Interaction between a laminar boundary layer and an elastic layer

Abstract

A two‐dimensional viscous flow field is bounded on both sides by a purely elastic layer of finite thickness that supports both shear and longitudinal waves. The mean flow field is assumed to be a Poiseuille flow. The boundary layer equations are linearized to the Orr‐Sommerfeld equation and the interaction between the fluid and the elastic layer occurs through the continuity of velocity and stress at the fluid‐solid interface. The stability of the disturbances in the fluid and the elastic layer is analyzed. Kaplan [“The Stability of a Laminar Incompressible Boundary Layer in the Presence of Compliant Boundaries,” MIT Report No. ASRL‐TR‐116‐1 (1964)] analyzed the stability characteristics of the problem by solving for the eigenvalues of the problem. The numerical solution of the eigenfunctions of the Orr—Sommerfeld equation has always been a very difficult and tricky problem. Davey [“An Automatic Orthonormalization Method for Solving Stiff Boundary Valve Problems,” J. Comput. Phys. 51, 343–356 (1983)] has developed a technique which allows for quick solution for the eigenvalues and the eigenfunction of the problem at any Reynolds number. This technique has been applied to the problem at hand and it has given solutions in a quick and efficient manner. The results for the stability of the boundary layer disturbances are presented in the form of velocity profiles (eigenfunctions) within the fluid and the solid layer.