Geodesic
Exact Solutions to the Navier – Stokes Equations
Russian journal of nonlinear dynamics, Tome 18 (2022) no. 3, pp. 397-410

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In this paper, we report on several classes of exact solutions for describing the convective flows of multilayer fluids. We show that the class of exact Lin – Sidorov – Aristov solutions is an exact solution to the Oberbeck – Boussinesq system for a fluid discretely stratified in density and viscosity. This class of exact solutions is characterized by the linear dependence of the velocity field on part of coordinates. In this case, the pressure field and the temperature field are quadratic forms. The application of the velocity field with nonlinear dependence on two coordinates has stimulated further development of the Lin – Sidorov – Aristov class. The values of the degrees of the forms of hydrodynamical fields satisfying the Oberbeck – Boussinesq equation are determined. Special attention is given to convective shear flows since the reduced Oberbeck – Boussinesq system will be overdetermined. Conditions for solvability within the framework of these classes are formulated.
Keywords: multilayer fluids, Oberbeck – Boussinesq equations, shear flows, self-similar flows with spatial acceleration.
Mots-clés : exact solution 
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     author = {N. V. Burmasheva and E. Yu. Prosviryakov},
     title = {Exact {Solutions} to the {Navier} {\textendash} {Stokes} {Equations}},
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     url = {http://geodesic.mathdoc.fr/item/ND_2022_18_3_a5/}
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N. V. Burmasheva; E. Yu. Prosviryakov. Exact Solutions to the Navier – Stokes Equations. Russian journal of nonlinear dynamics, Tome 18 (2022) no. 3, pp. 397-410. http://geodesic.mathdoc.fr/item/ND_2022_18_3_a5/