The singularity in particle-laden boundary layers

The classical ``dusty gas'' equations have been used recently in a number of investigations by the authors\footnote{See, most recently, Foster, Duck \& Hewitt (2006) {\it Proc. Roy. Soc A} {\bf 462}, 1145} to model boundary-layer flows of dilute suspensions of heavy particles. Though none of the difficulties of well-posedness that so often occur in more complicated particle-laden flow models seems to arise for this equation set, what does nearly always appear, and is now well documented in a variety of boundary layers, is a wall singularity that occurs at a finite distance from the leading edge, where the volume fraction is unbounded. The dusty-gas approximation replaces the quantity ``$1-\alpha$'' everywhere in the particle-laden equations by ``$1$''. One is forced to seek a more complicated model in order to remove the unphysical singularity, and there are plenty of candidates--including particle pressure, added mass, particle-particle interactions. From the point of view of modifying the theory in the simplest possible way, we restore ``$1-\alpha$'' where it had been replaced by ``1,'' and do nothing more. Such a procedure removes the singularity in boundary-layer flows, and we present computational and analytical results under such a change