Abstract
We propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω⊂ℝ3 for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6π ν j where j is the current density of the particles, and of a friction term 6π ν ρ u where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius ε=1/N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure \(\frac{1}{N}\sum_{1\le k\le N}\delta_{x_{k},v_{k}}\) , where x k is the center of the k-th ball and v k its instantaneous velocity. This can be seen as a generalization of Allaire’s result in [Arch. Ration. Mech. Anal. 113:209–259, [1991]] who considered the case of periodically distributed x k s with v k =0, and our proof is based on slightly simpler though similar homogenization arguments. Similar equations are used for describing the fluid phase in various models for sprays.
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References
Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. Arch. Ration. Mech. Anal. 113, 209–259 (1991)
Amsden, A.A., O’Rourke, P.J., Butler, T.D.: A computer program for chemically reactive flows with sprays. Report # LA-11560-MS, Los Alamos National Laboratory (1989)
Batchelor, G.K.: Sedimentation in a dilute suspension of spheres. J. Fluid Mech. 52, 245–268 (1972)
Baranger, C., Desvillettes, L.: Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions. J. Hyperbolic Differ. Equ. 3, 1–26 (2006)
Caflisch, R.E., Luke, J.H.C.: Variance in the sedimentation speed of a suspension. Phys. Fluids 28, 759–760 (1985)
Caflisch, R.E., Rubinstein, J.: Lectures on the Mathematical Theory of Multi-phase Flows. Courant Institute Lecture Notes, New York (1984)
Cioranescu, D., Murat, F.: Une terme étrange venu d’ailleurs. In Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. 2. Research Notes in Mathematics, vol. 60, pp. 98–138 (1982)
Feuillebois, F.: Sedimentation in a dispersion with vertical inhomogeneities. J. Fluid Mech. 139, 145–171 (1984)
Jabin, P.-E.: Various levels of models for aerosols. Math. Models Methods Appl. Sci. 12, 903–919 (2002)
Jabin, P.-E., Otto, F.: Identification of the dilute regime in particle sedimentation. Commun. Math. Phys. 250, 415–432 (2004)
Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Mathematics and its Applications, vol. 2. Gordon and Breach/Science Publishers, New York (1969)
Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Course of Theoretical Physics, vol. 6. Pergamon Press, Oxford (1987)
L’vov, V.A., Khruslov, E.Ya.: Perturbation of a viscous incompressible fluid by small particles. Theor. Appl. Quest. Differ. Equ. Algebra 267, 173–177 (1978) (in Russian)
O’Rourke, P.J.: Collective drop effects on vaporizing liquid sprays. PhD thesis, Los Alamos National Laboratory (1981)
Rubinstein, J.: On the macroscopic description of slow viscous flow past a random array of spheres. J. Stat. Phys. 44, 849–863 (1986)
Rubinstein, J., Keller, J.: Particle distribution functions in suspensions. Phys. Fluids A 1, 1632–1641 (1989)
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Desvillettes, L., Golse, F. & Ricci, V. The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow. J Stat Phys 131, 941–967 (2008). https://doi.org/10.1007/s10955-008-9521-3
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DOI: https://doi.org/10.1007/s10955-008-9521-3