The linear stability of plane Couette flow with a compliant boundary

Abstract

The linear stability of plane Couette flow subject to one rigid boundary and one flexible boundary is considered at both finite and asymptotically large Reynolds number. The wall flexibility is modelled using a very simple Hooke-type law involving a spring constant K and is incorporated into a boundary condition on the appropriate Orr–Sommerfeld eigenvalue problem. This problem is analyzed at large Reynolds number by the method of matched asymptotic expansions and eigenrelations are derived that demonstrate the existence of neutral modes at finite spring stiffness, propagating with speeds close to that of the rigid wall and possessing wavelengths comparable to the channel width. A large critical value of K is identified at which a new short wavelength asymptotic structure comes into play that describes the entirety of the linear neutral curve. The asymptotic theories compare well with finite Reynolds number Orr–Sommerfeld calculations and demonstrate that only the tiniest amount of wall flexibility is required to destabilize the flow, with the linear neutral curve for the instability emerging as a bifurcation from infinity.