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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.

Measure displacement downward from the release point and take downward speed $v\ge 0$. Newton's second law with quadratic drag givesWritingseparation of variables gives the terminal velocity solutionThe distance fallen by time $t$ is thereforeSetting $y(t)=h$ and solving for $t$ yields

The sediment entrainment threshold is the bed shear stress at which resting grains begin moving. An elementary force balance using spherical grains, tangential quadratic drag, and static friction gives ${\Theta }_{\mathrm{th},0}=4{\mu }_s/(3C_D)$ when the drag coefficient is defined relative to the bed shear velocity.