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Analysis of the Brinkman equation as a model for flow in porous media. (English) Zbl 0636.76098
The fundamental solution or Green’s function for flow in porous media is determined using Stokesian dynamics, a molecular-dynamics-like simulation method capable of describing the motions and forces of hydrodynamically interacting particles in Stokes flow. By evaluating the velocity disturbance caused by a source particle on field particles located throughout a monodisperse porous medium at a given value of volume fraction of solids $$\Phi$$, and by considering many such realizations of the (random) porous medium, the fundamental solution is determined.
Comparison of this fundamental solution with the Green’s function of the Brinkman equation shows that the Brinkman equation accurately describes the flow in porous media for volume fractions below 0.05. For larger volume fractions significant differences between the two exist, indicating that the Brinkman equation has lost detailed predictive value, although it still describes qualitatively the behavior in moderately concentrated porous media. At low $$\Phi$$ where the Brinkman equation is known to be valid, the agreement between the simulation results and the Brinkman equation demonstrates that the Stokesian dynamics method correctly captures the screening characteristic of porous media. The simulation results for $$\Phi\geq 0.05$$ may be useful as a basis of comparison for future theoretical work.

##### MSC:
 76S05 Flows in porous media; filtration; seepage 76M99 Basic methods in fluid mechanics 35Q99 Partial differential equations of mathematical physics and other areas of application
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##### References:
 [1] H. P. G. Darcy,Les fontanes publiques de la ville de Dijon(Dalmont, Paris, 1856). [2] J. B. Keller, inStatistical Mechanics and Statistical Methods in Theory and Applications, edited by U. Landman (Plenum, New York, 1977), pp. 631–644. [3] Whitaker, Transport in Porous Media 1 pp 3– (1986) [4] Brenner, Philos. Trans. R. Soc. London Ser. A 297 pp 81– (1980) [5] Adler, Phys. Chem. Hydrodyn. 5 pp 245– (1984) [6] Brinkman, Appl. Sci. Res. A 1 pp 27– (1947) [7] Tam, J. Fluid Mech. 38 pp 537– (1969) [8] Childress, J. Chem. Phys. 56 pp 2527– (1972) [9] Howells, J. Fluid Mech. 64 pp 449– (1974) [10] Hinch, J. Fluid Mech. 83 pp 695– (1977) [11] Freed, J. Chem. Phys. 68 pp 2088– (1978) [12] Muthukumar, J. Chem. Phys. 70 pp 5875– (1979) [13] Rubenstein, J. Fluid Mech. 170 pp 379– (1986) [14] Kim, J. Fluid Mech. 154 pp 269– (1985) [15] Durlofsky, J. Fluid Mech. 180 pp 21– (1987) [16] J. F. Brady, R. Phillips, J. Lester, and G. Bossis, submitted to J. Fluid Mech. [17] Beenakker, J. Chem. Phys. 85 pp 1581– (1986) [18] O’Brien, J. Fluid Mech. 91 pp 17– (1979) [19] Kim, J. Fluid Mech. 154 pp 253– (1985) [20] J. F. Brady and L. Durlofsky, submitted to Phys. Fluids. [21] Rubenstein, J. Stat. Phys. 44 pp 849– (1986)
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