752:
experimentally to validate whether it would indeed work or not. This was successful and since then, many variations and applications of stream power have surfaced. The lack of fixed guidelines on how to define stream power in this early stage lead to many authors publishing work under the name "stream power" while not always measuring the entity in the same way; this led to partially failed efforts to establish naming conventions for the various forms of the formula by Rhoads two decades later in 1986. Today stream power is still used and new ways of applying it are still being discovered and researched, with a large integration into modern numerical models utilizing
2008:
26:
2043:
Stream power has also been used as a criterion to determine whether a river is in a state of reshaping itself or whether it is stable. A value of unit stream power between 30 and 35 W m in which this transition occurs has been found by multiple studies. Another technique gaining popularity is using a
722:
in the 1960s, is the amount of energy the water in a river or stream is exerting on the sides and bottom of the river. Stream power is the result of multiplying the density of the water, the acceleration of the water due to gravity, the volume of water flowing through the river, and the slope of that
1692:
is the width of the channel. Normalizing the stream power by the width of the river allows for a better comparison between rivers of various widths. This also provides a better estimation of the sediment carrying capacity of the river as wide rivers with high stream power are exerting less force per
2134:
Stream power can be used as an indicator of potential damages to bridges as a result of large rain events and how strong bridges should be designed in order to avoid damage during these events. Stream power can also be used to guide culvert and bridge design in order to maintain healthy stream
751:
Although many authors had suggested the use of power formulas in sediment transport in the decades preceding
Bagnold's work, and in fact Bagnold himself suggested it a decade before putting it into practice in one of his other works, it wasn't until 1966 that R. A. Bagnold tested this theory
1378:
is the cross-sectional area, which can often be reasonably approximated as a rectangle with the characteristic width and depth. This absorbs velocity, width, and depth. We define stream power per unit channel length, so that term goes to 1, and the derivation is
723:
water. There are many forms of the stream power formula with varying utilities, such as comparing rivers of various widths or quantifying the energy required to move sediment of a certain size. Stream power is closely related to other criteria such as
1786:
of a river, which is a measure to determine the largest grain size that will be moved by a river. In rivers with large sediment sizes the relationship between critical unit stream power and sediment diameter displaced can be reduced
846:
1017:
764:
It can be derived by the fact that if the water is not accelerating and the river cross-section stays constant (generally good assumptions for an averaged reach of a stream over a modest distance), all of the
2124:
2126:) to identify patterns such as sudden jumps or drops in stream power, these features can help identify locations where the local terrain controls the flow or widens out as well as areas prone to erosion.
1206:
1096:
1429:
1154:
1617:
2019:
and river incision. Unit stream power is often used for this, because simple models use and evolve a 1-dimensional downstream profile of the river channel. It is also used with relation to
1890:
1836:
1752:
1679:
2035:
By plotting stream power along the length of a river course as a second-order exponential curve, you are able to identify areas where flood plains may form and why they will form there.
153:
1903:
is another variable used in erosion and sediment transport models representing the force applied on a surface by a perpendicular force, and can be calculated using the following formula
887:
1989:
1484:
1939:
1693:
surface area than a narrow river with the same stream power, as they are losing the same amount of energy but in the narrow river it is concentrated into a smaller area.
1247:
934:
1353:
914:
1376:
1314:
1290:
1270:
1040:
1766:
2452:"The Use of Stream Power as an Indicator of Channel Sensitivity to Erosion and Deposition Processes: SP AS AN INDICATOR OF EROSION AND DEPOSITION"
782:
1529:
Total stream power often refers simply to stream power, but some authors use it as the rate of energy dissipation against the bed and banks of a
942:
702:
1159:
Remembering that power is energy per time and using the equivalence between work against the bed and loss in potential energy, we can write:
1761:
is the critical shear stress of the grain size that will be moved which can be found in the literature or experimentally determined while v
2047:
851:
where water mass and gravitational acceleration are constant. We can use the channel slope and the stream velocity as a stand-in for
1701:
Critical unit stream power is the amount of stream power needed to displace a grain of a specific size, it is given by the equation:
2756:"Where do floodplains begin? The role of total stream power and longitudinal profile form on floodplain initiation processes"
1165:
1956:
is acceleration due to gravity (9.8 m/s). Shear stress can be used to compute the unit stream power using the formula
1045:
1385:
1104:
2536:"Stream Power Application for Bridge-Damage Probability Mapping Based on Empirical Evidence from Tropical Storm Irene"
1211:
Finally, we know that mass is equal to density times volume. From this, we can rewrite the mass on the right hand side
1539:
695:
2135:
morphology in which fish are able to continuing traversing the water course and no erosion processes are initiated.
1847:
2011:
An example of a map displaying a stream power index (SPI) alongside an index displaying how wet the ground is (TWI)
1793:
1707:
1640:
773:. Therefore, the potential energy drop is equal to the work done to the bed and banks, which is the stream power.
109:
2492:
Gartner, John D.; Dade, William B.; Renshaw, Carl E.; Magilligan, Francis J.; Buraas, Eirik M. (November 2015).
668:
2755:
2710:
769:
lost as the water flows downstream must be used up in friction or work against the bed: none can be added to
369:
206:
854:
688:
409:
295:
2590:
364:
273:
156:
2493:
2335:
Philosophical
Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
1502:
81:
2695:"Paleohydraulic reconstruction of flash-flood peaks from boulder deposits in the Colorado Front Range"
2655:
2613:
1962:
1454:
2044:
gradient of stream power by comparing the unit stream power upstream to the local unit stream power (
2016:
280:
2535:
1909:
575:
570:
359:
352:
185:
2825:
2794:
2020:
638:
633:
302:
1217:
2921:
2916:
190:
2574:
2007:
1510:
919:
613:
231:
2551:
2174:
1326:
892:
451:
268:
248:
236:
180:
8:
753:
653:
501:
394:
100:
2892:
2471:
2451:
2358:
2169:
2159:
2024:
1634:
Unit stream power is stream power per unit channel width, and is given by the equation:
1361:
1299:
1275:
1255:
1025:
673:
307:
263:
258:
2736:
2405:
2386:
889:: the water will lose elevation at a rate given by the downward component of velocity
2896:
2884:
2845:
2775:
2714:
2675:
2633:
2594:
2555:
2513:
2475:
2425:
2421:
2362:
2350:
2311:
2272:
2233:
1783:
776:
We know that change in potential energy over change in time is given by the equation:
743:, who use it in the planning and construction of roads, bridges, dams, and culverts.
740:
724:
290:
241:
2694:
2534:
Anderson, Ian; Rizzo, Donna M.; Huston, Dryver R.; Dewoolkar, Mandar M. (May 2017).
2494:"Gradients in stream power influence lateral and downstream sediment flux in floods"
2876:
2837:
2810:
2806:
2767:
2706:
2671:
2667:
2629:
2625:
2586:
2547:
2505:
2463:
2417:
2342:
2303:
2264:
2228:
2223:
2213:
1626:
is the stream power, per unit downstream length and L is the length of the stream.
766:
628:
603:
516:
491:
486:
441:
2864:
618:
542:
506:
456:
387:
376:
321:
223:
25:
770:
623:
481:
446:
347:
253:
1444:
Stream power is the rate of energy dissipation against the bed and banks of a
2910:
2888:
2849:
2779:
2718:
2679:
2637:
2598:
2559:
2517:
2429:
2354:
2315:
2291:
2276:
2252:
2237:
2149:
736:
719:
663:
496:
2869:
Transportation
Research Record: Journal of the Transportation Research Board
2330:
2307:
2268:
841:{\displaystyle {\frac {\Delta PE}{\Delta t}}=mg{\frac {\Delta z}{\Delta t}}}
2841:
2346:
2164:
1900:
1012:{\displaystyle {\frac {\Delta z}{\Delta t}}=u_{z}=u\sin(\alpha )\approx uS}
728:
648:
643:
608:
340:
57:
2179:
658:
561:
1841:
While in intermediate-sized rivers the relationship was found to follow:
580:
476:
2771:
2509:
2467:
2144:
732:
552:
547:
381:
2880:
2737:"Stream Power: Origins, Geomorphic Applications, and GIS Procedures"
2387:"Stream Power: Origins, Geomorphic Applications, and GIS Procedures"
2206:"An approach to the sediment transport problem from general physics"
1042:
is the downstream flow velocity. It is noted that for small angles,
2826:"The distribution and management of channelized streams in Denmark"
2793:
Orr, H.G.; Large, A.R.G.; Newson, M.D.; Walsh, C.L. (August 2008).
2218:
2205:
531:
436:
416:
402:
1448:
or stream per unit downstream length. It is given by the equation:
2865:"Consideration of Stream Morphology in Culvert and Bridge Design"
2154:
2119:{\displaystyle \Delta \omega =\omega _{local}-\omega _{upstream}}
285:
46:
2654:
Petit, F.; Gob, F.; Houbrechts, G.; Assani, A. A. (2005-07-01).
2612:
Petit, F.; Gob, F.; Houbrechts, G.; Assani, A. A. (2005-07-01).
1533:
or stream per entire stream length. It is given by the equation:
1406:
426:
1411:
1530:
1518:
1445:
330:
2533:
2491:
916:. For a channel slope (as measured from the horizontal) of
2795:"A predictive typology for characterising hydromorphology"
2711:
10.1130/0016-7606(1983)94<986:PROFPF>2.0.CO;2
466:
2653:
2611:
2292:"Energy-balance in stream-flows carrying suspended load"
2863:
Kosicki, Andrzej J.; Davis, Stanley R. (January 2001).
2656:"Critical specific stream power in gravel-bed rivers"
2614:"Critical specific stream power in gravel-bed rivers"
2050:
1965:
1912:
1850:
1796:
1710:
1643:
1542:
1457:
1388:
1364:
1329:
1302:
1278:
1258:
1220:
1201:{\displaystyle \Omega ={\frac {\Delta PE}{\Delta t}}}
1168:
1107:
1048:
1028:
945:
922:
895:
857:
785:
112:
1994:
Where V is the velocity of the water in the stream.
1091:{\displaystyle \sin(\alpha )\approx \tan(\alpha )=S}
2792:
2030:
1782:Critical stream power can be used to determine the
1772:
2753:
2118:
1983:
1948:is the shear stress, S is the slope of the water,
1933:
1884:
1830:
1746:
1673:
1629:
1611:
1478:
1424:{\displaystyle \Omega =\rho gQ{\cancelto {1}{L}}S}
1423:
1370:
1347:
1308:
1284:
1264:
1241:
1200:
1149:{\displaystyle {\frac {\Delta PE}{\Delta t}}=mguS}
1148:
1090:
1034:
1011:
928:
908:
881:
840:
739:tackling sediment transport issues as well as for
147:
2754:Jain, V.; Fryirs, K.; Brierley, G. (2008-01-01).
2908:
2253:"Equilibrium-conditions in debris-laden streams"
1612:{\displaystyle Total\ stream\ power=\Omega \ L}
1885:{\displaystyle \omega _{0}=0.130D_{i}^{1.438}}
1831:{\displaystyle \omega _{0}=0.030D_{i}^{1.69}}
1777:
1747:{\displaystyle \omega _{0}=\tau _{0}\nu _{0}}
1696:
1674:{\displaystyle \omega ={\frac {\rho gQS}{b}}}
731:. Stream power is a valuable measurement for
696:
2862:
2575:"Critical Shear Stress of Natural Sediments"
2129:
1098:. Rewriting the first equation, we now have:
2830:Regulated Rivers: Research & Management
2449:
2331:"The flow of cohesionless grains in fluids"
148:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
1952:is the density of water (1000 kg/m),
1497:is the density of water (1000 kg/m),
1320:eight). We use the definition of discharge
703:
689:
24:
2450:Bizzi, S.; Lerner, D. N. (January 2015).
2227:
2217:
2038:
2296:Transactions, American Geophysical Union
2257:Transactions, American Geophysical Union
2006:
2823:
2734:
2591:10.1061/(asce)0733-9429(1993)119:4(491)
2572:
2384:
2328:
2203:
1439:
2909:
2760:Geological Society of America Bulletin
2649:
2647:
2403:
2002:
2730:
2728:
2692:
2445:
2443:
2441:
2439:
2380:
2378:
2376:
2374:
2372:
2289:
2250:
1524:
882:{\displaystyle {\Delta z}/{\Delta t}}
2529:
2527:
2487:
2485:
2199:
2197:
2195:
2015:Stream power is used extensively in
2644:
2605:
13:
2725:
2552:10.1061/(ASCE)BE.1943-5592.0001022
2436:
2369:
2051:
2023:, and in some cases is applied to
1600:
1458:
1389:
1189:
1178:
1169:
1122:
1111:
957:
949:
872:
859:
829:
821:
800:
789:
14:
2933:
2524:
2482:
2192:
2579:Journal of Hydraulic Engineering
2573:Wilcock, Peter R. (April 1993).
2422:10.1111/j.0033-0124.1987.00189.x
2329:Bagnold, Ralph A. (1956-12-18).
2031:Predicting flood plain formation
1984:{\displaystyle \omega =\tau \ V}
1773:Relationships to other variables
1479:{\displaystyle \Omega =\rho gQS}
1434:
2856:
2817:
2786:
2747:
2686:
2566:
2456:River Research and Applications
1997:
1895:
1630:Unit (or Specific) Stream power
2811:10.1016/j.geomorph.2007.10.022
2672:10.1016/j.geomorph.2004.12.004
2630:10.1016/j.geomorph.2004.12.004
2397:
2322:
2283:
2244:
1934:{\displaystyle \tau =hS\rho g}
1688:is the unit stream power, and
1079:
1073:
1061:
1055:
997:
991:
1:
2693:COSTA, JOHN E. (1983-08-01).
2540:Journal of Bridge Engineering
2404:Rhoads, Bruce L. (May 1987).
2185:
2017:models of landscape evolution
759:
2735:Gartner, John (2016-01-01).
2385:Gartner, John (2016-01-01).
7:
2410:The Professional Geographer
2138:
1503:acceleration due to gravity
10:
2938:
2406:"Stream Power Terminology"
2204:Bagnold, Ralph A. (1966).
1778:Size of displaced sediment
1697:Critical Unit Stream Power
1242:{\displaystyle m=\rho Lbh}
746:
2290:Knapp, Robert T. (1938).
2130:Bridge and culvert design
80:
68:
55:
45:
35:
23:
18:
2824:Brookes, Andrew (1987).
718:, originally derived by
207:ClausiusâDuhem (entropy)
157:Fick's laws of diffusion
2308:10.1029/tr019i001p00501
2269:10.1029/tr014i001p00497
2229:2027/uc1.31210020748099
2021:river channel migration
1272:is the channel length,
929:{\displaystyle \alpha }
365:NavierâStokes equations
303:Material failure theory
2842:10.1002/rrr.3450010103
2347:10.1098/rsta.1956.0020
2120:
2039:Sensitivity to erosion
2012:
1992:
1985:
1942:
1935:
1893:
1886:
1839:
1832:
1755:
1748:
1682:
1675:
1620:
1613:
1487:
1480:
1432:
1425:
1372:
1356:
1349:
1316:is the channel depth (
1310:
1292:is the channel width (
1286:
1266:
1250:
1243:
1209:
1202:
1157:
1150:
1092:
1036:
1020:
1013:
930:
910:
883:
849:
842:
149:
30:Water flowing in creek
2251:Rubey, W. W. (1933).
2121:
2010:
1986:
1958:
1936:
1905:
1887:
1843:
1833:
1789:
1749:
1703:
1676:
1636:
1614:
1535:
1493:is the stream power,
1481:
1450:
1426:
1381:
1373:
1350:
1348:{\displaystyle Q=ubh}
1322:
1311:
1287:
1267:
1244:
1213:
1203:
1161:
1151:
1100:
1093:
1037:
1014:
938:
931:
911:
909:{\displaystyle u_{z}}
884:
843:
778:
360:Bernoulli's principle
353:Archimedes' principle
150:
2175:Deposition (geology)
2048:
1963:
1910:
1848:
1794:
1708:
1641:
1540:
1455:
1440:(Total) Stream power
1410:
1386:
1362:
1327:
1300:
1276:
1256:
1218:
1166:
1105:
1046:
1026:
943:
920:
893:
855:
783:
754:computer simulations
452:Cohesion (chemistry)
274:Infinitesimal strain
110:
2003:Landscape evolution
1881:
1827:
1415:
370:Poiseuille equation
101:Continuum mechanics
95:Part of a series on
2741:Water Publications
2391:Water Publications
2210:Professional Paper
2170:Hydrogeomorphology
2160:Sediment transport
2116:
2025:sediment transport
2013:
1981:
1931:
1882:
1867:
1828:
1813:
1767:mobilization speed
1744:
1671:
1609:
1525:Total Stream Power
1476:
1421:
1368:
1345:
1306:
1282:
1262:
1239:
1198:
1146:
1088:
1032:
1009:
926:
906:
879:
838:
576:Magnetorheological
571:Electrorheological
308:Fracture mechanics
145:
59:SI base units
1977:
1784:stream competency
1669:
1605:
1581:
1560:
1371:{\displaystyle A}
1309:{\displaystyle h}
1285:{\displaystyle b}
1265:{\displaystyle L}
1196:
1129:
1035:{\displaystyle u}
964:
836:
807:
725:stream competency
713:
712:
588:
587:
522:
521:
291:Contact mechanics
214:
213:
143:
90:
89:
2929:
2901:
2900:
2860:
2854:
2853:
2821:
2815:
2814:
2790:
2784:
2783:
2772:10.1130/B26092.1
2766:(1â2): 127â141.
2751:
2745:
2744:
2732:
2723:
2722:
2690:
2684:
2683:
2651:
2642:
2641:
2609:
2603:
2602:
2570:
2564:
2563:
2531:
2522:
2521:
2510:10.1130/G36969.1
2489:
2480:
2479:
2468:10.1002/rra.2717
2447:
2434:
2433:
2401:
2395:
2394:
2382:
2367:
2366:
2341:(964): 235â297.
2326:
2320:
2319:
2287:
2281:
2280:
2248:
2242:
2241:
2231:
2221:
2201:
2125:
2123:
2122:
2117:
2115:
2114:
2081:
2080:
1990:
1988:
1987:
1982:
1975:
1947:
1940:
1938:
1937:
1932:
1891:
1889:
1888:
1883:
1880:
1875:
1860:
1859:
1837:
1835:
1834:
1829:
1826:
1821:
1806:
1805:
1765:is the critical
1753:
1751:
1750:
1745:
1743:
1742:
1733:
1732:
1720:
1719:
1680:
1678:
1677:
1672:
1670:
1665:
1651:
1618:
1616:
1615:
1610:
1603:
1579:
1558:
1505:(9.8 m/s),
1485:
1483:
1482:
1477:
1430:
1428:
1427:
1422:
1417:
1416:
1377:
1375:
1374:
1369:
1354:
1352:
1351:
1346:
1315:
1313:
1312:
1307:
1291:
1289:
1288:
1283:
1271:
1269:
1268:
1263:
1248:
1246:
1245:
1240:
1207:
1205:
1204:
1199:
1197:
1195:
1187:
1176:
1155:
1153:
1152:
1147:
1130:
1128:
1120:
1109:
1097:
1095:
1094:
1089:
1041:
1039:
1038:
1033:
1018:
1016:
1015:
1010:
978:
977:
965:
963:
955:
947:
935:
933:
932:
927:
915:
913:
912:
907:
905:
904:
888:
886:
885:
880:
878:
870:
865:
847:
845:
844:
839:
837:
835:
827:
819:
808:
806:
798:
787:
767:potential energy
737:geomorphologists
705:
698:
691:
537:
536:
502:Gay-Lussac's law
492:Combined gas law
442:Capillary action
327:
326:
170:
169:
154:
152:
151:
146:
144:
142:
134:
126:
92:
91:
72:other quantities
70:Derivations from
60:
28:
16:
15:
2937:
2936:
2932:
2931:
2930:
2928:
2927:
2926:
2907:
2906:
2905:
2904:
2881:10.3141/1743-08
2861:
2857:
2822:
2818:
2791:
2787:
2752:
2748:
2733:
2726:
2705:(8): 986â1004.
2691:
2687:
2652:
2645:
2610:
2606:
2571:
2567:
2546:(5): 05017001.
2532:
2525:
2504:(11): 983â986.
2490:
2483:
2448:
2437:
2402:
2398:
2383:
2370:
2327:
2323:
2288:
2284:
2249:
2245:
2202:
2193:
2188:
2141:
2132:
2089:
2085:
2064:
2060:
2049:
2046:
2045:
2041:
2033:
2005:
2000:
1964:
1961:
1960:
1945:
1911:
1908:
1907:
1898:
1876:
1871:
1855:
1851:
1849:
1846:
1845:
1822:
1817:
1801:
1797:
1795:
1792:
1791:
1780:
1775:
1764:
1760:
1738:
1734:
1728:
1724:
1715:
1711:
1709:
1706:
1705:
1699:
1652:
1650:
1642:
1639:
1638:
1632:
1541:
1538:
1537:
1527:
1517:is the channel
1456:
1453:
1452:
1442:
1437:
1405:
1404:
1387:
1384:
1383:
1363:
1360:
1359:
1328:
1325:
1324:
1301:
1298:
1297:
1277:
1274:
1273:
1257:
1254:
1253:
1219:
1216:
1215:
1188:
1177:
1175:
1167:
1164:
1163:
1121:
1110:
1108:
1106:
1103:
1102:
1047:
1044:
1043:
1027:
1024:
1023:
973:
969:
956:
948:
946:
944:
941:
940:
921:
918:
917:
900:
896:
894:
891:
890:
871:
866:
858:
856:
853:
852:
828:
820:
818:
799:
788:
786:
784:
781:
780:
762:
749:
741:civil engineers
709:
680:
679:
678:
598:
590:
589:
543:Viscoelasticity
534:
524:
523:
511:
461:
457:Surface tension
421:
324:
322:Fluid mechanics
314:
313:
312:
226:
224:Solid mechanics
216:
215:
167:
159:
135:
127:
125:
111:
108:
107:
73:
71:
58:
38:
31:
12:
11:
5:
2935:
2925:
2924:
2919:
2903:
2902:
2855:
2816:
2805:(1â2): 32â40.
2785:
2746:
2724:
2685:
2643:
2604:
2585:(4): 491â505.
2565:
2523:
2481:
2435:
2416:(2): 189â195.
2396:
2368:
2321:
2282:
2243:
2219:10.3133/pp422i
2190:
2189:
2187:
2184:
2183:
2182:
2177:
2172:
2167:
2162:
2157:
2152:
2147:
2140:
2137:
2131:
2128:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2088:
2084:
2079:
2076:
2073:
2070:
2067:
2063:
2059:
2056:
2053:
2040:
2037:
2032:
2029:
2004:
2001:
1999:
1996:
1980:
1974:
1971:
1968:
1930:
1927:
1924:
1921:
1918:
1915:
1897:
1894:
1879:
1874:
1870:
1866:
1863:
1858:
1854:
1825:
1820:
1816:
1812:
1809:
1804:
1800:
1779:
1776:
1774:
1771:
1762:
1758:
1741:
1737:
1731:
1727:
1723:
1718:
1714:
1698:
1695:
1668:
1664:
1661:
1658:
1655:
1649:
1646:
1631:
1628:
1608:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1578:
1575:
1572:
1569:
1566:
1563:
1557:
1554:
1551:
1548:
1545:
1526:
1523:
1475:
1472:
1469:
1466:
1463:
1460:
1441:
1438:
1436:
1433:
1420:
1414:
1409:
1403:
1400:
1397:
1394:
1391:
1367:
1344:
1341:
1338:
1335:
1332:
1305:
1281:
1261:
1238:
1235:
1232:
1229:
1226:
1223:
1194:
1191:
1186:
1183:
1180:
1174:
1171:
1145:
1142:
1139:
1136:
1133:
1127:
1124:
1119:
1116:
1113:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1031:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
976:
972:
968:
962:
959:
954:
951:
925:
903:
899:
877:
874:
869:
864:
861:
834:
831:
826:
823:
817:
814:
811:
805:
802:
797:
794:
791:
771:kinetic energy
761:
758:
748:
745:
711:
710:
708:
707:
700:
693:
685:
682:
681:
677:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
626:
621:
616:
611:
606:
600:
599:
596:
595:
592:
591:
586:
585:
584:
583:
578:
573:
565:
564:
558:
557:
556:
555:
550:
545:
535:
530:
529:
526:
525:
520:
519:
513:
512:
510:
509:
504:
499:
494:
489:
484:
479:
473:
470:
469:
463:
462:
460:
459:
454:
449:
447:Chromatography
444:
439:
433:
430:
429:
423:
422:
420:
419:
400:
399:
398:
379:
367:
362:
350:
337:
334:
333:
325:
320:
319:
316:
315:
311:
310:
305:
300:
299:
298:
288:
283:
278:
277:
276:
271:
261:
256:
251:
246:
245:
244:
234:
228:
227:
222:
221:
218:
217:
212:
211:
210:
209:
201:
200:
196:
195:
194:
193:
188:
183:
175:
174:
168:
165:
164:
161:
160:
155:
141:
138:
133:
130:
124:
121:
118:
115:
104:
103:
97:
96:
88:
87:
84:
78:
77:
74:
69:
66:
65:
62:
53:
52:
49:
43:
42:
39:
37:Common symbols
36:
33:
32:
29:
21:
20:
9:
6:
4:
3:
2:
2934:
2923:
2922:Water streams
2920:
2918:
2917:Geomorphology
2915:
2914:
2912:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2866:
2859:
2851:
2847:
2843:
2839:
2835:
2831:
2827:
2820:
2812:
2808:
2804:
2800:
2799:Geomorphology
2796:
2789:
2781:
2777:
2773:
2769:
2765:
2761:
2757:
2750:
2742:
2738:
2731:
2729:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2689:
2681:
2677:
2673:
2669:
2666:(1): 92â101.
2665:
2661:
2660:Geomorphology
2657:
2650:
2648:
2639:
2635:
2631:
2627:
2624:(1): 92â101.
2623:
2619:
2618:Geomorphology
2615:
2608:
2600:
2596:
2592:
2588:
2584:
2580:
2576:
2569:
2561:
2557:
2553:
2549:
2545:
2541:
2537:
2530:
2528:
2519:
2515:
2511:
2507:
2503:
2499:
2495:
2488:
2486:
2477:
2473:
2469:
2465:
2461:
2457:
2453:
2446:
2444:
2442:
2440:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2400:
2392:
2388:
2381:
2379:
2377:
2375:
2373:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2336:
2332:
2325:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2286:
2278:
2274:
2270:
2266:
2262:
2258:
2254:
2247:
2239:
2235:
2230:
2225:
2220:
2215:
2211:
2207:
2200:
2198:
2196:
2191:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2150:Geomorphology
2148:
2146:
2143:
2142:
2136:
2127:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2086:
2082:
2077:
2074:
2071:
2068:
2065:
2061:
2057:
2054:
2036:
2028:
2026:
2022:
2018:
2009:
1995:
1991:
1978:
1972:
1969:
1966:
1957:
1955:
1951:
1941:
1928:
1925:
1922:
1919:
1916:
1913:
1904:
1902:
1892:
1877:
1872:
1868:
1864:
1861:
1856:
1852:
1842:
1838:
1823:
1818:
1814:
1810:
1807:
1802:
1798:
1788:
1785:
1770:
1768:
1754:
1739:
1735:
1729:
1725:
1721:
1716:
1712:
1702:
1694:
1691:
1687:
1681:
1666:
1662:
1659:
1656:
1653:
1647:
1644:
1635:
1627:
1625:
1619:
1606:
1597:
1594:
1591:
1588:
1585:
1582:
1576:
1573:
1570:
1567:
1564:
1561:
1555:
1552:
1549:
1546:
1543:
1534:
1532:
1522:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1486:
1473:
1470:
1467:
1464:
1461:
1449:
1447:
1435:Various Forms
1431:
1418:
1412:
1407:
1401:
1398:
1395:
1392:
1380:
1365:
1355:
1342:
1339:
1336:
1333:
1330:
1321:
1319:
1303:
1296:readth), and
1295:
1279:
1259:
1249:
1236:
1233:
1230:
1227:
1224:
1221:
1212:
1208:
1192:
1184:
1181:
1172:
1160:
1156:
1143:
1140:
1137:
1134:
1131:
1125:
1117:
1114:
1099:
1085:
1082:
1076:
1070:
1067:
1064:
1058:
1052:
1049:
1029:
1019:
1006:
1003:
1000:
994:
988:
985:
982:
979:
974:
970:
966:
960:
952:
937:
923:
901:
897:
875:
867:
862:
848:
832:
824:
815:
812:
809:
803:
795:
792:
777:
774:
772:
768:
757:
755:
744:
742:
738:
734:
730:
726:
721:
720:R. A. Bagnold
717:
706:
701:
699:
694:
692:
687:
686:
684:
683:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
635:
632:
630:
627:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
601:
594:
593:
582:
579:
577:
574:
572:
569:
568:
567:
566:
563:
560:
559:
554:
551:
549:
546:
544:
541:
540:
539:
538:
533:
528:
527:
518:
515:
514:
508:
505:
503:
500:
498:
495:
493:
490:
488:
487:Charles's law
485:
483:
480:
478:
475:
474:
472:
471:
468:
465:
464:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
434:
432:
431:
428:
425:
424:
418:
415:
411:
408:
404:
401:
396:
395:non-Newtonian
393:
389:
385:
384:
383:
380:
378:
375:
371:
368:
366:
363:
361:
358:
354:
351:
349:
346:
342:
339:
338:
336:
335:
332:
329:
328:
323:
318:
317:
309:
306:
304:
301:
297:
294:
293:
292:
289:
287:
284:
282:
281:Compatibility
279:
275:
272:
270:
269:Finite strain
267:
266:
265:
262:
260:
257:
255:
252:
250:
247:
243:
240:
239:
238:
235:
233:
230:
229:
225:
220:
219:
208:
205:
204:
203:
202:
198:
197:
192:
189:
187:
184:
182:
179:
178:
177:
176:
173:Conservations
172:
171:
163:
162:
158:
139:
136:
131:
128:
122:
119:
116:
113:
106:
105:
102:
99:
98:
94:
93:
85:
83:
79:
75:
67:
63:
61:
54:
50:
48:
44:
40:
34:
27:
22:
17:
2875:(1): 57â59.
2872:
2868:
2858:
2833:
2829:
2819:
2802:
2798:
2788:
2763:
2759:
2749:
2740:
2702:
2699:GSA Bulletin
2698:
2688:
2663:
2659:
2621:
2617:
2607:
2582:
2578:
2568:
2543:
2539:
2501:
2497:
2462:(1): 16â27.
2459:
2455:
2413:
2409:
2399:
2390:
2338:
2334:
2324:
2299:
2295:
2285:
2260:
2256:
2246:
2209:
2165:Shear stress
2133:
2042:
2034:
2014:
1998:Applications
1993:
1959:
1953:
1949:
1943:
1906:
1901:Shear stress
1899:
1896:Shear stress
1844:
1840:
1790:
1781:
1756:
1704:
1700:
1689:
1685:
1683:
1637:
1633:
1623:
1621:
1536:
1528:
1514:
1506:
1498:
1494:
1490:
1488:
1451:
1443:
1382:
1357:
1323:
1317:
1293:
1251:
1214:
1210:
1162:
1158:
1101:
1021:
939:
850:
779:
775:
763:
750:
733:hydrologists
729:shear stress
716:Stream power
715:
714:
562:Smart fluids
507:Graham's law
413:
406:
391:
377:Pascal's law
373:
356:
344:
199:Inequalities
47:SI unit
19:Stream power
2836:(1): 3â16.
2180:Water slope
1513:(m/s), and
581:Ferrofluids
482:Boyle's law
254:Hooke's law
232:Deformation
2911:Categories
2302:(1): 501.
2263:(1): 497.
2186:References
760:Derivation
634:Gay-Lussac
597:Scientists
497:Fick's law
477:Atmosphere
296:frictional
249:Plasticity
237:Elasticity
2897:109792586
2889:0361-1981
2850:1099-1646
2780:0016-7606
2719:0016-7606
2680:0169-555X
2638:0169-555X
2599:0733-9429
2560:1084-0702
2518:0091-7613
2476:129164405
2430:0033-0124
2363:124012787
2355:0080-4614
2316:0002-8606
2277:0002-8606
2238:2330-7102
2145:Hydrology
2087:ω
2083:−
2062:ω
2055:ω
2052:Δ
1973:τ
1967:ω
1926:ρ
1914:τ
1853:ω
1799:ω
1736:ν
1726:τ
1713:ω
1654:ρ
1645:ω
1601:Ω
1511:discharge
1465:ρ
1459:Ω
1396:ρ
1390:Ω
1379:complete.
1228:ρ
1190:Δ
1179:Δ
1170:Ω
1123:Δ
1112:Δ
1077:α
1071:
1065:≈
1059:α
1053:
1001:≈
995:α
989:
958:Δ
950:Δ
924:α
873:Δ
860:Δ
830:Δ
822:Δ
801:Δ
790:Δ
674:Truesdell
604:Bernoulli
553:Rheometer
548:Rheometry
388:Newtonian
382:Viscosity
132:φ
120:−
82:Dimension
2139:See also
532:Rheology
437:Adhesion
417:Pressure
403:Buoyancy
348:Dynamics
186:Momentum
2498:Geology
2155:Erosion
1757:where Ï
747:History
619:Charles
427:Liquids
341:Statics
286:Bending
2895:
2887:
2848:
2778:
2717:
2678:
2636:
2597:
2558:
2516:
2474:
2428:
2361:
2353:
2314:
2275:
2236:
1976:
1944:Where
1684:where
1622:where
1604:
1580:
1559:
1489:where
1358:where
1252:where
1022:where
669:Stokes
664:Pascal
654:Navier
649:Newton
639:Graham
614:Cauchy
517:Plasma
412:
410:Mixing
405:
390:
372:
355:
343:
331:Fluids
264:Strain
259:Stress
242:linear
191:Energy
76:Ω=ÏgQS
64:kg m s
2893:S2CID
2472:S2CID
2359:S2CID
1878:1.438
1865:0.130
1811:0.030
1531:river
1519:slope
1446:river
644:Hooke
624:Euler
609:Boyle
467:Gases
86:M L T
51:Watts
2885:ISSN
2873:1743
2846:ISSN
2776:ISSN
2715:ISSN
2676:ISSN
2634:ISSN
2595:ISSN
2556:ISSN
2514:ISSN
2426:ISSN
2351:ISSN
2312:ISSN
2273:ISSN
2234:ISSN
1824:1.69
735:and
727:and
659:Noll
629:Fick
181:Mass
166:Laws
41:Ω, Ï
2877:doi
2838:doi
2807:doi
2803:100
2768:doi
2764:120
2707:doi
2668:doi
2626:doi
2587:doi
2583:119
2548:doi
2506:doi
2464:doi
2418:doi
2343:doi
2339:249
2304:doi
2265:doi
2224:hdl
2214:doi
1787:to:
1509:is
1501:is
1068:tan
1050:sin
986:sin
56:In
2913::
2891:.
2883:.
2871:.
2867:.
2844:.
2832:.
2828:.
2801:.
2797:.
2774:.
2762:.
2758:.
2739:.
2727:^
2713:.
2703:94
2701:.
2697:.
2674:.
2664:69
2662:.
2658:.
2646:^
2632:.
2622:69
2620:.
2616:.
2593:.
2581:.
2577:.
2554:.
2544:22
2542:.
2538:.
2526:^
2512:.
2502:43
2500:.
2496:.
2484:^
2470:.
2460:31
2458:.
2454:.
2438:^
2424:.
2414:39
2412:.
2408:.
2389:.
2371:^
2357:.
2349:.
2337:.
2333:.
2310:.
2300:19
2298:.
2294:.
2271:.
2261:14
2259:.
2255:.
2232:.
2222:.
2212:.
2208:.
2194:^
2027:.
1769:.
1521:.
756:.
2899:.
2879::
2852:.
2840::
2834:1
2813:.
2809::
2782:.
2770::
2743:.
2721:.
2709::
2682:.
2670::
2640:.
2628::
2601:.
2589::
2562:.
2550::
2520:.
2508::
2478:.
2466::
2432:.
2420::
2393:.
2365:.
2345::
2318:.
2306::
2279:.
2267::
2240:.
2226::
2216::
2112:m
2109:a
2106:e
2103:r
2100:t
2097:s
2094:p
2091:u
2078:l
2075:a
2072:c
2069:o
2066:l
2058:=
1979:V
1970:=
1954:g
1950:Ï
1946:Ï
1929:g
1923:S
1920:h
1917:=
1873:i
1869:D
1862:=
1857:0
1819:i
1815:D
1808:=
1803:0
1763:0
1759:0
1740:0
1730:0
1722:=
1717:0
1690:b
1686:Ï
1667:b
1663:S
1660:Q
1657:g
1648:=
1624:Ω
1607:L
1598:=
1595:r
1592:e
1589:w
1586:o
1583:p
1577:m
1574:a
1571:e
1568:r
1565:t
1562:s
1556:l
1553:a
1550:t
1547:o
1544:T
1515:S
1507:Q
1499:g
1495:Ï
1491:Ω
1474:S
1471:Q
1468:g
1462:=
1419:S
1413:1
1408:L
1402:Q
1399:g
1393:=
1366:A
1343:h
1340:b
1337:u
1334:=
1331:Q
1318:h
1304:h
1294:b
1280:b
1260:L
1237:h
1234:b
1231:L
1225:=
1222:m
1193:t
1185:E
1182:P
1173:=
1144:S
1141:u
1138:g
1135:m
1132:=
1126:t
1118:E
1115:P
1086:S
1083:=
1080:)
1074:(
1062:)
1056:(
1030:u
1007:S
1004:u
998:)
992:(
983:u
980:=
975:z
971:u
967:=
961:t
953:z
936::
902:z
898:u
876:t
868:/
863:z
833:t
825:z
816:g
813:m
810:=
804:t
796:E
793:P
704:e
697:t
690:v
414:·
407:·
397:)
392:·
386:(
374:·
357:·
345:·
140:x
137:d
129:d
123:D
117:=
114:J
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