New Class of Exact Solutions of Navier–Stokes Equations with Exponential Dependence of Velocity on Two Spatial Coordinates
Tóm tắt
A new class of exact solutions of nonlinear and linearized Navier–Stokes equations has been proposed, which generalize the well-known family of exact solutions in which the velocity is linear in some coordinates. The case of the quadratic dependence of the velocities on two horizontal (longitudinal) coordinates with coefficients that are the functions of the vertical (transverse) coordinate and time was considered in detail. The solutions were generalized for rotating liquids. Equations for constructing exact solutions with an arbitrary dependence of velocities on the horizontal coordinates were derived.
Tài liệu tham khảo
Aristov, S.N., Knyazev, D.V., and Polyanin, A.D., Exact solutions of the Navier–Stokes equations with the linear dependence of velocity components on two space variables, Theor. Found. Chem. Eng., 2009, vol. 43, no. 5, pp. 642–662.https://doi.org/10.1134/S0040579509050066
Lin, C.C., Note on a class of exact solutions in magneto-hydrodynamics, Arch. Ration. Mech. Anal., 1957, vol. 1, no. 1, pp. 391–395. https://doi.org/10.1007/BF00298016
Sidorov, A.F., Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory, J. Appl. Mech. Tech. Phys., 1989, vol. 30, no. 2, pp. 197–203. https://doi.org/10.1007/BF00852164
Aristov, S.N., Vortex flows in thin liquid layers, Extended Abstract of Doctoral (Phys.–Math.) Dissertation, Perm: Inst. of Continuous Media Mechanics, Ural Branch, Academy of Sciences of the USSR, 1990.
Aristov, S.N. and Prosviryakov, E.Yu., A new class of exact solutions for three-dimensional thermal diffusion equations, Theor. Found. Chem. Eng., 2016, vol. 50, no. 3, pp. 286–293. https://doi.org/10.1134/S0040579516030027
Pukhnachev, V.V., Mathematical model of an incompressible viscoelastic Maxwell medium, J. Appl. Mech. Tech. Phys., 2010, vol. 51, no. 4, pp. 546–554. https://doi.org/10.1007/s10808-010-0071-5
Pukhnachev, V.V. and Pukhnacheva, T.P., The Couette problem for a Kelvin–Voigt medium, J. Math. Sci., 2012, vol. 186, no. 3, pp. 495–510. https://doi.org/10.1007/s10958-012-1003-0
Pukhnachev, V.V., Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium, J. Appl. Mech. Tech. Phys., 2009, vol. 50, no. 2, pp. 181–187. https://doi.org/10.1007/s10808-009-0025-y
Polyanin, A.D. and Vyazmin, A.V., Decomposition of three-dimensional linearized equations for Maxwell and Oldroyd viscoelastic fluids and their generalizations, Theor. Found. Chem. Eng., 2013, vol. 47, no. 4, pp. 321–329.https://doi.org/10.1134/S004057951304026X
Polyanin, A.D., Exact solutions to new classes of reaction-diffusion equations containing delay and arbitrary functions, Theor. Found. Chem. Eng., 2015, vol. 49, no. 2, pp. 169–175. https://doi.org/10.1134/S0040579515020104
Polyanin, A.D., Sorokin, V.G., and Vyazmin, A.V., Exact solutions and qualitative features of nonlinear hyperbolic reaction–diffusion equations with delay, Theor. Found. Chem. Eng., 2015, vol. 49, no. 5, pp. 622–635. https://doi.org/10.1134/S0040579515050243
Aristov, S.N. and Prosviryakov, E.Yu., Large-scale flows of viscous incompressible vortical fluid, Russ. Aeronaut., 2015, vol. 58, no. 4, pp. 413–418. https://doi.org/10.3103/S1068799815040091
Aristov, S.N. and Prosviryakov, E.Yu., Unsteady layered vortical fluid flows, Fluid Dyn., 2016, vol. 51, no. 2, pp. 148–154. https://doi.org/10.1134/S0015462816020034
Aristov, S.N., Prosviryakov, E.Yu., and Spevak, L.F., Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows, Theor. Found. Chem. Eng., 2016, vol. 50, no. 2, pp. 132–141. https://doi.org/10.1134/S0040579516020019
Knyazev, D.V., Two-dimensional flows of a viscous binary fluid between moving solid boundaries, J. Appl. Mech. Tech. Phys., 2011, vol. 52, no. 2, pp. 212–217. https://doi.org/10.1134/S0021894411020088
Aristov, S.N. and Prosviryakov, E.Yu., Nonuniform convective Couette flow, Fluid Dyn., 2016, vol. 51, no. 5, pp. 581–587. https://doi.org/10.1134/S001546281605001X
Aristov, S.N. and Shvarts, K.G., Convective heat transfer in a locally heated plane incompressible fluid layer, Fluid Dyn., 2013, vol. 48, no. 3, pp. 330–335. https://doi.org/10.1134/S001546281303006X
Andreev, V.K. and Bekezhanova, V.B., Stability of non-isothermal fluids (Review), J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 2, pp. 171–184. https://doi.org/10.1134/S0021894413020016
Goncharova, O.N. and Rezanova, E.V., Modeling of two-layer fluid flows with evaporation at the interface in the presence of the anomalous thermocapillary effect, J. Sib. Fed. Univ., Math. Phys., 2016, vol. 9, no. 1, pp. 48–59. https://doi.org/10.17516/1997-1397-2016-9-1-48-59
Efimova, M.V., On one two-dimensional stationary flow of a binary mixture and viscous fluid in a plane layer, J. Sib. Fed. Univ., Math. Phys., 2016, vol. 9, no. 1, pp. 30–36.https://doi.org/10.17516/1997-1397-2016-9-1-30-36
Goncharova, O.N., Kabov, O.A., and Pukhnachov, V.V., Solutions of special type describing the three dimensional thermocapillary flows with an interface, Int. J. Heat Mass Transfer, 2012, vol. 55, no. 4, pp. 715–725. https://doi.org/10.1016/j.ijheatmasstransfer.2011.10.038
Goncharova, O. and Rezanova, E., Mathematical modelling of the evaporating liquid films on the basis of the generalized interface conditions, MATEC Web Conf., 2016, vol. 84, article no. 00013. https://doi.org/10.1051/matecconf/2016840001310.1051/matecconf/20168400013