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The biharmonic equation is ${\nabla }^{4}f=0$, where the biharmonic operator is the square of the Laplacian. In planar Stokes flow, taking the curl of the momentum equation and using a Cartesian streamfunction produces this equation. Its decaying spatial Fourier series modes have form $(A+By)e^{-|k|y}{e}^{ikx}$.

The Brinkman equation augments Stokes flow with a local drag against a background matrix of velocity ${\mathbf{u}}_m$. The inverse screening length $\alpha$ has units of inverse length. With equal viscous coefficients the permeability is ${\alpha }^{-2}$. Taking the curl produces a screened vorticity equation.

A control volume is a chosen region over which fluid mass conservation and momentum balance are integrated. It may be fixed or move with a prescribed velocity. Boundary fluxes must use velocity relative to the moving boundary. In Stokes flow, negligible inertia reduces the momentum balance to the sum of body forces and boundary tractions, making a carefully chosen control volume useful for finding drag without integrating directly over a curved body.

A slender axisymmetric column of a Newtonian fluid, with radius $a(z,t)$, axial velocity $w$, external pressure $p_e$ and axial body-force density $f$, obeys $(a^2{)}_t+(a^{2}w{)}_z=0$ and $3\mu (a^2{w}_z{)}_z/a^2=p_{e,z}-f$ when inertia, surface tension and leading external shear are negligible. The factor three is the Trouton ratio. Radial mass conservation gives $u_r=-r{w}_z/2$; the Newtonian fluid stress tensor gives $p=p_e-\mu w_z$ and axial excess stress $3\mu w_z$. This reduces a three-dimensional Stokes flow to one-dimensional evolving geometry.

For a sphere of radius $a$ in a slowly varying ambient Stokes flow of vorticity ${\mathbf{\omega }}_{\infty }$, its angular velocity and applied torque obeyat the sphere centre.

A fixed sphere in a slowly varying incident Stokes flow requires an external force $\mathbf{F}=-6\pi \mu a{\mathbf{u}}_{\infty }$ and external couple $\mathbf{G}=-4\pi \mu a^3{\mathbf{\omega }}_{\infty }$ to leading order. These follow from Faxén translation law and Faxén rotation law; the incident Stokeslet supplies ${\mathbf{u}}_{\infty }$ and ${\mathbf{\omega }}_{\infty }$. Finite-radius Laplacian terms enter the translational force at the next nonzero order. The signs here refer to applied holding loads, not hydrodynamic loads.

Short-wavelength corrugations on the underside of a highly viscous ice shelf relax through local Stokes flow. For a wavelength much smaller than the shelf thickness, the deformation occupies a depth of order the inverse wavenumber and the ice may be treated as a viscous half-space.

In thermal-time units, taking the Prandtl number large removes momentum inertia at leading order. The Stokes flow momentum equation then determines velocity instantaneously from the temperature perturbation, while the nonlinear heat equation retains its time derivative. Small Prandtl number does not generally give this model. An exact contrasting shear in a free-slip layer is $u=U_0{e}^{-\mathrm{Pr}{\pi }^{2}t}\cos \pi z$, $w=\theta =0$: it decays slowly as the Prandtl number tends to zero but is excluded by the instantaneous Stokes equation.

A viscous current with density between those of two ambient fluids spreads along their interface. Vertical buoyancy divides its thickness above and below the undisturbed interface. When its internal shear and in-plane extension are both negligible, resistance from the two ambient Stokes flows controls its spreading.

The dimensional linear friction coefficient relates drag force and velocity, $F_d=-\zeta v$, and has units of force divided by speed. It differs from the dimensionless aerodynamic drag coefficient. For an isolated sphere in Stokes flow, the Stokes drag law gives $\zeta =6\pi \eta a$.

For an almost spherical inviscid bubble in exterior Stokes flow $\mathbf{Ex}$, let $\gamma ={\gamma }_0-{\gamma }_1(C-C_0)$ and $K=C_0{a}^2/(2D_s)$. The quadrupolar surfactant distribution on a spherical interface makes the tangential velocity ${\mathbf{u}}_s=\alpha {\mathbf{I}}_s\mathbf{Ex}$ satisfyThe Marangoni stress approaches the stress needed to suppress surface motion as $M$ grows. The quadrupolar shape perturbation $r=a(1+\mathbf{n}\cdot \mathbf{Dn})$ isThese expressions require small deformation and small surface Péclet number; large $M$ alone does not justify a small concentration perturbation.

A Taylor expansion of the no-slip boundary condition about the flat sheet makes the second-order mean tangential velocity $⟨u_2(0)⟩=-⟨y_1{\partial }_y{u}_1(0)⟩$. The mean mode of Stokes flow is linear in height when no pressure gradient is imposed. In an unbounded fluid, bounded velocity excludes mean shear; in a confined fluid, the force-free condition excludes it. The remaining mean velocity is uniform and equals the swimming speed in the sheet frame. Thus the leading speed can be found from the first-order field without solving the oscillatory second-order field.

A microswimmer is a microscopic body that propels itself through a fluid. At small Reynolds number its motion is governed by Stokes flow, and self-propulsion must respect the force-free and torque-free conditions.

Among all incompressible velocity fields with the same prescribed boundary velocity and far-field behavior, the Stokes flow minimizes viscous dissipation. If $\mathbf{v}=\mathbf{u}+\mathbf{w}$, where $\mathbf{u}$ is the Stokes solution and $\mathbf{w}$ has homogeneous boundary data, integration by parts and the Stokes equation eliminate the cross term, leaving