84:
458:
across this layer. This could be due to the liquid evaporating or being heated from below. There is a surface tension at the surface of a liquid that depends on temperature, typically as the temperature increases the surface tension decreases. Thus if due to a small fluctuation temperature, one part
35:, with the rate of transport of diffusion. The Marangoni effect is flow of a liquid due to gradients in the surface tension of the liquid. Diffusion is of whatever is creating the gradient in the surface tension. Thus as the Marangoni number compares flow and diffusion timescales it is a type of
88:
A common example is surface tension gradients caused by temperature gradients. Then the relevant diffusion process is that of thermal energy (heat). Another is surface gradients caused by variations in the concentration of surfactants, where the diffusion is now that of surfactant molecules.
45:
664:
426:
668:
When Ma is small thermal diffusion dominates and there is no flow, but for large Ma, flow (convection) occurs, driven by the gradients in the surface tension. This is called BĂ©nard-Marangoni convection.
578:
463:. This flow will transport thermal energy, and the Marangoni number compares the rate at which thermal energy is transported by this flow to the rate at which thermal energy diffuses.
833:
206:, where the fluid velocity is obtained by equating the stress gradient to the viscous dissipation. A surface tension is a force per unit length, so the resulting stress must scale as
583:
323:
235:
140:
459:
of the surface is hotter than another, there will be flow from the hotter part to the colder part, driven by this difference in surface tension, this flow is called the
456:
544:
524:
358:
266:
504:
117:
79:{\displaystyle \mathrm {Ma} ={\dfrac {\mbox{advective transport rate, due to surface tension gradient}}{\mbox{diffusive transport rate, of source of gradient}}}}
484:
351:
286:
200:
180:
160:
826:
1178:
819:
1183:
842:
739:
Block, Myron J. (1956). "Surface
Tension as the Cause of BĂ©nard Cells and Surface Deformation in a Liquid Film".
549:
988:
291:
209:
435:
A common application is to a layer of liquid, such as water, when there is a temperature difference
122:
1065:
1048:
96:, although its use dates from the 1950s and it was neither discovered nor used by Carlo Marangoni.
791:
353:, Here this is the diffusion constant of whatever is causing the surface tension difference. So,
1188:
1043:
943:
659:{\displaystyle \mathrm {Ma} =-(\partial \gamma /\partial T).{\frac {L.\Delta T}{\mu .\alpha }}}
182:
is the only length scale in the problem, which in practice implies that the liquid be at least
438:
529:
509:
240:
898:
811:
748:
697:
28:
903:
489:
102:
8:
968:
752:
701:
1152:
938:
878:
772:
721:
469:
336:
330:
271:
185:
165:
145:
1107:
1072:
853:
764:
725:
713:
1157:
1028:
978:
888:
776:
756:
705:
460:
32:
1033:
326:
36:
1122:
1102:
1060:
1055:
883:
93:
421:{\displaystyle \mathrm {Ma} ={\dfrac {uL}{D}}={\dfrac {\Delta \gamma L}{\mu D}}}
1132:
1112:
1097:
1092:
1038:
1023:
998:
993:
983:
963:
953:
918:
863:
162:
parallel to the surface, can be estimated as follows. Note that we assume that
709:
1172:
1142:
1137:
1127:
1117:
1082:
1077:
1018:
958:
948:
928:
923:
893:
858:
768:
717:
688:
Pearson, J. R. A. (1958). "On convection cells induced by surface tension".
1147:
1087:
1003:
973:
933:
908:
868:
1008:
913:
873:
203:
288:
the speed of the
Marangoni flow. Equating the two we have a flow speed
580:, the Marangoni number can be calculated using the following formula:
760:
202:
deep. The transport rate is usually estimated using the equations of
841:
430:
65:
advective transport rate, due to surface tension gradient
99:
The
Marangoni number for a simple liquid of viscosity
68:
63:
586:
552:
532:
512:
492:
472:
441:
394:
374:
361:
339:
294:
274:
243:
212:
188:
168:
148:
125:
105:
61:
48:
658:
572:
538:
518:
498:
478:
450:
420:
345:
317:
280:
260:
229:
194:
174:
154:
134:
111:
78:
1170:
329:, it is a velocity times a length, divided by a
70:diffusive transport rate, of source of gradient
827:
789:
92:The number is named after Italian scientist
573:{\displaystyle \partial \gamma /\partial T}
31:that compares the rate of transport due to
834:
820:
546:which changes with temperature at a rate
431:Marangoni number due to thermal gradients
1179:Dimensionless numbers of fluid mechanics
843:Dimensionless numbers in fluid mechanics
1184:Dimensionless numbers of thermodynamics
687:
1171:
815:
738:
318:{\displaystyle u=\Delta \gamma /\mu }
237:, while the viscous stress scales as
683:
681:
42:The Marangoni number is defined as:
13:
636:
615:
604:
591:
588:
564:
553:
442:
397:
366:
363:
301:
213:
126:
53:
50:
14:
1200:
678:
466:For a liquid layer of thickness
230:{\displaystyle \Delta \gamma /L}
783:
732:
621:
601:
135:{\displaystyle \Delta \gamma }
119:with a surface tension change
27:) is, as usually defined, the
1:
792:"Marangoni Number Calculator"
672:
7:
10:
1205:
690:Journal of Fluid Mechanics
849:
710:10.1017/S0022112058000616
526:, with a surface tension
16:Concept in fluid dynamics
506:and thermal diffusivity
451:{\displaystyle \Delta T}
539:{\displaystyle \gamma }
519:{\displaystyle \alpha }
261:{\displaystyle \mu u/L}
660:
574:
540:
520:
500:
480:
452:
422:
347:
319:
282:
262:
231:
196:
176:
156:
136:
113:
80:
661:
575:
541:
521:
501:
481:
453:
423:
348:
325:. As Ma is a type of
320:
283:
263:
232:
197:
177:
157:
137:
114:
81:
584:
550:
530:
510:
499:{\displaystyle \mu }
490:
470:
439:
359:
337:
292:
272:
241:
210:
186:
166:
146:
123:
112:{\displaystyle \mu }
103:
46:
29:dimensionless number
790:Pr. Steven Abbott.
753:1956Natur.178..650B
702:1958JFM.....4..489P
989:Keulegan–Carpenter
796:stevenabbott.co.uk
656:
570:
536:
516:
496:
476:
448:
418:
416:
388:
343:
331:diffusion constant
315:
278:
258:
227:
192:
172:
152:
132:
109:
76:
74:
72:
67:
1166:
1165:
747:(4534): 650–651.
654:
479:{\displaystyle L}
415:
387:
346:{\displaystyle D}
281:{\displaystyle u}
195:{\displaystyle L}
175:{\displaystyle L}
155:{\displaystyle L}
73:
71:
66:
1196:
836:
829:
822:
813:
812:
807:
806:
804:
802:
787:
781:
780:
761:10.1038/178650a0
736:
730:
729:
685:
665:
663:
662:
657:
655:
653:
642:
628:
614:
594:
579:
577:
576:
571:
563:
545:
543:
542:
537:
525:
523:
522:
517:
505:
503:
502:
497:
485:
483:
482:
477:
461:Marangoni effect
457:
455:
454:
449:
427:
425:
424:
419:
417:
414:
406:
395:
389:
383:
375:
369:
352:
350:
349:
344:
324:
322:
321:
316:
311:
287:
285:
284:
279:
267:
265:
264:
259:
254:
236:
234:
233:
228:
223:
201:
199:
198:
193:
181:
179:
178:
173:
161:
159:
158:
153:
142:over a distance
141:
139:
138:
133:
118:
116:
115:
110:
85:
83:
82:
77:
75:
69:
64:
62:
56:
21:Marangoni number
1204:
1203:
1199:
1198:
1197:
1195:
1194:
1193:
1169:
1168:
1167:
1162:
845:
840:
810:
800:
798:
788:
784:
737:
733:
686:
679:
675:
643:
629:
627:
610:
587:
585:
582:
581:
559:
551:
548:
547:
531:
528:
527:
511:
508:
507:
491:
488:
487:
471:
468:
467:
440:
437:
436:
433:
407:
396:
393:
376:
373:
362:
360:
357:
356:
338:
335:
334:
307:
293:
290:
289:
273:
270:
269:
250:
242:
239:
238:
219:
211:
208:
207:
187:
184:
183:
167:
164:
163:
147:
144:
143:
124:
121:
120:
104:
101:
100:
94:Carlo Marangoni
60:
49:
47:
44:
43:
33:Marangoni flows
17:
12:
11:
5:
1202:
1192:
1191:
1189:Fluid dynamics
1186:
1181:
1164:
1163:
1161:
1160:
1155:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1115:
1110:
1105:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1069:
1068:
1058:
1053:
1052:
1051:
1046:
1036:
1031:
1026:
1021:
1016:
1011:
1006:
1001:
996:
991:
986:
981:
976:
971:
966:
961:
956:
951:
946:
941:
936:
931:
926:
921:
916:
911:
906:
901:
896:
891:
886:
881:
876:
871:
866:
861:
856:
850:
847:
846:
839:
838:
831:
824:
816:
809:
808:
782:
731:
696:(5): 489–500.
676:
674:
671:
652:
649:
646:
641:
638:
635:
632:
626:
623:
620:
617:
613:
609:
606:
603:
600:
597:
593:
590:
569:
566:
562:
558:
555:
535:
515:
495:
475:
447:
444:
432:
429:
413:
410:
405:
402:
399:
392:
386:
382:
379:
372:
368:
365:
342:
314:
310:
306:
303:
300:
297:
277:
257:
253:
249:
246:
226:
222:
218:
215:
191:
171:
151:
131:
128:
108:
59:
55:
52:
15:
9:
6:
4:
3:
2:
1201:
1190:
1187:
1185:
1182:
1180:
1177:
1176:
1174:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1067:
1064:
1063:
1062:
1059:
1057:
1054:
1050:
1047:
1045:
1042:
1041:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
935:
932:
930:
927:
925:
922:
920:
917:
915:
912:
910:
907:
905:
902:
900:
899:Chandrasekhar
897:
895:
892:
890:
887:
885:
882:
880:
877:
875:
872:
870:
867:
865:
862:
860:
857:
855:
852:
851:
848:
844:
837:
832:
830:
825:
823:
818:
817:
814:
797:
793:
786:
778:
774:
770:
766:
762:
758:
754:
750:
746:
742:
735:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
684:
682:
677:
670:
666:
650:
647:
644:
639:
633:
630:
624:
618:
611:
607:
598:
595:
567:
560:
556:
533:
513:
493:
473:
464:
462:
445:
428:
411:
408:
403:
400:
390:
384:
380:
377:
370:
354:
340:
332:
328:
327:PĂ©clet number
312:
308:
304:
298:
295:
275:
255:
251:
247:
244:
224:
220:
216:
205:
189:
169:
149:
129:
106:
97:
95:
90:
86:
57:
40:
38:
37:PĂ©clet number
34:
30:
26:
22:
1013:
799:. Retrieved
795:
785:
744:
740:
734:
693:
689:
667:
486:, viscosity
465:
434:
355:
98:
91:
87:
41:
24:
20:
18:
1153:Weissenberg
204:Stokes flow
1173:Categories
1073:Richardson
854:Archimedes
673:References
1158:Womersley
1049:turbulent
1029:Ohnesorge
1014:Marangoni
979:Iribarren
904:Damköhler
889:Capillary
769:0028-0836
726:123404447
718:0022-1120
651:α
645:μ
637:Δ
616:∂
608:γ
605:∂
599:−
565:∂
557:γ
554:∂
534:γ
514:α
494:μ
443:Δ
409:μ
401:γ
398:Δ
313:μ
305:γ
302:Δ
245:μ
217:γ
214:Δ
130:γ
127:Δ
107:μ
1133:Suratman
1123:Strouhal
1103:Sherwood
1066:magnetic
1061:Reynolds
1056:Rayleigh
1044:magnetic
884:Brinkman
1113:Stanton
1108:Shields
1098:Scruton
1093:Schmidt
1039:Prandtl
1024:Nusselt
999:Laplace
994:Knudsen
984:Kapitza
969:Görtler
964:Grashof
954:Galilei
919:Deborah
864:Bagnold
801:2 March
777:4273633
749:Bibcode
698:Bibcode
1143:Ursell
1138:Taylor
1128:Stuart
1118:Stokes
1083:Rossby
1078:Roshko
1034:PĂ©clet
1019:Morton
959:Graetz
949:Froude
939:Eötvös
929:Eckert
924:Dukhin
894:Cauchy
859:Atwood
775:
767:
741:Nature
724:
716:
268:, for
1148:Weber
1088:Rouse
1004:Lewis
974:Hagen
944:Euler
934:Ekman
909:Darcy
869:Bejan
773:S2CID
722:S2CID
1009:Mach
914:Dean
879:Bond
874:Biot
803:2019
765:ISSN
714:ISSN
19:The
757:doi
745:178
706:doi
1175::
794:.
771:.
763:.
755:.
743:.
720:.
712:.
704:.
692:.
680:^
333:,
39:.
25:Ma
835:e
828:t
821:v
805:.
779:.
759::
751::
728:.
708::
700::
694:4
648:.
640:T
634:.
631:L
625:.
622:)
619:T
612:/
602:(
596:=
592:a
589:M
568:T
561:/
474:L
446:T
412:D
404:L
391:=
385:D
381:L
378:u
371:=
367:a
364:M
341:D
309:/
299:=
296:u
276:u
256:L
252:/
248:u
225:L
221:/
190:L
170:L
150:L
58:=
54:a
51:M
23:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.