Geodesic
Exact solutions for steady convective layered flows with a spatial acceleration
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2021), pp. 12-22.

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The article considers non-one-dimensional convective layered flows of a viscous incompressible fluid with a spatial acceleration. The simulation is based on the equations of thermal convection in the Boussinesq approximation. The solution to these equations is sought in a generalized class of exact solutions in which all components of the velocity vector, pressure and temperature are presented in the form of complete linear forms along two Cartesian coordinates with non-linear (relative to the third Cartesian coordinate) coefficients. It is shown that for layered flows the system of defining relations reduces to an overdetermined system of ordinary differential equations. Two theorems, that justify the existence (under a special algebraic condition) and the uniqueness of the solution of the resulting overdetermined system, are formulated and proved.
Mots-clés : exact solution
Keywords: layered flow, overdetermined system, compatibility condition. 
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N. V. Burmasheva; E. Yu. Prosviryakov. Exact solutions for steady convective layered flows with a spatial acceleration. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2021), pp. 12-22. http://geodesic.mathdoc.fr/item/IVM_2021_7_a1/

[1] Gershuni G. Z., Zhukhovitskii E. M., Nepomnyaschii A. A., Ustoichivost konvektivnykh techenii, Nauka, M., 1989 | MR

[2] Landau L. D., Lifshits E. M., Gidrodinamika, 6-e izd., Fizmatlit, M., 2006

[3] Burmasheva N. V., Prosviryakov E. Yu., “Krupnomasshtabnaya sloistaya statsionarnaya konvektsiya vyazkoi neszhimaemoi zhidkosti pod deistviem kasatelnykh napryazhenii na verkhnei granitse. Issledovanie polya skorostei”, Vestn. Samarsk. gos. tekh. un-ta, Ser. fiz.-matem. nauki, 21:1 (2017), 180–196

[4] Burmasheva N. V., Prosviryakov E. Yu., “Krupnomasshtabnaya sloistaya statsionarnaya konvektsiya vyazkoi neszhimaemoi zhidkosti pod deistviem kasatelnykh napryazhenii na verkhnei granitse. Issledovanie polei temperatury i davleniya”, Vestn. Samarsk. gos. tekh. un-ta, Ser. fiz.-matem. nauki, 21:4 (2017), 736–751 | Zbl

[5] Prosviryakov E.Yu., “Non-helical exact solutions to the Euler equations for swirling axisymmetric fluid flows”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 23:4 (2019), 764–770 | DOI | Zbl

[6] Burmasheva N. V., Prosviryakov E. Yu., “Termokapillyarnaya konvektsiya vertikalno zavikhrennoi zhidkosti”, Teor. osn. khim. tekhn., 54:1 (2020), 114–124 | MR

[7] Pukhnachev V. V., “Ierarkhiya modelei v teorii konvektsii”, Zap. nauchn. sem. POMI, 288, S.-Peterburg, 2002, 152–177 | Zbl

[8] Hillebrandt W., Müller E., Springel V., “Numerical fluid dynamics in astrophysics”, v. 100, Notes on numerical fluid mechanics and multidisciplinary design, Springer, Berlin–Heidelberg, 2009, 409–420 | DOI

[9] Wölbing R., Baschung B., “Three-dimensional numerical fluid flow simulation of the interior and transitional ballistics process”, Proceedings of $31$ International Symposium on Ballistics (Hyderabad, India, November $4$–$8$, $2019$), ed. Dr. V.K. Saraswat, The Aeronautical Society of India (Hyderabad Branch), The International Ballistics Society, Hyderabad, 2019

[10] Childs E., “The sonification of numerical fluid flow simulations”, Proceedings of the $7$th International Conference on Auditory Display, ICAD2001 (Espoo, Finland, July $29$–August $1$, $2001$), eds. J. Hiipakka, N. Zacharov, T. Takala, International Community for Auditory Display, Espoo, 2001

[11] Severin T., Brück T., Weuster-Botz D., “Validated numerical fluid simulation of a thin-layer cascade photobioreactor in OpenFOAM”, Engineer. in life sci., 19:2 (2019), 97–103 | DOI

[12] Abe H., Kawamura H., Matsuo Yu., “Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence”, Trans. of the ASME, 123 (2001), 382–393

[13] Wang X., Wache P., Navidbakhsh M., Lucius M., Stoltz J. F., “Three-dimensional numerical simulation of blood flow through a modeled aneurysm”, Rus. J. of Biomech., 1 (1999), 26–36

[14] Joseph D. D., Stability of fluid motions, Springer-Verlag, Berlin–Heidelberg–New York, 1976 | MR

[15] Monin A. S., Teoreticheskie osnovy geofizicheskoi gidrodinamiki, Gidrometeoizdat, L., 1988

[16] Pedloski Dzh., Geofizicheskaya gidrodinamika, Mir, M., 1984

[17] Reinhart W., Häusler K., Schaller P., Erhart S., Stetter M., Dual J., Sayir M., “Rheological properties of blood as assessed with a newly designed oscillating viscometer”, Clinic. hemorheol. and microcircul., 18 (1998), 59–65

[18] Skadsem H., Saasen A., “Concentric cylinder viscometer flows of Herschel-Bulkley fluid”, Appl. rheol., 29 (2019), 173–181 | DOI

[19] Scherson D. A., Tolmachev Yu., Wang Zh., Wang J., Palencsar A., “Extensions of the Koutecky–Levich equation to channel electrodes”, Electrochem. and solid state lett., 11:2 (2007)

[20] Kanzaki Ya., Tokuda K., Bruckenstein S., “Dissociation rates of weak acids using sinusoidal hydrodynamic modulated rotating disk electrode employing Koutecky–Levich equation”, J. of the Electrochem. Soc., 161:12 (2014), H770–H779 | DOI

[21] Treimer S., Tang A., Johnson D. C., “A Consideration of the application of Koutecky–Levich plots in the diagnoses of charge-transfer mechanisms at rotated disk electrodes”, Electroanalysis, 14:3 (2002), 165–171 | 3.0.CO;2-6 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[22] Miranda D., Knook M., Paalvast F., Rossi A., Hop W., Oei F., van Bommel J., Gommers D., “Experimental validation of frequent used echocardiographic right ventricular impedance parameters”, Minerva anestesiologica, 80:11 (2014), 1169–1177

[23] Lin C. C., “Note on a class of exact solutions in magneto-hydrodynamics”, Arch. for Rational Mech. and Anal., 1 (1958), 391–395 | DOI | MR | Zbl

[24] Frolovskaya O. A., Pukhnachev V. V., “Analysis of the models of motion of aqueous solutions of polymers on the basis of their exact solutions”, Polymers, 10:6 (2018), 684-1–684-13 | DOI | MR

[25] Desale B., Vivek Sharma, “Exact solutions superimposed with nonlinear plane waves”, Int. J. of Different. Equat., 2016 (2016), 1846341-1–1846341-7 | MR

[26] Burmasheva N. V., Prosviryakov E. Yu., “Tochnoe reshenie uravnenii Nave–Stoksa, opisyvayuschee prostranstvenno neodnorodnye techeniya vraschayuscheisya zhidkosti”, Tr. In-ta matem. i mekhan. UrO RAN, 26, no. 2, 2020, 79–87 | MR

[27] Burmasheva N. V., Prosviryakov E. Yu., “Klass tochnykh reshenii dlya dvumernykh uravnenii geofizicheskoi gidrodinamiki s dvumya parametrami Koriolisa”, Izv. Irkutsk. gos. un-ta, Ser. Matem., 32 (2020), 33–48 | MR | Zbl

[28] Aristov S. N., Prosviryakov E. Yu., “Krupnomasshtabnye techeniya zavikhrennoi vyazkoi neszhimaemoi zhidkosti”, Izv. vuzov. Aviatsionnaya tekhn., 2015, no. 4, 50–54

[29] Aristov S. N., Prosviryakov E. Yu., “Neodnorodnye techeniya Kuetta”, Nelin. dinam., 10:2 (2014), 177–182 | Zbl

[30] Zubarev N. M., Prosviryakov E. Yu., “O tochnykh resheniyakh dlya sloistykh trekhmernykh nestatsionarnykh izobaricheskikh techenii vyazkoi neszhimaemoi zhidkosti”, Prikl. mekhan. i tekhn. fiz., 60:6 (2019), 65–71 | Zbl

[31] Varsakelis Ch., Papalexandris M., “Existence of solutions to a continuum model for hydrostatics of fluid-saturated granular materials”, Appl. Math. Lett., 35 (2014), 77–81 | DOI | MR | Zbl

[32] Berker R., Sur quelques cas d'integration des equations du movement d'un fluide visqueux incompressible, Lille–Paris, 1936 | MR

[33] Shmyglevskii Yu. D., “Ob izobaricheskikh ploskikh techeniyakh vyazkoi neszhimaemoi zhidkosti”, Zhurn. vychisl. matem. i matem. fiziki, 25:12 (1985), 1895–1898 | MR | Zbl

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

[35] Troncoso J., “Isobaric heat capacity of ionic liquids in aqueous solutions. A review”, J. Chem. Eng. Data, 64:11 (2019), 4611–4618 | DOI

[36] Gorshkov A., Prosviryakov E., “Isobaric vortex flow of a viscous incompressible fluid with the Navier boundary condition”, AIP Conf. Proc., 2053 (2018), 040030-1–040030-5 | DOI

[37] Privalova V. V., Prosviryakov E.Yu., “An inhomogeneous Couette-type flow with a perfect slip condition at the lower boundary of an infinite fluid layer”, AIP Conf. Proc., 2176 (2019), 030012-1–030012-4 | DOI

[38] Sidorov A. F., “O dvukh klassakh reshenii uravnenii mekhaniki zhidkosti i gaza i ikh svyazi s teoriei beguschikh voln”, Prikl. mekhan. i tekhn. fiz., 2 (1989), 34–40

[39] Aristov S. N., Prosviryakov E. Yu., “Novyi klass tochnykh reshenii trekhmernykh uravnenii termodiffuzii”, Teor. osn. khim. tekhn., 50:3 (2016), 294–301

[40] Prosviryakov E. Yu., “Novyi klass tochnykh reshenii uravnenii Nave–Stoksa so stepennoi zavisimostyu skorostei ot dvukh prostranstvennykh koordinat”, Teor. osn. khim. tekhn., 53:1 (2019), 112–120

[41] Aristov S. N., Prosviryakov E. Yu., “Neodnorodnoe konvektivnoe techenie Kuetta”, Izv. RAN. MZhG, 2016, no. 5, 3–9 | Zbl