= Solution
For a dilute <particle suspension>, take $W_s>0$ as the downward speed relative to the carrier fluid. The <particle deposition flux> through a horizontal absorbing bed is $W_s\phi_b$, where $\phi_b$ is the near-bed <particle volume fraction>. Without stirring, settling can create a particle-free upper layer and a descending concentration front, so uniform concentration throughout the remaining fluid is not automatic. Weak <turbulence> can maintain a nearly uniform suspended concentration while leaving the mean <settling velocity> approximately equal to the isolated-particle value; this justifies using $W_s\phi$ in a mixed-layer balance. Strong stirring can produce resuspension and requires a separate erosion model.
For a bed inclined at angle $\vartheta$ to the horizontal, the normal settling speed is $W_s\cos\vartheta$, giving deposition flux $W_s\phi_b\cos\vartheta$ per unit actual bed area. There is also downslope particle motion, and near-wall effects or resuspension can alter this simplest absorbing-boundary model. Integrating the normal projection over a sloping bed is equivalent to integrating $W_s\phi_b$ over its horizontal projection.
The mixture <mass density> is $\rho=\rho_0+(\rho_p-\rho_0)\phi$, so its <reduced gravity> is
$$
\boxed{g'=d\phi,\qquad d=\frac{g(\rho_p-\rho_0)}{\rho_0}.}
$$
The <Boussinesq approximation> requires $(\rho_p-\rho_0)\phi/\rho_0\ll1$, in addition to the dilution assumption.
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