Abstract
In this paper we report on our investigation of the instability of a liquid layer flowing along a heated inclined plane. We develop and implement a theoretical model with a power-law constitutive relation which captures the temperature variation in the rheology of the fluid. We carry out a linear stability analysis and obtain Orr-Sommerfeld type equations for the evolution of infinitesimal perturbations imposed on the equilibrium flow. Numerical solutions were obtained, as well as asymptotic approximations based on the assumption of perturbations of long wavelength and small variation in the consistency of the fluid with respect to temperature. We investigate the critical conditions for the onset of instability and determine the effect of a non-Newtonian rheology and the dependence of the fluid properties on temperature. Nonlinear effects were considered by employing a reduced dimensionality model. Calculations of permanent waves arising from unstable uniform flows were made by carrying out numerical simulations of these equations.
Highlights
Gravity-driven flow with a planar free surface is subject to instability due to inertial effects resulting in the formation of waves propagating along the surface
The model incorporates the realistic Newton’s law of cooling at the surface of the liquid layer, which allows it to capture the enhancement to inertial instability provided by thermocapillarity
We carried out a linear stability analysis to determine the critical conditions for the onset of longwave instability in the steady flow uniform in the streamwise direction
Summary
Gravity-driven flow with a planar free surface is subject to instability due to inertial effects resulting in the formation of waves propagating along the surface. Both linear and nonlinear, have been recently undertaken to study the stability of Newtonian flows along heated inclines.[1,2,3,4] In these studies the heat transfer from the liquid to the ambient gas is modelled by Newton’s law of cooling This provides a realistic description of real world situations and allows variation in temperature along the surface capturing the occurrence of thermocapillarity. Hu et al.[24] studied the instability of thermocapillary convection in a horizontal layer with a non-deformable surface consisting of a shear-thinning fluid with the rheology described by the Carreau relation They have included buoyancy effects by assuming a temperature-dependent density. Bernabeu et al.[26] have established a model for lava flow consisting of a viscoplastic fluid with a power-law relation for the viscosity This investigation does not include the Marangoni effect but a temperature dependence is incorporated into the rheological relation.
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