N. V. Burmasheva, E. Yu. Prosviryakov

ISOTHERMAL LAYERED FLOWS OF A VISCOUS INCOMPRESSIBLE FLUID WITH SPATIAL ACCELERATION IN THE CASE OF THREE CORIOLIS PARAMETERS

We study the solvability of the overdetermined system of Navier–Stokes equations, supplemented by the incompressibility equation, which is used to describe isothermal large-scale shear flows of a rotating viscous incompressible fluid. Large–scale flows are studied in a thin-layer approximation (the vertical velocity of the fluid is assumed to be zero). The rotation of a continuous fluid medium is described by three Coriolis parameters. The solution of the reduced system of Navier–Stokes equations is constructed in the Lin–Sidorov–Aristov class. In this case, both nonzero components of the velocity vector, the pressure and temperature fields are assumed to be full linear forms of two Cartesian coordinates, and the dependence on the third Cartesian coordinate has an arbitrary form (including non-polynomial). It is shown that the nonlinear overdetermined system of Navier–Stokes equations and of the incompressibility equation in the framework of the Lin–Sidorov–Aristov class reduces to the equivalent nonlinear overdetermined system of ordinary differential equations, in which the components of the hydrodynamic fields act as unknown functions. The compatibility condition for the equations of the resulting system is derived. It is shown that, if this compatibility condition is fulfilled, the system has a unique solution, and the spatial accelerations in both variables (the linearity with respect to them was postulated when choosing the solution class) prove to be constant functions. These results are a generalization of similar results obtained earlier in the study of solvability in the cases of one and two Coriolis parameters.

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