Boundary layers in dilute particle suspensions

Boundary layers in dilute particle suspensions have been found to have a number of interesting features. The development of a singularity at the wall has recently been found to be common to many of these flows, \footnote{See Foster, Duck \& Hewitt, {\it J. Fluid Mech.} {\bf 474} (2003) and Duck, Hewitt \& Foster, {\it J. Fluid Mech.} {\bf 514}, (2004)} and we note here that Falkner-Skan-type boundary layers (layers with `edge' velocity proportional to $x^m$) and the boundary layer under a linearly decelerating flow \footnote{Howarth (1934)} also break down at the wall in the absence of gravity, but can be singularity-free for heavy particles. In addition, we find that matching of the Falkner-Skan profile to an outer flow is problematic for some values of $m$, though the case most studied heretofore---the Blasius case (for $m=0$)---does not feature this difficulty. Finally, a boundary layer that does not develop a singularity takes on a the typical Falkner-Skan self-similarity far downstream, in the absence of gravity. For heavy particles, however, gravity causes a constant drift of particles toward the wall, and a constant-thickness far-downstream layer. The far-downstream behavior in a light-particle suspension is different, with a particle-free zone between the wall and a particle `shock' that grows like $x^{(1-m)}$.