Exact coherent structures with dilute particle suspensions

The physics of settling suspensions under shear are investigated by theoretical and numerical analyses of unstable equilibrium solutions to the incompressible Navier-Stokes equations, coupled with an advection-diffusion-settling equation for a dilute phase of particles. Two cases are considered: the 'passive scalar' regime, in which the sediment is advected by the fluid motion, but concentrations are too dilute to affect the flow; and the 'stratified' regime, where nonuniform vertical distribution of sediment due to particle settling leads to a bulk stratification that feeds back on the flow via buoyancy. In the passive regime, we characterise the structure of the resultant sediment concentration fields and derive formulae for transport fluxes of sediment with asymptotically low and high settling velocities. In the stratified regime, parametric continuation is employed to explore the dependence of states upon the bulk Richardson number $Ri_b$. Symmetry breaking in the governing equations leads to travelling wave solutions with a rich bifurcation structure. The maximum $Ri_b$ attained by these states depends non-monotonically on settling velocity and obeys asymptotic scalings that have also been observed to capture the dependence of the laminar-turbulent boundary in direct numerical simulations.