Linear stability involving the Bingham model when the yield stress approaches zero

The plane Poiseuille flow of Bingham fluid is characterized by a plug zone around the axis of the channel, where τ, the second invariant of the deviatoric stress tensor, is less than or equal to the yield stress τ0. According to the Bingham model, the plug zone moves as a rigid body with a constant velocity. The dimension of the plug zone, scaled with the width of the channel, depends only on the Bingham number, B (the ratio of the yield stress to a nominal viscous stress). The linear stability analysis of this flow as well as the Rayleigh–Bénard Poiseuille flow is performed. The numerical results are discussed essentially for the case B⪡1. By comparison with the Newtonian fluid (B=0), a discontinuous behavior of the critical conditions is observed. This discontinuity is a consequence of the linear stability analysis that allows the plug zone to remain intact.