Instability due to second normal stress jump in two-layer shear flow of the Giesekus fluid

The two-layer Couette flow of superposed Giesekus liquids is examined. In order to emphasize the effect of a jump in the second normal stress difference, the analysis is focused on flows where the shear rate and first normal stress difference are continuous across the interface. In this case, the flow is neutrally stable to streamwise disturbances, but can be unstable for spanwise disturbances driven by a jump in the second normal stress difference. Whether the long and order one waves are stable or not depends on the sign of this difference. Short waves are unstable. In the case of order one wave instability, the mode of maximum growth rate gives rise to stationary ripples perpendicular to the flow. The eigenvalue problem for purely spanwise wave vectors can in principle be solved analytically, although, in general, the analytical solution is too complicated to obtain. In most cases, however, a simplifying assumption can be made which makes analytical solutions feasible. We present such solutions and compare them with purely numerical solutions.