Non-modal stability in Hagen-Poiseuille flow of a Bingham fluid

Abstract

Linear stability in Hagen-Poiseuille flow of a Bingham fluid is considered. Bingham fluid exhibits a yield stress in addition to a plastic viscosity. A Bingham number B, which describes the ratio of yield and viscous stresses, is used to characterize the behavior of Bingham-Hagen-Poiseuille flow. The effects of B on the stability are investigated using the energy method and the non-modal stability theory. The energy analysis shows that the non-axisymmetric disturbance has the lowest critical energy Reynolds number for all B. The global critical energy Reynolds number Reg increases with B. At sufficient large B, Reg has the order of B1/2. For the non-modal stability, we focus on response to external excitations and initial conditions. The former is studied by examining the ε-pseudospectrum, and the latter is by examining the energy growth function G(t). For the problem of response to external excitations, the maximum response is achieved by non-axisymmetric and streamwise uniform disturbances at the frequency of ω = 0, with a possible choice of the azimuthal wavenumbers of n = 1, 2, or 3. For the problem of response to initial conditions, it is found that there can be a rather large transient growth even though the linear operator of the Bingham-Hagen-Poiseuille flow has no unstable eigenvalue. For small B, the optimal disturbance is in the form of streamwise uniform vortices and streaks. For large B, the optimal disturbance is in the form of oblique waves. The optimal energy growth decreases and the optimal azimuthal wavenumber increases with the increase of B.