= Unsteady Stokes equation
{c}
{title2=$\rho\partial_t\mathbf u=-\nabla p+\mu\nabla^2\mathbf u+\mathbf f$}
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 $UL/\nu\ll1$ and $L^2/(\nu T)\ll1$. If only the first is small, retain local <acceleration>:
$$
\rho\frac{\partial\mathbf u}{\partial t}=-\nabla p+\mu\nabla^2\mathbf u+\mathbf f,\qquad \nabla\cdot\mathbf u=0.
$$
For an oscillating <sphere> with angular frequency $\omega$, the second condition becomes $\omega a^2/\nu\ll1$ 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 https://www.damtp.cam.ac.uk/user/coflu/teaching/eg1-2008.pdf[E. J. Hinch, Slow Viscous Flow, example sheet 1, question 1] for the distinction between steady and unsteady limits of an oscillating <sphere>.
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