The effects of inertia on the viscoelastic Dean and Taylor–Couette flow instabilities with application to coating flows

The effects of inertia on the elastic instabilities in Dean and Taylor–Couette flows are investigated through a linear stability analysis. The critical conditions and the structure of the vortex flow at the onset of these instabilities are presented. The results reveal that the purely elastic Dean flow is destabilized by inertial effects. It is also found that inertia destabilizes elastic Taylor–Couette flow if the rotation of the inner cylinder is the flow driving force, while it stabilizes the flow driven by rotation of the outer cylinder. The mechanism of destabilization or stabilization of these viscoelastic instabilities is investigated through an examination of the disturbance‐energy equation. It is shown that Dean flow is destabilized by two separate mechanisms: a purely elastic mechanism discussed previously (i.e., energy production due to the coupling of a perturbation velocity to the polymeric stress gradient in the base state) [see Phys. Fluids A 3, 1691 (1991)] and a purely inertial mechanism discussed by Dean [Proc. R. Soc. London Ser. A 121, 402 (1928)] (i.e., energy production from Reynolds stresses). It is also shown that, when rotation of the inner cylinder drives Taylor–Couette flow, the Reynolds stresses produce energy, and thus are destablizing, while for the flow driven by the rotation of the outer cylinder alone, the Reynolds stresses dissipate energy, thus stabilizing the flow. The elastic forces remain destabilizing in both modes of operation. In a second study, a pressure‐driven viscoelastic coating flow over a curved surface is examined. The results demonstrate the existence of a purely elastic stationary instability in the coating flow on a concave wall which is very similar to that which occurs in viscoelastic Dean flow. It is demonstrated that the mechanisms of instability in Dean flow and the coating flow are the same, again through an examination of the disturbance‐energy equation.