Continuous Local Saturation of a Semi-Bounded Isotropic Porous Medium by a Heated Flow in the Ideal Displacement Regime

A model of local saturation of a "hot" viscous incompressible fluid under the action of a constant pressure gradient from the "free" surface of a "cold" porous matrix is presented using the Darcy-Brinkman equations and a single-temperature energy formulation. It is shown that for laminar pore flow, the velocity is virtually constant, with the exception of an insignificant initial time interval. This allowed us to apply a hydrodynamic idealization of displacement and consider a non-conjugate thermal initial-boundary value subproblem for a semi-bounded isotropic fine-grained porous medium, replacing the infinitely extending "free" surface with a bounded region whose scale significantly exceeds the local saturation zone. An analytical solution is obtained by combining the one-sided integral Laplace transform and the finite integral cosine Fourier transform. A calculation examples is presented quantitatively characterizing the migration of temperature inhomogeneity in a porous layer.