Dynamics of a two-layer flow with an interfacial heat source/sink: viscosity stratification

Abstract

The temporal stability of a two-layer plane Couette flow with an interfacial heat source/sink is studied, in the presence of a viscosity stratification, predicting unstable streamwise and spanwise perturbations. Based on the wavenumber at onset, the perturbations are further classified as long-wave and finite-wave, where the long-wave mode is investigated using asymptotic and numerical analyses. The finite-wave instability is expected to be relevant for highly viscous fluids. Generally, a heat source (sink) at the interface is found to have a stabilising (destabilising) effect on the flow. However, if the rate of heat release exceeds a certain threshold, a new instability emerges, referred to here as an ‘explosive’ instability, characterised by an unbounded growth rate of the perturbations. The interaction between the viscosity stratification, thermocapillarity and inertia leads to the emergence of another new mode of streamwise instability, termed the ‘interaction’ instability. The spanwise perturbations possess higher growth rates than the streamwise perturbations for the heat sink along the interface when inertia has a stabilising influence, and vice versa. The heat sink leads to the emergence of thermocapillary instability by generating a negative temperature gradient across the fluids. In contrast, a heat source increases the temperature of the interface, resulting in a positive temperature gradient across the fluids, leading to the suppression of the instabilities by the thermocapillary effect. If the heat source provides energy at a faster rate than the energy lost due to Marangoni convection and heat diffusion, then it leads to an accumulation of energy, which presumably leads to the explosive instability.

Spatio-temporal dynamics of a two-layer pressure-driven flow subjected to a wall-normal temperature gradient

“…(1994), Patne, Agnon & Oron (2020, 2021); Patne et al. (2022). The MATLAB routine eig is used to solve the constructed generalized eigenvalue problem equation (3.2).…”

“…It must be noted that the numerical and analytical approaches employed here are similar to the one used for studying the stability of a non-isothermal two-layer plane Couette flow by Patne et al. (2022). Additionally, for an isothermal two-layer Poiseuille flow, we have validated the asymptotic approach against the standard procedure developed by Yih (1967) and Yiantsios & Higgins (1988) as follows.…”

“…The details of the discretisation of the governing equations and boundary conditions, and construction of A and B can be found in the standard procedure described by Trefethen (2000) and Schmid & Henningson (2001). Application of the pseudo-spectral method for similar problems can be found in the study of Boomkamp et al (1997), Barmak et al (1994), Patne, Agnon & Oron (2020, 2021; Patne et al (2022). The MATLAB routine eig is used to solve the constructed generalized eigenvalue problem equation (3.2).…”

“…For a liquid layer supported by a heated substrate and a free surface, thermocapillary instabilities exist as the Marangoni number crosses a threshold or critical value Ma c . As explained in the literature (Smith & Davis 1983a,b;Patne et al 2022), for such a system to exhibit a thermocapillary instability, the free surface temperature of the liquid must be lower than that of the substrate. Here, we have a two-layer system with plate 1 at y * = 0 maintained at temperature T * 1 and plate 2 at y * = R maintained at temperature T * 2 , where the asterisk implies a dimensional quantity.…”

“…As explained in the literature (Smith & Davis 1983 a , b ; Patne et al. 2022), for such a system to exhibit a thermocapillary instability, the free surface temperature of the liquid must be lower than that of the substrate. Here, we have a two-layer system with plate 1 at maintained at temperature and plate 2 at maintained at temperature , where the asterisk implies a dimensional quantity.…”