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

Abstract

The present study investigates the linear spatio-temporal and weakly nonlinear stability of a pressure-driven two-layer channel flow subjected to a wall-normal temperature gradient commonly encountered in industrial applications. The liquid–liquid interface tension is assumed to be a linearly decreasing function of temperature. The study employs both numerical (pseudo-spectral method) and long-wave approaches. The general linear stability analysis (GLSA) predicts shear-flow and thermocapillary modes that arise due to the imposed pressure and temperature gradients, respectively. The previous stability analyses of the same problem predicted a negligible effect of the pressure-driven flow on the linear stability of the system. However, the GLSA reveals stabilising and destabilising effects of the pressure-driven flow depending on the viscosity ratio ( $\mu _r$ ), thermal conductivity ratio ( $\kappa _r$ ), interface position ( $H$ ) and the sign of the imposed temperature gradient ( $\beta _1$ ). The analysis predicts a range of $H$ for given $\mu _r$ and $\kappa _r$ , which can not be stabilised by the thermocapillarity. The numerically predicted long-wave instability is then captured using the long-wave asymptotic approach. The arguments based on the physical mechanism further successfully explain the role of $\mu _r$ , $\kappa _r$ , $H$ , the sign of $\beta _1$ and the interaction between the velocity and temperature perturbations in stabilising/destabilising the flow. The spatio-temporal analysis reveals the dominance of the spanwise mode in causing the absolutely unstable flow. The weakly nonlinear analysis reveals a subcritical pitchfork bifurcation without shear flow. However, with the shear flow, the streamwise mode undergoes a supercritical Hopf bifurcation.

Linear dynamics of a thick liquid layer subjected to an oblique temperature gradient

“…The application of the pseudospectral method for similar problems can be found in Patne, Agnon & Oron (2020 a ); Patne et al. (2020 b , 2021 a , b ), Patne & Oron (2022), Patne & Chandarana (2023) and Patne (2024). To solve the eigenvalue problem (3.2), we use the eig MATLAB routine.…”

“…The standard procedure to discretise the governing equations and boundary conditions using Chebyshev polynomials can be found in Trefethen (2000) and Schmid & Henningson (2001). The application of the pseudospectral method for similar problems can be found in Patne, Agnon & Oron (2020a); Patne et al (2020bPatne et al ( , 2021a, Patne & Oron (2022), Patne & Chandarana (2023) and Patne (2024). To solve the eigenvalue problem (3.2), we use the eig MATLAB routine.…”