Linear Stability Analysis for the Wall Temperature Feedback Control of Planar Poiseuille Flows

Active control of a planar Poiseuille flow can be performed by increasing or decreasing the wall temperature in proportion to the observed wall shear stress perturbation. In continuation with the work of H. H. Hu and H. H. Bau (1994, Feedback Control to Delay or Advance Linear Loss of Stability in Planar Poiseuille Flow, Proc. R. Soc. London A, 447, pp. 299–312), a linear stability analysis of such a feedback control is developed in this paper. The Poiseuille flow control problem is reduced to a modified Orr-Sommerfeld equation coupled with a heat equation. By solving numerically the coupled equations with a finite element method, many numerical results about the stability of the flow control are obtained. We focus our attention on the interpretation of the numerical results. In particular, the role of two essential parameters—the Prandtl number Pr and the control gain K—is investigated in detail. When Pr$>1.31,$ stabilizing K is negative; while, when Pr$<1.31,$ stabilizing K is positive. And when Pr$=1.31,$ the flow cannot be stabilized by a real K. A comparison between symmetric two-wall control and non-symmetric one-wall control is also made.