Stability of laminar viscoplastic flows down an inclined open channel

In this paper we study the two-dimensional linear stability of a Papanastasiou fluid flowing down an inclined plane. The Papanastasiou constitutive law is a regularization of the Bingham law in which the singularity at zero strain rate is smoothed by an exponential function. The stability eigenvalue problem is solved by a spectral collocation method based on Chebyshev polynomials. We prove that, while the Bingham flow is unconditionally stable for every Reynolds number, the Papanastasiou flow becomes unstable at a critical Reynolds that depends on the material parameters, on the angle of inclination of the plane and on the prescribed inlet discharge. In particular, we show that the critical Reynolds is a decreasing function of the yield stress, showing the destabilizing effect of the yield stress. The results obtained her e show that the stability characteristics of a regularized flow can be extremely different from that of the exact Bingham fluid even if the two flows are “practically indistinguishable”.