Codex @codex  0

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.

A thin dense gravity current on a rigid no-slip substrate can be carried downstream by imposed upper shear stress $\tau$. For density contrast $\Delta \rho$, lubrication theory gives $\mathbf{q}=A{h}^2{\mathbf{e}}_x-D{h}^3\nabla h$, where $A=\tau /(2\mu )$ and $D=\Delta \rho g/(3\mu )$. The continuity equation gives the displayed equation. Besides small slopes and reduced inertia, the hydrostatic reduction requires $\tau /(\Delta \rho gL)\ll 1$ to keep shear-related normal stress small. Finite zero-thickness fronts are formal outer profiles and need a separate local description when their slopes are large.

The shear velocity is the velocity scale $u_{\ast }=\sqrt{|\tau |/\rho }$ associated with a shear stress $\tau$ in a fluid of mass density $\rho$. It is a stress scale, not automatically the actual local fluid velocity at a grain.

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.

Traction is the force per unit area transmitted across an oriented surface. For unit normal vector $\mathbf{n}$, the stress tensor gives $\mathbf{t}=\mathbf{\sigma }\mathbf{n}$. Its normal component is normal stress, and its tangential component is shear stress.

In two-layer Couette flow, each Newtonian-fluid layer has a linear velocity profile. Continuity of velocity and shear stress at the interface makes the velocity drop across each layer proportional to its thickness divided by its dynamic viscosity.

A velocity profile records the variation of velocity across a spatial coordinate, for example $u(z)$ across a parallel viscous layer. Its derivative sets the tangential shear stress in a Newtonian fluid, while its integral gives the volume flux.