The unidirectional motion of two heat-conducting liquids in a flat channel

V K Andreev,E N Cheremnykh

doi:10.1088/1742-6596/894/1/012106

Abstract

The unidirectional motion of two viscous incompressible liquids in a flat channel is studied. Liquids contact on a flat interface. External boundaries are fixed solid walls, on which the non-stationary temperature gradients are given. The motion is induced by a joint action of thermogravitational and thermocapillary forces and given total non - stationary fluid flow rate in layers. The corresponding initial boundary value problem is conjugate and inverse because the pressure gradients along axes channel have to be determined together with the velocity and temperature field. For this problem the exact stationary solution is found and a priori estimates of non - stationary solutions are obtained. In Laplace images the solution of the non - stationary problem is found in quadratures. It is proved, that the solution converges to a steady regime with time, if the temperature on the walls and the fluid flow rate are stabilized. The numerical calculations for specific liquid media good agree with the theoretical results.

Highlights

Tjt = χj Tjyy + aj wj, Pjy = ρj gβj Tj, where uj is the projection of the velocity vector on the axis x, pj is the deviation of pressure from hydrostatic one, g = const is the acceleration of gravity force, θj is the temperature
The functions Tj is the solution of the conjugate problem that is analogous to the problem for aj
The aim of this paper is to study the inverse problem

Summary

Introduction

The conjugate boundary value problem for the functions aj(y, t) has the form ajt = χj ajyy, (3) A1(0, t) = a2(0, t), k1a1y(0, t) = k2a2y(0, t) with given functions aj0(y), Aj(t), j = 1, 2, which for the smooth solutions must satisfy the matching conditions a10(−h1) = A1(0), a20(h2) = A2(0), a10(0) = a20(0), k1a10y(0) = k2a20y(0). If the function C(t) is given, the equation (8) with the boundary conditions (10) –

Objectives

Results

Conclusion