The stability of spiral Poiseuille flows of Newtonian and Bingham fluids in an annular gap

Abstract

We consider the linear stability of both Newtonian and Bingham fluids in spiral Poiseuille flow in the annular gap between two co-rotating cylinders using the method of normal modes. Only axisymmetric disturbances are considered. We find that for the Newtonian case, linear instability does occur but the margin of stability increases with increasing Reθ. For the Bingham fluid case, we find the eigenvalue problem to be linearly stable over the range [Rez, Reθ, η]∈[0, 10000]×[0, 5000]×[0.75, 0.9], where Rez is axial Reynolds number, Reθ is the tangential Reynolds number and η is the ratio of inner to outer radius of the annular gap and we believe that the flow is linearly stable for all B>0 where B is Bingham number. In the limit of B→0, we demonstrate that we cannot recover the results for the Newtonian fluid. The stability behaviour is singular in this limit and we show that this arises from imposition of boundary conditions for the Bingham fluid eigenvalue-problem at the unperturbed yield surface position, rather than any other effect of the yield stress.