On three-dimensional linear stability of Poiseuille flow of Bingham fluids

Plane channel Poiseuille flow of a Bingham fluid is characterized by the Bingham number, B, which describes the ratio of yield and viscous stresses. Unlike purely viscous non-Newtonian fluids, which modify hydrodynamic stability studies only through the dissipation and the basic flow, inclusion of a yield stress additionally results in a modified domain and boundary conditions for the stability problem. We investigate the effects of increasing B on the stability of the flow, using eigenvalue bounds that incorporate these features. As B-->infinity we show that three-dimensional linear stability can be achieved for a Reynolds number bound of form Re=O(B-3/4), for all wavelengths. For long wavelengths this can be improved to Re=O(B), which compares well with computed linear stability results for two-dimensional disturbances [J. Fluid Mech. 263, 133 (1994)]. It is also possible to find bounds of form Re=O(B-1/2), which derive from purely viscous dissipation acting over the reduced domain and are comparable with the nonlinear stability bounds in J. Non-Newt. Fluid Mech. 100, 127 (2001). We also show that a Squire-like result can be derived for the plane channel flow. Namely, if the equivalent eigenvalue bounds for a Newtonian fluid yield a stability criterion, then the same stability criterion is valid for the Bingham fluid flow, but with reduced wavenumbers and Reynolds numbers. An application of these results is to bound the regions of parameter space in which computational methods need to be used. (C) 2003 American Institute of Physics.