20:
3080:
1502:
531:
2853:
3250:
3638:. The conventional Stokes number will significantly underestimate the drag force for large particle free-stream Reynolds numbers. Thus overestimating the tendency for particles to depart from the fluid flow direction. This will lead to errors in subsequent calculations or experimental comparisons.
542:
The Stokes number provides a means of estimating the quality of PIV data sets, as previously discussed. However, a definition of a characteristic velocity or length scale may not be evident in all applications. Thus, a deeper insight of how a tracking delay arises could be drawn by simply defining
23:
Illustration of the effect of varying the Stokes number. Orange and green trajectories are for small and large Stokes numbers, respectively. Orange curve is trajectory of particle with Stokes number less than one that follows the streamlines (blue), while green curve is for a Stokes number greater
3608:
1982:
2715:
890:
2709:
The preceding analysis will not be accurate in the ultra-Stokesian regime. i.e. if the particle
Reynolds number is much greater than unity. Assuming a Mach number much less than unity, a generalized form of the Stokes number was demonstrated by Israel & Rosner.
3108:
1037:
2104:
3796:
1514:
The bias error in particle tracking discussed in the previous section is evident in the frequency domain, but it can be difficult to appreciate in cases where the particle motion is being tracked to perform flow field measurements (like in
3446:
2662:
Although a shock wave is the worst-case scenario of abrupt deceleration of a flow, it illustrates the effect of particle tracking error in PIV, which results in a blurring of the velocity fields acquired at the length scales of order
2275:
2969:
1497:
The cut-off frequency and the particle transfer function, plotted on the side panel, allows for the assessment of PIV error in unsteady flow applications and its effect on turbulence spectral quantities and kinetic energy.
215:
is the characteristic dimension of the obstacle (typically its diameter) or a characteristic length scale in the flow (like boundary layer thickness). A particle with a low Stokes number follows fluid streamlines (perfect
1801:
3072:
2360:
1207:
1396:
1744:
1116:
312:
3355:
125:
2657:
1492:
2564:
3907:
417:
of the fluid). For acceptable tracing accuracy, the particle response time should be faster than the smallest time scale of the flow. Smaller Stokes numbers represent better tracing accuracy; for
764:
2848:{\displaystyle {\text{Stk}}_{\text{e}}={\text{Stk}}{\frac {24}{{\text{Re}}_{o}}}\int _{0}^{{\text{Re}}_{o}}{\frac {d{\text{Re}}^{\prime }}{C_{D}({\text{Re}}^{\prime }){\text{Re}}^{\prime }}}}
1301:
692:
753:
1587:
538:
advecting in a stagnation point flow field (gray streamlines). Note that the 1 mm particles crash onto the stagnation plate whereas the 0.1 mm particles follow the streamlines.
2699:
2439:
2401:
4409:
4086:
Israel, R.; Rosner, D. E. (1982-09-20). "Use of a
Generalized Stokes Number to Determine the Aerodynamic Capture Efficiency of Non-Stokesian Particles from a Compressible Gas Flow".
2608:
901:
3009:
2515:
517:
3287:
3245:{\displaystyle \psi ({\text{Re}}_{o})={\frac {24}{{\text{Re}}_{o}}}\int _{0}^{{\text{Re}}_{o}}{\frac {d{\text{Re}}^{\prime }}{C_{D}({\text{Re}}^{\prime }){\text{Re}}^{\prime }}}}
483:
449:
3841:
is speed, and the subscript 0 indicates conditions far upstream of the nozzle. The characteristic distance is the diameter of the nozzle. Here the Stokes number is calculated,
3439:
2884:
2146:
1990:
3381:
413:
experiments where very small particles are entrained in turbulent flows and optically observed to determine the speed and direction of fluid movement (also known as the
342:
3652:
1637:
613:
577:
400:
3636:
2891:
3937:
3401:
3100:
1614:
1426:
373:
213:
186:
155:
2471:
1793:
3977:
3957:
3839:
3819:
2154:
1767:
4402:
3016:
1749:
To convert the velocity as a function of time to a particle velocity distribution as a function of distance, let's assume a 1-dimensional velocity jump in the
579:
will encounter a variable fluid velocity field as it advects. Let's assume the velocity of the fluid, in the
Lagrangian frame of reference of the particle, is
24:
than one, and so the particle does not follow the streamlines. That particle collides with one of the obstacles (brown circles) at point shown in yellow.
3603:{\displaystyle \psi ({\text{Re}}_{o})={\frac {3({\sqrt {c}}{\text{Re}}_{o}^{1/3}-\arctan({\sqrt {c}}{\text{Re}}_{o}^{1/3}))}{c^{3/2}{\text{Re}}_{o}}}}
4764:
4395:
1977:{\displaystyle x_{\text{particle}}=\int _{0}^{\Delta t}v_{p}(t)dt=\int _{0}^{\Delta t}(V_{u}-\Delta V)dt+\int _{0}^{\Delta t}\Delta Ve^{-t/t_{0}}dt}
2283:
1124:
1307:
1648:
1045:
241:
4418:
235:
is inversely proportional to the
Reynolds number itself. In that case, the characteristic time of the particle can be written as
3292:
62:
2613:
1434:
2520:
4306:
4287:
4067:
4017:
3847:
3646:
For example, the selective capture of particles by an aligned, thin-walled circular nozzle is given by
Belyaev and Levin as:
885:{\displaystyle m_{p}=\rho _{p}{\frac {4}{3}}\pi {\bigg (}{\frac {d_{p}}{2}}{\bigg )}^{3}=\rho _{p}{\frac {\pi d_{p}^{3}}{6}}}
615:. It is the difference between these velocities that will generate the drag force necessary to correct the particle path:
4316:
Snyder, WH; Lumley, JL (1971). "Some
Measurements of Particle Velocity Autocorrelation Functions in a Turbulent Flow".
1519:). A simple but insightful solution to the above-mentioned differential equation is possible when the forcing function
220:), while a particle with a large Stokes number is dominated by its inertia and continues along its initial trajectory.
4042:
19:
1222:
621:
703:
534:
Comparison between two different particle sizes for tracking accuracy for PIV. Simulated particles (blue dots) of
1522:
543:
the differential equations of a particle in the Stokes regime. A particle moving with the fluid at some velocity
2666:
2406:
2403:
from the shock. In practice, this means a shock wave would look, to a PIV system, blurred by approximately this
2368:
1032:{\displaystyle {\frac {dv_{p}(t)}{dt}}={\frac {F_{D}}{m_{p}}}={\frac {18\mu }{{d_{p}}^{2}\rho _{p}}}\Delta v(t)}
3383:. Thus as expected there correction factor is unity in the Stokesian drag regime. Wessel & Righi evaluated
37:
1401:
The solution above, in the frequency domain, characterizes a first-order system with a characteristic time of
4564:
2569:
2976:
2476:
4754:
488:
4178:
L, Schiller & Z. Naumann (1935). "Uber die grundlegenden
Berechnung bei der Schwerkraftaufbereitung".
3257:
454:
420:
3406:
1516:
410:
2860:
2112:
2099:{\displaystyle x_{\text{particle}}=\Delta t(V_{u}-\Delta V)+\Delta t\Delta V(1-e^{-\Delta t/t_{0}})}
4641:
4624:
161:
of the particle (the time constant in the exponential decay of the particle velocity due to drag),
4769:
4619:
3791:{\displaystyle c/c_{0}=1+(u_{0}/u-1)\left(1-{\frac {1}{1+\mathrm {Stk} (2+0.617u/u_{0})}}\right)}
3360:
4519:
4008:
Raffel, M.; Willert, C. E.; Scarano, F.; Kahler, C. J.; Wereley, S. T.; Kompenhans, J. (2018).
320:
45:
4197:
Belyaev, SP; Levin, LM (1974). "Techniques for collection of representative aerosol samples".
1619:
1589:
is a
Heaviside step function; representing particles going through a shockwave. In this case,
582:
546:
378:
3615:
2365:
This means the particle velocity would be settled to within 5% of the downstream velocity at
4474:
4387:
4366:
4325:
4206:
4142:
4095:
3915:
3386:
3085:
1592:
1404:
351:
191:
164:
133:
41:
4479:
2270:{\displaystyle x_{{\text{particle}},95\%}=3t_{0}(V_{u}-\Delta V)+3t_{0}\Delta V(1-e^{-3})}
409:
In experimental fluid dynamics, the Stokes number is a measure of flow tracer fidelity in
52:. The Stokes number is defined as the ratio of the characteristic time of a particle (or
8:
4234:"Terminal fall velocity: the legacy of Stokes from the perspective of fluvial hydraulics"
2450:
1772:
525:
4544:
4370:
4329:
4210:
4146:
4099:
1795:
is positioned where the shock wave is, and then integrate the previous equation to get:
451:, particles will detach from a flow especially where the flow decelerates abruptly. For
4728:
4514:
4454:
4341:
4260:
4233:
3962:
3942:
3824:
3804:
3079:
1752:
16:
Dimensionless number characterising the behavior of particles suspended in a fluid flow
4759:
4683:
4648:
4429:
4345:
4302:
4283:
4218:
4160:
4131:"Generalized Correlations for Inertial Impaction of Particles on a Circular Cylinder"
4111:
4063:
4038:
4013:
1213:
403:
4379:
4354:
4733:
4604:
4589:
4554:
4464:
4374:
4333:
4255:
4245:
4214:
4150:
4103:
535:
232:
4609:
3991:– For the drag force in fluids on particles whose Reynolds number is less than one
1505:
Bode plot of a propylene glycol particle in air for different particle diameters.
4698:
4678:
4636:
4631:
4459:
3441:
from the empirical correlation for drag on a sphere from
Schiller & Naumann.
228:
158:
2964:{\displaystyle {\text{Re}}_{o}={\frac {\rho _{g}|\mathbf {u} |d_{p}}{\mu _{g}}}}
4708:
4688:
4673:
4668:
4614:
4599:
4574:
4569:
4559:
4539:
4529:
4494:
4439:
4299:
Aerosol technology: properties, behavior, and measurement of airborne particles
3988:
4337:
4155:
4130:
4107:
4748:
4718:
4713:
4703:
4658:
4653:
4594:
4534:
4524:
4504:
4499:
4469:
4434:
4164:
4115:
53:
4723:
4663:
4579:
4549:
4509:
4484:
4444:
4250:
3102:
describes the non-Stokesian drag correction factor for a spherical particle
414:
2473:
at a stagnation temperature of 298 K. A propylene glycol particle of
895:
Thus, the particle acceleration can be found through Newton's second law:
4584:
4489:
4449:
3067:{\displaystyle {\text{Stk}}_{e}={\text{Stk}}\cdot \psi ({\text{Re}}_{o})}
3011:
was defined by; this describes the non-Stokesian drag correction factor,
2445:
224:
49:
4355:"Reynolds number scaling of particle clustering in turbulent aerosols"
4057:
1212:
The first-order differential equation above can be solved through the
526:
Relaxation time and tracking error in particle image velocimetry (PIV)
2355:{\displaystyle x_{{\text{particle}},95\%}=3t_{0}(V_{u}-0.05\Delta V)}
1202:{\displaystyle {\frac {dv_{p}(t)}{dt}}={\frac {1}{t_{0}}}\Delta v(t)}
217:
3641:
3254:
Considering the limiting particle free-stream
Reynolds numbers, as
188:
is the fluid velocity of the flow well away from the obstacle, and
1391:{\displaystyle {\frac {v_{p}(s)}{v_{f}(s)}}={\frac {1}{t_{0}s+1}}}
530:
4012:(3rd ed.). Switzerland : Springer International Publishing.
345:
1501:
4058:
Cameron Tropea; Alexander Yarin; John Foss, eds. (2007-10-09).
2659:(which would, in most cases, yield unacceptable PIV results).
1739:{\displaystyle v_{p}(t)=(V_{u}-\Delta V)+\Delta Ve^{-t/t_{0}}}
1111:{\displaystyle t_{0}={\frac {\rho _{p}d_{p}^{2}}{18\mu _{g}}}}
307:{\displaystyle t_{0}={\frac {\rho _{p}d_{p}^{2}}{18\mu _{g}}}}
56:) to a characteristic time of the flow or of an obstacle, or
4417:
4177:
3350:{\displaystyle C_{D}({\text{Re}}_{o})\to 24/{\text{Re}}_{o}}
120:{\displaystyle \mathrm {Stk} ={\frac {t_{0}\,u_{0}}{l_{0}}}}
4007:
2652:{\displaystyle x_{{\text{particle}},95\%}=500{\text{ mm}}}
1642:
The step response for a particle is a simple exponential:
1487:{\displaystyle f_{-3{\text{ dB}}}={\frac {1}{2\pi t_{0}}}}
1428:. Thus, the −3 dB gain (cut-off) frequency will be:
4037:(Reprint. ed.). Cambridge : Cambridge Univ. Press.
2559:{\displaystyle x_{{\text{particle}},95\%}=5{\text{ mm}}}
3902:{\displaystyle \mathrm {Stk} ={\frac {u_{0}V_{s}}{dg}}}
3965:
3945:
3918:
3850:
3827:
3807:
3655:
3618:
3449:
3409:
3389:
3363:
3295:
3260:
3111:
3088:
3019:
2979:
2894:
2863:
2718:
2669:
2616:
2572:
2523:
2479:
2453:
2409:
2371:
2286:
2157:
2115:
1993:
1804:
1775:
1755:
1651:
1622:
1595:
1525:
1437:
1407:
1310:
1225:
1127:
1048:
904:
767:
706:
624:
585:
549:
491:
457:
423:
381:
354:
323:
244:
194:
167:
136:
65:
1616:
is the flow velocity upstream of the shock; whereas
3971:
3951:
3931:
3901:
3833:
3813:
3790:
3630:
3602:
3433:
3395:
3375:
3349:
3281:
3244:
3094:
3066:
3003:
2963:
2878:
2847:
2693:
2651:
2602:
2558:
2509:
2465:
2433:
2395:
2354:
2269:
2140:
2098:
1976:
1787:
1761:
1738:
1631:
1608:
1581:
1486:
1420:
1390:
1295:
1201:
1110:
1031:
884:
747:
686:
607:
571:
511:
477:
443:
394:
367:
336:
306:
207:
180:
149:
119:
4296:
4060:Springer Handbook of Experimental Fluid Mechanics
3642:Application to anisokinetic sampling of particles
1509:
831:
806:
485:, particles follow fluid streamlines closely. If
4746:
2886:is the "particle free-stream Reynolds number",
4352:
4403:
4180:Zeitschrift des Vereines Deutscher Ingenieure
3076:It follows that this function is defined by,
2444:For example, consider a normal shock wave of
1296:{\displaystyle t_{0}sv_{p}(s)=v_{f}-v_{p}(s)}
687:{\displaystyle \Delta v(t)=v_{f}(t)-v_{p}(t)}
4315:
4277:
4196:
4085:
2704:
748:{\displaystyle F_{D}=3\pi \mu d_{p}\Delta v}
4231:
4128:
3959:is the sampling tube's inner diameter, and
1582:{\displaystyle v_{f}(t)=V_{u}-\Delta VH(t)}
4410:
4396:
2694:{\displaystyle x_{{\text{particle}},95\%}}
2434:{\displaystyle x_{{\text{particle}},95\%}}
2396:{\displaystyle x_{{\text{particle}},95\%}}
227:, which is when the particle (or droplet)
4378:
4259:
4249:
4154:
519:, tracing accuracy errors are below 1%.
93:
44:characterising the behavior of particles
4765:Dimensionless numbers of fluid mechanics
4419:Dimensionless numbers in fluid mechanics
3078:
2148:(time to 95% velocity change), we have:
1500:
529:
18:
4129:Wessel, R. A.; Righi, J. (1988-01-01).
4032:
2603:{\displaystyle d_{p}=10~\mu {\text{m}}}
1639:is the velocity drop across the shock.
4747:
3004:{\displaystyle \psi ({\text{Re}}_{o})}
2510:{\displaystyle d_{p}=1~\mu {\text{m}}}
4391:
3939:is the particle's settling velocity,
512:{\displaystyle \mathrm {Stk} <0.1}
231:is less than about one, the particle
4081:
4079:
3282:{\displaystyle {\text{Re}}_{o}\to 0}
478:{\displaystyle \mathrm {Stk} \ll 1}
444:{\displaystyle \mathrm {Stk} \gg 1}
13:
4271:
4238:Proceedings of the Royal Society A
4232:Dey, S; Ali, SZ; Padhi, E (2019).
3858:
3855:
3852:
3743:
3740:
3737:
3434:{\displaystyle C_{D}({\text{Re}})}
3234:
3219:
3192:
2837:
2822:
2795:
2686:
2633:
2540:
2426:
2388:
2343:
2303:
2236:
2211:
2174:
2116:
2070:
2047:
2041:
2029:
2007:
1934:
1926:
1898:
1874:
1828:
1702:
1690:
1623:
1561:
1184:
1014:
739:
625:
499:
496:
493:
465:
462:
459:
431:
428:
425:
73:
70:
67:
14:
4781:
4076:
2109:Considering a relaxation time of
4353:Collins, LR; Keswani, A (2004).
4282:. New York: Dover Publications.
4033:Brennen, Christopher E. (2005).
3979:is the acceleration of gravity.
2929:
411:particle image velocimetry (PIV)
4035:Fundamentals of multiphase flow
2879:{\displaystyle {\text{Re}}_{o}}
2141:{\displaystyle \Delta t=3t_{0}}
697:The stokes drag force is then:
4225:
4190:
4171:
4135:Aerosol Science and Technology
4122:
4088:Aerosol Science and Technology
4051:
4026:
4001:
3777:
3747:
3710:
3683:
3562:
3559:
3524:
3480:
3468:
3453:
3428:
3420:
3367:
3324:
3321:
3306:
3273:
3224:
3209:
3130:
3115:
3061:
3046:
2998:
2983:
2934:
2924:
2827:
2812:
2349:
2324:
2264:
2242:
2217:
2195:
2093:
2053:
2035:
2013:
1904:
1882:
1852:
1846:
1696:
1674:
1668:
1662:
1576:
1570:
1542:
1536:
1510:Particles through a shock wave
1351:
1345:
1330:
1324:
1290:
1284:
1255:
1249:
1196:
1190:
1150:
1144:
1026:
1020:
927:
921:
681:
675:
659:
653:
637:
631:
602:
596:
566:
560:
1:
3995:
375:is the particle diameter and
4219:10.1016/0021-8502(74)90130-X
7:
3982:
3821:is particle concentration,
10:
4786:
4318:Journal of Fluid Mechanics
4297:Hinds, William C. (1999).
4010:Particle Image Velocimetry
3376:{\displaystyle \psi \to 1}
1517:particle image velocimetry
1118:can be replaced to yield:
4425:
4380:10.1088/1367-2630/6/1/119
4338:10.1017/S0022112071001460
4280:The mechanics of aerosols
4156:10.1080/02786828808959193
4108:10.1080/02786828308958612
2705:Non-Stokesian drag regime
1042:Note the relaxation time
337:{\displaystyle \rho _{p}}
1769:direction. Let's assume
1632:{\displaystyle \Delta V}
608:{\displaystyle v_{f}(t)}
572:{\displaystyle v_{p}(t)}
395:{\displaystyle \mu _{g}}
3631:{\displaystyle c=0.158}
2973:An additional function
2610:would blur the flow by
2517:would blur the flow by
4359:New Journal of Physics
4251:10.1098/rspa.2019.0277
3973:
3953:
3933:
3903:
3835:
3815:
3792:
3632:
3604:
3435:
3397:
3377:
3351:
3283:
3246:
3103:
3096:
3068:
3005:
2965:
2880:
2849:
2695:
2653:
2604:
2560:
2511:
2467:
2435:
2397:
2356:
2271:
2142:
2100:
1978:
1789:
1763:
1740:
1633:
1610:
1583:
1506:
1488:
1422:
1392:
1297:
1203:
1112:
1033:
886:
758:The particle mass is:
749:
688:
609:
573:
539:
513:
479:
445:
396:
369:
338:
308:
209:
182:
151:
121:
25:
4278:Fuchs, N. A. (1989).
3974:
3954:
3934:
3932:{\displaystyle V_{s}}
3904:
3836:
3816:
3793:
3633:
3605:
3436:
3398:
3396:{\displaystyle \psi }
3378:
3352:
3284:
3247:
3097:
3095:{\displaystyle \psi }
3082:
3069:
3006:
2966:
2881:
2850:
2696:
2654:
2605:
2561:
2512:
2468:
2436:
2398:
2357:
2272:
2143:
2101:
1979:
1790:
1764:
1741:
1634:
1611:
1609:{\displaystyle V_{u}}
1584:
1504:
1489:
1423:
1421:{\displaystyle t_{0}}
1393:
1298:
1204:
1113:
1034:
887:
750:
689:
610:
574:
533:
514:
480:
446:
397:
370:
368:{\displaystyle d_{p}}
339:
309:
210:
208:{\displaystyle l_{0}}
183:
181:{\displaystyle u_{0}}
152:
150:{\displaystyle t_{0}}
122:
38:George Gabriel Stokes
22:
3963:
3943:
3916:
3848:
3825:
3805:
3653:
3616:
3447:
3407:
3387:
3361:
3293:
3258:
3109:
3086:
3017:
2977:
2892:
2861:
2716:
2667:
2614:
2570:
2521:
2477:
2451:
2407:
2369:
2284:
2155:
2113:
1991:
1802:
1773:
1753:
1649:
1620:
1593:
1523:
1435:
1405:
1308:
1223:
1125:
1046:
902:
765:
704:
622:
583:
547:
489:
455:
421:
379:
352:
321:
242:
192:
165:
134:
63:
42:dimensionless number
4755:Discrete-phase flow
4371:2004NJPh....6..119C
4330:1971JFM....48...41S
4301:. New York: Wiley.
4211:1974JAerS...5..325B
4147:1988AerST...9...29W
4100:1982AerST...2...45I
3612:Where the constant
3558:
3514:
3178:
2781:
2466:{\displaystyle M=2}
1933:
1881:
1835:
1788:{\displaystyle x=0}
1089:
875:
285:
4565:Keulegan–Carpenter
3969:
3949:
3929:
3899:
3831:
3811:
3788:
3628:
3600:
3534:
3490:
3431:
3393:
3373:
3347:
3279:
3242:
3155:
3104:
3092:
3064:
3001:
2961:
2876:
2845:
2758:
2691:
2649:
2600:
2556:
2507:
2463:
2431:
2393:
2352:
2267:
2138:
2096:
1974:
1916:
1864:
1818:
1785:
1759:
1736:
1629:
1606:
1579:
1507:
1484:
1418:
1388:
1293:
1199:
1108:
1075:
1029:
882:
861:
745:
684:
605:
569:
540:
509:
475:
441:
392:
365:
334:
304:
271:
205:
178:
147:
117:
26:
4742:
4741:
4308:978-0-471-19410-1
4289:978-0-486-66055-4
4069:978-3-540-25141-5
4019:978-3-319-68851-0
3972:{\displaystyle g}
3952:{\displaystyle d}
3897:
3834:{\displaystyle u}
3814:{\displaystyle c}
3781:
3598:
3589:
3538:
3532:
3494:
3488:
3460:
3426:
3339:
3313:
3265:
3240:
3231:
3216:
3189:
3169:
3153:
3145:
3122:
3053:
3038:
3024:
2990:
2959:
2899:
2868:
2843:
2834:
2819:
2792:
2772:
2756:
2748:
2737:
2728:
2723:
2678:
2647:
2625:
2598:
2591:
2554:
2532:
2505:
2498:
2418:
2380:
2295:
2166:
2001:
1812:
1762:{\displaystyle x}
1482:
1452:
1386:
1355:
1214:Laplace transform
1182:
1162:
1106:
1012:
966:
939:
880:
826:
799:
404:dynamic viscosity
302:
115:
4777:
4412:
4405:
4398:
4389:
4388:
4384:
4382:
4349:
4312:
4293:
4266:
4265:
4263:
4253:
4229:
4223:
4222:
4194:
4188:
4187:
4175:
4169:
4168:
4158:
4126:
4120:
4119:
4083:
4074:
4073:
4055:
4049:
4048:
4030:
4024:
4023:
4005:
3978:
3976:
3975:
3970:
3958:
3956:
3955:
3950:
3938:
3936:
3935:
3930:
3928:
3927:
3908:
3906:
3905:
3900:
3898:
3896:
3888:
3887:
3886:
3877:
3876:
3866:
3861:
3840:
3838:
3837:
3832:
3820:
3818:
3817:
3812:
3797:
3795:
3794:
3789:
3787:
3783:
3782:
3780:
3776:
3775:
3766:
3746:
3725:
3700:
3695:
3694:
3673:
3672:
3663:
3637:
3635:
3634:
3629:
3609:
3607:
3606:
3601:
3599:
3597:
3596:
3595:
3590:
3587:
3584:
3583:
3579:
3565:
3557:
3553:
3544:
3539:
3536:
3533:
3528:
3513:
3509:
3500:
3495:
3492:
3489:
3484:
3475:
3467:
3466:
3461:
3458:
3440:
3438:
3437:
3432:
3427:
3424:
3419:
3418:
3402:
3400:
3399:
3394:
3382:
3380:
3379:
3374:
3356:
3354:
3353:
3348:
3346:
3345:
3340:
3337:
3334:
3320:
3319:
3314:
3311:
3305:
3304:
3288:
3286:
3285:
3280:
3272:
3271:
3266:
3263:
3251:
3249:
3248:
3243:
3241:
3239:
3238:
3237:
3232:
3229:
3223:
3222:
3217:
3214:
3208:
3207:
3197:
3196:
3195:
3190:
3187:
3180:
3177:
3176:
3175:
3170:
3167:
3163:
3154:
3152:
3151:
3146:
3143:
3137:
3129:
3128:
3123:
3120:
3101:
3099:
3098:
3093:
3073:
3071:
3070:
3065:
3060:
3059:
3054:
3051:
3039:
3036:
3031:
3030:
3025:
3022:
3010:
3008:
3007:
3002:
2997:
2996:
2991:
2988:
2970:
2968:
2967:
2962:
2960:
2958:
2957:
2948:
2947:
2946:
2937:
2932:
2927:
2922:
2921:
2911:
2906:
2905:
2900:
2897:
2885:
2883:
2882:
2877:
2875:
2874:
2869:
2866:
2854:
2852:
2851:
2846:
2844:
2842:
2841:
2840:
2835:
2832:
2826:
2825:
2820:
2817:
2811:
2810:
2800:
2799:
2798:
2793:
2790:
2783:
2780:
2779:
2778:
2773:
2770:
2766:
2757:
2755:
2754:
2749:
2746:
2740:
2738:
2735:
2730:
2729:
2726:
2724:
2721:
2700:
2698:
2697:
2692:
2690:
2689:
2679:
2676:
2658:
2656:
2655:
2650:
2648:
2645:
2637:
2636:
2626:
2623:
2609:
2607:
2606:
2601:
2599:
2596:
2589:
2582:
2581:
2565:
2563:
2562:
2557:
2555:
2552:
2544:
2543:
2533:
2530:
2516:
2514:
2513:
2508:
2506:
2503:
2496:
2489:
2488:
2472:
2470:
2469:
2464:
2440:
2438:
2437:
2432:
2430:
2429:
2419:
2416:
2402:
2400:
2399:
2394:
2392:
2391:
2381:
2378:
2361:
2359:
2358:
2353:
2336:
2335:
2323:
2322:
2307:
2306:
2296:
2293:
2276:
2274:
2273:
2268:
2263:
2262:
2235:
2234:
2207:
2206:
2194:
2193:
2178:
2177:
2167:
2164:
2147:
2145:
2144:
2139:
2137:
2136:
2105:
2103:
2102:
2097:
2092:
2091:
2090:
2089:
2080:
2025:
2024:
2003:
2002:
1999:
1983:
1981:
1980:
1975:
1967:
1966:
1965:
1964:
1955:
1932:
1924:
1894:
1893:
1880:
1872:
1845:
1844:
1834:
1826:
1814:
1813:
1810:
1794:
1792:
1791:
1786:
1768:
1766:
1765:
1760:
1745:
1743:
1742:
1737:
1735:
1734:
1733:
1732:
1723:
1686:
1685:
1661:
1660:
1638:
1636:
1635:
1630:
1615:
1613:
1612:
1607:
1605:
1604:
1588:
1586:
1585:
1580:
1557:
1556:
1535:
1534:
1493:
1491:
1490:
1485:
1483:
1481:
1480:
1479:
1460:
1455:
1454:
1453:
1450:
1427:
1425:
1424:
1419:
1417:
1416:
1397:
1395:
1394:
1389:
1387:
1385:
1375:
1374:
1361:
1356:
1354:
1344:
1343:
1333:
1323:
1322:
1312:
1302:
1300:
1299:
1294:
1283:
1282:
1270:
1269:
1248:
1247:
1235:
1234:
1208:
1206:
1205:
1200:
1183:
1181:
1180:
1168:
1163:
1161:
1153:
1143:
1142:
1129:
1117:
1115:
1114:
1109:
1107:
1105:
1104:
1103:
1090:
1088:
1083:
1074:
1073:
1063:
1058:
1057:
1038:
1036:
1035:
1030:
1013:
1011:
1010:
1009:
1000:
999:
994:
993:
992:
980:
972:
967:
965:
964:
955:
954:
945:
940:
938:
930:
920:
919:
906:
891:
889:
888:
883:
881:
876:
874:
869:
856:
854:
853:
841:
840:
835:
834:
827:
822:
821:
812:
810:
809:
800:
792:
790:
789:
777:
776:
754:
752:
751:
746:
738:
737:
716:
715:
693:
691:
690:
685:
674:
673:
652:
651:
614:
612:
611:
606:
595:
594:
578:
576:
575:
570:
559:
558:
536:propylene glycol
518:
516:
515:
510:
502:
484:
482:
481:
476:
468:
450:
448:
447:
442:
434:
401:
399:
398:
393:
391:
390:
374:
372:
371:
366:
364:
363:
344:is the particle
343:
341:
340:
335:
333:
332:
313:
311:
310:
305:
303:
301:
300:
299:
286:
284:
279:
270:
269:
259:
254:
253:
233:drag coefficient
214:
212:
211:
206:
204:
203:
187:
185:
184:
179:
177:
176:
156:
154:
153:
148:
146:
145:
126:
124:
123:
118:
116:
114:
113:
104:
103:
102:
92:
91:
81:
76:
4785:
4784:
4780:
4779:
4778:
4776:
4775:
4774:
4745:
4744:
4743:
4738:
4421:
4416:
4309:
4290:
4274:
4272:Further reading
4269:
4230:
4226:
4199:Aerosol Science
4195:
4191:
4176:
4172:
4127:
4123:
4084:
4077:
4070:
4056:
4052:
4045:
4031:
4027:
4020:
4006:
4002:
3998:
3985:
3964:
3961:
3960:
3944:
3941:
3940:
3923:
3919:
3917:
3914:
3913:
3889:
3882:
3878:
3872:
3868:
3867:
3865:
3851:
3849:
3846:
3845:
3826:
3823:
3822:
3806:
3803:
3802:
3771:
3767:
3762:
3736:
3729:
3724:
3717:
3713:
3696:
3690:
3686:
3668:
3664:
3659:
3654:
3651:
3650:
3644:
3617:
3614:
3613:
3591:
3586:
3585:
3575:
3571:
3567:
3566:
3549:
3545:
3540:
3535:
3527:
3505:
3501:
3496:
3491:
3483:
3476:
3474:
3462:
3457:
3456:
3448:
3445:
3444:
3423:
3414:
3410:
3408:
3405:
3404:
3388:
3385:
3384:
3362:
3359:
3358:
3341:
3336:
3335:
3330:
3315:
3310:
3309:
3300:
3296:
3294:
3291:
3290:
3267:
3262:
3261:
3259:
3256:
3255:
3233:
3228:
3227:
3218:
3213:
3212:
3203:
3199:
3198:
3191:
3186:
3185:
3181:
3179:
3171:
3166:
3165:
3164:
3159:
3147:
3142:
3141:
3136:
3124:
3119:
3118:
3110:
3107:
3106:
3087:
3084:
3083:
3055:
3050:
3049:
3035:
3026:
3021:
3020:
3018:
3015:
3014:
2992:
2987:
2986:
2978:
2975:
2974:
2953:
2949:
2942:
2938:
2933:
2928:
2923:
2917:
2913:
2912:
2910:
2901:
2896:
2895:
2893:
2890:
2889:
2870:
2865:
2864:
2862:
2859:
2858:
2836:
2831:
2830:
2821:
2816:
2815:
2806:
2802:
2801:
2794:
2789:
2788:
2784:
2782:
2774:
2769:
2768:
2767:
2762:
2750:
2745:
2744:
2739:
2734:
2725:
2720:
2719:
2717:
2714:
2713:
2707:
2675:
2674:
2670:
2668:
2665:
2664:
2644:
2622:
2621:
2617:
2615:
2612:
2611:
2595:
2577:
2573:
2571:
2568:
2567:
2551:
2529:
2528:
2524:
2522:
2519:
2518:
2502:
2484:
2480:
2478:
2475:
2474:
2452:
2449:
2448:
2415:
2414:
2410:
2408:
2405:
2404:
2377:
2376:
2372:
2370:
2367:
2366:
2331:
2327:
2318:
2314:
2292:
2291:
2287:
2285:
2282:
2281:
2255:
2251:
2230:
2226:
2202:
2198:
2189:
2185:
2163:
2162:
2158:
2156:
2153:
2152:
2132:
2128:
2114:
2111:
2110:
2085:
2081:
2076:
2066:
2062:
2020:
2016:
1998:
1994:
1992:
1989:
1988:
1960:
1956:
1951:
1944:
1940:
1925:
1920:
1889:
1885:
1873:
1868:
1840:
1836:
1827:
1822:
1809:
1805:
1803:
1800:
1799:
1774:
1771:
1770:
1754:
1751:
1750:
1728:
1724:
1719:
1712:
1708:
1681:
1677:
1656:
1652:
1650:
1647:
1646:
1621:
1618:
1617:
1600:
1596:
1594:
1591:
1590:
1552:
1548:
1530:
1526:
1524:
1521:
1520:
1512:
1475:
1471:
1464:
1459:
1449:
1442:
1438:
1436:
1433:
1432:
1412:
1408:
1406:
1403:
1402:
1370:
1366:
1365:
1360:
1339:
1335:
1334:
1318:
1314:
1313:
1311:
1309:
1306:
1305:
1278:
1274:
1265:
1261:
1243:
1239:
1230:
1226:
1224:
1221:
1220:
1176:
1172:
1167:
1154:
1138:
1134:
1130:
1128:
1126:
1123:
1122:
1099:
1095:
1091:
1084:
1079:
1069:
1065:
1064:
1062:
1053:
1049:
1047:
1044:
1043:
1005:
1001:
995:
988:
984:
983:
982:
981:
973:
971:
960:
956:
950:
946:
944:
931:
915:
911:
907:
905:
903:
900:
899:
870:
865:
857:
855:
849:
845:
836:
830:
829:
828:
817:
813:
811:
805:
804:
791:
785:
781:
772:
768:
766:
763:
762:
733:
729:
711:
707:
705:
702:
701:
669:
665:
647:
643:
623:
620:
619:
590:
586:
584:
581:
580:
554:
550:
548:
545:
544:
528:
522:
492:
490:
487:
486:
458:
456:
453:
452:
424:
422:
419:
418:
386:
382:
380:
377:
376:
359:
355:
353:
350:
349:
328:
324:
322:
319:
318:
295:
291:
287:
280:
275:
265:
261:
260:
258:
249:
245:
243:
240:
239:
229:Reynolds number
223:In the case of
199:
195:
193:
190:
189:
172:
168:
166:
163:
162:
159:relaxation time
141:
137:
135:
132:
131:
109:
105:
98:
94:
87:
83:
82:
80:
66:
64:
61:
60:
36:), named after
17:
12:
11:
5:
4783:
4773:
4772:
4770:Fluid dynamics
4767:
4762:
4757:
4740:
4739:
4737:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4696:
4691:
4686:
4681:
4676:
4671:
4666:
4661:
4656:
4651:
4646:
4645:
4644:
4634:
4629:
4628:
4627:
4622:
4612:
4607:
4602:
4597:
4592:
4587:
4582:
4577:
4572:
4567:
4562:
4557:
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4512:
4507:
4502:
4497:
4492:
4487:
4482:
4477:
4472:
4467:
4462:
4457:
4452:
4447:
4442:
4437:
4432:
4426:
4423:
4422:
4415:
4414:
4407:
4400:
4392:
4386:
4385:
4350:
4313:
4307:
4294:
4288:
4273:
4270:
4268:
4267:
4224:
4205:(4): 325–338.
4189:
4170:
4121:
4075:
4068:
4050:
4043:
4025:
4018:
3999:
3997:
3994:
3993:
3992:
3984:
3981:
3968:
3948:
3926:
3922:
3910:
3909:
3895:
3892:
3885:
3881:
3875:
3871:
3864:
3860:
3857:
3854:
3830:
3810:
3799:
3798:
3786:
3779:
3774:
3770:
3765:
3761:
3758:
3755:
3752:
3749:
3745:
3742:
3739:
3735:
3732:
3728:
3723:
3720:
3716:
3712:
3709:
3706:
3703:
3699:
3693:
3689:
3685:
3682:
3679:
3676:
3671:
3667:
3662:
3658:
3643:
3640:
3627:
3624:
3621:
3594:
3582:
3578:
3574:
3570:
3564:
3561:
3556:
3552:
3548:
3543:
3531:
3526:
3523:
3520:
3517:
3512:
3508:
3504:
3499:
3487:
3482:
3479:
3473:
3470:
3465:
3455:
3452:
3430:
3422:
3417:
3413:
3392:
3372:
3369:
3366:
3357:and therefore
3344:
3333:
3329:
3326:
3323:
3318:
3308:
3303:
3299:
3278:
3275:
3270:
3236:
3226:
3221:
3211:
3206:
3202:
3194:
3184:
3174:
3162:
3158:
3150:
3140:
3135:
3132:
3127:
3117:
3114:
3091:
3063:
3058:
3048:
3045:
3042:
3034:
3029:
3000:
2995:
2985:
2982:
2956:
2952:
2945:
2941:
2936:
2931:
2926:
2920:
2916:
2909:
2904:
2873:
2839:
2829:
2824:
2814:
2809:
2805:
2797:
2787:
2777:
2765:
2761:
2753:
2743:
2733:
2706:
2703:
2688:
2685:
2682:
2673:
2643:
2640:
2635:
2632:
2629:
2620:
2594:
2588:
2585:
2580:
2576:
2550:
2547:
2542:
2539:
2536:
2527:
2501:
2495:
2492:
2487:
2483:
2462:
2459:
2456:
2428:
2425:
2422:
2413:
2390:
2387:
2384:
2375:
2363:
2362:
2351:
2348:
2345:
2342:
2339:
2334:
2330:
2326:
2321:
2317:
2313:
2310:
2305:
2302:
2299:
2290:
2278:
2277:
2266:
2261:
2258:
2254:
2250:
2247:
2244:
2241:
2238:
2233:
2229:
2225:
2222:
2219:
2216:
2213:
2210:
2205:
2201:
2197:
2192:
2188:
2184:
2181:
2176:
2173:
2170:
2161:
2135:
2131:
2127:
2124:
2121:
2118:
2107:
2106:
2095:
2088:
2084:
2079:
2075:
2072:
2069:
2065:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2023:
2019:
2015:
2012:
2009:
2006:
1997:
1985:
1984:
1973:
1970:
1963:
1959:
1954:
1950:
1947:
1943:
1939:
1936:
1931:
1928:
1923:
1919:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1892:
1888:
1884:
1879:
1876:
1871:
1867:
1863:
1860:
1857:
1854:
1851:
1848:
1843:
1839:
1833:
1830:
1825:
1821:
1817:
1808:
1784:
1781:
1778:
1758:
1747:
1746:
1731:
1727:
1722:
1718:
1715:
1711:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1684:
1680:
1676:
1673:
1670:
1667:
1664:
1659:
1655:
1628:
1625:
1603:
1599:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1555:
1551:
1547:
1544:
1541:
1538:
1533:
1529:
1511:
1508:
1495:
1494:
1478:
1474:
1470:
1467:
1463:
1458:
1448:
1445:
1441:
1415:
1411:
1399:
1398:
1384:
1381:
1378:
1373:
1369:
1364:
1359:
1353:
1350:
1347:
1342:
1338:
1332:
1329:
1326:
1321:
1317:
1303:
1292:
1289:
1286:
1281:
1277:
1273:
1268:
1264:
1260:
1257:
1254:
1251:
1246:
1242:
1238:
1233:
1229:
1210:
1209:
1198:
1195:
1192:
1189:
1186:
1179:
1175:
1171:
1166:
1160:
1157:
1152:
1149:
1146:
1141:
1137:
1133:
1102:
1098:
1094:
1087:
1082:
1078:
1072:
1068:
1061:
1056:
1052:
1040:
1039:
1028:
1025:
1022:
1019:
1016:
1008:
1004:
998:
991:
987:
979:
976:
970:
963:
959:
953:
949:
943:
937:
934:
929:
926:
923:
918:
914:
910:
893:
892:
879:
873:
868:
864:
860:
852:
848:
844:
839:
833:
825:
820:
816:
808:
803:
798:
795:
788:
784:
780:
775:
771:
756:
755:
744:
741:
736:
732:
728:
725:
722:
719:
714:
710:
695:
694:
683:
680:
677:
672:
668:
664:
661:
658:
655:
650:
646:
642:
639:
636:
633:
630:
627:
604:
601:
598:
593:
589:
568:
565:
562:
557:
553:
527:
524:
508:
505:
501:
498:
495:
474:
471:
467:
464:
461:
440:
437:
433:
430:
427:
415:velocity field
389:
385:
362:
358:
331:
327:
315:
314:
298:
294:
290:
283:
278:
274:
268:
264:
257:
252:
248:
202:
198:
175:
171:
144:
140:
128:
127:
112:
108:
101:
97:
90:
86:
79:
75:
72:
69:
15:
9:
6:
4:
3:
2:
4782:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4752:
4750:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4643:
4640:
4639:
4638:
4635:
4633:
4630:
4626:
4623:
4621:
4618:
4617:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4475:Chandrasekhar
4473:
4471:
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4446:
4443:
4441:
4438:
4436:
4433:
4431:
4428:
4427:
4424:
4420:
4413:
4408:
4406:
4401:
4399:
4394:
4393:
4390:
4381:
4376:
4372:
4368:
4364:
4360:
4356:
4351:
4347:
4343:
4339:
4335:
4331:
4327:
4323:
4319:
4314:
4310:
4304:
4300:
4295:
4291:
4285:
4281:
4276:
4275:
4262:
4257:
4252:
4247:
4243:
4239:
4235:
4228:
4220:
4216:
4212:
4208:
4204:
4200:
4193:
4185:
4181:
4174:
4166:
4162:
4157:
4152:
4148:
4144:
4140:
4136:
4132:
4125:
4117:
4113:
4109:
4105:
4101:
4097:
4093:
4089:
4082:
4080:
4071:
4065:
4061:
4054:
4046:
4044:9780521848046
4040:
4036:
4029:
4021:
4015:
4011:
4004:
4000:
3990:
3987:
3986:
3980:
3966:
3946:
3924:
3920:
3893:
3890:
3883:
3879:
3873:
3869:
3862:
3844:
3843:
3842:
3828:
3808:
3784:
3772:
3768:
3763:
3759:
3756:
3753:
3750:
3733:
3730:
3726:
3721:
3718:
3714:
3707:
3704:
3701:
3697:
3691:
3687:
3680:
3677:
3674:
3669:
3665:
3660:
3656:
3649:
3648:
3647:
3639:
3625:
3622:
3619:
3610:
3592:
3580:
3576:
3572:
3568:
3554:
3550:
3546:
3541:
3529:
3521:
3518:
3515:
3510:
3506:
3502:
3497:
3485:
3477:
3471:
3463:
3450:
3442:
3415:
3411:
3390:
3370:
3364:
3342:
3331:
3327:
3316:
3301:
3297:
3276:
3268:
3252:
3204:
3200:
3182:
3172:
3160:
3156:
3148:
3138:
3133:
3125:
3112:
3089:
3081:
3077:
3074:
3056:
3043:
3040:
3032:
3027:
3012:
2993:
2980:
2971:
2954:
2950:
2943:
2939:
2918:
2914:
2907:
2902:
2887:
2871:
2855:
2807:
2803:
2785:
2775:
2763:
2759:
2751:
2741:
2731:
2711:
2702:
2683:
2680:
2671:
2660:
2641:
2638:
2630:
2627:
2618:
2592:
2586:
2583:
2578:
2574:
2548:
2545:
2537:
2534:
2525:
2499:
2493:
2490:
2485:
2481:
2460:
2457:
2454:
2447:
2442:
2423:
2420:
2411:
2385:
2382:
2373:
2346:
2340:
2337:
2332:
2328:
2319:
2315:
2311:
2308:
2300:
2297:
2288:
2280:
2279:
2259:
2256:
2252:
2248:
2245:
2239:
2231:
2227:
2223:
2220:
2214:
2208:
2203:
2199:
2190:
2186:
2182:
2179:
2171:
2168:
2159:
2151:
2150:
2149:
2133:
2129:
2125:
2122:
2119:
2086:
2082:
2077:
2073:
2067:
2063:
2059:
2056:
2050:
2044:
2038:
2032:
2026:
2021:
2017:
2010:
2004:
1995:
1987:
1986:
1971:
1968:
1961:
1957:
1952:
1948:
1945:
1941:
1937:
1929:
1921:
1917:
1913:
1910:
1907:
1901:
1895:
1890:
1886:
1877:
1869:
1865:
1861:
1858:
1855:
1849:
1841:
1837:
1831:
1823:
1819:
1815:
1806:
1798:
1797:
1796:
1782:
1779:
1776:
1756:
1729:
1725:
1720:
1716:
1713:
1709:
1705:
1699:
1693:
1687:
1682:
1678:
1671:
1665:
1657:
1653:
1645:
1644:
1643:
1640:
1626:
1601:
1597:
1573:
1567:
1564:
1558:
1553:
1549:
1545:
1539:
1531:
1527:
1518:
1503:
1499:
1476:
1472:
1468:
1465:
1461:
1456:
1446:
1443:
1439:
1431:
1430:
1429:
1413:
1409:
1382:
1379:
1376:
1371:
1367:
1362:
1357:
1348:
1340:
1336:
1327:
1319:
1315:
1304:
1287:
1279:
1275:
1271:
1266:
1262:
1258:
1252:
1244:
1240:
1236:
1231:
1227:
1219:
1218:
1217:
1215:
1193:
1187:
1177:
1173:
1169:
1164:
1158:
1155:
1147:
1139:
1135:
1131:
1121:
1120:
1119:
1100:
1096:
1092:
1085:
1080:
1076:
1070:
1066:
1059:
1054:
1050:
1023:
1017:
1006:
1002:
996:
989:
985:
977:
974:
968:
961:
957:
951:
947:
941:
935:
932:
924:
916:
912:
908:
898:
897:
896:
877:
871:
866:
862:
858:
850:
846:
842:
837:
823:
818:
814:
801:
796:
793:
786:
782:
778:
773:
769:
761:
760:
759:
742:
734:
730:
726:
723:
720:
717:
712:
708:
700:
699:
698:
678:
670:
666:
662:
656:
648:
644:
640:
634:
628:
618:
617:
616:
599:
591:
587:
563:
555:
551:
537:
532:
523:
520:
506:
503:
472:
469:
438:
435:
416:
412:
407:
405:
402:is the fluid
387:
383:
360:
356:
347:
329:
325:
296:
292:
288:
281:
276:
272:
266:
262:
255:
250:
246:
238:
237:
236:
234:
230:
226:
221:
219:
200:
196:
173:
169:
160:
142:
138:
110:
106:
99:
95:
88:
84:
77:
59:
58:
57:
55:
51:
47:
43:
39:
35:
31:
30:Stokes number
21:
4693:
4365:(119): 119.
4362:
4358:
4321:
4317:
4298:
4279:
4241:
4237:
4227:
4202:
4198:
4192:
4183:
4179:
4173:
4141:(1): 29–60.
4138:
4134:
4124:
4094:(1): 45–51.
4091:
4087:
4062:. Springer.
4059:
4053:
4034:
4028:
4009:
4003:
3911:
3800:
3645:
3611:
3443:
3253:
3105:
3075:
3013:
2972:
2888:
2856:
2712:
2708:
2661:
2566:; whereas a
2443:
2364:
2108:
1748:
1641:
1513:
1496:
1400:
1211:
1041:
894:
757:
696:
541:
521:
408:
316:
222:
129:
33:
29:
27:
4729:Weissenberg
4264:. 20190277.
3989:Stokes' law
2446:Mach number
2441:distance.
225:Stokes flow
4749:Categories
4649:Richardson
4430:Archimedes
4186:: 318–320.
3996:References
50:fluid flow
4734:Womersley
4625:turbulent
4605:Ohnesorge
4590:Marangoni
4555:Iribarren
4480:Damköhler
4465:Capillary
4346:122731370
4324:: 41–71.
4165:0278-6826
4116:0278-6826
3722:−
3705:−
3522:
3516:−
3451:ψ
3391:ψ
3368:→
3365:ψ
3325:→
3274:→
3235:′
3220:′
3193:′
3157:∫
3113:ψ
3090:ψ
3044:ψ
3041:⋅
2981:ψ
2951:μ
2915:ρ
2838:′
2823:′
2796:′
2760:∫
2687:%
2634:%
2593:μ
2541:%
2500:μ
2427:%
2389:%
2344:Δ
2338:−
2304:%
2257:−
2249:−
2237:Δ
2212:Δ
2209:−
2175:%
2117:Δ
2071:Δ
2068:−
2060:−
2048:Δ
2042:Δ
2030:Δ
2027:−
2008:Δ
1946:−
1935:Δ
1927:Δ
1918:∫
1899:Δ
1896:−
1875:Δ
1866:∫
1829:Δ
1820:∫
1714:−
1703:Δ
1691:Δ
1688:−
1624:Δ
1562:Δ
1559:−
1469:π
1444:−
1272:−
1185:Δ
1097:μ
1067:ρ
1015:Δ
1003:ρ
978:μ
859:π
847:ρ
802:π
783:ρ
740:Δ
727:μ
724:π
663:−
626:Δ
470:≪
436:≫
384:μ
326:ρ
293:μ
263:ρ
218:advection
46:suspended
4760:Aerosols
4709:Suratman
4699:Strouhal
4679:Sherwood
4642:magnetic
4637:Reynolds
4632:Rayleigh
4620:magnetic
4460:Brinkman
4244:(2228).
3983:See also
2677:particle
2646: mm
2624:particle
2553: mm
2531:particle
2417:particle
2379:particle
2294:particle
2165:particle
2000:particle
1811:particle
1451: dB
1216:method:
4689:Stanton
4684:Shields
4674:Scruton
4669:Schmidt
4615:Prandtl
4600:Nusselt
4575:Laplace
4570:Knudsen
4560:Kapitza
4545:Görtler
4540:Grashof
4530:Galilei
4495:Deborah
4440:Bagnold
4367:Bibcode
4326:Bibcode
4261:6735480
4207:Bibcode
4143:Bibcode
4096:Bibcode
346:density
157:is the
54:droplet
40:, is a
4719:Ursell
4714:Taylor
4704:Stuart
4694:Stokes
4659:Rossby
4654:Roshko
4610:Péclet
4595:Morton
4535:Graetz
4525:Froude
4515:Eötvös
4505:Eckert
4500:Dukhin
4470:Cauchy
4435:Atwood
4344:
4305:
4286:
4258:
4163:
4114:
4066:
4041:
4016:
3912:where
3801:where
3519:arctan
2857:Where
2590:
2497:
317:where
130:where
4724:Weber
4664:Rouse
4580:Lewis
4550:Hagen
4520:Euler
4510:Ekman
4485:Darcy
4445:Bejan
4342:S2CID
3757:0.617
3626:0.158
3289:then
48:in a
4585:Mach
4490:Dean
4455:Bond
4450:Biot
4303:ISBN
4284:ISBN
4161:ISSN
4112:ISSN
4064:ISBN
4039:ISBN
4014:ISBN
3403:for
2341:0.05
504:<
28:The
4375:doi
4334:doi
4256:PMC
4246:doi
4242:475
4215:doi
4151:doi
4104:doi
3037:Stk
3023:Stk
2736:Stk
2722:Stk
2642:500
507:0.1
34:Stk
4751::
4373:.
4361:.
4357:.
4340:.
4332:.
4322:48
4320:.
4254:.
4240:.
4236:.
4213:.
4201:.
4184:77
4182:.
4159:.
4149:.
4137:.
4133:.
4110:.
4102:.
4090:.
4078:^
3588:Re
3537:Re
3493:Re
3459:Re
3425:Re
3338:Re
3328:24
3312:Re
3264:Re
3230:Re
3215:Re
3188:Re
3168:Re
3144:Re
3139:24
3121:Re
3052:Re
2989:Re
2898:Re
2867:Re
2833:Re
2818:Re
2791:Re
2771:Re
2747:Re
2742:24
2701:.
2684:95
2631:95
2587:10
2538:95
2424:95
2386:95
2301:95
2172:95
1093:18
975:18
406:.
348:,
289:18
4411:e
4404:t
4397:v
4383:.
4377::
4369::
4363:6
4348:.
4336::
4328::
4311:.
4292:.
4248::
4221:.
4217::
4209::
4203:5
4167:.
4153::
4145::
4139:9
4118:.
4106::
4098::
4092:2
4072:.
4047:.
4022:.
3967:g
3947:d
3925:s
3921:V
3894:g
3891:d
3884:s
3880:V
3874:0
3870:u
3863:=
3859:k
3856:t
3853:S
3829:u
3809:c
3785:)
3778:)
3773:0
3769:u
3764:/
3760:u
3754:+
3751:2
3748:(
3744:k
3741:t
3738:S
3734:+
3731:1
3727:1
3719:1
3715:(
3711:)
3708:1
3702:u
3698:/
3692:0
3688:u
3684:(
3681:+
3678:1
3675:=
3670:0
3666:c
3661:/
3657:c
3623:=
3620:c
3593:o
3581:2
3577:/
3573:3
3569:c
3563:)
3560:)
3555:3
3551:/
3547:1
3542:o
3530:c
3525:(
3511:3
3507:/
3503:1
3498:o
3486:c
3481:(
3478:3
3472:=
3469:)
3464:o
3454:(
3429:)
3421:(
3416:D
3412:C
3371:1
3343:o
3332:/
3322:)
3317:o
3307:(
3302:D
3298:C
3277:0
3269:o
3225:)
3210:(
3205:D
3201:C
3183:d
3173:o
3161:0
3149:o
3134:=
3131:)
3126:o
3116:(
3062:)
3057:o
3047:(
3033:=
3028:e
2999:)
2994:o
2984:(
2955:g
2944:p
2940:d
2935:|
2930:u
2925:|
2919:g
2908:=
2903:o
2872:o
2828:)
2813:(
2808:D
2804:C
2786:d
2776:o
2764:0
2752:o
2732:=
2727:e
2681:,
2672:x
2639:=
2628:,
2619:x
2597:m
2584:=
2579:p
2575:d
2549:5
2546:=
2535:,
2526:x
2504:m
2494:1
2491:=
2486:p
2482:d
2461:2
2458:=
2455:M
2421:,
2412:x
2383:,
2374:x
2350:)
2347:V
2333:u
2329:V
2325:(
2320:0
2316:t
2312:3
2309:=
2298:,
2289:x
2265:)
2260:3
2253:e
2246:1
2243:(
2240:V
2232:0
2228:t
2224:3
2221:+
2218:)
2215:V
2204:u
2200:V
2196:(
2191:0
2187:t
2183:3
2180:=
2169:,
2160:x
2134:0
2130:t
2126:3
2123:=
2120:t
2094:)
2087:0
2083:t
2078:/
2074:t
2064:e
2057:1
2054:(
2051:V
2045:t
2039:+
2036:)
2033:V
2022:u
2018:V
2014:(
2011:t
2005:=
1996:x
1972:t
1969:d
1962:0
1958:t
1953:/
1949:t
1942:e
1938:V
1930:t
1922:0
1914:+
1911:t
1908:d
1905:)
1902:V
1891:u
1887:V
1883:(
1878:t
1870:0
1862:=
1859:t
1856:d
1853:)
1850:t
1847:(
1842:p
1838:v
1832:t
1824:0
1816:=
1807:x
1783:0
1780:=
1777:x
1757:x
1730:0
1726:t
1721:/
1717:t
1710:e
1706:V
1700:+
1697:)
1694:V
1683:u
1679:V
1675:(
1672:=
1669:)
1666:t
1663:(
1658:p
1654:v
1627:V
1602:u
1598:V
1577:)
1574:t
1571:(
1568:H
1565:V
1554:u
1550:V
1546:=
1543:)
1540:t
1537:(
1532:f
1528:v
1477:0
1473:t
1466:2
1462:1
1457:=
1447:3
1440:f
1414:0
1410:t
1383:1
1380:+
1377:s
1372:0
1368:t
1363:1
1358:=
1352:)
1349:s
1346:(
1341:f
1337:v
1331:)
1328:s
1325:(
1320:p
1316:v
1291:)
1288:s
1285:(
1280:p
1276:v
1267:f
1263:v
1259:=
1256:)
1253:s
1250:(
1245:p
1241:v
1237:s
1232:0
1228:t
1197:)
1194:t
1191:(
1188:v
1178:0
1174:t
1170:1
1165:=
1159:t
1156:d
1151:)
1148:t
1145:(
1140:p
1136:v
1132:d
1101:g
1086:2
1081:p
1077:d
1071:p
1060:=
1055:0
1051:t
1027:)
1024:t
1021:(
1018:v
1007:p
997:2
990:p
986:d
969:=
962:p
958:m
952:D
948:F
942:=
936:t
933:d
928:)
925:t
922:(
917:p
913:v
909:d
878:6
872:3
867:p
863:d
851:p
843:=
838:3
832:)
824:2
819:p
815:d
807:(
797:3
794:4
787:p
779:=
774:p
770:m
743:v
735:p
731:d
721:3
718:=
713:D
709:F
682:)
679:t
676:(
671:p
667:v
660:)
657:t
654:(
649:f
645:v
641:=
638:)
635:t
632:(
629:v
603:)
600:t
597:(
592:f
588:v
567:)
564:t
561:(
556:p
552:v
500:k
497:t
494:S
473:1
466:k
463:t
460:S
439:1
432:k
429:t
426:S
388:g
361:p
357:d
330:p
297:g
282:2
277:p
273:d
267:p
256:=
251:0
247:t
201:0
197:l
174:0
170:u
143:0
139:t
111:0
107:l
100:0
96:u
89:0
85:t
78:=
74:k
71:t
68:S
32:(
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