Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 83 / 3 / ii

For a continuous size distribution, let $c_0(w)dw$ be initial particle volume fraction in a settling-speed interval. The settling exposure for a particle size distribution givesThe spectrum continually shifts toward slower-settling particles. The deposit becomes progressively finer outward, with a smooth range of depositional distances instead of two distinct size components. Size-resolved deposition is ${\rho }_p(w)w{\int }_{t_a(r)}^{\infty }{c}_0(w)e^{-wJ(t)}dt$ per speed interval.

There is no universal monodisperse runout formula with an arbitrarily chosen mean settling speed. The box relation isA spectrum bounded away from zero settling speed has a finite limiting radius, but an arbitrarily slow-settling tail can extend the reach substantially or remove a finite runout bound. For example, if $b(w)c_0(w)\sim C{w}^p$ near $w=0$, $p>-1$, its Laplace transform behaves as $g^{\prime }(J)\sim C\Gamma (p+1)J^{-(p+1)}$. The runout integral converges only for $p>1$. A positive population with exactly zero settling speed retains buoyancy indefinitely and spreads without a finite radius limit in this no-entrainment model. Thus the fine tail of the actual distribution, and possible lofting when thermal buoyancy is restored, control the distal outcome.