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# Effective Velocity and Pressure Interface Laws Between aViscous Fluid and a Porous Medium Using Homogenization

Tuesday, April 17, 3:30PM – 5PM
ACE 6.304

### Andro Mikelic

In this talk we present rigorous justification of the interface law describing contact between the flow in an unconfined fluid and a porous bed. The velocity of the free fluid dominates the filtration velocity, but the pressures are of the same order. Main results are the following:

1. We confirm Saffman’s form of the Beavers and Joseph law in a new, more general, setting.

2. We show that a perturbation of the interface position, which is an artificial mathematical boundary, of the order \$O(varepsilon)\$ implies a perturbation in the solution of order \$O(varepsilon^2)\$ . Consequently, there is a freedom in fixing position of the interface. It influences the result only at the next order of the asymptotic expansion.

3. We obtain a uniform bound on the pressure approximation. Furthermore, we prove that there is a jump of the effective pressure on the interface and that it is proportional to the free fluid shear at the interface. Together with the mathematical proof, we will present a numerical confirmation of the pressure jump, obtained by a pore scale simulation of the Stokes system.

4. At the end we will present the interface law for the coupling between an elastic body and a poroelastic domain and make comments on modeling soil infiltration using homogenization approach.

References:

[1] A. Marciniak-Czochra, A. Mikelic: Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization, accepted for publication in " Multiscale modeling and simulation", 2012.

[2] A. Mikelic, M. F. Wheeler: On the interface law between a deformable porous medium containing a viscous fluid and an elastic body, accepted for publication in "M3AS: Mathematical Models and Methods in Applied Sciences", 2011.

[3] W.Jaeger, A.Mikelic: On the interface boundary conditions by Beavers, Joseph and Saffman, SIAM J. Appl. Math., 60 (2000), pp. 1111 - 1127.

Hosted by Mary Wheeler