Viscoelastic instability in viscosity-stratified channel flow with viscous heating effects

Abstract

The linear instability analysis of the isothermal pressure-driven flow of an Oldroyd-B fluid through a plane channel with viscous heating effects is carried out. The temperature dependence of the viscosity of solute, solvent, and polymeric solution is described using the Nahme law. There is no external temperature imposed in the system, the temperature gradient arises purely from frictional dissipation. The Reynolds number $\left(Re\right)$, Nahme number $\left(Na\right)$, Peclet number $\left(Pe\right)$, Deborah number $\left(De\right)$, the ratio of solvent to solution viscosity $\left(\beta \right)$, and the dimensionless heating coefficient for polymeric solution $\left(\alpha \right)$ are the dimensionless parameters that modify the instability characteristics of the flow. The Chebyshev spectral collocation method is employed to numerically solve the generalised eigenvalue problem. The viscous heating in the flow is characterised by Nahme number. It is observed that the unstable area under the neutral stability curves increases as we increase the viscous heating effects. Moreover, it is found that there exist a minimum and maximum threshold value of Deborah number for which linear instability persists. An increase in the $\beta$ value expands the parameter range for instability. The Peclet number $\left(Pe\right)$ and dimensionless heating coefficient $\left(\alpha \right)$ do not alter the linear stability characteristics.