Stability of Cylindrical Domain under Poiseuille Flow

Abstract

We investigate stability of a cylindrical domain in a phase-separating binary fluid under a Poiseuille flow. Linearization for a couple of the Navier–Stokes and Cahn–Hilliard equations around stationary solutions leads to an eigenvalue problem, a solution of which yields a stability eigenvalue. The stability eigenvalue is composed of a term originated from the hydrodynamic Rayleigh instability and that from the diffusive instability. Concerning the stability eigenvalue of the hydrodynamic instability, a set of equations is obtained from boundary conditions at an interface and a solution of the equations yields the stability eigenvalue. In case of equal viscosity, the analytic formula of the solution is obtained without an external flow. The Tomotika's stability eigenvalue with the equal viscosity [Proc. R. Soc. London, Ser. A 150 (1935) 322] is shown to agree with the Stone and Brenner's result [J. Fluid Mech. 318 (1996) 373] explicitly. The Poiseuille flow has an effect of mixing an unstable axisymmetric mode with stable nonaxisymmetric modes, so that the two instability are suppressed. A radius of a stable cylindrical domain depends on a distance from the center of the Poiseuille flow. We estimate a Reynolds number under an external flow. Validity of the Stokes' approximation for the Navier–Stokes equation is confirmed.