Thermophoresis migration of an aerosol spherical particle embedded in a Brinkman medium at small non-zero Peclet numbers

The method of matched asymptotic expansions is used to investigate the problem of thermophoresis migration of an aerosol spherical particle immersed in a porous medium saturated by a viscous fluid at a small non-zero Peclet number Pe. A uniform temperature gradient is imposed on the system parallel to the diameter of the particle in the opposite direction of z axis. It is assumed that the Knudsen number is in the range of the slip fluid flow through the pores of the porous medium and is compatible with the assumption of the continuum model. The porous medium is modeled by the Brinkman equation and is assumed to be homogenous and isotropic, and the solid matrix is in thermal equilibrium with the fluid through the voids of the medium. In the analysis of motion, the thermal stress slip is considered in addition to the temperature jump, the thermal creep, and the frictional slip. The thermophoretic velocity of the particle is obtained in the closed form up to order Pe(3) as a function of the thermal properties of the system and the permeability of the porous medium. The present asymptotic analytical solutions can be viewed as an effective method for checking the numerical schemes for future work on arbitrary values of the Peclet number. The limiting case of the thermophoretic velocity for the Stokes clear fluid is recovered.