Effect of inertia on thermoelastic flow instability

Abstract

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T 0 to span the parameter space. In this work, the influence of T 0 on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number ( We)–Nahme number ( Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T 0 is increased, the critical Weissenberg number, We c, associated with this upper branch increases. Increasing T 0 beyond a certain value T ∗ causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/ Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T 0 = T ∗ represents a turning point in the We c– Na c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T 0, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T 0 is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.