The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow

Abstract

The viscoelastic analogue to the Newtonian Orr amplification mechanism is examined using linear theory. A weak, two-dimensional Gaussian vortex is superposed onto a uniform viscoelastic shear flow. Whilst in the Newtonian solution the spanwise vorticity perturbations are simply advected, the viscoelastic behaviour is markedly different. When the polymer relaxation rate is much slower than the rate of deformation by the shear, the vortex splits into a new pair of co-rotating but counter-propagating vortices. Furthermore, the disturbance exhibits a significant amplification in its spanwise vorticity as it is tilted forward by the shear. Asymptotic solutions for an Oldroyd-B fluid in the limits of high and low elasticity isolate and explain these two effects. The splitting of the vortex is a manifestation of vorticity wave propagation along the tensioned mean-flow streamlines, while the spanwise vorticity growth is driven by the amplification of a polymer torque perturbation. The analysis explicitly demonstrates that the polymer torque amplifies as the disturbance becomes aligned with the shear. This behaviour is opposite to the Orr mechanism for energy amplification in Newtonian flows, and is therefore labelled a ‘reverse-Orr’ mechanism. Numerical evaluations of vortex evolutions using the more realistic FENE-P model, which takes into account the finite extensibility of the polymer chains, show the same qualitative behaviour. However, a new form of stress perturbation is established in regions where the polymer is significantly stretched, and results in an earlier onset of decay.