Scientific publications

An exact solution of unsteady Navier – Stokes equations is presented for a threedimensional isobaric flow of a viscous incompressible fluid in a horizontal layer with permeable boundaries. The velocity field is sought in the Lin class, which reduces the original system to one-dimensional equations for amplitude functions depending on the vertical coordinate and time. It is shown that the vertical velocity component is constant, corresponding to uniform suction or injection. Analytical expressions for the evolution of the velocity fields are obtained, including the nonlinear component describing the vertical vorticity. The Sturm – Liouville problem for the convection–diffusion operator is solved analytically taking into account the Navier slip condition on the lower boundary. For the component excited by a nonlinear source, the Duhamel integral is used. Parametric analysis is performed for the impact of the vertical Reynolds number, the modified Taylor number, the angle of the velocity direction on the upper boundary, and the slip parameter on the flow dynamics. Regimes with developed counterflows and stagnation points are identified. It is shown that for a negative vertical Reynolds number (suction), the relaxation time decreases with increasing slip, which differs qualitatively from the classical no-slip case. The results obtained can be used for verifying numerical algorithms and in modelling hydrodynamic processes in porous media, membrane devices an catalytic layers, where unsteady effects and the presence of permeable boundaries are important.