The linear stability of plane Poiseuille flow of Burgers fluid at very low Reynolds numbers

Abstract

The linear stability of plane Poiseuille flow of Burgers fluid at very low Reynolds numbers is studied. In the previous works, it was shown that plane Poiseuille flows of UCM and Oldroyd-B fluids at very low Reynolds numbers are linearly stable. However, in the present work, unstable modes for Burges fluid are detected with some special value of parameters. In critical stable condition, there are two kinds of perturbation waves. One travels downstream with the name of forward wave, while the other travels upstream with the name of reverse wave. The mean value of the two wave speeds is independent of the characteristic velocity of flow, while the difference between the wave speeds reflects the equivalent velocity of flow. By comparing the perturbation modes for Newtonian fluid and Burgers fluid under critical stable conditions, it can be found that although the shapes of their amplitudes are basically the same, their phase diagrams are significantly different. The neutral curve of Burgers fluid is also different from that of Newtonian fluid. Unstable modes can still be found at high wave numbers. The flow instability is induced due to the phase difference between the perturbation strain and stress. The perturbation energy caused by the forward wave, which is dissipated by the effects of the component of perturbation normal stress in the streamwise direction, is transported by the effects of both the perturbation normal stress in the wall-normal direction and the perturbation shear stress. The converse is true for the reverse wave.