Consider the motion of a solute molecule in a homogeneous isotropic porous medium saturated with a pure liquid. The following assumptions are made on the velocity of the molecule: the time intervals between successive collisions of the molecule with the solid matter of the medium are i. i. d. random variables; the molecule is scattered by these collisions in random directions which are i. i. d. uniform and independent of the collison times; in between two successive collisions with the solid phase the velocity of the molecule is governed by the Langevin equation. Under these and mild additional assumptions it is proved that the position left brace x(t):t greater than equivalent to 0 right brace of the molecule is approximately a Brownian motion. If the solute molecules are weakly interacting among themselves, then the above result leads to a macroscopic parabolic equation governing solute concentration. If the successive collision times with the solid phase are assumed to be exponential, then the velocity left brace v(t):t greater than equivalent to 0 right brace as well as left brace (v(t),x(t)):t greater than equivalent to 0 right brace are Markovian. This leads to laws of mass, momentum, and energy conservation for solute transport.
ASJC Scopus subject areas
- Applied Mathematics