Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 330 / 3 / d

For a steady no-source particle-laden gravity current, $B_x=-v_s{g}^{\prime }<0$. Simply entraining particle-free ambient fluid increases volume and dilutes particle volume fraction; it cannot create particle buoyancy flux.

A steady increase can occur if sediment resuspension or bed erosion adds more suspended particle buoyancy than settling removes, if a tributary or lateral particle source supplies a positive flux, or if the buoyancy per unit particle amount increases through a change of the surrounding fluid's density. Changes of particle properties or another buoyancy source can also alter the flux, but they go beyond the fixed-particle, fixed-reference model. In a model with a source $S_B$,Thus the buoyancy production in a particle-laden current requires $S_B>v_s{g}^{\prime }$ in a steady current. In an unsteady current, $B_x$ can be positive even without external addition: local depletion has ${\partial }_t(g^{\prime }h)<-v_s{g}^{\prime }$, so more buoyancy leaves a section than enters it while stored particles are being released. An instantaneous spatial flux increase is not by itself proof of particle production.