1191:
1635:
2070:
937:
1481:
1913:
1041:
445:
1521:
2154:
268:
1953:
1034:
2223:
659:
1386:
468:- ignoring the area spanned by the holes in the filter/membrane. The spaces would be cross-sectional areas. For liquids passing through a pipe, the area is the cross-section of the pipe, at the section considered.
809:
1820:
1308:
When describing fluid flow (i.e. flow of matter), mass flux is appropriate. When describing particle transport (movement of a large number of particles), it is useful to use an analogous quantity, called the
1246:
176:
1749:
1401:
814:
550:
1833:
1941:
1509:
344:
721:. Suppose the pipe has a constant cross section and we consider a straight section of it (not at any bends/junctions), and the water is flowing steadily at a constant rate, under
593:
510:
2090:
181:
773:
950:
2170:
598:
1186:{\displaystyle j_{m}={\frac {1000\times \left(1.5\times 10^{-3}\right)}{\pi \times \left(2\times 10^{-2}\right)^{2}\times 2}}={\frac {3}{16\pi }}\times 10^{4},}
1323:
1630:{\displaystyle \langle \mathbf {u} \rangle ={\frac {1}{\rho }}\sum _{i}\rho _{i}\mathbf {u} _{i}={\frac {1}{\rho }}\sum _{i}\mathbf {j} _{{\rm {m}},\,i}}
1760:
1305:
of substances (technically contains a number of component substances), the mass fluxes must be considered separately for each component of the mixture.
2065:{\displaystyle \langle \mathbf {u} \rangle ={\frac {1}{n}}\sum _{i}c_{i}\mathbf {u} _{i}={\frac {1}{c}}\sum _{i}\mathbf {j} _{{\rm {n}},\,i}.}
1209:
115:
1709:
2267:
932:{\displaystyle {\begin{aligned}\Delta m&=\rho \Delta V\\m_{2}-m_{1}&=\rho (V_{2}-V_{1})\\m&=\rho V\\\end{aligned}}}
2160:
2311:
2156:
which is a statement of the mass conservation of fluid. In hydrodynamics, mass can only flow from one place to another.
784:(magnitude), we also need the amount of mass of water transferred through the area and the time taken. Suppose a volume
93:
Less commonly the defining equation for mass flux in this article is used interchangeably with the defining equation in
515:
2327:
2262:
453:
required to calculate the flux is real or imaginary, flat or curved, either as a cross-sectional area or a surface.
1269:
of mass elements flowing (i.e. at each point in space the velocity of an element of matter is some velocity vector
1918:
1486:
796:
739:
1476:{\displaystyle \mathbf {j} _{{\rm {m}},\,i}=\rho \left(\mathbf {u} _{i}-\langle \mathbf {u} \rangle \right),}
671:
of the unit vectors. That is, the component of mass flux passing through the surface (i.e. normal to it) is
2345:
2287:
1908:{\displaystyle \mathbf {j} _{{\rm {n}},\,i}=c\left(\mathbf {u} _{i}-\langle \mathbf {u} \rangle \right),}
569:
486:
2350:
440:{\displaystyle m=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} _{m}\cdot \mathbf {\hat {n}} \,dA\,dt.}
74:
to indicate mass is the flowing quantity. Mass flux can also refer to an alternate form of flux in
2322:
Vectors, Tensors, and the basic
Equations of Fluid Mechanics, R. Aris, Dover Publications, 1989,
2230:
301:
2149:{\displaystyle \nabla \cdot \mathbf {j} _{\rm {m}}+{\frac {\partial \rho }{\partial t}}=0,}
8:
2084:
722:
461:
110:
263:{\displaystyle I_{m}=\lim _{\Delta t\to 0}{\frac {\Delta m}{\Delta t}}={\frac {dm}{dt}}}
2242:
2306:
Fluid
Mechanics, M. Potter, D.C. Wiggart, Schuam's outlines, McGraw Hill (USA), 2008,
2323:
2307:
475:
is a combination of the magnitude of the area through which the mass passes through,
465:
297:
1029:{\displaystyle j_{m}={\frac {\Delta m}{A\Delta t}}={\frac {\rho V}{\pi r^{2}t}}.}
101:
uses the definition of mass flux as the equation in the mass flow rate article.
2257:
1266:
94:
83:
79:
63:
35:
1706:
The molar flux is the number of moles per unit time per unit area, generally:
684:. While the component of mass flux passing tangential to the area is given by
2339:
2252:
2080:
1693:
75:
2218:{\displaystyle \nabla \cdot \mathbf {j} _{\rm {n}}=-\nabla \cdot D\nabla n}
709:
component of mass flux passing normal to the area is the cosine component.
654:{\displaystyle \mathbf {j} _{m}\cdot \mathbf {\hat {n}} =j_{m}\cos \theta }
457:
325:, gives the total amount of mass flowing through the surface in that time (
668:
480:
472:
464:, the real surface is the (generally curved) surface area of the filter,
24:
1381:{\displaystyle \mathbf {j} _{{\rm {m}},\,i}=\rho _{i}\mathbf {u} _{i}.}
2164:
729:
is the cross-sectional area of the pipe. Suppose the pipe has radius
39:
1815:{\displaystyle \mathbf {j} _{{\rm {n}},\,i}=c_{i}\mathbf {u} _{i}}
939:(since initial volume passing through the area was zero, final is
1944:
1512:
1302:
87:
20:
16:
Vector quantity describing mass flow rate through a given area
1291:
718:
1680:
The average is taken over the velocities of the components.
2247:
450:
30:
1206:
Using the vector definition, mass flux is also equal to:
566:
passes through the area at an angle θ to the area normal
67:
1241:{\displaystyle \mathbf {j} _{\rm {m}}=\rho \mathbf {u} }
171:{\displaystyle j_{m}=\lim _{A\to 0}{\frac {I_{m}}{A}},}
2173:
2093:
1956:
1921:
1836:
1763:
1744:{\displaystyle \mathbf {j} _{\rm {n}}=c\mathbf {u} .}
1712:
1524:
1489:
1404:
1326:
1212:
1044:
953:
812:
742:
601:
572:
518:
489:
347:
184:
118:
2217:
2148:
2064:
1935:
1907:
1814:
1757:is (number of moles per unit time per unit area):
1743:
1629:
1503:
1475:
1380:
1240:
1185:
1028:
931:
767:
653:
587:
544:
504:
439:
262:
170:
623:
579:
545:{\displaystyle \mathbf {A} =A\mathbf {\hat {n}} }
536:
496:
414:
307:, followed by an integral over the time duration
2337:
1950:of all the components in the mixture, given by:
199:
133:
1518:of all the components in the mixture, given by
1279:Sometimes this equation may be used to define
456:For example, for substances passing through a
1965:
1957:
1936:{\displaystyle \langle \mathbf {u} \rangle }
1930:
1922:
1894:
1886:
1533:
1525:
1504:{\displaystyle \langle \mathbf {u} \rangle }
1498:
1490:
1462:
1454:
109:Mathematically, mass flux is defined as the
1292:Mass and molar fluxes for composite fluids
1193:which is approximately 596.8 kg s m.
705:the area in the tangential direction. The
282:is the area through which the mass flows.
2053:
1854:
1781:
1621:
1422:
1344:
1301:In the case fluid is not pure, i.e. is a
427:
420:
2079:Mass flux appears in some equations in
1316:Using mass, the mass flux of component
1201:
1196:
2338:
2268:Defining equation (physical chemistry)
1644:= mass density of the entire mixture,
588:{\displaystyle \mathbf {\hat {n}} }
505:{\displaystyle \mathbf {\hat {n}} }
42:are kg m s. The common symbols are
13:
2209:
2200:
2188:
2174:
2128:
2120:
2108:
2094:
2045:
1846:
1773:
1721:
1613:
1414:
1336:
1221:
981:
970:
833:
817:
270:is the mass current (flow of mass
228:
220:
203:
14:
2362:
2263:Wave mass flux and wave momentum
2182:
2102:
2038:
2003:
1961:
1926:
1890:
1873:
1839:
1802:
1766:
1734:
1715:
1606:
1571:
1529:
1494:
1458:
1441:
1407:
1365:
1329:
1234:
1215:
1038:Substituting the numbers gives:
620:
604:
576:
533:
520:
493:
411:
395:
1753:So the molar flux of component
1683:
99:Fluid Mechanics, Schaum's et al
2316:
2300:
2280:
1296:
902:
876:
209:
140:
1:
2273:
104:
1655:= mass density of component
768:{\displaystyle A=\pi r^{2}.}
70:), sometimes with subscript
7:
2236:
943:, so corresponding mass is
775:To calculate the mass flux
717:Consider a pipe of flowing
701:mass flux actually passing
10:
2367:
712:
285:For mass flux as a vector
2074:
1692:by the "molar density",
1672:= velocity of component
947:), so the mass flux is
791:passes through in time
86:that includes the mass
2288:"Thesaurus: Mass flux"
2219:
2150:
2066:
1937:
1909:
1824:barycentric molar flux
1816:
1745:
1688:If we replace density
1631:
1505:
1477:
1382:
1242:
1187:
1030:
933:
769:
655:
589:
546:
506:
441:
264:
172:
38:per unit of area. Its
2231:diffusion coefficient
2220:
2159:Molar flux occurs in
2151:
2067:
1938:
1910:
1817:
1746:
1632:
1506:
1478:
1392:barycentric mass flux
1383:
1243:
1188:
1031:
934:
770:
656:
590:
547:
507:
442:
265:
173:
66:lowercase or capital
2171:
2091:
2083:, in particular the
1954:
1919:
1834:
1761:
1710:
1522:
1487:
1402:
1324:
1210:
1202:Alternative equation
1197:Equations for fluids
1042:
951:
810:
795:= 2 s. Assuming the
789:= 1.5 L = 1.5 × 10 m
740:
736:. The area is then
599:
570:
516:
487:
483:normal to the area,
345:
182:
116:
2346:Physical quantities
2085:continuity equation
723:standard conditions
382:
2243:Mass-flux fraction
2215:
2146:
2062:
2035:
1990:
1933:
1905:
1812:
1741:
1627:
1603:
1558:
1501:
1473:
1378:
1238:
1183:
1026:
929:
927:
765:
651:
585:
542:
512:. The relation is
502:
437:
354:
260:
216:
168:
147:
78:that includes the
2312:978-0-07-148781-8
2135:
2026:
2024:
1981:
1979:
1943:this time is the
1594:
1592:
1549:
1547:
1165:
1147:
1021:
988:
734:= 2 cm = 2 × 10 m
626:
582:
555:If the mass flux
539:
499:
417:
258:
235:
198:
163:
132:
36:rate of mass flow
2358:
2330:
2320:
2314:
2304:
2298:
2297:
2295:
2294:
2284:
2228:
2224:
2222:
2221:
2216:
2193:
2192:
2191:
2185:
2161:Fick's first law
2155:
2153:
2152:
2147:
2136:
2134:
2126:
2118:
2113:
2112:
2111:
2105:
2071:
2069:
2068:
2063:
2058:
2057:
2049:
2048:
2041:
2034:
2025:
2017:
2012:
2011:
2006:
2000:
1999:
1989:
1980:
1972:
1964:
1942:
1940:
1939:
1934:
1929:
1914:
1912:
1911:
1906:
1901:
1897:
1893:
1882:
1881:
1876:
1859:
1858:
1850:
1849:
1842:
1821:
1819:
1818:
1813:
1811:
1810:
1805:
1799:
1798:
1786:
1785:
1777:
1776:
1769:
1750:
1748:
1747:
1742:
1737:
1726:
1725:
1724:
1718:
1698:
1691:
1671:
1654:
1643:
1636:
1634:
1633:
1628:
1626:
1625:
1617:
1616:
1609:
1602:
1593:
1585:
1580:
1579:
1574:
1568:
1567:
1557:
1548:
1540:
1532:
1510:
1508:
1507:
1502:
1497:
1482:
1480:
1479:
1474:
1469:
1465:
1461:
1450:
1449:
1444:
1427:
1426:
1418:
1417:
1410:
1387:
1385:
1384:
1379:
1374:
1373:
1368:
1362:
1361:
1349:
1348:
1340:
1339:
1332:
1287:
1274:
1264:
1256:
1247:
1245:
1244:
1239:
1237:
1226:
1225:
1224:
1218:
1192:
1190:
1189:
1184:
1179:
1178:
1166:
1164:
1153:
1148:
1146:
1139:
1138:
1133:
1129:
1128:
1127:
1096:
1095:
1091:
1090:
1089:
1059:
1054:
1053:
1035:
1033:
1032:
1027:
1022:
1020:
1016:
1015:
1002:
994:
989:
987:
976:
968:
963:
962:
946:
942:
938:
936:
935:
930:
928:
901:
900:
888:
887:
865:
864:
852:
851:
805:
797:density of water
790:
783:
774:
772:
771:
766:
761:
760:
735:
696:
683:
666:
660:
658:
657:
652:
641:
640:
628:
627:
619:
613:
612:
607:
594:
592:
591:
586:
584:
583:
575:
565:
551:
549:
548:
543:
541:
540:
532:
523:
511:
509:
508:
503:
501:
500:
492:
446:
444:
443:
438:
419:
418:
410:
404:
403:
398:
392:
391:
381:
380:
379:
369:
368:
367:
340:
324:
315:
298:surface integral
295:
281:
277:
273:
269:
267:
266:
261:
259:
257:
249:
241:
236:
234:
226:
218:
215:
194:
193:
177:
175:
174:
169:
164:
159:
158:
149:
146:
128:
127:
2366:
2365:
2361:
2360:
2359:
2357:
2356:
2355:
2351:Vector calculus
2336:
2335:
2334:
2333:
2321:
2317:
2305:
2301:
2292:
2290:
2286:
2285:
2281:
2276:
2239:
2226:
2187:
2186:
2181:
2180:
2172:
2169:
2168:
2127:
2119:
2117:
2107:
2106:
2101:
2100:
2092:
2089:
2088:
2077:
2044:
2043:
2042:
2037:
2036:
2030:
2016:
2007:
2002:
2001:
1995:
1991:
1985:
1971:
1960:
1955:
1952:
1951:
1925:
1920:
1917:
1916:
1889:
1877:
1872:
1871:
1870:
1866:
1845:
1844:
1843:
1838:
1837:
1835:
1832:
1831:
1806:
1801:
1800:
1794:
1790:
1772:
1771:
1770:
1765:
1764:
1762:
1759:
1758:
1733:
1720:
1719:
1714:
1713:
1711:
1708:
1707:
1696:
1689:
1686:
1670:
1662:
1652:
1647:
1641:
1612:
1611:
1610:
1605:
1604:
1598:
1584:
1575:
1570:
1569:
1563:
1559:
1553:
1539:
1528:
1523:
1520:
1519:
1493:
1488:
1485:
1484:
1457:
1445:
1440:
1439:
1438:
1434:
1413:
1412:
1411:
1406:
1405:
1403:
1400:
1399:
1369:
1364:
1363:
1357:
1353:
1335:
1334:
1333:
1328:
1327:
1325:
1322:
1321:
1299:
1294:
1286:
1280:
1270:
1260:
1257:= mass density,
1254:
1233:
1220:
1219:
1214:
1213:
1211:
1208:
1207:
1204:
1199:
1174:
1170:
1157:
1152:
1134:
1120:
1116:
1109:
1105:
1104:
1097:
1082:
1078:
1071:
1067:
1060:
1058:
1049:
1045:
1043:
1040:
1039:
1011:
1007:
1003:
995:
993:
977:
969:
967:
958:
954:
952:
949:
948:
944:
940:
926:
925:
912:
906:
905:
896:
892:
883:
879:
866:
860:
856:
847:
843:
840:
839:
823:
813:
811:
808:
807:
800:
785:
781:
776:
756:
752:
741:
738:
737:
730:
715:
690:
685:
677:
672:
662:
636:
632:
618:
617:
608:
603:
602:
600:
597:
596:
574:
573:
571:
568:
567:
564:
556:
531:
530:
519:
517:
514:
513:
491:
490:
488:
485:
484:
466:macroscopically
409:
408:
399:
394:
393:
387:
383:
375:
371:
370:
363:
359:
358:
346:
343:
342:
339:
332:
326:
323:
317:
314:
308:
294:
286:
279:
275:
271:
250:
242:
240:
227:
219:
217:
202:
189:
185:
183:
180:
179:
154:
150:
148:
136:
123:
119:
117:
114:
113:
107:
97:. For example,
17:
12:
11:
5:
2364:
2354:
2353:
2348:
2332:
2331:
2315:
2299:
2278:
2277:
2275:
2272:
2271:
2270:
2265:
2260:
2255:
2250:
2245:
2238:
2235:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2190:
2184:
2179:
2176:
2145:
2142:
2139:
2133:
2130:
2125:
2122:
2116:
2110:
2104:
2099:
2096:
2076:
2073:
2061:
2056:
2052:
2047:
2040:
2033:
2029:
2023:
2020:
2015:
2010:
2005:
1998:
1994:
1988:
1984:
1978:
1975:
1970:
1967:
1963:
1959:
1948:molar velocity
1932:
1928:
1924:
1904:
1900:
1896:
1892:
1888:
1885:
1880:
1875:
1869:
1865:
1862:
1857:
1853:
1848:
1841:
1809:
1804:
1797:
1793:
1789:
1784:
1780:
1775:
1768:
1740:
1736:
1732:
1729:
1723:
1717:
1699:, we have the
1685:
1682:
1678:
1677:
1666:
1660:
1650:
1645:
1624:
1620:
1615:
1608:
1601:
1597:
1591:
1588:
1583:
1578:
1573:
1566:
1562:
1556:
1552:
1546:
1543:
1538:
1535:
1531:
1527:
1500:
1496:
1492:
1472:
1468:
1464:
1460:
1456:
1453:
1448:
1443:
1437:
1433:
1430:
1425:
1421:
1416:
1409:
1377:
1372:
1367:
1360:
1356:
1352:
1347:
1343:
1338:
1331:
1298:
1295:
1293:
1290:
1284:
1277:
1276:
1267:velocity field
1258:
1236:
1232:
1229:
1223:
1217:
1203:
1200:
1198:
1195:
1182:
1177:
1173:
1169:
1163:
1160:
1156:
1151:
1145:
1142:
1137:
1132:
1126:
1123:
1119:
1115:
1112:
1108:
1103:
1100:
1094:
1088:
1085:
1081:
1077:
1074:
1070:
1066:
1063:
1057:
1052:
1048:
1025:
1019:
1014:
1010:
1006:
1001:
998:
992:
986:
983:
980:
975:
972:
966:
961:
957:
924:
921:
918:
915:
913:
911:
908:
907:
904:
899:
895:
891:
886:
882:
878:
875:
872:
869:
867:
863:
859:
855:
850:
846:
842:
841:
838:
835:
832:
829:
826:
824:
822:
819:
816:
815:
779:
764:
759:
755:
751:
748:
745:
714:
711:
688:
675:
650:
647:
644:
639:
635:
631:
625:
622:
616:
611:
606:
581:
578:
560:
538:
535:
529:
526:
522:
498:
495:
436:
433:
430:
426:
423:
416:
413:
407:
402:
397:
390:
386:
378:
374:
366:
362:
357:
353:
350:
337:
330:
321:
312:
290:
274:per unit time
256:
253:
248:
245:
239:
233:
230:
225:
222:
214:
211:
208:
205:
201:
197:
192:
188:
167:
162:
157:
153:
145:
142:
139:
135:
131:
126:
122:
106:
103:
95:mass flow rate
80:molecular mass
15:
9:
6:
4:
3:
2:
2363:
2352:
2349:
2347:
2344:
2343:
2341:
2329:
2328:0-486-66110-5
2325:
2319:
2313:
2309:
2303:
2289:
2283:
2279:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2240:
2234:
2232:
2212:
2206:
2203:
2197:
2194:
2177:
2166:
2162:
2157:
2143:
2140:
2137:
2131:
2123:
2114:
2097:
2086:
2082:
2081:hydrodynamics
2072:
2059:
2054:
2050:
2031:
2027:
2021:
2018:
2013:
2008:
1996:
1992:
1986:
1982:
1976:
1973:
1968:
1949:
1946:
1902:
1898:
1883:
1878:
1867:
1863:
1860:
1855:
1851:
1829:
1826:of component
1825:
1807:
1795:
1791:
1787:
1782:
1778:
1756:
1751:
1738:
1730:
1727:
1704:
1702:
1695:
1694:concentration
1681:
1675:
1669:
1665:
1661:
1658:
1653:
1646:
1640:
1639:
1638:
1622:
1618:
1599:
1595:
1589:
1586:
1581:
1576:
1564:
1560:
1554:
1550:
1544:
1541:
1536:
1517:
1516:mass velocity
1514:
1470:
1466:
1451:
1446:
1435:
1431:
1428:
1423:
1419:
1397:
1394:of component
1393:
1388:
1375:
1370:
1358:
1354:
1350:
1345:
1341:
1319:
1314:
1312:
1306:
1304:
1289:
1288:as a vector.
1283:
1273:
1268:
1263:
1259:
1253:
1252:
1251:
1248:
1230:
1227:
1194:
1180:
1175:
1171:
1167:
1161:
1158:
1154:
1149:
1143:
1140:
1135:
1130:
1124:
1121:
1117:
1113:
1110:
1106:
1101:
1098:
1092:
1086:
1083:
1079:
1075:
1072:
1068:
1064:
1061:
1055:
1050:
1046:
1036:
1023:
1017:
1012:
1008:
1004:
999:
996:
990:
984:
978:
973:
964:
959:
955:
922:
919:
916:
914:
909:
897:
893:
889:
884:
880:
873:
870:
868:
861:
857:
853:
848:
844:
836:
830:
827:
825:
820:
803:
798:
794:
788:
782:
762:
757:
753:
749:
746:
743:
733:
728:
724:
720:
710:
708:
704:
700:
695:
691:
682:
678:
670:
665:
648:
645:
642:
637:
633:
629:
614:
609:
563:
559:
553:
527:
524:
482:
478:
474:
469:
467:
463:
459:
454:
452:
447:
434:
431:
428:
424:
421:
405:
400:
388:
384:
376:
372:
364:
360:
355:
351:
348:
336:
329:
320:
311:
306:
303:
300:of it over a
299:
293:
289:
283:
254:
251:
246:
243:
237:
231:
223:
212:
206:
195:
190:
186:
165:
160:
155:
151:
143:
137:
129:
124:
120:
112:
102:
100:
96:
91:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
32:
26:
22:
2318:
2302:
2291:. Retrieved
2282:
2158:
2078:
1947:
1827:
1823:
1754:
1752:
1705:
1700:
1687:
1684:Molar fluxes
1679:
1673:
1667:
1663:
1656:
1648:
1515:
1395:
1391:
1389:
1317:
1315:
1310:
1307:
1300:
1281:
1278:
1271:
1261:
1249:
1205:
1037:
801:
792:
786:
777:
731:
726:
716:
706:
702:
698:
693:
686:
680:
673:
663:
561:
557:
554:
476:
470:
455:
448:
334:
327:
318:
309:
304:
291:
287:
284:
108:
98:
92:
71:
59:
55:
51:
47:
43:
28:
18:
2258:Darcy's law
1703:analogues.
1297:Mass fluxes
806:, we have:
804:= 1000 kg m
725:. The area
697:, there is
669:dot product
481:unit vector
473:vector area
84:Darcy's law
25:engineering
2340:Categories
2293:2008-12-24
2274:References
2253:Fick's law
1701:molar flux
1311:molar flux
105:Definition
76:Fick's law
2210:∇
2204:⋅
2201:∇
2198:−
2178:⋅
2175:∇
2165:diffusion
2129:∂
2124:ρ
2121:∂
2098:⋅
2095:∇
2028:∑
1983:∑
1966:⟩
1958:⟨
1931:⟩
1923:⟨
1895:⟩
1887:⟨
1884:−
1596:∑
1590:ρ
1561:ρ
1551:∑
1545:ρ
1534:⟩
1526:⟨
1499:⟩
1491:⟨
1463:⟩
1455:⟨
1452:−
1432:ρ
1355:ρ
1231:ρ
1168:×
1162:π
1141:×
1122:−
1114:×
1102:×
1099:π
1084:−
1076:×
1065:×
1005:π
997:ρ
982:Δ
971:Δ
920:ρ
890:−
874:ρ
854:−
834:Δ
831:ρ
818:Δ
750:π
649:θ
646:
624:^
615:⋅
580:^
537:^
497:^
415:^
406:⋅
385:∬
356:∫
229:Δ
221:Δ
210:→
204:Δ
141:→
2237:See also
1822:and the
479:, and a
462:membrane
82:, or in
62:, or Φ (
40:SI units
2229:is the
1945:average
1513:average
1511:is the
1303:mixture
1250:where:
713:Example
703:through
667:is the
595:, then
302:surface
88:density
34:is the
21:physics
2326:
2310:
2225:where
1915:where
1637:where
1483:where
661:where
458:filter
296:, the
278:) and
178:where
2075:Usage
719:water
460:or a
111:limit
64:Greek
29:mass
2324:ISBN
2308:ISBN
2248:Flux
1390:The
1062:1000
707:only
692:sin
679:cos
471:The
451:area
449:The
31:flux
23:and
2163:of
1830:is
1398:is
1320:is
1073:1.5
799:is
643:cos
341:):
316:to
200:lim
134:lim
68:Phi
19:In
2342::
2233:.
2167::
2087::
1313:.
1275:).
1265:=
1172:10
1159:16
1118:10
1080:10
699:no
552:.
333:−
90:.
58:,
54:,
50:,
46:,
27:,
2296:.
2227:D
2213:n
2207:D
2195:=
2189:n
2183:j
2144:,
2141:0
2138:=
2132:t
2115:+
2109:m
2103:j
2060:.
2055:i
2051:,
2046:n
2039:j
2032:i
2022:c
2019:1
2014:=
2009:i
2004:u
1997:i
1993:c
1987:i
1977:n
1974:1
1969:=
1962:u
1927:u
1903:,
1899:)
1891:u
1879:i
1874:u
1868:(
1864:c
1861:=
1856:i
1852:,
1847:n
1840:j
1828:i
1808:i
1803:u
1796:i
1792:c
1788:=
1783:i
1779:,
1774:n
1767:j
1755:i
1739:.
1735:u
1731:c
1728:=
1722:n
1716:j
1697:c
1690:ρ
1676:.
1674:i
1668:i
1664:u
1659:,
1657:i
1651:i
1649:ρ
1642:ρ
1623:i
1619:,
1614:m
1607:j
1600:i
1587:1
1582:=
1577:i
1572:u
1565:i
1555:i
1542:1
1537:=
1530:u
1495:u
1471:,
1467:)
1459:u
1447:i
1442:u
1436:(
1429:=
1424:i
1420:,
1415:m
1408:j
1396:i
1376:.
1371:i
1366:u
1359:i
1351:=
1346:i
1342:,
1337:m
1330:j
1318:i
1285:m
1282:j
1272:u
1262:u
1255:ρ
1235:u
1228:=
1222:m
1216:j
1181:,
1176:4
1155:3
1150:=
1144:2
1136:2
1131:)
1125:2
1111:2
1107:(
1093:)
1087:3
1069:(
1056:=
1051:m
1047:j
1024:.
1018:t
1013:2
1009:r
1000:V
991:=
985:t
979:A
974:m
965:=
960:m
956:j
945:m
941:V
923:V
917:=
910:m
903:)
898:1
894:V
885:2
881:V
877:(
871:=
862:1
858:m
849:2
845:m
837:V
828:=
821:m
802:ρ
793:t
787:V
780:m
778:j
763:.
758:2
754:r
747:=
744:A
732:r
727:A
694:θ
689:m
687:j
681:θ
676:m
674:j
664:·
638:m
634:j
630:=
621:n
610:m
605:j
577:n
562:m
558:j
534:n
528:A
525:=
521:A
494:n
477:A
435:.
432:t
429:d
425:A
422:d
412:n
401:m
396:j
389:S
377:2
373:t
365:1
361:t
352:=
349:m
338:1
335:t
331:2
328:t
322:2
319:t
313:1
310:t
305:S
292:m
288:j
280:A
276:t
272:m
255:t
252:d
247:m
244:d
238:=
232:t
224:m
213:0
207:t
196:=
191:m
187:I
166:,
161:A
156:m
152:I
144:0
138:A
130:=
125:m
121:j
72:m
60:φ
56:Q
52:q
48:J
44:j
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