Modal stability and Squire's theorem for an inhomogeneous viscoelastic suspension

We study the linear stability of the Poiseuille flow of a viscoelastic upper convected Maxwell fluid in which the rheological parameters depend on the concentration of particles suspended in the fluid (dense suspension with negligible diffusion). After determining the basic flow and basic concentration profile we consider a temporal three dimensional perturbation in the form of a stream-wise and span-wise wave. We derive the linearized perturbed equation and prove the validity of Squire's theorem, extending the result of Tlapa and Bernstein (1970) in which the theorem was proved for constant rheological parameters. We discuss the relation between the Weissenberg and the Reynolds numbers. We finally study the 2D eigenvalue problem for the case of constant coefficients and for non-constant coefficients with low Weissenberg number. We solve the problem numerically by means of a spectral collocation method and we plot the marginal stability curves discussing how stability depends on the fluid rheology.