TWO-SCALE EXPANSION METHOD IN THE PROBLEM OF TEMPERATURE OSCILLATIONS IN FROZEN SOIL

Abstract

The paper investigates the problem of the dynamics of frozen soil with a change in the external temperature at the boundary of the physical domain under consideration. According to the generally accepted scheme, independently proposed by R. Barridge and J. Keller and E. Sanchez-Palencia in 1980, we first formulate a microscopic mathematical model, that describes the physical process at the microscopic level by the equations of classical Newtonian mechanics of continuous media. A natural small parameter here is the average dimensionless diameter of the pores of the solid skeleton. In this model the change in the temperature of the medium is controlled by the Stefan problem, and the dynamics of fluid in the pores of an absolutely rigid skeleton obeys the Stokes equations for a viscous incompressible fluid. The second and main point of the method is the derivation of the macroscopic equations of the physical process, which are obtained by passing to the limit as a small parameter tends to zero (homogenization). The purpose of this work is to derive macroscopic (homogenized) equations describing the dynamics of frozen soil using the two-scale expansion method.