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We propose a mathematical model of the pressure flow of a viscous incompressible fluid in a porous horizontal channel with a rectangular cross section having an anisotropic structure described by an orthotropic tensor. The motion of the Newtonian fluid was assumed to be laminar (internal Reynolds number Re<50) and inertialess, as well as unidirectional along the axial direction of the porous channel, which allowed us to use the Darcy-Brinkman phenomenological equation, for which we formulated an initial-boundary-value problem, the solution of which we obtained analytically using a one-sided integral Laplace transformation and the finite integral Fourier transformation. A comparative analysis with known experimental data showed the correctness of the physical linearization of the Darcy-Brinkman equations. We show that mathematically filtering in a porous isotropic and anisotropic layer is described identically. The difference is in the values of the Darcy numbers, which characterize the internal structure of the granular layers.

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The precise analytical solution for the heat transfer problem in a porous flat channel with a laminar flow of a viscous incompressible coolant with boundary conditions of the first kind is obtained based on the assumption that the flow is unidirectional, phase transitions are absent, and the thermos-physical properties are constant. The model is based on the Schumann equations as an initial-boundary value task for the system of parabolic equations referring to the local temperature of the coolant and the porous matrix. The paper reveals the relation of the local Nusselt number and the conditions necessary to access the compactness of heat transfer systems with porous fillers.