Inhomogeneous Poiseuille flow

N. V. Burmasheva   ; A. V. Dyachkova   ; E. Yu. Prosviryakov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 77 (2022), pp. 68-85

Résumé

The paper presents an investigation of the isothermal steady flow of a viscous incompressible fluid in an extended flat layer using hydrodynamic equations. The bottom of the layer under consideration is limited by a stationary solid hydrophilic surface. At the upper boundary of the layer, the pressure field, which is inhomogeneous in both horizontal coordinates, and the velocity field are specified. These boundary conditions allow one to generalize the classical Poiseuille flow. The exact solution, satisfying the set boundary value problem, is described by a series of polynomials of different orders. The highest (fifth) degree of the polynomials corresponds to a homogeneous component of the horizontal velocity. Here, the pressure field depends only on the horizontal coordinates; the dependence is linear. The detailed analysis of the velocity field is carried out. The obtained results confirm that the determined exact solution can describe multiple stratification of the velocity field and the corresponding field of tangent stresses. The analysis of spectral properties of the velocity field is performed for a general case without specifying the values of physical constants that unambiguously identify the studied fluid. Therefore, the presented results are applicable to viscous fluids of various nature.

Détail

Keywords: vertically swirling fluid, isothermal flow, inhomogeneous Poiseuille flow, Navier-Stokes equations, countercurrent
Mots-clés : exact solution, stagnation point.

Bibliographie

[1] Hagen G., “Uber die Bewegung des Wasser in engen zylindrischen Rohren”, Pogg. Ann., 46 (1839), 423–442

[2] Poiseuille J.-L.-M., “Recherches experimented sur le mouvement des liquides dans les tubes de tres petits diametres”, Comptes rendus hebdomadaires des seances de l'Academie des sciences, 11 (1840), 961–967; 1041–1048

[3] Poiseuille J.-L.-M., “Recherches experimented sur le mouvement des liquides dans les tubes de tres petits diametres (suite)”, Comptes rendus hebdomadaires des seances de l'Academie des sciences, 12 (1841), 112–115

[4] Volarovich M.P., “Raboty Puazeilya o techenii zhidkosti v trubakh”, Izvestiya Akademii nauk SSSR. Ser. fizicheskaya, XI:1 (1947), 7–18

[5] Pedley T.J., The fluid mechanics of large blood vessels, Cambridge monographs on mechanics and applied mathematics, Cambridge University Press, Cambridge–New York–Melbourne, 1980, 446 pp.

[6] Regirer S.A., “Kvaziodnomernaya teoriya peristalticheskikh potokov”, Izvestiya Akademii nauk SSSR. Mekhanika zhidkosti i gaza, 1984, no. 5, 89–97

[7] Polyanin A.D., Kutepov A.M., Vyazmin A.V., Kazenin D.A., Hydrodynamics, mass and heat transfer in chemical engineering, Taylor Francis, London, 2002, 406 pp.

[8] Knyazev D.V., Kolpakov I.Yu., “Tochnye resheniya zadachi o techenii vyazkoi zhidkosti v tsilindricheskoi oblasti s menyayuschimsya radiusom”, Nelineinaya dinamika, 11:1 (2015), 89–97 | MR

[9] Aristov S.N., Knyazev D.V., “Novoe tochnoe reshenie zadachi o vraschatelno-simmetrichnom techenii Kuetta-Puazeilya”, Prikladnaya mekhanika i tekhnicheskaya fizika, 48:5 (285) (2007), 71–77 | DOI | MR

[10] Fomin A.A., Fomina L.N., “Chislennoe modelirovanie techeniya zhidkosti v ploskoi kaverne pri bolshikh chislakh Reinoldsa”, Vychislitelnaya mekhanika sploshnykh sred, 7:4 (2014), 363–377 | DOI

[11] Boiko A.V., Kirilovskii S.V., Maslov A.A., Poplavskaya T.V., “Inzhenernoe modelirovanie laminarno-turbulentnogo perekhoda: dostizheniya i problemy (obzor)”, Prikladnaya mekhanika i tekhnicheskaya fizika, 56:5 (2015), 30–49 | DOI

[12] Stankevich Yu.A., Fisenko S.P., “Perestroika profilya Puazeilya v neizotermicheskikh techeniyakh v reaktore”, Inzhenerno-fizicheskii zhurnal, 84:6 (2011), 1225–1228

[13] Savenkov I.V., “O neustoichivosti ploskogo techeniya Puazeilya mezhdu uprugimi plastinami”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 51:12 (2011), 2288–2295 | MR

[14] Troshkin O.V., “K nelineinoi ustoichivosti techenii Kuetta, Puazeilya i Kolmogorova v ploskom kanale”, Doklady Akademii nauk, 443:1 (2012), 29 | MR

[15] Proskurin A.V., Sagalakov A.M., “Ustoichivost techeniya Puazeilya pri nalichii prodolnogo magnitnogo polya”, Zhurnal tekhnicheskoi fiziki, 82:5 (2012), 29–35

[16] Chefranov S.G., Chefranov A.G., “Ustranenie paradoksa lineinoi ustoichivosti techeniya Khagena-Puazeilya i vyazkii dissipativnyi mekhanizm vozniknoveniya turbulentnosti v pogranichnom sloe”, Zhurnal eksperimentalnoi i teoreticheskoi fiziki, 146:2 (2014), 373–383 | MR

[17] Savenkov I.V., “Ob osesimmetrichnoi neustoichivosti techeniya Puazeilya-Kuetta mezhdu kontsentricheskimi tsilindrami pri vysokikh chislakh Reinoldsa”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 55:2 (2015), 295 | DOI | MR

[18] Skulskii O.I., Aristov S.N., Mekhanika anomalno vyazkikh zhidkostei, Izd-vo UrO RAN, Ekaterinburg, 2004, 156 pp.

[19] Altukhov Yu.A., Gusev A.S., Makarova M.A., Pyshnograi G.V., “Obobschenie zakona Puazeilya dlya ploskoparallelnogo techeniya vyazkouprugikh sred”, Mekhanika kompozitsionnykh materialov i konstruktsii, 13:4 (2007), 581–590

[20] Aristov S.N., Shvarts K.G., Vikhrevye techeniya advektivnoi prirody vo vraschayuschemsya sloe zhidkosti, Permskii gos. un-t, Perm, 2006, 155 pp.

[21] Aristov S.N., Shvarts K.G., “Advektivnoe techenie vo vraschayuscheisya zhidkoi plenke”, Prikladnaya mekhanika i tekhnicheskaya fizika, 57:1 (2016) | DOI

[22] Ostroumov G.A., Svobodnaya konvektsiya v usloviyakh vnutrennei zadachi, Gos. izd-vo tekhniko-teoret. lit., M., 1952, 256 pp. | MR

[23] Birikh R.V., “O termokapillyarnoi konvektsii v gorizontalnom sloe zhidkosti”, Prikladnaya mekhanika i tekhnicheskaya fizika, 3 (1966), 69–72

[24] Birikh R.V., Pukhnachev V.V., “Osevoe konvektivnoe techenie vo vraschayuscheisya trube s prodolnym gradientom temperatury”, Doklady RAN, 436:3 (2011), 323–327 | MR

[25] Birikh R.V., Pukhnachev V.V., “Konvektivnoe techenie v gorizontalnom kanale s nenyutonovskoi reologiei poverkhnosti pri nestatsionarnom prodolnom gradiente temperatury”, Izvestiya Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza, 2015, no. 1, 192–198 | MR

[26] Pukhnachev V.V., “Nestatsionarnye analogi resheniya Birikha”, Izvestiya Izvestiya Altaiskogo gosudarstvennogo universiteta, 2011, no. 1-2, 62–69

[27] Andreev V.K., Resheniya Birikha uravnenii konvektsii i nekotorye ego obobscheniya, IVM SO RAN, Krasnoyarsk, 2010, 68 pp.

[28] Andreev V.K., Bekezhanova V.B., “Ustoichivost neizotermicheskikh zhidkostei (obzor)”, Prikladnaya mekhanika i tekhnicheskaya fizika, 54:2 (2013), 3–20 | DOI

[29] Shvarts K.G., “Ploskoparallelnoe advektivnoe techenie v gorizontalnom sloe neszhimaemoi zhidkosti s tverdymi granitsami”, Izvestiya Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza, 2014, no. 4, 26–30

[30] Aristov S.N., Prosviryakov E.Yu., “O sloistykh techeniyakh ploskoi svobodnoi konvektsii”, Nelineinaya dinamika, 9:4 (2013), 651–657

[31] Burmasheva N.V., Prosviryakov E.Yu., “Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation”, Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences, 23:2 (2019), 341–360 | DOI

[32] Burmasheva N.V., Prosviryakov E.Yu., “Convective layered flows of a vertically whirling viscous incompressible fluid. Temperature field investigation”, Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences, 24:3 (2020), 528541 | DOI

[33] Burmasheva N.V., Prosviryakov E.Yu., “Termokapillyarnaya konvektsiya vertikalno zavikhrennoi zhidkosti”, Teoreticheskie osnovy khimicheskoi tekhnologii, 54:1 (2020), 114–124 | DOI | MR

[34] Aristov S.N., Prosviryakov E.Yu., Spevak L.F., “Nestatsionarnaya sloistaya teplovaya i kontsentratsionnaya konvektsiya marangoni vyazkoi neszhimaemoi zhidkosti”, Vychislitelnaya mekhanika sploshnykh sred, 8:4 (2015), 445–456 | DOI

[35] Aristov S.N., Prosviryakov E.Yu., Spevak L.F., “Nestatsionarnaya konvektsiya Benara-Marangoni sloistykh techenii vyazkoi neszhimaemoi zhidkosti”, Teoreticheskie osnovy khimicheskoi tekhnologii, 50:2 (2016), 132–141 | DOI

[36] Burmasheva N.V., Prosviryakov E.Yu., “Tochnoe reshenie uravnenii Nave-Stoksa, opisyvayuschee prostranstvenno neodnorodnye techeniya vraschayuscheisya zhidkosti”, Trudy Instituta matematiki i mekhaniki UrO RAN, 26, no. 2, 2020, 79–87 | DOI | MR

[37] Burmasheva N.V., Prosviryakov E.Yu., “Klass tochnykh reshenii dlya dvumernykh uravnenii geofizicheskoi gidrodinamiki s dvumya parametrami Koriolisa”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Cer. Matematika, 32 (2020), 33–48 | DOI | MR

[38] Burmasheva N.V., Prosviryakov E.Yu., “Isothermal layered flows of a viscous incompressible fluid with spatial acceleration in the case of three Coriolis parameters”, Diagnostics, Resource and Mechanics of materials and structures, 2020, no. 3, 29–46 | DOI

[39] Aristov S.N., Prosviryakov E.Yu., “Neodnorodnoe konvektivnoe techenie Kuetta”, Izvestiya Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza, 2016, no. 5, 3–9 | DOI

[40] Aristov S.N., Prosviryakov E.Yu., “Neodnorodnye techeniya Kuetta”, Nelineinaya dinamika, 10:2 (2014), 177–182

[41] Privalova V.V., Prosviryakov E.Yu., Simonov M.A., “Nonlinear gradient flow of a vertical vortex fluid in a thin layer”, Rus. J. Nonlin. Dyn., 15:3 (2019), 271–283 | MR

[42] Marshall J., Hill C., Perelman L., Adroft A., “Hydrostatic, quasi-hydrostatic and nonhydrostatic ocean modeling”, Geophys. Res. J., 102:S.3 (1997), 5733–5752 | DOI

[43] Gill A., Dinamika atmosfery i okeana, v 2 t., Mir, M., 1986

[44] Zubarev N.M., Prosviryakov E.Yu., “O tochnykh resheniyakh dlya sloistykh trekhmernykh nestatsionarnykh izobaricheskikh techenii vyazkoi neszhimaemoi zhidkosti”, Prikladnaya mekhanika i tekhnicheskaya fizika, 2019, no. 6, 65–71 | DOI | MR

[45] Privalova V.V., Prosviryakov E.Yu., “Nelineinoe izobaricheskoe techenie vyazkoi neszhimaemoi zhidkosti v tonkom sloe s pronitsaemymi granitsami”, Vychislitelnaya mekhanika sploshnykh sred, 12:2 (2019), 230–242 | DOI | MR

[46] Burmasheva N.V., Prosviryakov E.Yu., “Issledovanie stratifikatsii gidrodinamicheskikh polei dlya sloistykh techenii vertikalno zavikhrennoi zhidkosti”, DReaM, 2020, no. 4, 62–78 | DOI