Stability of plane Couette flow of a viscoelastic fluid with uniform cross-flow

Abstract

The linear stability of plane Couette flow of viscoelastic fluids has been the subject of several investigations in recent years. Giesekus (1) has discussed the stability of viscoelastic fluids for circular and plane Couette flows theoretically as well as experimentally. One of the most important facts observed by him is that viscoelasticity gives rise to cellular type of instabilities even in the absence of inertial forces if the second normal stress difference is chosen as positive. In other words inertia only modifies the critical value of the characteristic number associated with the neutral stability. Recently, in a series of subsequent papers Bhatnagar and Giesekus (2, 3) confirmed these ideas for the case of plane channel flow (2) and plane Poiseuille flow (3). For these flows, they also pointed out that two types of disturbances with different cell widths may exist simultaneously provided the parameter representing the ratio of inertial to elastic forces lies in a certain range. While investigating the overstability of plane Couette flow, Giesekus and Bhatnagar (4) found that, in general, the overstable mode is higher than the stationary mode but that both can come close to each other if certain conditions are satisfied.