Stokes equation ($-\nabla p+\mu {\nabla }^2\mathbf{u}+\mathbf{f}=0$)

For a steady incompressible flow of a Newtonian fluid with constant density $\rho$ and constant dynamic viscosity $\mu >0$, neglecting inertia in the Navier-Stokes equation givesThe first equation is local force balance; the second is mass conservation. The term $\mu {\nabla }^2\mathbf{u}$ is the divergence of the viscous part of the Newtonian fluid stress tensor. The body force $\mathbf{f}$ has units of force per unit volume, $\mathrm{N}{\mathrm{m}}^{-3}$, and equals $\rho \mathbf{g}$ for uniform gravity. Appropriate boundary data for velocity or traction, together with suitable distant conditions for an exterior flow, complete the problem; a rigid surface normally imposes the no-slip boundary condition.

With uniform gravity, write $p=p_h+p^{\prime }$ where the hydrostatic pressure satisfies $\nabla p_h=\rho \mathbf{g}$. Then the disturbance equation is $-\nabla p^{\prime }+\mu {\nabla }^2\mathbf{u}=0$. This removes gravity from the fluid disturbance equation, while the hydrostatic pressure still supplies buoyancy on an immersed body. Omitting it from the total force balance would give the wrong Stokes settling velocity.

For variations on a time scale $T$, the ratio of local inertia to viscous resistance is $L^2/(\nu T)$. Steady Stokes flow at each instant requires both $UL/\nu \ll 1$ and $L^2/(\nu T)\ll 1$. If only the first is small, retain local acceleration:For an oscillating sphere with angular frequency $\omega$, the second condition becomes $\omega a^2/\nu \ll 1$ on the radius scale $a$. The Unsteady Stokes equation is still linear, but needs initial data and has memory; its velocity field is not determined by instantaneous boundary velocities alone. See E. J. Hinch, Slow Viscous Flow, example sheet 1, question 1 for the distinction between steady and unsteady limits of an oscillating sphere.