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Analytical Solution of Brinkman Hydrodynamics in Filtration Problems

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Abstract

The problem of Brinkman hydrodynamics is considered. A variational generalized Sedov model is constructed to simulate the problem of filtration of a viscous and, in the general case, compressible fluid in a porous medium, taking into account the adhesion interactions between the fluid and the porous structure, considered as a solid deformable body. It is proposed to use a variant of the model of surface interactions, characterized by a reversible component in the dynamic process and a dissipative component, for the introduction of which the corresponding dissipation channel is formulated in the Sedov variational principle. An analytical solution of the problem of the flow of an incompressible fluid around a solitary inclusion of a spherical or cylindrical shape is constructed under the condition of a polynomial behavior of the velocity at infinity. As an application, the problem of determining the effective properties of filtration in a flow around a periodic system of spheres or cylinders is considered. To solve this problem, it is proposed to use the Trefftz approximation scheme developed earlier for the problems of the gradient theory of elasticity. As an approximating basis, it is proposed to use an analytically constructed system of solutions for the flow of an incompressible fluid with polynomial behavior at infinity around a solitary inclusion. A constructive method is proposed for constructing this system of functions in a closed finite form using the generalized Papkovich–Neiber representation and a system of canonical potentials constructed using radial multipliers (functions that depend only on the radius) based on harmonic polynomials. In this case, the Gauss theorem on the representation of an arbitrary homogeneous polynomial in the form of an expansion in terms of a system of homogeneous harmonic polynomials is used. It is shown that due to the analytical construction of the used functions, the generalized Trefftz scheme can be effectively applied in the cell problem when we determine the effective permeability.

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Funding

This work was supported by the Russian Science Foundation under the grant 20-41-04404 issued to the Institute of Applied Mechanics of Russian Academy of Sciences.

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  1. Institute of Applied Mechanics of Russian Academy of Sciences, 125040, Moscow, Russia

    S. A. Lurie, D. B. Volkov-Bogorodskiy & P. A. Belov

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  1. S. A. Lurie
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  2. D. B. Volkov-Bogorodskiy
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  3. P. A. Belov
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Correspondence to S. A. Lurie, D. B. Volkov-Bogorodskiy or P. A. Belov.

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(Submitted by A. M. Elizarov)

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Lurie, S.A., Volkov-Bogorodskiy, D.B. & Belov, P.A. Analytical Solution of Brinkman Hydrodynamics in Filtration Problems. Lobachevskii J Math 43, 1894–1907 (2022). https://doi.org/10.1134/S1995080222100237

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  • DOI: https://doi.org/10.1134/S1995080222100237

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