Stability of a sheared particle suspension

A stability analysis is carried out for a gas–particle suspension in which the energy dissipation occurs due to the viscous drag force exerted on the particles. The flow is driven by two types of energy sources, an imposed mean shear and fluid velocity fluctuations, in the limit where the time between collisions $\tau_c$ is small compared to the viscous relaxation time $\tau_v$, so that the dissipation of energy between collisions is small compared to the energy of a particle. Constitutive relations from the kinetic theory of dense gases are used when the flow is driven by the mean shear. The effect of fluid velocity fluctuations is incorporated using an additional diffusive term in the Boltzmann equation for the particle velocity distribution, and this leads to an additional "diffusion" stress. For a suspension driven by fluid velocity fluctuations, it is found that perturbations are always stable. For a suspension driven by mean shear, the viscous relaxation time is large compared to the collision time for $\bar{G} \tau_v \gg 1$, where \bar{G} is the mean strain rate. The rate of diffusion of energy is small compared to the rate of dissipation for $k^* \ll (\bar{G} \tau_v)^{–1}$, where $k^*$ is the wave number scaled by the mean free path. In this regime, it is found that density perturbations are unstable in all three directions in the limit of low volume fraction, but become stable when the volume fraction is increased beyond a critical value. For $k^* \gg (\bar{G} \tau_v)^{–1}$, the rate of diffusion of energy is large compared to the rate of dissipation, and it is found that perturbations are always stable. The transition between these two regimes is obtained numerically in the dilute limit, and the neutral stability curves for the density perturbations are obtained. It is found that in the gradient-vorticity plane, the transition wave number is proportional to $(\bar{G} \tau_v)^{–1}$ in the limit $(\bar{G} \tau_v) \gg 1$