On solving problems of heat and mass transfer in piecewise homogeneous regions with a weakly permeable film

S. E. Kholodovskii

Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 3, pp. 312-320

Résumé

Boundary value problems for the equations of thermal conductivity in a band $D(x\in R,\,0$ divided by a weakly permeable film $x=0$ into two half-bands $D_1(x0,\,0$ and $D_2(x>0,\,0$ with different permeabilities $k_i$ in $D_i$, $i=1,2$, under different types of boundary conditions are considered. A weakly permeable film is modeled as an infinitely thin layer with an infinitesimal permeability. Generalized conjugation conditions on the film are derived for the potentials $u_i(x,y,t)$, $i=1,2$. Problems with a weakly permeable film $x=0$ are considered for steady-state processes in a piecewise homogeneous band $D$ (at $k_1\neq k_2$), for unsteady processes in a homogeneous band $D$ (at $k_1=k_2$), and for unsteady processes in a piecewise homogeneous rod $D(x\in R)=D_1(x0)\cup\{x=0\}\cup D_2(x>0)$ (at $k_1\neq k_2$ for one-dimensional thermal conductivity equations). General formulas are derived that express the solutions of the considered problems through the solutions of similar classical problems in the corresponding homogeneous domain $D$ (without film) in the form of rapidly converging nonconforming integrals. The existence and uniqueness theorem is proved for the considered class of problems.

Détail

Keywords: boundary value problems for the heat equation, weakly permeable film.