1069:
645:, 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,
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388:
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2007:, 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.
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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
1064:{\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}.}
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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.
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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:
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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
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834:{\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.}
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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
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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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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
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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
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The stringent nature of incompressible flow equations means that specific mathematical techniques have been devised to solve them. Some of these methods include:
560:{\displaystyle {\iiint \limits _{V}{\partial \rho \over \partial t}\,\mathrm {d} V}={-\iiint \limits _{V}\left(\nabla \cdot \mathbf {J} \right)\,\mathrm {d} V},}
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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
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1871:{\displaystyle {\frac {D\rho }{Dt}}={\frac {\partial \rho }{\partial t}}+\mathbf {u} \cdot \nabla \rho =0\ \Rightarrow \ \nabla \cdot \mathbf {u} =0}
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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,
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1647:(convection term for scalar field). For a flow to be accounted as bearing incompressibility, the accretion sum of these terms should vanish.
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These methods make differing assumptions about the flow, but all take into account the general form of the constraint
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of the flow velocity is zero (see the derivation below, which illustrates why these conditions are equivalent).
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describes the changes in the density as the material element moves from one point to another. This is the
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1555:{\displaystyle {\frac {D\rho }{Dt}}={\frac {\partial \rho }{\partial t}}+\mathbf {u} \cdot \nabla \rho =0}
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1184:{\displaystyle {D\rho \over Dt}={\partial \rho \over \partial t}+{\nabla \rho \cdot \mathbf {u} }.}
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describes how the density of the material element changes with time. This term is also known as the
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The partial derivative of the density with respect to time need not vanish to ensure incompressible
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we can derive the relationship between the flux and the partial time derivative of the density:
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So if we choose a control volume that is moving at the same rate as the fluid (i.e. (
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1386:{\displaystyle \beta ={\frac {1}{\rho }}{\frac {\mathrm {d} \rho }{\mathrm {d} p}}.}
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If the compressibility is acceptably small, the flow is considered incompressible.
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Otherwise, if an incompressible flow also has a curl of zero, so that it is also
1262:{\displaystyle {D\rho \over Dt}={-\rho \left(\nabla \cdot \mathbf {u} \right)}.}
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The conservation of mass requires that the time derivative of the mass inside a
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623:{\displaystyle {\partial \rho \over \partial t}=-\nabla \cdot \mathbf {J} .}
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As defined earlier, an incompressible (isochoric) flow is the one in which
649:. The flux is related to the flow velocity through the following function:
142:. For strings which cannot be reduced by a given compression algorithm, see
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208:
The fundamental requirement for incompressible flow is that the density,
134:"Incompressible" redirects here. For the property of vector fields, see
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170:
2118:{\displaystyle \nabla \cdot \left(\alpha \mathbf {u} \right)=\beta }
2057:{\displaystyle \nabla \cdot \left(\alpha \mathbf {u} \right)=\beta }
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flow velocity field. But a solenoidal field, besides having a zero
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1996:{\displaystyle {\nabla \cdot \left(\rho _{o}\mathbf {u} \right)=0}}
2279:"Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics"
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is one that has constant density throughout. For such a material,
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to generate the necessary relations. The mass is calculated by a
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can, to a good approximation, be modelled as incompressible flow.
1937:, pressure and/or temperature fields, and can allow for pressure
1194:
And so using the continuity equation derived above, we see that:
2277:
Almgren, A.S.; Bell, J.B.; Rendleman, C.A.; Zingale, M. (2006).
317:{\displaystyle {m}={\iiint \limits _{V}\!\rho \,\mathrm {d} V}.}
26:
185:— is time-invariant. An equivalent statement that implies
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The previous relation (where we have used the appropriate
439:{\displaystyle \mathbf {J} \cdot \mathrm {d} \mathbf {S} }
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10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2
1412:, also has the additional connotation of having non-zero
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236:. Mathematically, this constraint implies that the
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685:{\displaystyle {\mathbf {J} }={\rho \mathbf {u} }.}
57:. Unsourced material may be challenged and removed.
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852:. Now, we need the following relation about the
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695:So that the conservation of mass implies that:
379:{\displaystyle {\partial m \over \partial t}=-}
1766:From the continuity equation it follows that
1636:{\displaystyle \mathbf {u} \cdot \nabla \rho }
1466:{\displaystyle \nabla \cdot \mathbf {u} =0.\,}
1308:{\displaystyle {\nabla \cdot \mathbf {u} }=0.}
1926:{\displaystyle {\nabla \cdot \mathbf {u} =0}}
228:, is constant within a small element volume,
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131:Fluid flow in which density remains constant
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1404:An incompressible flow is described by a
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117:Learn how and when to remove this message
2229:"Improving the Anelastic Approximation"
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1675:{\displaystyle \rho ={\text{constant}}}
2125:for general flow dependent functions
7:
1652:homogeneous, incompressible material
138:. For the topological property, see
55:adding citations to reliable sources
2236:Journal of the Atmospheric Sciences
2003:. Principally used in the field of
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1476:This is equivalent to saying that
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1752:{\displaystyle \nabla \rho =0}
1416:(i.e., rotational component).
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2181:(both approximate and exact)
1400:Relation to solenoidal field
1323:Relation to compressibility
331:be equal to the mass flux,
181:volume that moves with the
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133:
2169:Numerical approximations
2015:pseudo-incompressibility
1885:Related flow constraints
1431:Difference from material
2208:Navier–Stokes equations
2138:{\displaystyle \alpha }
136:Solenoidal vector field
2159:
2158:{\displaystyle \beta }
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1682:. This implies that,
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140:Incompressible surface
2286:Astrophysical Journal
2227:Durran, D.R. (1989).
2198:Bernoulli's principle
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144:Incompressible string
66:"Incompressible flow"
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2011:Low Mach-number flow
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18:Incompressible fluid
2308:2006ApJ...637..922A
2248:1989JAtS...46.1453D
1895:Incompressible flow
1612:. The second term,
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1764:
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1748:
1745:
1742:
1739:
1729:
1717:
1714:
1708:
1705:
1700:
1697:
1666:
1663:
1645:advection term
1632:
1629:
1626:
1622:
1593:
1590:
1585:
1582:
1563:
1562:
1551:
1548:
1545:
1542:
1539:
1535:
1531:
1525:
1522:
1517:
1514:
1508:
1502:
1499:
1494:
1491:
1474:
1473:
1461:
1458:
1454:
1450:
1447:
1432:
1429:
1401:
1398:
1394:
1393:
1382:
1376:
1372:
1366:
1362:
1353:
1350:
1345:
1342:
1324:
1321:
1316:
1315:
1304:
1301:
1296:
1292:
1289:
1270:
1269:
1258:
1253:
1248:
1244:
1241:
1237:
1233:
1230:
1226:
1220:
1217:
1212:
1209:
1192:
1191:
1180:
1175:
1171:
1168:
1165:
1161:
1155:
1152:
1147:
1144:
1138:
1132:
1129:
1124:
1121:
1098:) =
1072:
1071:
1060:
1054:
1050:
1044:
1040:
1030:
1027:
1022:
1019:
1013:
1007:
1003:
997:
993:
983:
980:
975:
972:
966:
960:
956:
950:
946:
936:
933:
928:
925:
919:
913:
910:
905:
902:
896:
890:
886:
880:
876:
842:
841:
830:
827:
822:
817:
813:
810:
806:
802:
798:
793:
789:
786:
783:
779:
773:
770:
765:
762:
756:
751:
746:
742:
738:
734:
731:
727:
721:
718:
713:
710:
693:
692:
681:
676:
672:
668:
663:
639:fixed position
631:
630:
619:
615:
611:
608:
605:
602:
596:
593:
588:
585:
568:
567:
556:
552:
548:
542:
537:
533:
530:
526:
520:
516:
512:
508:
504:
500:
492:
489:
484:
481:
473:
469:
448:
447:
434:
429:
425:
421:
400:
375:
372:
366:
363:
358:
355:
329:control volume
325:
324:
313:
309:
305:
300:
294:
290:
285:
281:
257:
217:
205:
202:
165:) refers to a
163:isochoric flow
130:
125:
124:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2362:
2351:
2348:
2347:
2345:
2325:on 2008-10-31
2321:
2317:
2313:
2309:
2305:
2300:
2295:
2291:
2287:
2280:
2273:
2270:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2230:
2223:
2220:
2213:
2209:
2206:
2204:
2201:
2199:
2196:
2195:
2191:
2186:
2183:
2180:
2176:
2175:
2174:
2168:
2166:
2152:
2132:
2112:
2109:
2105:
2096:
2092:
2088:
2072:
2067:
2051:
2048:
2044:
2035:
2031:
2027:
2016:
2012:
2009:
2006:
1989:
1986:
1982:
1971:
1967:
1962:
1958:
1946:
1943:
1940:
1936:
1919:
1916:
1908:
1896:
1893:
1892:
1891:
1884:
1882:
1865:
1862:
1854:
1839:
1836:
1833:
1827:
1819:
1813:
1805:
1796:
1790:
1787:
1782:
1779:
1769:
1768:
1767:
1761:
1760:independently
1746:
1743:
1740:
1730:
1715:
1712:
1706:
1698:
1685:
1684:
1683:
1664:
1661:
1653:
1648:
1646:
1630:
1624:
1611:
1610:unsteady term
1591:
1583:
1568:
1549:
1546:
1543:
1537:
1529:
1523:
1515:
1506:
1500:
1497:
1492:
1489:
1479:
1478:
1477:
1459:
1456:
1448:
1438:
1437:
1436:
1430:
1428:
1426:
1422:
1417:
1415:
1411:
1407:
1399:
1397:
1380:
1374:
1364:
1351:
1348:
1343:
1340:
1333:
1332:
1331:
1330:
1322:
1320:
1302:
1299:
1290:
1279:
1278:
1277:
1275:
1256:
1251:
1242:
1235:
1231:
1228:
1224:
1218:
1215:
1210:
1207:
1197:
1196:
1195:
1178:
1169:
1166:
1159:
1153:
1145:
1136:
1130:
1127:
1122:
1119:
1109:
1108:
1107:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1058:
1052:
1042:
1028:
1020:
1011:
1005:
995:
981:
973:
964:
958:
948:
934:
926:
917:
911:
903:
894:
888:
878:
863:
862:
861:
859:
855:
851:
847:
828:
825:
820:
811:
804:
800:
796:
787:
784:
777:
771:
763:
754:
749:
740:
736:
732:
725:
719:
711:
698:
697:
696:
679:
670:
666:
652:
651:
650:
648:
644:
640:
636:
617:
609:
603:
600:
594:
586:
573:
572:
571:
554:
550:
540:
531:
524:
518:
514:
510:
506:
502:
490:
482:
471:
467:
458:
457:
456:
454:
423:
398:
373:
370:
364:
356:
342:
341:
340:
338:
334:
330:
311:
307:
298:
292:
288:
283:
279:
271:
270:
269:
255:
247:
243:
239:
235:
231:
215:
203:
201:
199:
194:
192:
188:
184:
183:flow velocity
180:
179:infinitesimal
176:
172:
168:
164:
160:
156:
152:
145:
141:
137:
129:
121:
118:
110:
107:December 2019
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
2327:. Retrieved
2320:the original
2289:
2285:
2272:
2239:
2235:
2222:
2172:
2077:
2070:
2014:
2010:
1944:
1894:
1888:
1880:
1765:
1759:
1651:
1649:
1644:
1609:
1564:
1475:
1434:
1421:irrotational
1418:
1403:
1395:
1326:
1317:
1273:
1271:
1193:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1073:
846:product rule
843:
694:
646:
642:
638:
634:
632:
569:
449:
332:
326:
233:
229:
207:
195:
189:is that the
186:
175:fluid parcel
158:
148:
128:
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
2066:Mach-number
570:therefore:
2329:2008-12-04
2214:References
2064:. The low
1410:divergence
1406:solenoidal
858:chain rule
204:Derivation
191:divergence
77:newspapers
2264:1520-0469
2153:β
2133:α
2113:β
2097:α
2089:⋅
2086:∇
2052:β
2036:α
2028:⋅
2025:∇
1968:ρ
1959:⋅
1956:∇
1909:⋅
1906:∇
1855:⋅
1852:∇
1846:⇒
1834:ρ
1831:∇
1828:⋅
1811:∂
1806:ρ
1803:∂
1783:ρ
1741:ρ
1738:∇
1704:∂
1699:ρ
1696:∂
1662:ρ
1631:ρ
1628:∇
1625:⋅
1589:∂
1584:ρ
1581:∂
1565:i.e. the
1544:ρ
1541:∇
1538:⋅
1521:∂
1516:ρ
1513:∂
1493:ρ
1449:⋅
1446:∇
1425:Laplacian
1365:ρ
1352:ρ
1341:β
1291:⋅
1288:∇
1243:⋅
1240:∇
1232:ρ
1229:−
1211:ρ
1170:⋅
1167:ρ
1164:∇
1151:∂
1146:ρ
1143:∂
1123:ρ
1026:∂
1021:ρ
1018:∂
979:∂
974:ρ
971:∂
932:∂
927:ρ
924:∂
909:∂
904:ρ
901:∂
879:ρ
812:⋅
809:∇
801:ρ
788:⋅
785:ρ
782:∇
769:∂
764:ρ
761:∂
741:ρ
733:⋅
730:∇
717:∂
712:ρ
709:∂
671:ρ
610:⋅
607:∇
604:−
592:∂
587:ρ
584:∂
532:⋅
529:∇
515:∭
511:−
488:∂
483:ρ
480:∂
468:∭
424:⋅
374:−
362:∂
354:∂
299:ρ
289:∭
256:ρ
216:ρ
2344:Category
2192:See also
1669:constant
173:of each
2304:Bibcode
2244:Bibcode
1935:density
1090:,
1082:,
171:density
91:scholar
2262:
1849:
1843:
643:fluids
93:
86:
79:
72:
64:
2323:(PDF)
2294:arXiv
2282:(PDF)
2232:(PDF)
2071:large
2013:, or
177:— an
98:JSTOR
84:books
2260:ISSN
2177:The
2145:and
1414:curl
635:flow
167:flow
70:news
2312:doi
2290:637
2252:doi
1728:and
860:):
149:In
53:by
2346::
2310:.
2302:.
2288:.
2284:.
2258:.
2250:.
2240:46
2238:.
2234:.
2165:.
2017::
1947::
1897::
1460:0.
1427:.
1303:0.
1274:dV
1106::
1096:dt
1092:dz
1088:dt
1084:dy
1080:dt
1076:dx
829:0.
339::
268::
230:dV
157:,
2332:.
2314::
2306::
2296::
2266:.
2254::
2246::
2110:=
2106:)
2101:u
2093:(
2049:=
2045:)
2040:u
2032:(
1990:0
1987:=
1983:)
1978:u
1972:o
1963:(
1920:0
1917:=
1913:u
1866:0
1863:=
1859:u
1840:0
1837:=
1824:u
1820:+
1814:t
1797:=
1791:t
1788:D
1780:D
1762:.
1747:0
1744:=
1716:0
1713:=
1707:t
1665:=
1621:u
1592:t
1550:0
1547:=
1534:u
1530:+
1524:t
1507:=
1501:t
1498:D
1490:D
1457:=
1453:u
1381:.
1375:p
1371:d
1361:d
1349:1
1344:=
1300:=
1295:u
1257:.
1252:)
1247:u
1236:(
1225:=
1219:t
1216:D
1208:D
1179:.
1174:u
1160:+
1154:t
1137:=
1131:t
1128:D
1120:D
1100:u
1094:/
1086:/
1078:/
1059:.
1053:t
1049:d
1043:z
1039:d
1029:z
1012:+
1006:t
1002:d
996:y
992:d
982:y
965:+
959:t
955:d
949:x
945:d
935:x
918:+
912:t
895:=
889:t
885:d
875:d
826:=
821:)
816:u
805:(
797:+
792:u
778:+
772:t
755:=
750:)
745:u
737:(
726:+
720:t
680:.
675:u
667:=
662:J
647:u
618:.
614:J
601:=
595:t
555:,
551:V
547:d
541:)
536:J
525:(
519:V
507:=
503:V
499:d
491:t
472:V
433:S
428:d
420:J
399:S
371:=
365:t
357:m
333:J
312:.
308:V
304:d
293:V
284:=
280:m
234:u
161:(
146:.
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
47:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
