 Solution

The turbulent Schmidt number compares turbulent momentum and concentration diffusivities:The mean fluid has no vertical velocity. Taking $z$ positive upward and $W_s>0$, the settling flux is $-W_s\varphi$, while the given turbulent particle flux is $-{\kappa }_T{\varphi }_z$. Local particle-volume conservation therefore givesConsequently the evolution equation isFor constant settling speed this is ${\varphi }_t=W_s{\varphi }_z+{\partial }_z({\kappa }_T{\varphi }_z)$. In the logarithmic region, ${\kappa }_T=Cz/2$ from the preceding part. The sign is consistent with downward settling: without diffusion, a profile evolves as $\varphi (z,t)=\varphi (z+W_{s}t,0)$. The dilute suspension approximation neglects feedback on the mean flow and particle interactions; deposition or erosion must be imposed through the bed boundary flux.