus if one follows a material element, its mass density remains constant. Note that the material derivative consists of two terms. The first term
1101:
2132:
1922:. This can assume either constant density (strict incompressible) or varying density flow. The varying density set accepts solutions involving small perturbations in
2152:
1316:
In some fields, a measure of the incompressibility of a flow is the change in density as a result of the pressure variations. This is best expressed in terms of the
255:
215:
2162:
The stringent nature of incompressible flow equations means that specific mathematical techniques have been devised to solve them. Some of these methods include:
549:{\displaystyle {\iiint \limits _{V}{\partial \rho \over \partial t}\,\mathrm {d} V}={-\iiint \limits _{V}\left(\nabla \cdot \mathbf {J} \right)\,\mathrm {d} V},}
185:
Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that under the right conditions even the flow of
398:
1325:
1189:
838:
1860:{\displaystyle {\frac {D\rho }{Dt}}={\frac {\partial \rho }{\partial t}}+\mathbf {u} \cdot \nabla \rho =0\ \Rightarrow \ \nabla \cdot \mathbf {u} =0}
1261:
A change in the density over time would imply that the fluid had either compressed or expanded (or that the mass contained in our constant volume,
565:
1636:(convection term for scalar field). For a flow to be accounted as bearing incompressibility, the accretion sum of these terms should vanish.
626:. When we speak of the partial derivative of the density with respect to time, we refer to this rate of change within a control volume of
2070:
2009:
2267:
1939:
263:
229:(discussed below) of the density must vanish to ensure incompressible flow. Before introducing this constraint, we must apply the
86:
403:
58:
1677:
1561:
65:
39:
644:
630:. By letting the partial time derivative of the density be non-zero, we are not restricting ourselves to incompressible
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2067:
These methods make differing assumptions about the flow, but all take into account the general form of the constraint
335:
105:
72:
1604:
1430:
1271:
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of the flow velocity is zero (see the derivation below, which illustrates why these conditions are equivalent).
54:
43:
2244:
1646:
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describes the changes in the density as the material element moves from one point to another. This is the
1544:{\displaystyle {\frac {D\rho }{Dt}}={\frac {\partial \rho }{\partial t}}+\mathbf {u} \cdot \nabla \rho =0}
2196:
1927:
834:
1173:{\displaystyle {D\rho \over Dt}={\partial \rho \over \partial t}+{\nabla \rho \cdot \mathbf {u} }.}
1722:
1597:
describes how the density of the material element changes with time. This term is also known as the
622:
The partial derivative of the density with respect to time need not vanish to ensure incompressible
2338:
2186:
124:
79:
32:
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we can derive the relationship between the flux and the partial time derivative of the density:
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1993:
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8:
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324:, across its boundaries. Mathematically, we can represent this constraint in terms of a
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186:
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So if we choose a control volume that is moving at the same rate as the fluid (i.e. (
151:
1375:{\displaystyle \beta ={\frac {1}{\rho }}{\frac {\mathrm {d} \rho }{\mathrm {d} p}}.}
2300:
2240:
1385:
If the compressibility is acceptably small, the flow is considered incompressible.
842:
325:
1408:
Otherwise, if an incompressible flow also has a curl of zero, so that it is also
1317:
1251:{\displaystyle {D\rho \over Dt}={-\rho \left(\nabla \cdot \mathbf {u} \right)}.}
234:
139:
316:
The conservation of mass requires that the time derivative of the mass inside a
317:
2332:
2252:
612:{\displaystyle {\partial \rho \over \partial t}=-\nabla \cdot \mathbf {J} .}
171:
167:
1424:
As defined earlier, an incompressible (isochoric) flow is the one in which
638:. The flux is related to the flow velocity through the following function:
131:. For strings which cannot be reduced by a given compression algorithm, see
163:
2287:
2054:
197:
The fundamental requirement for incompressible flow is that the density,
123:"Incompressible" redirects here. For the property of vector fields, see
1398:
1394:
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179:
155:
2107:{\displaystyle \nabla \cdot \left(\alpha \mathbf {u} \right)=\beta }
2046:{\displaystyle \nabla \cdot \left(\alpha \mathbf {u} \right)=\beta }
1397:
flow velocity field. But a solenoidal field, besides having a zero
21:
2304:
1985:{\displaystyle {\nabla \cdot \left(\rho _{o}\mathbf {u} \right)=0}}
2268:"Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics"
1643:
is one that has constant density throughout. For such a material,
233:
to generate the necessary relations. The mass is calculated by a
189:
can, to a good approximation, be modelled as incompressible flow.
1926:, pressure and/or temperature fields, and can allow for pressure
1923:
1183:
And so using the continuity equation derived above, we see that:
159:
2266:
Almgren, A.S.; Bell, J.B.; Rendleman, C.A.; Zingale, M. (2006).
306:{\displaystyle {m}={\iiint \limits _{V}\!\rho \,\mathrm {d} V}.}
2265:
174:— is time-invariant. An equivalent statement that implies
833:
The previous relation (where we have used the appropriate
428:{\displaystyle \mathbf {J} \cdot \mathrm {d} \mathbf {S} }
2245:
10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2
1401:, also has the additional connotation of having non-zero
1710:{\displaystyle {\frac {\partial \rho }{\partial t}}=0}
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1590:{\displaystyle {\tfrac {\partial \rho }{\partial t}}}
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225:. Mathematically, this constraint implies that the
203:
674:{\displaystyle {\mathbf {J} }={\rho \mathbf {u} }.}
46:. Unsourced material may be challenged and removed.
2173:Artificial compressibility technique (approximate)
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841:. Now, we need the following relation about the
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1388:
684:So that the conservation of mass implies that:
368:{\displaystyle {\partial m \over \partial t}=-}
1755:From the continuity equation it follows that
1625:{\displaystyle \mathbf {u} \cdot \nabla \rho }
1455:{\displaystyle \nabla \cdot \mathbf {u} =0.\,}
1311:
1297:{\displaystyle {\nabla \cdot \mathbf {u} }=0.}
1915:{\displaystyle {\nabla \cdot \mathbf {u} =0}}
217:, is constant within a small element volume,
120:Fluid flow in which density remains constant
2157:
1873:
1419:
1412:, then the flow velocity field is actually
1091:), then this expression simplifies to the
2286:
1451:
1393:An incompressible flow is described by a
533:
485:
290:
106:Learn how and when to remove this message
2218:"Improving the Anelastic Approximation"
1664:{\displaystyle \rho ={\text{constant}}}
2331:
2215:
2114:for general flow dependent functions
1641:homogeneous, incompressible material
127:. For the topological property, see
44:adding citations to reliable sources
15:
2225:Journal of the Atmospheric Sciences
1992:. Principally used in the field of
845:of the density (where we apply the
221:, which moves at the flow velocity
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1465:This is equivalent to saying that
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2192:Euler equations (fluid dynamics)
2176:Compressibility pre-conditioning
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31:needs additional citations for
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1834:
1741:{\displaystyle \nabla \rho =0}
1405:(i.e., rotational component).
1:
2202:
192:
2170:(both approximate and exact)
1389:Relation to solenoidal field
7:
2180:
1312:Relation to compressibility
320:be equal to the mass flux,
170:volume that moves with the
10:
2355:
122:
2158:Numerical approximations
2004:pseudo-incompressibility
1874:Related flow constraints
1420:Difference from material
2197:Navier–Stokes equations
2127:{\displaystyle \alpha }
125:Solenoidal vector field
2148:
2147:{\displaystyle \beta }
2128:
2108:
2047:
1986:
1916:
1861:
1742:
1711:
1671:. This implies that,
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1252:
1174:
1054:
824:
675:
613:
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394:
369:
307:
251:
211:
158:in which the material
129:Incompressible surface
2275:Astrophysical Journal
2216:Durran, D.R. (1989).
2187:Bernoulli's principle
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2129:
2109:
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1987:
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1639:On the other hand, a
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252:
250:{\displaystyle \rho }
212:
210:{\displaystyle \rho }
133:Incompressible string
55:"Incompressible flow"
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2118:
2071:
2063:pressure base state.
2010:
2000:Low Mach-number flow
1994:atmospheric sciences
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231:conservation of mass
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142:, or more generally
40:improve this article
2297:2006ApJ...637..922A
2237:1989JAtS...46.1453D
1884:Incompressible flow
1601:. The second term,
1556:material derivative
1093:material derivative
839:continuity equation
227:material derivative
187:compressible fluids
176:incompressible flow
148:incompressible flow
144:continuum mechanics
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837:) is known as the
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442:divergence theorem
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2231:(11): 1453–1461.
2168:projection method
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393:{\displaystyle S}
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2307:. Archived from
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2288:astro-ph/0509892
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843:total derivative
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1318:compressibility
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2281:(2): 922–936.
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1934:Anelastic flow
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1770:
1753:
1752:
1737:
1734:
1731:
1728:
1718:
1706:
1703:
1697:
1694:
1689:
1686:
1655:
1652:
1634:advection term
1621:
1618:
1615:
1611:
1582:
1579:
1574:
1571:
1552:
1551:
1540:
1537:
1534:
1531:
1528:
1524:
1520:
1514:
1511:
1506:
1503:
1497:
1491:
1488:
1483:
1480:
1463:
1462:
1450:
1447:
1443:
1439:
1436:
1421:
1418:
1390:
1387:
1383:
1382:
1371:
1365:
1361:
1355:
1351:
1342:
1339:
1334:
1331:
1313:
1310:
1305:
1304:
1293:
1290:
1285:
1281:
1278:
1259:
1258:
1247:
1242:
1237:
1233:
1230:
1226:
1222:
1219:
1215:
1209:
1206:
1201:
1198:
1181:
1180:
1169:
1164:
1160:
1157:
1154:
1150:
1144:
1141:
1136:
1133:
1127:
1121:
1118:
1113:
1110:
1087:) =
1061:
1060:
1049:
1043:
1039:
1033:
1029:
1019:
1016:
1011:
1008:
1002:
996:
992:
986:
982:
972:
969:
964:
961:
955:
949:
945:
939:
935:
925:
922:
917:
914:
908:
902:
899:
894:
891:
885:
879:
875:
869:
865:
831:
830:
819:
816:
811:
806:
802:
799:
795:
791:
787:
782:
778:
775:
772:
768:
762:
759:
754:
751:
745:
740:
735:
731:
727:
723:
720:
716:
710:
707:
702:
699:
682:
681:
670:
665:
661:
657:
652:
628:fixed position
620:
619:
608:
604:
600:
597:
594:
591:
585:
582:
577:
574:
557:
556:
545:
541:
537:
531:
526:
522:
519:
515:
509:
505:
501:
497:
493:
489:
481:
478:
473:
470:
462:
458:
437:
436:
423:
418:
414:
410:
389:
364:
361:
355:
352:
347:
344:
318:control volume
314:
313:
302:
298:
294:
289:
283:
279:
274:
270:
246:
206:
194:
191:
154:) refers to a
152:isochoric flow
119:
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
2351:
2340:
2337:
2336:
2334:
2314:on 2008-10-31
2310:
2306:
2302:
2298:
2294:
2289:
2284:
2280:
2276:
2269:
2262:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2219:
2212:
2208:
2198:
2195:
2193:
2190:
2188:
2185:
2184:
2175:
2172:
2169:
2165:
2164:
2163:
2155:
2141:
2121:
2101:
2098:
2094:
2085:
2081:
2077:
2061:
2056:
2040:
2037:
2033:
2024:
2020:
2016:
2005:
2001:
1998:
1995:
1978:
1975:
1971:
1960:
1956:
1951:
1947:
1935:
1932:
1929:
1925:
1908:
1905:
1897:
1885:
1882:
1881:
1880:
1871:
1854:
1851:
1843:
1828:
1825:
1822:
1816:
1808:
1802:
1794:
1785:
1779:
1776:
1771:
1768:
1758:
1757:
1756:
1750:
1749:independently
1735:
1732:
1729:
1719:
1704:
1701:
1695:
1687:
1674:
1673:
1672:
1653:
1650:
1642:
1637:
1635:
1619:
1613:
1600:
1599:unsteady term
1580:
1572:
1557:
1538:
1535:
1532:
1526:
1518:
1512:
1504:
1495:
1489:
1486:
1481:
1478:
1468:
1467:
1466:
1448:
1445:
1437:
1427:
1426:
1425:
1417:
1415:
1411:
1406:
1404:
1400:
1396:
1386:
1369:
1363:
1353:
1340:
1337:
1332:
1329:
1322:
1321:
1320:
1319:
1309:
1291:
1288:
1279:
1268:
1267:
1266:
1264:
1245:
1240:
1231:
1224:
1220:
1217:
1213:
1207:
1204:
1199:
1196:
1186:
1185:
1184:
1167:
1158:
1155:
1148:
1142:
1134:
1125:
1119:
1116:
1111:
1108:
1098:
1097:
1096:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1047:
1041:
1031:
1017:
1009:
1000:
994:
984:
970:
962:
953:
947:
937:
923:
915:
906:
900:
892:
883:
877:
867:
852:
851:
850:
848:
844:
840:
836:
817:
814:
809:
800:
793:
789:
785:
776:
773:
766:
760:
752:
743:
738:
729:
725:
721:
714:
708:
700:
687:
686:
685:
668:
659:
655:
641:
640:
639:
637:
633:
629:
625:
606:
598:
592:
589:
583:
575:
562:
561:
560:
543:
539:
529:
520:
513:
507:
503:
499:
495:
491:
479:
471:
460:
456:
447:
446:
445:
443:
412:
387:
362:
359:
353:
345:
331:
330:
329:
327:
323:
319:
300:
296:
287:
281:
277:
272:
268:
260:
259:
258:
244:
236:
232:
228:
224:
220:
204:
190:
188:
183:
181:
177:
173:
172:flow velocity
169:
168:infinitesimal
165:
161:
157:
153:
149:
145:
141:
134:
130:
126:
118:
110:
107:
99:
96:December 2019
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
2316:. Retrieved
2309:the original
2278:
2274:
2261:
2228:
2224:
2211:
2161:
2066:
2059:
2003:
1999:
1933:
1883:
1877:
1869:
1754:
1748:
1640:
1638:
1633:
1598:
1553:
1464:
1423:
1410:irrotational
1407:
1392:
1384:
1315:
1306:
1262:
1260:
1182:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1062:
835:product rule
832:
683:
635:
631:
627:
623:
621:
558:
438:
321:
315:
222:
218:
196:
184:
178:is that the
175:
164:fluid parcel
147:
137:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
2055:Mach-number
559:therefore:
2318:2008-12-04
2203:References
2053:. The low
1399:divergence
1395:solenoidal
847:chain rule
193:Derivation
180:divergence
66:newspapers
2253:1520-0469
2142:β
2122:α
2102:β
2086:α
2078:⋅
2075:∇
2041:β
2025:α
2017:⋅
2014:∇
1957:ρ
1948:⋅
1945:∇
1898:⋅
1895:∇
1844:⋅
1841:∇
1835:⇒
1823:ρ
1820:∇
1817:⋅
1800:∂
1795:ρ
1792:∂
1772:ρ
1730:ρ
1727:∇
1693:∂
1688:ρ
1685:∂
1651:ρ
1620:ρ
1617:∇
1614:⋅
1578:∂
1573:ρ
1570:∂
1554:i.e. the
1533:ρ
1530:∇
1527:⋅
1510:∂
1505:ρ
1502:∂
1482:ρ
1438:⋅
1435:∇
1414:Laplacian
1354:ρ
1341:ρ
1330:β
1280:⋅
1277:∇
1232:⋅
1229:∇
1221:ρ
1218:−
1200:ρ
1159:⋅
1156:ρ
1153:∇
1140:∂
1135:ρ
1132:∂
1112:ρ
1015:∂
1010:ρ
1007:∂
968:∂
963:ρ
960:∂
921:∂
916:ρ
913:∂
898:∂
893:ρ
890:∂
868:ρ
801:⋅
798:∇
790:ρ
777:⋅
774:ρ
771:∇
758:∂
753:ρ
750:∂
730:ρ
722:⋅
719:∇
706:∂
701:ρ
698:∂
660:ρ
599:⋅
596:∇
593:−
581:∂
576:ρ
573:∂
521:⋅
518:∇
504:∭
500:−
477:∂
472:ρ
469:∂
457:∭
413:⋅
363:−
351:∂
343:∂
288:ρ
278:∭
245:ρ
205:ρ
2333:Category
2181:See also
1658:constant
162:of each
2293:Bibcode
2233:Bibcode
1924:density
1079:,
1071:,
160:density
80:scholar
2251:
1838:
1832:
632:fluids
82:
75:
68:
61:
53:
2312:(PDF)
2283:arXiv
2271:(PDF)
2221:(PDF)
2060:large
2002:, or
166:— an
87:JSTOR
73:books
2249:ISSN
2166:The
2134:and
1403:curl
624:flow
156:flow
59:news
2301:doi
2279:637
2241:doi
1717:and
849:):
138:In
42:by
2335::
2299:.
2291:.
2277:.
2273:.
2247:.
2239:.
2229:46
2227:.
2223:.
2154:.
2006::
1936::
1886::
1449:0.
1416:.
1292:0.
1263:dV
1095::
1085:dt
1081:dz
1077:dt
1073:dy
1069:dt
1065:dx
818:0.
328::
257::
219:dV
146:,
2321:.
2303::
2295::
2285::
2255:.
2243::
2235::
2099:=
2095:)
2090:u
2082:(
2038:=
2034:)
2029:u
2021:(
1979:0
1976:=
1972:)
1967:u
1961:o
1952:(
1909:0
1906:=
1902:u
1855:0
1852:=
1848:u
1829:0
1826:=
1813:u
1809:+
1803:t
1786:=
1780:t
1777:D
1769:D
1751:.
1736:0
1733:=
1705:0
1702:=
1696:t
1654:=
1610:u
1581:t
1539:0
1536:=
1523:u
1519:+
1513:t
1496:=
1490:t
1487:D
1479:D
1446:=
1442:u
1370:.
1364:p
1360:d
1350:d
1338:1
1333:=
1289:=
1284:u
1246:.
1241:)
1236:u
1225:(
1214:=
1208:t
1205:D
1197:D
1168:.
1163:u
1149:+
1143:t
1126:=
1120:t
1117:D
1109:D
1089:u
1083:/
1075:/
1067:/
1048:.
1042:t
1038:d
1032:z
1028:d
1018:z
1001:+
995:t
991:d
985:y
981:d
971:y
954:+
948:t
944:d
938:x
934:d
924:x
907:+
901:t
884:=
878:t
874:d
864:d
815:=
810:)
805:u
794:(
786:+
781:u
767:+
761:t
744:=
739:)
734:u
726:(
715:+
709:t
669:.
664:u
656:=
651:J
636:u
607:.
603:J
590:=
584:t
544:,
540:V
536:d
530:)
525:J
514:(
508:V
496:=
492:V
488:d
480:t
461:V
422:S
417:d
409:J
388:S
360:=
354:t
346:m
322:J
301:.
297:V
293:d
282:V
273:=
269:m
223:u
150:(
135:.
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
