A survey of analytical studies of steady viscous incompressible flows (2000–2004)

V. N. Koterov   ; Yu. D. Shmyglevskii   ; A. V. Shcheprov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 5, pp. 899-920

Résumé

General studies of the Navier–Stokes equations and their analytical solutions describing new vortex structures in viscous flows are surveyed. In the case of plane-parallel steady flows, homogeneous forms of the Navier–Stokes equations are derived, the properties of boundary value problems are examined, the domain of analytical solutions to Cauchy problems is estimated, solutions to Cauchy problems that describe separated flows are found, and the interaction between vortex systems and channel flows is analyzed in the Stokes approximation. In the axisymmetric case, the Kovalevskaya theorem is generalized to Cauchy problems with data lying on the axis of symmetry, which is a singular line for the Navier–Stokes equations; the domain of analytical solutions for such a problem is estimated; in the special case, a class of exact solutions is obtained that describe, in particular, systems of axial vortex structures such as Hill spherical vortices; an Cauchy problem is solved analytically in the general case; and a flow past a sphere with the no-slip condition set on the sphere is considered.

Bibliographie

[1] Berker R., “Intégration des équations du mouvement d'un fluide visqueux incompressible”, Handbuch der Physik, v. 8/2, Springer, Berlin, 1963, 1–384 | MR

[2] Shmyglevskii Yu. D., Analiticheskie issledovaniya dinamiki gaza i zhidkosti, Editorial URSS, M., 1999

[3] Hill M. J. M., “On a spherical vortex”, Brit. Assoc. Rept., 1893, 696–698

[4] Shmyglevskii Yu. D., “Odnorodnye formy uravneniya Nave–Stoksa v sluchae ustanovivshikhsya ploskoparallelnykh techenii neszhimaemoi zhidkosti”, Dokl. RAN, 384:3 (2002), 316–317 ; 389:6 (2003) | MR | Zbl | MR

[5] Paltsev B. V., Shmyglevskii Yu. D., “O podkhode razdelyayuschikh linii toka k obtekaemomu konturu v ploskoparallelnom potoke vyazkoi zhidkosti”, Izv. RAN. Mekhan. zhidkosti i gaza, 2002, no. 2, 76–89 | MR

[6] Strutt J. W. baron Rayleigh, “351 Hydrodynamic notes. Steady motion in a corner of a viscous fluid”, Scient. Papers, v. 6, Univ. Press, Cambridge, 1920, 18–21

[7] Petrovskii I. G., Lektsii ob uravneniyakh s chastnymi proizvodnymi, Fizmatgiz, M., 1961 | MR

[8] Shmyglevskii Yu. D., Scheprov A. V., “O vikhrevykh sistemakh v vyazkoi zhidkosti vblizi uglovoi tochki granitsy”, Izv. RAN. Mekhan. zhidkosti i gaza, 2002, no. 1, 62–73 | MR

[9] Kopchenov V. I., Kraiko A. N., Levin M. P., “K ispolzovaniyu suschestvenno neravnomernykh setok pri chislennom reshenii uravnenii Nave–Stoksa”, Zh. vychisl. matem. i matem. fiz., 22:6 (1982), 1457–1467 | MR

[10] Scheprov A. V., “Raschet odnogo techeniya vyazkoi zhidkosti na adaptivnykh setkakh”, Zh. vychisl. matem. i matem. fiz., 32:6 (1992), 939–951 | MR

[11] Kondratev V. A., “Asimptotika resheniya uravneniya Nave–Stoksa v okrestnosti uglovoi tochki granitsy”, Prikl. matem. i mekhan., 31:1 (1967), 119–123

[12] Dean W. R., Montagnon P. E., “On the steady motion of viscous liquid in a corner”, Proc. Cambridge Philos. Soc., 45:3 (1949), 389–394 | DOI | MR | Zbl

[13] Grinberg G. A., “O metode, predlozhennom P. F. Papkovichem dlya resheniya ploskoi zadachi teorii uprugosti dlya pryamougolnoi oblasti i zadachi izgiba pryamougolnoi tonkoi plity s dvumya zakreplennymi kromkami, i o nekotorykh ego obobscheniyakh”, Prikl. matem. i mekhan., 17:2 (1953), 211–228 | MR | Zbl

[14] Moffatt H. K., “Viscous and resistive eddies near a sharp corner”, J. Fluid Mech., 18:1 (1964), 1–18 | DOI | Zbl

[15] Koterov B. H., Shmyglevskii Yu. D., “O stoksovskikh ploskoparallelnykh vikhrevykh sistemakh v kanalakh”, Izv. RAN. Mekhan. zhidkosti i gaza, 2000, no. 5, 57–65 | MR | Zbl

[16] Papkovich P. F., “Dva voprosa teorii izgiba tonkikh uprugikh plit”, Prikl. matem. i mekhan., 5:3 (1941), 359–374

[17] Shmyglevskii Yu. D., “Dva primera obtekaniya plastinok v priblizhenii Stoksa”, Zh. vychisl. matem. i matem. fiz., 35:6 (1995), 997–1000 | MR

[18] Poiseuille J.-L. M., “Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres”, Compt. Rend., 11:24 (1840), 961–967; 26, 1041–1048; 12:2 (1841), 112–115; Mém. Savans Étrang., 1846, no. 60, 433–544; Poggend. Annal., 1843, no. 43, 424–447; Ann. Fis. Chim., 1843, no. 10, 113–122

[19] Kochin N. E., Kibel I. A., Roze N. V., Teoreticheskaya gidromekhanika, Fizmatgiz, M., 1963 | Zbl

[20] Scheprov A. V., “Poluchenie analiticheskikh reshenii uravneniya Nave–Stoksa dlya osesimmetrichnykh i ploskikh techenii vyazkoi neszhimaemoi zhidkosti”, Dokl. RAN, 394:5 (2004), 626–630

[21] Shmyglevskii Yu. D., Scheprov A. V., “O ploskoparallelnykh otryvnykh techeniyakh neszhimaemoi zhidkosti u ploskikh granits”, Izv. RAN. Mekhan. zhidkosti i gaza, 2002, no. 5, 12–21 | MR

[22] O'Brien V., “Steady spheroidal vortices — more exact solutions to the Navier–Stokes equation”, Quart. Appl. Math., 19:2 (1961), 163–168 | MR

[23] Werlé H., “Visualisation en tunnel hydrodynamique”, Rech. Áeronaut., 1953, no. 33, 3–7

[24] Gartshore I. S., Recent work in swirling incompressible flow, Nat. Res. Coun. Canada. Aeronaut. Rept. LR-343, 1962, 58 pp.

[25] Hall M. G., “Vortex breakdown”, Ann. Rev. Fluid Mech., 4 (1972), 195–218 | DOI | Zbl

[26] Leibovich S., “The structure of vortex breakdown”, Ann. Rev. Fluid Mech., 10 (1978), 221–246 | DOI

[27] Leibovich S., “Vortex stability and breakdown: survey and extensions”, AIAA Journal, 22:9 (1984), 1192–1206 | DOI

[28] Escudier M., “Vortex breakdown: observations and explanations”, Progr. Aerospace Sci., 25:2 (1988), 189–229 | DOI

[29] Spall R. E., Gatski T. B., Ash R. L., “The structure and dynamics of bubble-type vortex breakdown”, Proc. Roy. Soc. A, 429:1877 (1990), 313–637

[30] Van Dyke M., An album of fluid motion, Parabolic Press, Stanford, 1982

[31] Trigub V. H., “K voprosu o razrushenii vikhrevoi niti”, Prikl. matem. i mekhan., 49:2 (1985), 220–226 | MR | Zbl

[32] Sychev Vik. V., “Asimptoticheskaya teoriya pazrusheniya vikhrya”, Izv. RAN. Mekhan. zhidk. i gaza, 1993, no. 3, 78–90 | MR | Zbl

[33] Shmyglevskii Yu. D., “O “razrushenii vikhrya””, Izv. RAN. Mekhan. zhidkosti i gaza, 1995, no. 3, 167–169 | MR

[34] Shmyglevskii Yu. D., Scheprov A. V., “Tochnoe predstavlenie nekotorykh osesimmetrichnykh vikhrevykh obrazovanii v vyazkoi neszhimaemoi zhidkosti”, Dokl. RAN, 393:4 (2003), 489–492 | MR

[35] Crudeli U., “Sui moti di un liquido viscoso (omogeno) simmetrici rispetto ad un asse”, Atti Reale Accad. Naz. dei Lincei. Rendiconti. Ser. 6, 5:7 (1927), 500–504 | Zbl

[36] Crudeli U., “Una nuova categoria di moti stazionari dei liquidi (pesanti) viscosi entro tubi cilindrici (rotondi) verticali”, Atti Reale Accad. Naz. dei Lincei. Rendiconti. Ser. 6, 5:10 (1927), 783–789 | Zbl

[37] Crudeli U., “Sopra una categoria di moti stazionari dei liquidi (pesanti) viscosi entro tubi cilindrici (rotondi) verticali”, Atti Reale Accad. Naz. dei Lincei. Rendiconti. Ser. 6, 6:10 (1927), 397–401 | Zbl

[38] Scheprov A. V., “Primer vneshnego obtekaniya sfery vyazkoi neszhimaemoi zhidkostyu pri proizvolnom chisle Reinoldsa”, Dokl. RAN, 395:4 (2004), 485–486

[39] Stokes G. G., “On the effect of the internal friction of fluids on the motion of pendulums”, Philos. Mag., 1 (1851), 337–339; Trans. Cambridge Philos. Soc., 9 (1856), 8–106

[40] Hadamard J., “Mouvement permanent lent d'une sphère liquide et visqueuse dans un liquide visqueux”, ØE uvres de Jacques Hadamard, v. 3, CNRS, P., 1968, 1311–1314