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Particles settle toward the bed under their submerged weight. A concentration increasing downward gives an upward turbulent diffusive flux, which can oppose that settling. If turbulence cannot sustain the particles, they enter the weakly mixed near-wall region and can deposit. Sediment resuspension requires bed forces strong enough to entrain grains, not merely a specified diffusivity far above the bed.

For noncohesive grains, an appropriate criterion uses the Shields parameterwhere $d$ is grain size, ${\rho }_s$ its density and ${\Theta }_c$ the sediment entrainment threshold expressed dimensionlessly. The threshold depends on grain geometry, contacts and viscous effects; cohesion changes the criterion. Sustained suspension additionally requires sufficiently vigorous upward turbulent motions relative to $W_s$. Bed motion and well-mixed suspension are therefore not identical conditions.

At sedimentation-resuspension equilibrium, the total vertical particle flux is constant. Set ${\kappa }_T=Kz$ with $K=C/2$. The stationary equation givesSince $W_s>0$ and $\varphi \to 0$ at infinity, necessarily $J=0$. Settling and upward diffusion then balance pointwise. Taking a reference height $z_1=1$ in the specified length units gives the equilibrium settling-diffusion profileIt is nonnegative and decays to zero. The profile describes the region $z\gg \delta$; it does not assert that this power law continues through the viscous layer down to the physical bed. The supplied reference concentration summarizes the bed exchange and near-wall physics that the outer diffusion closure does not resolve.

Let $\Delta \rho ={\rho }_p-{\rho }_f>0$. A spherical grain has submerged weightNeglect lift and contact torque, and use a sliding static friction model with normal reaction $R=G$. The inertial quadratic drag iswhere the last equality fixes the convention $\tau ={\rho }_f{U}^2$. Equivalently $U$ is the bed shear velocity and $C_D$ is an effective drag coefficient referred to it. A literal grain-level flow speed can differ from the shear velocity; that conversion must then be absorbed into $C_D$.

Downstream sliding begins when $F_D>{\mu }_{s}R$. Defining the Shields parameter by $\Theta =\tau /(\Delta \rho gd)$, the threshold force balance isThe grain moves downstream above this threshold in the stated sliding model. Real grain motion can instead involve lift, rolling, irregular contacts, or viscous drag; the printed constant belongs to the particular inertial-drag convention and force balance above.

The normal contact force and resisting static friction balance the drag force and submerged weight at impending motion. The right panel uses locally bed-tangent drag force.

The Shields parameter is $\Theta =\tau /[({\rho }_p-{\rho }_f)gd]$, comparing bed shear stress with a grain-scale submerged weight per area. Its critical value depends on the assumed drag coefficient, static friction, lift, and contact geometry.