Abstract
An exact solution describing a Couette-Poiseuille-type gradient flow in an infinite horizontal layer of a viscous incompressible fluid is proposed. The horizontal velocity component of the proposed exact solution is given as a linear function of the longitudinal (horizontal) coordinate with coefficients depending on the transverse (vertical) coordinate. The vertical velocity component is assumed to depend only on the transverse coordinate. The pressure and temperature fields are specified by an incomplete quadratic form along the longitudinal coordinate with coefficients depending on the transverse (vertical) coordinate. On one side the fluid layer is bounded by a solid surface, where the no-slip conditions and parabolic heating are set in the boundary-value problem. On the other side the fluid is bounded by a free permeable surface. On this surface, the velocity field is determined in the form of parabolic wind, as well as the pressure field with a nonzero longitudinal surface pressure gradient and the zero (reference) temperature value. The obtained exact particular solution is analyzed, and the conditions for the existence of stagnation points in the fluid layer are obtained. The areas of occurrence of counterflows depending on the change in the values of the longitudinal pressure gradient are shown.
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