The microscopic dynamics of frozen ground near hot pipes.

In the present paper we consider a phase transition in a viscous liquid in a solid skeleton near a single pipe filled with hot water. We suppose that the tube itself is part of a solid skeleton. First, we consider this physical process at the microscopic level (the characteristic pore size ε' is approximately 5-20 microns) governed by Stokes equations for incompressible viscous liquid coupled with heat equations. We also assume that the liquid in the frozen state becomes the part of the unmoved solid skeleton. Note that the boundary between the solid and liquid parts of the pore fluid is unknown (free boundary) and must be found together with the solution to the Stokes equations. The suggested mathematical model at the microscopic level is physically correct, because based on the classical Newtonian continuum mechanics. But any physically correct mathematical model at the microscopic level is practically useless due to its average size. Indeed, for the numerical implementation of any such model, a step in spatial variables of several microns is required. On the other hand-side, the problem must be calculated for domains of several tens meters. No modern supercomputers or cluster computing systems are yet capable of such computations. To make it practically valuable, we must use homogenization method, which is, roughly speaking, the mathematically rigorous approximation of the microscopic mathematical model in consideration by some macroscopic mathematical model. This part will be done in the next publication. [ABSTRACT FROM AUTHOR]