Hydrodynamic stability of plane Poiseuille flow in Maxwell fluid with cross-flow

This paper deals with the linear stability analysis of a plane Poiseuille in a Maxwell fluid in the presence of a uniform cross-flow with respect to two-dimensional wave disturbances. The physical problem is reduced to a modified Orr-Sommerfeld equation with nonlinear eigenvalues and solved numerically using the Chebyshev spectral collocation method. Attention is focused on the combined effects of uniform cross-flow and the relaxation time of the fluid (Deborah number) on the flow stability. Results obtained in this framework show that for a given Reynolds number the cross-flow can either delay or advances the stability of this system depending on the magnitude of the wave number. This result is observed in the case of a Newtonian fluid as well as in the case of a Maxwell one. An important feature concerns the existence of bicritical states that occur when two different instability modes with incommensurate wave numbers bifurcate simultaneously from the basis state is observed. In addition, it is attempted to determine the critical parameters of instability when one varies the cross-flow in the presence of fluid's elasticity. It turns out that the cross-flow has a stabilizing effect which becomes less pronounced when the fluid's elasticity is present.