Self-similar solutions of the problem of thermal convection, averaged over a thin layer of the evaporating liquid

Abstract

We obtain some types of self-similar substitutions for thermal convection problem, averaged over a thin layer, extended wave case. The problem is the asymptotic model of dry non-viscous, no temperature wire drops and is averaging approach Oberbeck-Boussinesq on a thin layer of an evaporating liquid. The problem is the use of simulation in modern processes involving heat and mass transfer processes: in medical diagnosis, pharmacological studies, crystallography, printing, etc. For the construction of self-similar solutions of the original problem moves to the Riemann invariants. To the resulting problem characteristic relations that determine the self-substitution were obtained. These substitutions allowed to reduce the dimension of the problem and reduce it to a system of ordinary differential equations (ODE). Built ODE solutions - self-similar solution of the problem of thermal convection for extended drops that determine height drops by osredennuyu toschine mass transfer rate and heat flux as a function of time and location. Established the applicability of the self-similar solutions to the simulation of different situations: the evaporation of the droplet and its condensation, pinning, ie, securing the borders of three-phase contact on a rough surface, and depinning, ie her separation; different geometric configurations drop at the initial time. Built similar solution exponential type describes a situation where the liquid layer thickness decreases exponentially (evaporation) or increases (condensing) with time for a given law. In particular, the rate of decrease (increase) may be determined by the flow of fluid through a fixed boundary phase contact (drain). The resulting solution of traveling wave type corresponds to the case when the liquid is drying up direction of the external mechanical action, causing fluid flow at a constant rate.