Thermal convection in a HeleShaw cell with the dependence of thermal diffusivity on temperature

TL;DR

This paper theoretically investigates thermal convection in a Hele-Shaw cell with temperature-dependent thermal diffusivity, deriving analytical temperature distribution, analyzing stability, and confirming findings with numerical simulations consistent with experimental data.

Contribution

It introduces a novel theoretical model accounting for temperature-dependent thermal diffusivity and analyzes its effects on convection stability and symmetry breaking.

Findings

Temperature distribution follows a square root law.

Most dangerous perturbation involves a two vortex steady flow.

Inclusion of temperature-dependent diffusivity causes symmetry breaking.

Abstract

The investigation of thermal convection of a fluid with the dependence of thermal diffusivity on temperature in a vertical Hele Shaw cell heated from below has been fulfilled theoretically.The expression for equilibrium temperature distribution in a cavity has been derived analytically. It has been found that the dependence of temperature on the vertical coordinate looks like a square rootlaw.The linear stability of mechanical equilibrium state against small normal perturbations has been investigated by means of Galerkin method. It has been shown that the most dangerous perturbation in a cavity under considerationis described by the mode which corresponds to the two vortexsteady flow. The numerical simulation of overcritical steady and oscillatory flows has been carried out in the approximation of plane trajectories.This simplification of theoretical modelis consistent with all experimental data on thermal convection in similar cavities. It has been shown that the inclusion of the dependence of thermal diffusivity on temperature into the mathematical model leads to the up down symmetry breakdown for the small values of over criticality.