Marangoni instability of a Maxwell’s viscoelastic liquid with Cattaneo–Christov heat flux in the absence of gravity

Abstract

The Marangoni instability of a viscoelastic liquid film coating a thick wall with finite thermal conductivity is analyzed under the assumption that the liquid heat flux follows the Cattaneo–Christov constitutive equation. This analysis is conducted in the absence of gravity, as buoyancy forces can be neglected for thin liquid films. This study investigates the influence of the Prandtl number ($Pr$), the viscoelastic relaxation time ($F$), the thermal relaxation time (${\varepsilon }_t$), and the wall’s thermal and geometric properties on the onset of thermocapillary convection. The effect of $Pr$ is examined within the range of 5 to 100, which is characteristic of polymeric suspensions. The wall’s thermal conductivity is quantified using the fluid-to-wall conductivity ratio, varying from 10⁻⁷ to 10${}^7$, thus encompassing the limiting cases of an ideal heat conductor and an adiabatic wall, along with intermediate scenarios. The effect of wall thickness is analyzed by setting the ratio of wall thickness to fluid depth to 0.1 for a thin wall and 100 for a thick wall.

Methodology:

A linear stability analysis is performed by perturbing and linearizing the governing equations of motion and energy. A normal mode expansion is applied to velocity and temperature perturbations, leading to an eigenvalue problem that is analytically solvable for the stability parameter, the Marangoni number. This number is then computed numerically to determine the marginal and critical states under varying parameter sets.

Key findings:

For large values of ${\varepsilon }_t$ or $F$, oscillatory convection dominates over stationary modes. Increasing ${\varepsilon }_t$ enhances instability, while the effect of $Pr$ varies, either stabilizing or destabilizing the system depending on $F$. Competing oscillatory modes, characterized by distinct wavenumbers and frequencies, arise within specific parameter regimes. Additionally, wall properties — such as thermal conductivity, specific heat, and thickness — significantly influence the onset conditions for both monotonic and oscillatory instabilities.