4957:
4037:
2489:
5258:
3687:
4643:
1881:
5513:
646:
3799:
2289:
4242:
5043:
3472:
3173:
4952:{\displaystyle c_{2}(\xi ,\tau )=c_{2a}(\xi ,\tau )+{\frac {Pe}{4}}\left(r^{2}-{\frac {r^{4}}{2}}\right){\frac {\partial c_{1a}}{\partial \xi }}+{\frac {Pe^{2}}{32}}\left({\frac {r^{2}}{6}}+{\frac {r^{4}}{2}}-{\frac {5r^{6}}{8}}+{\frac {r^{8}}{8}}\right){\frac {\partial ^{2}c_{0}}{\partial \xi ^{2}}}}
1200:
Dispersion is also a function of channel geometry. An interesting phenomenon for example is that the dispersion of a flow between two infinite flat plates and a rectangular channel, which is infinitely thin, differs approximately 8.75 times. Here the very small side walls of the rectangular channel
1694:
5314:
446:
4032:{\displaystyle {\frac {\partial c_{0}}{\partial \tau }}=\left(1+{\frac {Pe^{2}}{48}}\right){\frac {\partial ^{2}c_{0}}{\partial \xi ^{2}}}\quad \Rightarrow \quad {\frac {\partial c_{0}}{\partial t}}=\left(1+{\frac {Pe^{2}}{48}}\right){\frac {\partial ^{2}c_{0}}{\partial x^{2}}}.}
1601:
2484:{\displaystyle \epsilon {\frac {\partial c}{\partial \tau }}+{\sqrt {\epsilon }}Pe(1-2r^{2}){\frac {\partial c}{\partial \xi }}=\epsilon {\frac {\partial ^{2}c}{\partial \xi ^{2}}}+{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c}{\partial r}}\right).}
977:
3355:
2819:
4048:
5253:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{3}}{\partial r}}\right)=Pe(1-2r^{2}){\frac {\partial c_{2}}{\partial \xi }}+{\frac {\partial c_{1}}{\partial \tau }}-{\frac {\partial ^{2}c_{1}}{\partial \xi ^{2}}}.}
3682:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{2}}{\partial r}}\right)=Pe(1-2r^{2}){\frac {\partial c_{1}}{\partial \xi }}+{\frac {\partial c_{0}}{\partial \tau }}-{\frac {\partial ^{2}c_{0}}{\partial \xi ^{2}}}.}
279:
178:
819:. The interesting result is that for large values of the PĂ©clet number, the effective diffusivity is inversely proportional to the molecular diffusivity. The effect of Taylor dispersion is therefore more pronounced at higher PĂ©clet numbers.
27:
on the small scale. Essentially, the shear acts to smear out the concentration distribution in the direction of the flow, enhancing the rate at which it spreads in that direction. The effect is named after the
British fluid dynamicist
3032:
737:
1876:{\displaystyle {\frac {\partial c}{\partial t}}+Pe(1-2r^{2}){\frac {\partial c}{\partial x}}={\frac {\partial ^{2}c}{\partial x^{2}}}+{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c}{\partial r}}\right).}
1221:
One may derive the Taylor equation using method of averages, first introduced by Aris. The result can also be derived from large-time asymptotics, which is more intuitively clear. In the dimensional coordinate system
2728:
5508:{\displaystyle {\frac {\partial c_{1a}}{\partial \tau }}-\left(1+{\frac {Pe^{2}}{48}}\right){\frac {\partial ^{2}c_{1a}}{\partial \xi ^{2}}}=-{\frac {Pe^{3}}{2880}}{\frac {\partial ^{3}c_{0}}{\partial \xi ^{3}}}.}
641:{\displaystyle {\frac {\partial {\bar {c}}}{\partial t}}+{\bar {w}}{\frac {\partial {\bar {c}}}{\partial z}}=D\left(1+{\frac {a^{2}{\bar {w}}^{2}}{48D^{2}}}\right){\frac {\partial ^{2}{\bar {c}}}{\partial z^{2}}}}
1431:. As long as this initial distribution is compact, for instance the species/solute is not released everywhere with finite concentration level, the species will be convected along the pipe with the mean velocity
1204:
While the exact formula will not hold in more general circumstances, the mechanism still applies, and the effect is stronger at higher PĂ©clet numbers. Taylor dispersion is of particular relevance for flows in
1194:
2278:
2226:
651:
Observe how the effective diffusivity multiplying the derivative on the right hand side is greater than the original value of diffusion coefficient, D. The effective diffusivity is often written as:
3793:. This is also the same as averaging the above equation over the radial direction. Using the boundary conditions and results obtained in the previous two orders, the solvability condition leads to
872:
4505:
3204:
4628:
435:
5035:
793:
1457:
349:
4552:
4390:
4237:{\displaystyle {\frac {\partial c_{0}}{\partial t'}}+U{\frac {\partial c_{0}}{\partial x'}}=D\left(1+{\frac {U^{2}a^{2}}{48D^{2}}}\right){\frac {\partial ^{2}c_{0}}{\partial x'^{2}}}.}
2736:
1268:
2877:
2598:
2532:
2111:
1023:
209:
3024:
1956:
5306:
5005:
4436:
4291:
2997:
2028:
864:
81:
2653:
2172:
4349:
2730:
Substituting this series into the governing equation and collecting terms of different orders will lead to series of equations. At leading order, the equation obtained is
3464:
1429:
5566:
2146:
1910:
1639:
3388:
1982:
4318:
3415:
3168:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{1}}{\partial r}}\right)=Pe(1-2r^{2}){\frac {\partial c_{0}}{\partial \xi }}.}
2931:
2904:
3739:
1338:
3791:
3765:
3441:
2624:
2558:
1682:
4572:
3710:
3196:
2951:
2068:
2048:
1659:
1449:
1398:
1378:
1358:
1123:
1103:
1083:
1063:
1043:
817:
657:
284:
The concentration and velocity are written as the sum of a cross-sectional average (indicated by an overbar) and a deviation (indicated by a prime), thus:
3712:, but what is required is the solvability condition of the above equation. The solvability condition is obtained by multiplying the above equation by
2030:), diffusion in the radial direction will make the concentration uniform across the pipe, although however the species is still diffusing in the
5591:
2658:
5853:
Taylor, Geoffrey (1954). "Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion".
23:
is an apparent or effective diffusion of some scalar field arising on the large scale due to the presence of a strong, confined, zero-mean
5658:
Liñán, Amable; Rajamanickam, Prabakaran; Weiss, Adam D.; Sánchez, Antonio L. (2020). "Taylor-diffusion-controlled combustion in ducts".
1131:
2234:
2177:
972:{\displaystyle {\frac {\partial {\bar {c}}}{\partial t}}=D_{\mathrm {eff} }{\frac {\partial ^{2}{\bar {c}}}{\partial \xi ^{2}}}}
4042:
This is the required diffusion equation. Going back to the laboratory frame and dimensional variables, the equation becomes
3350:{\displaystyle c_{1}(\xi ,r,\tau )=c_{1a}(\xi ,\tau )+{\frac {Pe}{8}}(2r^{2}-r^{4}){\frac {\partial c_{0}}{\partial \xi }}}
200:
4577:
1596:{\displaystyle t={\frac {t'}{a^{2}/D}},\quad x={\frac {x'-Ut'}{a}},\quad r={\frac {r'}{a}},\quad Pe={\frac {Ua}{D}}}
5552:
5013:
2814:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c_{0}}{\partial r}}\right)=0.}
745:
355:
440:
Under some assumptions (see below), it is possible to derive an equation just involving the average quantities:
5947:
274:{\displaystyle {\frac {\partial c}{\partial t}}+{\boldsymbol {w}}\cdot {\boldsymbol {\nabla }}c=D\nabla ^{2}c}
2827:
2563:
2497:
5952:
2076:
290:
4354:
29:
3005:
5266:
4965:
825:
173:{\displaystyle {\boldsymbol {u}}=w{\hat {\boldsymbol {z}}}=w_{0}(1-r^{2}/a^{2}){\hat {\boldsymbol {z}}}}
4441:
988:
5899:
5862:
5825:
5788:
5751:
5714:
5667:
5620:
4510:
1225:
2632:
2151:
1915:
4395:
4250:
2956:
1987:
5779:
Taylor, Geoffrey (1953). "Dispersion of soluble matter in solvent flowing slowly through a tube".
5915:
5878:
5841:
5804:
5767:
5730:
5683:
5560:
2999:, the radial diffusion will dominate first and make the concentration uniform across the pipe.
5742:
Frankel, I.; Brenner, H. (1989). "On the foundations of generalized Taylor dispersion theory".
3449:
5890:
Brenner, H. (1980). "Dispersion resulting from flow through spatially periodic porous media".
5587:
4323:
2116:
1889:
1609:
5907:
5870:
5833:
5796:
5759:
5722:
5675:
5638:
5628:
3363:
1961:
1451:. In a frame moving with the mean velocity and scaled with following non-dimensional scales
796:
192:
4296:
3393:
2909:
2882:
3715:
2174:
is a small number. Then at these times, the concentration would spread to an axial extent
1403:
1273:
52:
37:
3770:
3744:
3420:
2603:
2537:
5903:
5866:
5829:
5792:
5755:
5718:
5671:
5624:
1664:
1400:
with some arbitrary distribution is to be released at somewhere inside the pipe at time
4557:
3695:
3181:
2936:
2053:
2033:
1644:
1434:
1383:
1363:
1343:
1210:
1108:
1088:
1068:
1048:
1028:
802:
5816:
Taylor, Geoffrey (1954). "The dispersion of matter in turbulent flow through a pipe".
732:{\displaystyle D_{\mathrm {eff} }=D\left(1+{\frac {{\mathit {Pe}}^{2}}{48}}\right)\,,}
5941:
5919:
5771:
5687:
1685:
1206:
184:
41:
33:
5882:
5845:
5808:
5734:
5705:
Aris, R. (1956). "On the dispersion of a solute in a fluid flowing through a tube".
4247:
By the way in which this equation is derived, it can be seen that this is valid for
4438:, at any small length scale about some location that moves with the mean flow, say
2494:
If pipe walls do not absorb or react with the species, then the boundary condition
5679:
5263:
The solvability condition of the above equation yields the governing equation for
5927:
5609:"Shear dispersion in a porous medium. Part 1. An intrusion with a steady shape"
2050:
direction. Taylor dispersion quantifies this axial diffusion process for large
5763:
24:
2824:
Integrating this equation with boundary conditions defined before, one finds
1085:
direction. This can be translated into the requirement that the length scale
5911:
5874:
5837:
5800:
5726:
5633:
5608:
1641:
is the time required for the species to diffuse in the radial direction,
5643:
5581:
5583:
Micro- and
Nanoscale Fluid Mechanics: Transport in Microfluidic Devices
2723:{\displaystyle c=c_{0}+{\sqrt {\epsilon }}c_{1}+\epsilon c_{2}+\cdots }
51:
The canonical example is that of a simple diffusing species in uniform
71:
as the radial coordinate, and assume axisymmetry. The pipe has radius
5892:
Philosophical
Transactions of the Royal Society of London, Series A
55:
through a uniform circular pipe with no-flux boundary conditions.
1958:), the species will diffuse radially. It is clear then that when
1380:
is the average velocity of the fluid. A species of concentration
1189:{\displaystyle L\gg {\frac {a^{2}}{D}}{\bar {w}}=a{\mathit {Pe}}}
4637:
Integrating the equations obtained at the second order, we find
2273:{\displaystyle \tau =\epsilon t,\quad \xi ={\sqrt {\epsilon }}x}
2221:{\displaystyle x\sim {\sqrt {t}}\sim {\sqrt {1/\epsilon }}\gg 1}
2113:(i.e., times large in comparison with the radial diffusion time
822:
In a frame moving with the mean velocity, i.e., by introducing
44:. The dispersion process is sometimes also referred to as the
866:, the dispersion process becomes a purely diffusion process,
2228:. To quantify large-time behavior, the following rescalings
1181:
1178:
754:
751:
704:
701:
199:. The concentration is assumed to be governed by the linear
5549:
2655:, the solution can be expanded in an asymptotic series,
1065:
direction is long enough to smooth the gradient in the
32:, who described the shear-induced dispersion for large
2953:
is an expected result since as already said, at times
5317:
5269:
5046:
5016:
4968:
4646:
4580:
4560:
4513:
4444:
4398:
4357:
4326:
4299:
4253:
4051:
3802:
3773:
3747:
3718:
3698:
3692:
This equation can also be integrated with respect to
3475:
3452:
3423:
3396:
3366:
3207:
3184:
3035:
3008:
2959:
2939:
2912:
2885:
2830:
2739:
2661:
2635:
2606:
2566:
2540:
2500:
2292:
2237:
2180:
2154:
2119:
2079:
2056:
2036:
1990:
1964:
1918:
1892:
1697:
1667:
1647:
1612:
1460:
1437:
1406:
1386:
1366:
1346:
1276:
1228:
1134:
1111:
1091:
1071:
1051:
1031:
991:
982:
with diffusivity given by the effective diffusivity.
875:
828:
805:
748:
660:
449:
358:
293:
212:
84:
5855:
Proceedings of the Royal
Society of London, Series A
5818:
Proceedings of the Royal
Society of London, Series A
5781:
Proceedings of the Royal
Society of London, Series A
5707:
Proceedings of the Royal
Society of London, Series A
5507:
5300:
5252:
5029:
4999:
4951:
4622:
4566:
4546:
4499:
4430:
4384:
4343:
4312:
4285:
4236:
4031:
3785:
3759:
3733:
3704:
3681:
3458:
3435:
3409:
3382:
3349:
3190:
3167:
3018:
2991:
2945:
2925:
2898:
2871:
2813:
2722:
2647:
2618:
2592:
2552:
2526:
2483:
2272:
2220:
2166:
2140:
2105:
2062:
2042:
2022:
1976:
1950:
1904:
1875:
1676:
1653:
1633:
1595:
1443:
1423:
1392:
1372:
1352:
1332:
1262:
1188:
1117:
1097:
1077:
1057:
1037:
1017:
971:
858:
811:
787:
731:
640:
429:
343:
273:
172:
4623:{\displaystyle c=c_{0}+{\sqrt {\epsilon }}c_{1}.}
4554:, the concentration is no longer independent of
1661:is the diffusion coefficient of the species and
1270:, consider the fully-developed Poiseuille flow
1045:, which is the case if the length scale in the
1201:have an enormous influence on the dispersion.
5928:Taylor dispersion — shear augmented diffusion
2906:is still an unknown function. This fact that
2283:can be introduced. The equation then becomes
8:
5607:Hinton, Edward M.; Woods, Andrew W. (2020).
5565:: CS1 maint: multiple names: authors list (
5030:{\displaystyle \epsilon {\sqrt {\epsilon }}}
788:{\displaystyle {\mathit {Pe}}=a{\bar {w}}/D}
430:{\displaystyle c(r,z)={\bar {c}}(z)+c'(r,z)}
4320:changes significantly over a length scale
3178:Integrating this equation with respect to
5642:
5632:
5493:
5478:
5468:
5461:
5449:
5439:
5424:
5406:
5396:
5389:
5372:
5362:
5328:
5318:
5316:
5274:
5268:
5238:
5223:
5213:
5206:
5186:
5176:
5156:
5146:
5137:
5090:
5080:
5057:
5047:
5045:
5020:
5015:
4973:
4967:
4940:
4925:
4915:
4908:
4892:
4886:
4871:
4861:
4847:
4841:
4827:
4821:
4804:
4794:
4771:
4761:
4745:
4739:
4730:
4706:
4679:
4651:
4645:
4611:
4600:
4591:
4579:
4559:
4529:
4512:
4474:
4443:
4420:
4414:
4397:
4364:
4356:
4325:
4304:
4298:
4275:
4269:
4252:
4221:
4202:
4192:
4185:
4171:
4156:
4146:
4139:
4100:
4090:
4062:
4052:
4050:
4017:
4002:
3992:
3985:
3968:
3958:
3927:
3917:
3903:
3888:
3878:
3871:
3854:
3844:
3813:
3803:
3801:
3772:
3746:
3717:
3697:
3667:
3652:
3642:
3635:
3615:
3605:
3585:
3575:
3566:
3519:
3509:
3486:
3476:
3474:
3451:
3422:
3401:
3395:
3371:
3365:
3330:
3320:
3311:
3298:
3273:
3246:
3212:
3206:
3183:
3145:
3135:
3126:
3079:
3069:
3046:
3036:
3034:
3009:
3007:
2981:
2975:
2958:
2938:
2917:
2911:
2890:
2884:
2848:
2835:
2829:
2783:
2773:
2750:
2740:
2738:
2708:
2692:
2681:
2672:
2660:
2634:
2605:
2573:
2565:
2539:
2507:
2499:
2453:
2430:
2420:
2408:
2390:
2383:
2357:
2348:
2319:
2296:
2291:
2260:
2236:
2202:
2197:
2187:
2179:
2153:
2130:
2124:
2118:
2089:
2078:
2055:
2035:
2012:
2006:
1989:
1963:
1940:
1934:
1917:
1891:
1845:
1822:
1812:
1800:
1782:
1775:
1752:
1743:
1698:
1696:
1666:
1646:
1623:
1617:
1611:
1578:
1550:
1509:
1488:
1482:
1467:
1459:
1436:
1405:
1385:
1365:
1345:
1321:
1309:
1275:
1227:
1177:
1176:
1159:
1158:
1147:
1141:
1133:
1110:
1090:
1070:
1050:
1030:
1004:
1003:
990:
960:
940:
939:
933:
926:
913:
912:
883:
882:
876:
874:
842:
841:
827:
804:
777:
766:
765:
750:
749:
747:
725:
709:
700:
699:
696:
666:
665:
659:
629:
609:
608:
602:
595:
581:
566:
555:
554:
547:
540:
501:
500:
494:
483:
482:
457:
456:
450:
448:
381:
380:
357:
310:
309:
292:
262:
244:
236:
213:
211:
159:
158:
149:
140:
134:
115:
97:
96:
85:
83:
3741:and integrating the whole equation from
36:. The analysis was later generalized by
5524:
4218:
3198:using the boundary conditions leads to
2872:{\displaystyle c_{0}=c_{0}(\xi ,\tau )}
2593:{\displaystyle \partial c/\partial r=0}
2527:{\displaystyle \partial c/\partial r=0}
1688:, the governing equations are given by
237:
161:
99:
86:
5558:
2106:{\displaystyle t\sim 1/\epsilon \gg 1}
3443:, an unknown function at this order.
344:{\displaystyle w(r)={\bar {w}}+w'(r)}
7:
4385:{\displaystyle x\sim {\sqrt {Dt'}})}
1886:Thus in this moving frame, at times
187:of the diffusing species is denoted
3019:{\displaystyle {\sqrt {\epsilon }}}
5486:
5465:
5417:
5393:
5339:
5321:
5301:{\displaystyle c_{1a}(\xi ,\tau )}
5231:
5210:
5194:
5179:
5164:
5149:
5098:
5083:
5063:
5059:
5000:{\displaystyle c_{2a}(\xi ,\tau )}
4933:
4912:
4782:
4764:
4210:
4189:
4108:
4093:
4070:
4055:
4010:
3989:
3935:
3920:
3896:
3875:
3821:
3806:
3660:
3639:
3623:
3608:
3593:
3578:
3527:
3512:
3492:
3488:
3338:
3323:
3153:
3138:
3087:
3072:
3052:
3048:
2791:
2776:
2756:
2752:
2578:
2567:
2512:
2501:
2464:
2456:
2436:
2432:
2401:
2387:
2368:
2360:
2307:
2299:
1856:
1848:
1828:
1824:
1793:
1779:
1763:
1755:
1709:
1701:
953:
930:
920:
917:
914:
896:
879:
859:{\displaystyle \xi =z-{\bar {w}}t}
673:
670:
667:
622:
599:
514:
497:
470:
453:
259:
224:
216:
14:
4500:{\displaystyle x'-Ut'=x_{s}'-Ut'}
5932:Lecture Handout for Course M4A33
1340:flowing inside a pipe of radius
1018:{\displaystyle c'\ll {\bar {c}}}
245:
5660:Combustion Theory and Modelling
4547:{\displaystyle x'-x_{s}'\sim a}
3916:
3912:
2253:
1568:
1543:
1502:
1263:{\displaystyle (x',r',\theta )}
5586:. Cambridge University Press.
5295:
5283:
5143:
5121:
5010:Now collecting terms of order
4994:
4982:
4700:
4688:
4669:
4657:
4379:
4351:(or more precisely on a scale
3913:
3572:
3550:
3317:
3288:
3267:
3255:
3236:
3218:
3132:
3110:
2866:
2854:
2648:{\displaystyle \epsilon \ll 1}
2354:
2332:
2167:{\displaystyle \epsilon \ll 1}
1951:{\displaystyle t'\sim a^{2}/D}
1749:
1727:
1327:
1318:
1298:
1289:
1257:
1229:
1164:
1009:
945:
888:
847:
799:, based on the channel radius
771:
614:
560:
506:
488:
462:
424:
412:
398:
392:
386:
374:
362:
338:
332:
315:
303:
297:
164:
155:
121:
102:
1:
5680:10.1080/13647830.2020.1813335
5547:Chang, H.C., Yeo, L. (2009).
5534:Physicochemical Hydrodynamics
5007:is an unknown at this order.
4431:{\displaystyle t'\gg a^{2}/D}
4286:{\displaystyle t'\gg a^{2}/D}
2992:{\displaystyle t'\gg a^{2}/D}
2023:{\displaystyle t'\gg a^{2}/D}
75:, and the fluid velocity is:
4507:, i.e., on the length scale
201:advection–diffusion equation
40:for arbitrary values of the
1984:(in dimensional variables,
1912:(in dimensional variables,
67:as an axial coordinate and
5969:
5744:Journal of Fluid Mechanics
5613:Journal of Fluid Mechanics
5553:Cambridge University Press
5764:10.1017/S0022112089001679
4392:. At the same time scale
3459:{\displaystyle \epsilon }
4633:Higher order asymptotics
4344:{\displaystyle x'\gg a}
2141:{\displaystyle a^{2}/D}
1905:{\displaystyle t\sim 1}
1634:{\displaystyle a^{2}/D}
985:The assumption is that
5912:10.1098/rsta.1980.0205
5875:10.1098/rspa.1954.0216
5838:10.1098/rspa.1954.0130
5801:10.1098/rspa.1953.0139
5727:10.1098/rspa.1956.0065
5509:
5302:
5254:
5031:
5001:
4953:
4624:
4568:
4548:
4501:
4432:
4386:
4345:
4314:
4287:
4238:
4033:
3787:
3761:
3735:
3706:
3683:
3466:leads to the equation
3460:
3437:
3411:
3384:
3383:{\displaystyle c_{1a}}
3351:
3192:
3169:
3026:leads to the equation
3020:
2993:
2947:
2927:
2900:
2873:
2815:
2724:
2649:
2620:
2594:
2554:
2528:
2485:
2274:
2222:
2168:
2142:
2107:
2064:
2044:
2024:
1978:
1977:{\displaystyle t\gg 1}
1952:
1906:
1877:
1678:
1655:
1635:
1597:
1445:
1425:
1394:
1374:
1354:
1334:
1264:
1190:
1119:
1099:
1079:
1059:
1039:
1019:
973:
860:
813:
789:
733:
642:
431:
345:
275:
174:
46:Taylor-Aris dispersion
5510:
5303:
5255:
5032:
5002:
4954:
4625:
4569:
4549:
4502:
4433:
4387:
4346:
4315:
4313:{\displaystyle c_{0}}
4288:
4239:
4034:
3788:
3762:
3736:
3707:
3684:
3461:
3438:
3412:
3410:{\displaystyle c_{1}}
3385:
3352:
3193:
3170:
3021:
2994:
2948:
2928:
2926:{\displaystyle c_{0}}
2901:
2899:{\displaystyle c_{0}}
2874:
2816:
2725:
2650:
2621:
2595:
2555:
2534:must be satisfied at
2529:
2486:
2275:
2223:
2169:
2143:
2108:
2065:
2045:
2025:
1979:
1953:
1907:
1878:
1679:
1656:
1636:
1598:
1446:
1426:
1395:
1375:
1355:
1335:
1265:
1191:
1125:direction satisfies:
1120:
1100:
1080:
1060:
1040:
1020:
974:
861:
814:
790:
734:
643:
432:
346:
276:
175:
5634:10.1017/jfm.2020.478
5580:Kirby, B.J. (2010).
5532:Probstein R (1994).
5315:
5267:
5044:
5014:
4966:
4644:
4578:
4558:
4511:
4442:
4396:
4355:
4324:
4297:
4251:
4049:
3800:
3771:
3745:
3734:{\displaystyle 2rdr}
3716:
3696:
3473:
3450:
3421:
3394:
3364:
3205:
3182:
3033:
3006:
2957:
2937:
2910:
2883:
2828:
2737:
2659:
2633:
2604:
2564:
2538:
2498:
2290:
2235:
2178:
2152:
2117:
2077:
2054:
2034:
1988:
1962:
1916:
1890:
1695:
1665:
1645:
1610:
1458:
1435:
1424:{\displaystyle t'=0}
1404:
1384:
1364:
1344:
1333:{\displaystyle u=2U}
1274:
1226:
1132:
1109:
1089:
1069:
1049:
1029:
989:
873:
826:
803:
746:
658:
447:
356:
291:
210:
82:
5934:, Imperial College.
5904:1980RSPTA.297...81B
5867:1954RSPSA.225..473T
5830:1954RSPSA.223..446T
5793:1953RSPSA.219..186T
5756:1989JFM...204...97F
5719:1956RSPSA.235...67A
5672:2020CTM....24.1054L
5625:2020JFM...899A..38H
4537:
4482:
3786:{\displaystyle r=1}
3760:{\displaystyle r=0}
3436:{\displaystyle r=0}
2619:{\displaystyle r=0}
2560:. Due to symmetry,
2553:{\displaystyle r=1}
5505:
5298:
5250:
5027:
4997:
4949:
4620:
4574:, but is given by
4564:
4544:
4525:
4497:
4470:
4428:
4382:
4341:
4310:
4283:
4234:
4029:
3783:
3757:
3731:
3702:
3679:
3456:
3433:
3407:
3380:
3347:
3188:
3165:
3016:
2989:
2943:
2933:is independent of
2923:
2896:
2869:
2811:
2720:
2645:
2616:
2590:
2550:
2524:
2481:
2270:
2218:
2164:
2138:
2103:
2060:
2040:
2020:
1974:
1948:
1902:
1873:
1677:{\displaystyle Pe}
1674:
1651:
1631:
1593:
1441:
1421:
1390:
1370:
1350:
1330:
1260:
1186:
1115:
1095:
1075:
1055:
1035:
1015:
969:
856:
809:
785:
729:
638:
427:
341:
271:
170:
5861:(1163): 473–477.
5824:(1155): 446–468.
5787:(1137): 186–203.
5593:978-0-521-11903-0
5500:
5459:
5431:
5382:
5346:
5245:
5201:
5171:
5105:
5070:
5055:
5025:
4947:
4901:
4881:
4856:
4836:
4814:
4789:
4754:
4719:
4605:
4567:{\displaystyle r}
4377:
4229:
4178:
4120:
4082:
4024:
3978:
3942:
3910:
3864:
3828:
3705:{\displaystyle r}
3674:
3630:
3600:
3534:
3499:
3484:
3345:
3286:
3191:{\displaystyle r}
3160:
3094:
3059:
3044:
3014:
2946:{\displaystyle r}
2879:. At this order,
2798:
2763:
2748:
2686:
2471:
2443:
2428:
2415:
2375:
2324:
2314:
2265:
2210:
2192:
2063:{\displaystyle t}
2043:{\displaystyle x}
1863:
1835:
1820:
1807:
1770:
1716:
1654:{\displaystyle D}
1591:
1563:
1538:
1497:
1444:{\displaystyle U}
1393:{\displaystyle c}
1373:{\displaystyle U}
1353:{\displaystyle a}
1167:
1156:
1118:{\displaystyle z}
1098:{\displaystyle L}
1078:{\displaystyle r}
1058:{\displaystyle z}
1038:{\displaystyle z}
1012:
967:
948:
903:
891:
850:
812:{\displaystyle a}
774:
718:
636:
617:
588:
563:
521:
509:
491:
477:
465:
389:
318:
231:
167:
105:
17:Taylor dispersion
5960:
5923:
5898:(1430): 81–133.
5886:
5849:
5812:
5775:
5738:
5692:
5691:
5666:(6): 1054–1069.
5655:
5649:
5648:
5646:
5636:
5604:
5598:
5597:
5577:
5571:
5570:
5564:
5556:
5544:
5538:
5537:
5529:
5514:
5512:
5511:
5506:
5501:
5499:
5498:
5497:
5484:
5483:
5482:
5473:
5472:
5462:
5460:
5455:
5454:
5453:
5440:
5432:
5430:
5429:
5428:
5415:
5414:
5413:
5401:
5400:
5390:
5388:
5384:
5383:
5378:
5377:
5376:
5363:
5347:
5345:
5337:
5336:
5335:
5319:
5307:
5305:
5304:
5299:
5282:
5281:
5259:
5257:
5256:
5251:
5246:
5244:
5243:
5242:
5229:
5228:
5227:
5218:
5217:
5207:
5202:
5200:
5192:
5191:
5190:
5177:
5172:
5170:
5162:
5161:
5160:
5147:
5142:
5141:
5111:
5107:
5106:
5104:
5096:
5095:
5094:
5081:
5071:
5069:
5058:
5056:
5048:
5036:
5034:
5033:
5028:
5026:
5021:
5006:
5004:
5003:
4998:
4981:
4980:
4958:
4956:
4955:
4950:
4948:
4946:
4945:
4944:
4931:
4930:
4929:
4920:
4919:
4909:
4907:
4903:
4902:
4897:
4896:
4887:
4882:
4877:
4876:
4875:
4862:
4857:
4852:
4851:
4842:
4837:
4832:
4831:
4822:
4815:
4810:
4809:
4808:
4795:
4790:
4788:
4780:
4779:
4778:
4762:
4760:
4756:
4755:
4750:
4749:
4740:
4735:
4734:
4720:
4715:
4707:
4687:
4686:
4656:
4655:
4629:
4627:
4626:
4621:
4616:
4615:
4606:
4601:
4596:
4595:
4573:
4571:
4570:
4565:
4553:
4551:
4550:
4545:
4533:
4521:
4506:
4504:
4503:
4498:
4496:
4478:
4466:
4452:
4437:
4435:
4434:
4429:
4424:
4419:
4418:
4406:
4391:
4389:
4388:
4383:
4378:
4376:
4365:
4350:
4348:
4347:
4342:
4334:
4319:
4317:
4316:
4311:
4309:
4308:
4292:
4290:
4289:
4284:
4279:
4274:
4273:
4261:
4243:
4241:
4240:
4235:
4230:
4228:
4227:
4226:
4225:
4208:
4207:
4206:
4197:
4196:
4186:
4184:
4180:
4179:
4177:
4176:
4175:
4162:
4161:
4160:
4151:
4150:
4140:
4121:
4119:
4118:
4106:
4105:
4104:
4091:
4083:
4081:
4080:
4068:
4067:
4066:
4053:
4038:
4036:
4035:
4030:
4025:
4023:
4022:
4021:
4008:
4007:
4006:
3997:
3996:
3986:
3984:
3980:
3979:
3974:
3973:
3972:
3959:
3943:
3941:
3933:
3932:
3931:
3918:
3911:
3909:
3908:
3907:
3894:
3893:
3892:
3883:
3882:
3872:
3870:
3866:
3865:
3860:
3859:
3858:
3845:
3829:
3827:
3819:
3818:
3817:
3804:
3792:
3790:
3789:
3784:
3766:
3764:
3763:
3758:
3740:
3738:
3737:
3732:
3711:
3709:
3708:
3703:
3688:
3686:
3685:
3680:
3675:
3673:
3672:
3671:
3658:
3657:
3656:
3647:
3646:
3636:
3631:
3629:
3621:
3620:
3619:
3606:
3601:
3599:
3591:
3590:
3589:
3576:
3571:
3570:
3540:
3536:
3535:
3533:
3525:
3524:
3523:
3510:
3500:
3498:
3487:
3485:
3477:
3465:
3463:
3462:
3457:
3442:
3440:
3439:
3434:
3416:
3414:
3413:
3408:
3406:
3405:
3390:is the value of
3389:
3387:
3386:
3381:
3379:
3378:
3356:
3354:
3353:
3348:
3346:
3344:
3336:
3335:
3334:
3321:
3316:
3315:
3303:
3302:
3287:
3282:
3274:
3254:
3253:
3217:
3216:
3197:
3195:
3194:
3189:
3174:
3172:
3171:
3166:
3161:
3159:
3151:
3150:
3149:
3136:
3131:
3130:
3100:
3096:
3095:
3093:
3085:
3084:
3083:
3070:
3060:
3058:
3047:
3045:
3037:
3025:
3023:
3022:
3017:
3015:
3010:
2998:
2996:
2995:
2990:
2985:
2980:
2979:
2967:
2952:
2950:
2949:
2944:
2932:
2930:
2929:
2924:
2922:
2921:
2905:
2903:
2902:
2897:
2895:
2894:
2878:
2876:
2875:
2870:
2853:
2852:
2840:
2839:
2820:
2818:
2817:
2812:
2804:
2800:
2799:
2797:
2789:
2788:
2787:
2774:
2764:
2762:
2751:
2749:
2741:
2729:
2727:
2726:
2721:
2713:
2712:
2697:
2696:
2687:
2682:
2677:
2676:
2654:
2652:
2651:
2646:
2625:
2623:
2622:
2617:
2599:
2597:
2596:
2591:
2577:
2559:
2557:
2556:
2551:
2533:
2531:
2530:
2525:
2511:
2490:
2488:
2487:
2482:
2477:
2473:
2472:
2470:
2462:
2454:
2444:
2442:
2431:
2429:
2421:
2416:
2414:
2413:
2412:
2399:
2395:
2394:
2384:
2376:
2374:
2366:
2358:
2353:
2352:
2325:
2320:
2315:
2313:
2305:
2297:
2279:
2277:
2276:
2271:
2266:
2261:
2227:
2225:
2224:
2219:
2211:
2206:
2198:
2193:
2188:
2173:
2171:
2170:
2165:
2147:
2145:
2144:
2139:
2134:
2129:
2128:
2112:
2110:
2109:
2104:
2093:
2069:
2067:
2066:
2061:
2049:
2047:
2046:
2041:
2029:
2027:
2026:
2021:
2016:
2011:
2010:
1998:
1983:
1981:
1980:
1975:
1957:
1955:
1954:
1949:
1944:
1939:
1938:
1926:
1911:
1909:
1908:
1903:
1882:
1880:
1879:
1874:
1869:
1865:
1864:
1862:
1854:
1846:
1836:
1834:
1823:
1821:
1813:
1808:
1806:
1805:
1804:
1791:
1787:
1786:
1776:
1771:
1769:
1761:
1753:
1748:
1747:
1717:
1715:
1707:
1699:
1683:
1681:
1680:
1675:
1660:
1658:
1657:
1652:
1640:
1638:
1637:
1632:
1627:
1622:
1621:
1602:
1600:
1599:
1594:
1592:
1587:
1579:
1564:
1559:
1551:
1539:
1534:
1533:
1519:
1510:
1498:
1496:
1492:
1487:
1486:
1476:
1468:
1450:
1448:
1447:
1442:
1430:
1428:
1427:
1422:
1414:
1399:
1397:
1396:
1391:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1326:
1325:
1313:
1308:
1269:
1267:
1266:
1261:
1250:
1239:
1195:
1193:
1192:
1187:
1185:
1184:
1169:
1168:
1160:
1157:
1152:
1151:
1142:
1124:
1122:
1121:
1116:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
1014:
1013:
1005:
999:
978:
976:
975:
970:
968:
966:
965:
964:
951:
950:
949:
941:
938:
937:
927:
925:
924:
923:
904:
902:
894:
893:
892:
884:
877:
865:
863:
862:
857:
852:
851:
843:
818:
816:
815:
810:
794:
792:
791:
786:
781:
776:
775:
767:
758:
757:
738:
736:
735:
730:
724:
720:
719:
714:
713:
708:
707:
697:
678:
677:
676:
647:
645:
644:
639:
637:
635:
634:
633:
620:
619:
618:
610:
607:
606:
596:
594:
590:
589:
587:
586:
585:
572:
571:
570:
565:
564:
556:
552:
551:
541:
522:
520:
512:
511:
510:
502:
495:
493:
492:
484:
478:
476:
468:
467:
466:
458:
451:
436:
434:
433:
428:
411:
391:
390:
382:
350:
348:
347:
342:
331:
320:
319:
311:
280:
278:
277:
272:
267:
266:
248:
240:
232:
230:
222:
214:
179:
177:
176:
171:
169:
168:
160:
154:
153:
144:
139:
138:
120:
119:
107:
106:
98:
89:
21:Taylor diffusion
5968:
5967:
5963:
5962:
5961:
5959:
5958:
5957:
5948:Fluid mechanics
5938:
5937:
5889:
5852:
5815:
5778:
5741:
5713:(1200): 67–77.
5704:
5701:
5696:
5695:
5657:
5656:
5652:
5606:
5605:
5601:
5594:
5579:
5578:
5574:
5557:
5546:
5545:
5541:
5531:
5530:
5526:
5521:
5489:
5485:
5474:
5464:
5463:
5445:
5441:
5420:
5416:
5402:
5392:
5391:
5368:
5364:
5355:
5351:
5338:
5324:
5320:
5313:
5312:
5270:
5265:
5264:
5234:
5230:
5219:
5209:
5208:
5193:
5182:
5178:
5163:
5152:
5148:
5133:
5097:
5086:
5082:
5076:
5072:
5062:
5042:
5041:
5012:
5011:
4969:
4964:
4963:
4936:
4932:
4921:
4911:
4910:
4888:
4867:
4863:
4843:
4823:
4820:
4816:
4800:
4796:
4781:
4767:
4763:
4741:
4726:
4725:
4721:
4708:
4675:
4647:
4642:
4641:
4635:
4607:
4587:
4576:
4575:
4556:
4555:
4514:
4509:
4508:
4489:
4459:
4445:
4440:
4439:
4410:
4399:
4394:
4393:
4369:
4353:
4352:
4327:
4322:
4321:
4300:
4295:
4294:
4265:
4254:
4249:
4248:
4217:
4213:
4209:
4198:
4188:
4187:
4167:
4163:
4152:
4142:
4141:
4132:
4128:
4111:
4107:
4096:
4092:
4073:
4069:
4058:
4054:
4047:
4046:
4013:
4009:
3998:
3988:
3987:
3964:
3960:
3951:
3947:
3934:
3923:
3919:
3899:
3895:
3884:
3874:
3873:
3850:
3846:
3837:
3833:
3820:
3809:
3805:
3798:
3797:
3769:
3768:
3743:
3742:
3714:
3713:
3694:
3693:
3663:
3659:
3648:
3638:
3637:
3622:
3611:
3607:
3592:
3581:
3577:
3562:
3526:
3515:
3511:
3505:
3501:
3491:
3471:
3470:
3448:
3447:
3446:Terms of order
3419:
3418:
3397:
3392:
3391:
3367:
3362:
3361:
3337:
3326:
3322:
3307:
3294:
3275:
3242:
3208:
3203:
3202:
3180:
3179:
3152:
3141:
3137:
3122:
3086:
3075:
3071:
3065:
3061:
3051:
3031:
3030:
3004:
3003:
3002:Terms of order
2971:
2960:
2955:
2954:
2935:
2934:
2913:
2908:
2907:
2886:
2881:
2880:
2844:
2831:
2826:
2825:
2790:
2779:
2775:
2769:
2765:
2755:
2735:
2734:
2704:
2688:
2668:
2657:
2656:
2631:
2630:
2602:
2601:
2562:
2561:
2536:
2535:
2496:
2495:
2463:
2455:
2449:
2445:
2435:
2404:
2400:
2386:
2385:
2367:
2359:
2344:
2306:
2298:
2288:
2287:
2233:
2232:
2176:
2175:
2150:
2149:
2120:
2115:
2114:
2075:
2074:
2052:
2051:
2032:
2031:
2002:
1991:
1986:
1985:
1960:
1959:
1930:
1919:
1914:
1913:
1888:
1887:
1855:
1847:
1841:
1837:
1827:
1796:
1792:
1778:
1777:
1762:
1754:
1739:
1708:
1700:
1693:
1692:
1663:
1662:
1643:
1642:
1613:
1608:
1607:
1580:
1552:
1526:
1512:
1511:
1478:
1477:
1469:
1456:
1455:
1433:
1432:
1407:
1402:
1401:
1382:
1381:
1362:
1361:
1342:
1341:
1317:
1301:
1272:
1271:
1243:
1232:
1224:
1223:
1219:
1143:
1130:
1129:
1107:
1106:
1087:
1086:
1067:
1066:
1047:
1046:
1027:
1026:
992:
987:
986:
956:
952:
929:
928:
908:
895:
878:
871:
870:
824:
823:
801:
800:
744:
743:
698:
689:
685:
661:
656:
655:
625:
621:
598:
597:
577:
573:
553:
543:
542:
533:
529:
513:
496:
469:
452:
445:
444:
404:
354:
353:
324:
289:
288:
258:
223:
215:
208:
207:
145:
130:
111:
80:
79:
61:
53:Poiseuille flow
38:Rutherford Aris
12:
11:
5:
5966:
5964:
5956:
5955:
5953:Fluid dynamics
5950:
5940:
5939:
5936:
5935:
5924:
5887:
5850:
5813:
5776:
5739:
5700:
5697:
5694:
5693:
5650:
5599:
5592:
5572:
5539:
5523:
5522:
5520:
5517:
5516:
5515:
5504:
5496:
5492:
5488:
5481:
5477:
5471:
5467:
5458:
5452:
5448:
5444:
5438:
5435:
5427:
5423:
5419:
5412:
5409:
5405:
5399:
5395:
5387:
5381:
5375:
5371:
5367:
5361:
5358:
5354:
5350:
5344:
5341:
5334:
5331:
5327:
5323:
5297:
5294:
5291:
5288:
5285:
5280:
5277:
5273:
5261:
5260:
5249:
5241:
5237:
5233:
5226:
5222:
5216:
5212:
5205:
5199:
5196:
5189:
5185:
5181:
5175:
5169:
5166:
5159:
5155:
5151:
5145:
5140:
5136:
5132:
5129:
5126:
5123:
5120:
5117:
5114:
5110:
5103:
5100:
5093:
5089:
5085:
5079:
5075:
5068:
5065:
5061:
5054:
5051:
5024:
5019:
4996:
4993:
4990:
4987:
4984:
4979:
4976:
4972:
4960:
4959:
4943:
4939:
4935:
4928:
4924:
4918:
4914:
4906:
4900:
4895:
4891:
4885:
4880:
4874:
4870:
4866:
4860:
4855:
4850:
4846:
4840:
4835:
4830:
4826:
4819:
4813:
4807:
4803:
4799:
4793:
4787:
4784:
4777:
4774:
4770:
4766:
4759:
4753:
4748:
4744:
4738:
4733:
4729:
4724:
4718:
4714:
4711:
4705:
4702:
4699:
4696:
4693:
4690:
4685:
4682:
4678:
4674:
4671:
4668:
4665:
4662:
4659:
4654:
4650:
4634:
4631:
4619:
4614:
4610:
4604:
4599:
4594:
4590:
4586:
4583:
4563:
4543:
4540:
4536:
4532:
4528:
4524:
4520:
4517:
4495:
4492:
4488:
4485:
4481:
4477:
4473:
4469:
4465:
4462:
4458:
4455:
4451:
4448:
4427:
4423:
4417:
4413:
4409:
4405:
4402:
4381:
4375:
4372:
4368:
4363:
4360:
4340:
4337:
4333:
4330:
4307:
4303:
4282:
4278:
4272:
4268:
4264:
4260:
4257:
4245:
4244:
4233:
4224:
4220:
4216:
4212:
4205:
4201:
4195:
4191:
4183:
4174:
4170:
4166:
4159:
4155:
4149:
4145:
4138:
4135:
4131:
4127:
4124:
4117:
4114:
4110:
4103:
4099:
4095:
4089:
4086:
4079:
4076:
4072:
4065:
4061:
4057:
4040:
4039:
4028:
4020:
4016:
4012:
4005:
4001:
3995:
3991:
3983:
3977:
3971:
3967:
3963:
3957:
3954:
3950:
3946:
3940:
3937:
3930:
3926:
3922:
3915:
3906:
3902:
3898:
3891:
3887:
3881:
3877:
3869:
3863:
3857:
3853:
3849:
3843:
3840:
3836:
3832:
3826:
3823:
3816:
3812:
3808:
3782:
3779:
3776:
3756:
3753:
3750:
3730:
3727:
3724:
3721:
3701:
3690:
3689:
3678:
3670:
3666:
3662:
3655:
3651:
3645:
3641:
3634:
3628:
3625:
3618:
3614:
3610:
3604:
3598:
3595:
3588:
3584:
3580:
3574:
3569:
3565:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3539:
3532:
3529:
3522:
3518:
3514:
3508:
3504:
3497:
3494:
3490:
3483:
3480:
3455:
3432:
3429:
3426:
3404:
3400:
3377:
3374:
3370:
3358:
3357:
3343:
3340:
3333:
3329:
3325:
3319:
3314:
3310:
3306:
3301:
3297:
3293:
3290:
3285:
3281:
3278:
3272:
3269:
3266:
3263:
3260:
3257:
3252:
3249:
3245:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3215:
3211:
3187:
3176:
3175:
3164:
3158:
3155:
3148:
3144:
3140:
3134:
3129:
3125:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3099:
3092:
3089:
3082:
3078:
3074:
3068:
3064:
3057:
3054:
3050:
3043:
3040:
3013:
2988:
2984:
2978:
2974:
2970:
2966:
2963:
2942:
2920:
2916:
2893:
2889:
2868:
2865:
2862:
2859:
2856:
2851:
2847:
2843:
2838:
2834:
2822:
2821:
2810:
2807:
2803:
2796:
2793:
2786:
2782:
2778:
2772:
2768:
2761:
2758:
2754:
2747:
2744:
2719:
2716:
2711:
2707:
2703:
2700:
2695:
2691:
2685:
2680:
2675:
2671:
2667:
2664:
2644:
2641:
2638:
2615:
2612:
2609:
2589:
2586:
2583:
2580:
2576:
2572:
2569:
2549:
2546:
2543:
2523:
2520:
2517:
2514:
2510:
2506:
2503:
2492:
2491:
2480:
2476:
2469:
2466:
2461:
2458:
2452:
2448:
2441:
2438:
2434:
2427:
2424:
2419:
2411:
2407:
2403:
2398:
2393:
2389:
2382:
2379:
2373:
2370:
2365:
2362:
2356:
2351:
2347:
2343:
2340:
2337:
2334:
2331:
2328:
2323:
2318:
2312:
2309:
2304:
2301:
2295:
2281:
2280:
2269:
2264:
2259:
2256:
2252:
2249:
2246:
2243:
2240:
2217:
2214:
2209:
2205:
2201:
2196:
2191:
2186:
2183:
2163:
2160:
2157:
2137:
2133:
2127:
2123:
2102:
2099:
2096:
2092:
2088:
2085:
2082:
2059:
2039:
2019:
2015:
2009:
2005:
2001:
1997:
1994:
1973:
1970:
1967:
1947:
1943:
1937:
1933:
1929:
1925:
1922:
1901:
1898:
1895:
1884:
1883:
1872:
1868:
1861:
1858:
1853:
1850:
1844:
1840:
1833:
1830:
1826:
1819:
1816:
1811:
1803:
1799:
1795:
1790:
1785:
1781:
1774:
1768:
1765:
1760:
1757:
1751:
1746:
1742:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1714:
1711:
1706:
1703:
1673:
1670:
1650:
1630:
1626:
1620:
1616:
1604:
1603:
1590:
1586:
1583:
1577:
1574:
1571:
1567:
1562:
1558:
1555:
1549:
1546:
1542:
1537:
1532:
1529:
1525:
1522:
1518:
1515:
1508:
1505:
1501:
1495:
1491:
1485:
1481:
1475:
1472:
1466:
1463:
1440:
1420:
1417:
1413:
1410:
1389:
1369:
1349:
1329:
1324:
1320:
1316:
1312:
1307:
1304:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1259:
1256:
1253:
1249:
1246:
1242:
1238:
1235:
1231:
1218:
1215:
1198:
1197:
1183:
1180:
1175:
1172:
1166:
1163:
1155:
1150:
1146:
1140:
1137:
1114:
1094:
1074:
1054:
1034:
1011:
1008:
1002:
998:
995:
980:
979:
963:
959:
955:
947:
944:
936:
932:
922:
919:
916:
911:
907:
901:
898:
890:
887:
881:
855:
849:
846:
840:
837:
834:
831:
808:
784:
780:
773:
770:
764:
761:
756:
753:
740:
739:
728:
723:
717:
712:
706:
703:
695:
692:
688:
684:
681:
675:
672:
669:
664:
649:
648:
632:
628:
624:
616:
613:
605:
601:
593:
584:
580:
576:
569:
562:
559:
550:
546:
539:
536:
532:
528:
525:
519:
516:
508:
505:
499:
490:
487:
481:
475:
472:
464:
461:
455:
438:
437:
426:
423:
420:
417:
414:
410:
407:
403:
400:
397:
394:
388:
385:
379:
376:
373:
370:
367:
364:
361:
351:
340:
337:
334:
330:
327:
323:
317:
314:
308:
305:
302:
299:
296:
282:
281:
270:
265:
261:
257:
254:
251:
247:
243:
239:
235:
229:
226:
221:
218:
181:
180:
166:
163:
157:
152:
148:
143:
137:
133:
129:
126:
123:
118:
114:
110:
104:
101:
95:
92:
88:
60:
57:
34:Peclet numbers
13:
10:
9:
6:
4:
3:
2:
5965:
5954:
5951:
5949:
5946:
5945:
5943:
5933:
5929:
5925:
5921:
5917:
5913:
5909:
5905:
5901:
5897:
5893:
5888:
5884:
5880:
5876:
5872:
5868:
5864:
5860:
5856:
5851:
5847:
5843:
5839:
5835:
5831:
5827:
5823:
5819:
5814:
5810:
5806:
5802:
5798:
5794:
5790:
5786:
5782:
5777:
5773:
5769:
5765:
5761:
5757:
5753:
5749:
5745:
5740:
5736:
5732:
5728:
5724:
5720:
5716:
5712:
5708:
5703:
5702:
5699:Other sources
5698:
5689:
5685:
5681:
5677:
5673:
5669:
5665:
5661:
5654:
5651:
5645:
5640:
5635:
5630:
5626:
5622:
5618:
5614:
5610:
5603:
5600:
5595:
5589:
5585:
5584:
5576:
5573:
5568:
5562:
5554:
5550:
5543:
5540:
5535:
5528:
5525:
5518:
5502:
5494:
5490:
5479:
5475:
5469:
5456:
5450:
5446:
5442:
5436:
5433:
5425:
5421:
5410:
5407:
5403:
5397:
5385:
5379:
5373:
5369:
5365:
5359:
5356:
5352:
5348:
5342:
5332:
5329:
5325:
5311:
5310:
5309:
5292:
5289:
5286:
5278:
5275:
5271:
5247:
5239:
5235:
5224:
5220:
5214:
5203:
5197:
5187:
5183:
5173:
5167:
5157:
5153:
5138:
5134:
5130:
5127:
5124:
5118:
5115:
5112:
5108:
5101:
5091:
5087:
5077:
5073:
5066:
5052:
5049:
5040:
5039:
5038:
5022:
5017:
5008:
4991:
4988:
4985:
4977:
4974:
4970:
4941:
4937:
4926:
4922:
4916:
4904:
4898:
4893:
4889:
4883:
4878:
4872:
4868:
4864:
4858:
4853:
4848:
4844:
4838:
4833:
4828:
4824:
4817:
4811:
4805:
4801:
4797:
4791:
4785:
4775:
4772:
4768:
4757:
4751:
4746:
4742:
4736:
4731:
4727:
4722:
4716:
4712:
4709:
4703:
4697:
4694:
4691:
4683:
4680:
4676:
4672:
4666:
4663:
4660:
4652:
4648:
4640:
4639:
4638:
4632:
4630:
4617:
4612:
4608:
4602:
4597:
4592:
4588:
4584:
4581:
4561:
4541:
4538:
4534:
4530:
4526:
4522:
4518:
4515:
4493:
4490:
4486:
4483:
4479:
4475:
4471:
4467:
4463:
4460:
4456:
4453:
4449:
4446:
4425:
4421:
4415:
4411:
4407:
4403:
4400:
4373:
4370:
4366:
4361:
4358:
4338:
4335:
4331:
4328:
4305:
4301:
4280:
4276:
4270:
4266:
4262:
4258:
4255:
4231:
4222:
4214:
4203:
4199:
4193:
4181:
4172:
4168:
4164:
4157:
4153:
4147:
4143:
4136:
4133:
4129:
4125:
4122:
4115:
4112:
4101:
4097:
4087:
4084:
4077:
4074:
4063:
4059:
4045:
4044:
4043:
4026:
4018:
4014:
4003:
3999:
3993:
3981:
3975:
3969:
3965:
3961:
3955:
3952:
3948:
3944:
3938:
3928:
3924:
3904:
3900:
3889:
3885:
3879:
3867:
3861:
3855:
3851:
3847:
3841:
3838:
3834:
3830:
3824:
3814:
3810:
3796:
3795:
3794:
3780:
3777:
3774:
3754:
3751:
3748:
3728:
3725:
3722:
3719:
3699:
3676:
3668:
3664:
3653:
3649:
3643:
3632:
3626:
3616:
3612:
3602:
3596:
3586:
3582:
3567:
3563:
3559:
3556:
3553:
3547:
3544:
3541:
3537:
3530:
3520:
3516:
3506:
3502:
3495:
3481:
3478:
3469:
3468:
3467:
3453:
3444:
3430:
3427:
3424:
3402:
3398:
3375:
3372:
3368:
3341:
3331:
3327:
3312:
3308:
3304:
3299:
3295:
3291:
3283:
3279:
3276:
3270:
3264:
3261:
3258:
3250:
3247:
3243:
3239:
3233:
3230:
3227:
3224:
3221:
3213:
3209:
3201:
3200:
3199:
3185:
3162:
3156:
3146:
3142:
3127:
3123:
3119:
3116:
3113:
3107:
3104:
3101:
3097:
3090:
3080:
3076:
3066:
3062:
3055:
3041:
3038:
3029:
3028:
3027:
3011:
3000:
2986:
2982:
2976:
2972:
2968:
2964:
2961:
2940:
2918:
2914:
2891:
2887:
2863:
2860:
2857:
2849:
2845:
2841:
2836:
2832:
2808:
2805:
2801:
2794:
2784:
2780:
2770:
2766:
2759:
2745:
2742:
2733:
2732:
2731:
2717:
2714:
2709:
2705:
2701:
2698:
2693:
2689:
2683:
2678:
2673:
2669:
2665:
2662:
2642:
2639:
2636:
2627:
2613:
2610:
2607:
2587:
2584:
2581:
2574:
2570:
2547:
2544:
2541:
2521:
2518:
2515:
2508:
2504:
2478:
2474:
2467:
2459:
2450:
2446:
2439:
2425:
2422:
2417:
2409:
2405:
2396:
2391:
2380:
2377:
2371:
2363:
2349:
2345:
2341:
2338:
2335:
2329:
2326:
2321:
2316:
2310:
2302:
2293:
2286:
2285:
2284:
2267:
2262:
2257:
2254:
2250:
2247:
2244:
2241:
2238:
2231:
2230:
2229:
2215:
2212:
2207:
2203:
2199:
2194:
2189:
2184:
2181:
2161:
2158:
2155:
2135:
2131:
2125:
2121:
2100:
2097:
2094:
2090:
2086:
2083:
2080:
2071:
2057:
2037:
2017:
2013:
2007:
2003:
1999:
1995:
1992:
1971:
1968:
1965:
1945:
1941:
1935:
1931:
1927:
1923:
1920:
1899:
1896:
1893:
1870:
1866:
1859:
1851:
1842:
1838:
1831:
1817:
1814:
1809:
1801:
1797:
1788:
1783:
1772:
1766:
1758:
1744:
1740:
1736:
1733:
1730:
1724:
1721:
1718:
1712:
1704:
1691:
1690:
1689:
1687:
1686:Peclet number
1671:
1668:
1648:
1628:
1624:
1618:
1614:
1588:
1584:
1581:
1575:
1572:
1569:
1565:
1560:
1556:
1553:
1547:
1544:
1540:
1535:
1530:
1527:
1523:
1520:
1516:
1513:
1506:
1503:
1499:
1493:
1489:
1483:
1479:
1473:
1470:
1464:
1461:
1454:
1453:
1452:
1438:
1418:
1415:
1411:
1408:
1387:
1367:
1347:
1322:
1314:
1310:
1305:
1302:
1295:
1292:
1286:
1283:
1280:
1277:
1254:
1251:
1247:
1244:
1240:
1236:
1233:
1216:
1214:
1212:
1208:
1202:
1173:
1170:
1161:
1153:
1148:
1144:
1138:
1135:
1128:
1127:
1126:
1112:
1092:
1072:
1052:
1032:
1006:
1000:
996:
993:
983:
961:
957:
942:
934:
909:
905:
899:
885:
869:
868:
867:
853:
844:
838:
835:
832:
829:
820:
806:
798:
797:PĂ©clet number
782:
778:
768:
762:
759:
726:
721:
715:
710:
693:
690:
686:
682:
679:
662:
654:
653:
652:
630:
626:
611:
603:
591:
582:
578:
574:
567:
557:
548:
544:
537:
534:
530:
526:
523:
517:
503:
485:
479:
473:
459:
443:
442:
441:
421:
418:
415:
408:
405:
401:
395:
383:
377:
371:
368:
365:
359:
352:
335:
328:
325:
321:
312:
306:
300:
294:
287:
286:
285:
268:
263:
255:
252:
249:
241:
233:
227:
219:
206:
205:
204:
202:
198:
194:
190:
186:
185:concentration
150:
146:
141:
135:
131:
127:
124:
116:
112:
108:
93:
90:
78:
77:
76:
74:
70:
66:
58:
56:
54:
49:
47:
43:
42:Peclet number
39:
35:
31:
26:
22:
18:
5931:
5895:
5891:
5858:
5854:
5821:
5817:
5784:
5780:
5747:
5743:
5710:
5706:
5663:
5659:
5653:
5644:11343/274191
5616:
5612:
5602:
5582:
5575:
5548:
5542:
5533:
5527:
5262:
5009:
4961:
4636:
4246:
4041:
3691:
3445:
3359:
3177:
3001:
2823:
2628:
2493:
2282:
2072:
1885:
1605:
1220:
1209:modelled by
1207:porous media
1203:
1199:
984:
981:
821:
741:
650:
439:
283:
196:
188:
182:
72:
68:
64:
62:
50:
45:
30:G. I. Taylor
20:
16:
15:
5926:Mestel. J.
5308:as follows
1211:Darcy's law
193:diffusivity
59:Description
5942:Categories
5519:References
5037:, we find
1217:Derivation
1025:for given
25:shear flow
5920:121853893
5772:123719494
5688:225262971
5561:cite book
5491:ξ
5487:∂
5466:∂
5437:−
5422:ξ
5418:∂
5394:∂
5349:−
5343:τ
5340:∂
5322:∂
5293:τ
5287:ξ
5236:ξ
5232:∂
5211:∂
5204:−
5198:τ
5195:∂
5180:∂
5168:ξ
5165:∂
5150:∂
5128:−
5099:∂
5084:∂
5064:∂
5060:∂
5023:ϵ
5018:ϵ
4992:τ
4986:ξ
4938:ξ
4934:∂
4913:∂
4859:−
4786:ξ
4783:∂
4765:∂
4737:−
4698:τ
4692:ξ
4667:τ
4661:ξ
4603:ϵ
4539:∼
4523:−
4484:−
4454:−
4408:≫
4362:∼
4336:≫
4293:in which
4263:≫
4211:∂
4190:∂
4109:∂
4094:∂
4071:∂
4056:∂
4011:∂
3990:∂
3936:∂
3921:∂
3914:⇒
3901:ξ
3897:∂
3876:∂
3825:τ
3822:∂
3807:∂
3665:ξ
3661:∂
3640:∂
3633:−
3627:τ
3624:∂
3609:∂
3597:ξ
3594:∂
3579:∂
3557:−
3528:∂
3513:∂
3493:∂
3489:∂
3454:ϵ
3342:ξ
3339:∂
3324:∂
3305:−
3265:τ
3259:ξ
3234:τ
3222:ξ
3157:ξ
3154:∂
3139:∂
3117:−
3088:∂
3073:∂
3053:∂
3049:∂
3012:ϵ
2969:≫
2864:τ
2858:ξ
2792:∂
2777:∂
2757:∂
2753:∂
2718:⋯
2702:ϵ
2684:ϵ
2640:≪
2637:ϵ
2579:∂
2568:∂
2513:∂
2502:∂
2465:∂
2457:∂
2437:∂
2433:∂
2406:ξ
2402:∂
2388:∂
2381:ϵ
2372:ξ
2369:∂
2361:∂
2339:−
2322:ϵ
2311:τ
2308:∂
2300:∂
2294:ϵ
2263:ϵ
2255:ξ
2245:ϵ
2239:τ
2213:≫
2208:ϵ
2195:∼
2185:∼
2159:≪
2156:ϵ
2148:), where
2098:≫
2095:ϵ
2084:∼
2000:≫
1969:≫
1928:∼
1897:∼
1857:∂
1849:∂
1829:∂
1825:∂
1794:∂
1780:∂
1764:∂
1756:∂
1734:−
1710:∂
1702:∂
1521:−
1296:−
1255:θ
1165:¯
1139:≫
1010:¯
1001:≪
958:ξ
954:∂
946:¯
931:∂
897:∂
889:¯
880:∂
848:¯
839:−
830:ξ
772:¯
623:∂
615:¯
600:∂
561:¯
515:∂
507:¯
498:∂
489:¯
471:∂
463:¯
454:∂
387:¯
316:¯
260:∇
246:∇
242:⋅
225:∂
217:∂
165:^
128:−
103:^
5883:97701431
5846:96182418
5809:97372019
5735:95229777
4535:′
4519:′
4494:′
4480:′
4464:′
4450:′
4404:′
4374:′
4332:′
4259:′
4219:′
4116:′
4078:′
2965:′
2073:Suppose
1996:′
1924:′
1557:′
1531:′
1517:′
1474:′
1412:′
1360:, where
1306:′
1248:′
1237:′
997:′
409:′
329:′
191:and its
5900:Bibcode
5863:Bibcode
5826:Bibcode
5789:Bibcode
5752:Bibcode
5715:Bibcode
5668:Bibcode
5621:Bibcode
1684:is the
1105:in the
795:is the
63:We use
5918:
5881:
5844:
5807:
5770:
5750:: 97.
5733:
5686:
5590:
4962:where
3360:where
2629:Since
1606:where
742:where
5916:S2CID
5879:S2CID
5842:S2CID
5805:S2CID
5768:S2CID
5731:S2CID
5684:S2CID
5588:ISBN
5567:link
5457:2880
183:The
5908:doi
5896:297
5871:doi
5859:225
5834:doi
5822:223
5797:doi
5785:219
5760:doi
5748:204
5723:doi
5711:235
5676:doi
5639:hdl
5629:doi
5617:899
3767:to
3417:at
2600:at
195:is
19:or
5944::
5930:,
5914:.
5906:.
5894:.
5877:.
5869:.
5857:.
5840:.
5832:.
5820:.
5803:.
5795:.
5783:.
5766:.
5758:.
5746:.
5729:.
5721:.
5709:.
5682:.
5674:.
5664:24
5662:.
5637:.
5627:.
5619:.
5615:.
5611:.
5563:}}
5559:{{
5551:.
5380:48
4812:32
4165:48
3976:48
3862:48
2809:0.
2626:.
2070:.
1213:.
716:48
575:48
203::
48:.
5922:.
5910::
5902::
5885:.
5873::
5865::
5848:.
5836::
5828::
5811:.
5799::
5791::
5774:.
5762::
5754::
5737:.
5725::
5717::
5690:.
5678::
5670::
5647:.
5641::
5631::
5623::
5596:.
5569:)
5555:.
5536:.
5503:.
5495:3
5480:0
5476:c
5470:3
5451:3
5447:e
5443:P
5434:=
5426:2
5411:a
5408:1
5404:c
5398:2
5386:)
5374:2
5370:e
5366:P
5360:+
5357:1
5353:(
5333:a
5330:1
5326:c
5296:)
5290:,
5284:(
5279:a
5276:1
5272:c
5248:.
5240:2
5225:1
5221:c
5215:2
5188:1
5184:c
5174:+
5158:2
5154:c
5144:)
5139:2
5135:r
5131:2
5125:1
5122:(
5119:e
5116:P
5113:=
5109:)
5102:r
5092:3
5088:c
5078:r
5074:(
5067:r
5053:r
5050:1
4995:)
4989:,
4983:(
4978:a
4975:2
4971:c
4942:2
4927:0
4923:c
4917:2
4905:)
4899:8
4894:8
4890:r
4884:+
4879:8
4873:6
4869:r
4865:5
4854:2
4849:4
4845:r
4839:+
4834:6
4829:2
4825:r
4818:(
4806:2
4802:e
4798:P
4792:+
4776:a
4773:1
4769:c
4758:)
4752:2
4747:4
4743:r
4732:2
4728:r
4723:(
4717:4
4713:e
4710:P
4704:+
4701:)
4695:,
4689:(
4684:a
4681:2
4677:c
4673:=
4670:)
4664:,
4658:(
4653:2
4649:c
4618:.
4613:1
4609:c
4598:+
4593:0
4589:c
4585:=
4582:c
4562:r
4542:a
4531:s
4527:x
4516:x
4491:t
4487:U
4476:s
4472:x
4468:=
4461:t
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4447:x
4426:D
4422:/
4416:2
4412:a
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4380:)
4371:t
4367:D
4359:x
4339:a
4329:x
4306:0
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4281:D
4277:/
4271:2
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4256:t
4232:.
4223:2
4215:x
4204:0
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4173:2
4169:D
4158:2
4154:a
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4130:(
4126:D
4123:=
4113:x
4102:0
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4088:U
4085:+
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4027:.
4019:2
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4004:0
4000:c
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3856:2
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3831:=
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3781:1
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3752:=
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3729:r
3726:d
3723:r
3720:2
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3677:.
3669:2
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3644:2
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3603:+
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3531:r
3521:2
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3507:r
3503:(
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3318:)
3313:4
3309:r
3300:2
3296:r
3292:2
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3262:,
3256:(
3251:a
3248:1
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3231:,
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3219:(
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3163:.
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2806:=
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2767:(
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2699:+
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2679:+
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2666:=
2663:c
2643:1
2614:0
2611:=
2608:r
2588:0
2585:=
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2575:/
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2475:)
2468:r
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2364:c
2355:)
2350:2
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2014:/
2008:2
2004:a
1993:t
1972:1
1966:t
1946:D
1942:/
1936:2
1932:a
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1900:1
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1867:)
1860:r
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1832:r
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1750:)
1745:2
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1719:+
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1585:a
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1576:=
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1570:P
1566:,
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1500:,
1494:D
1490:/
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1465:=
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1179:P
1174:a
1171:=
1162:w
1154:D
1149:2
1145:a
1136:L
1113:z
1093:L
1073:r
1053:z
1033:z
1007:c
994:c
962:2
943:c
935:2
921:f
918:f
915:e
910:D
906:=
900:t
886:c
854:t
845:w
836:z
833:=
807:a
783:D
779:/
769:w
763:a
760:=
755:e
752:P
727:,
722:)
711:2
705:e
702:P
694:+
691:1
687:(
683:D
680:=
674:f
671:f
668:e
663:D
631:2
627:z
612:c
604:2
592:)
583:2
579:D
568:2
558:w
549:2
545:a
538:+
535:1
531:(
527:D
524:=
518:z
504:c
486:w
480:+
474:t
460:c
425:)
422:z
419:,
416:r
413:(
406:c
402:+
399:)
396:z
393:(
384:c
378:=
375:)
372:z
369:,
366:r
363:(
360:c
339:)
336:r
333:(
326:w
322:+
313:w
307:=
304:)
301:r
298:(
295:w
269:c
264:2
256:D
253:=
250:c
238:w
234:+
228:t
220:c
197:D
189:c
162:z
156:)
151:2
147:a
142:/
136:2
132:r
125:1
122:(
117:0
113:w
109:=
100:z
94:w
91:=
87:u
73:a
69:r
65:z
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