Stokeslet

A Stokeslet is the velocity field produced by a point force $\mathbf{F}$ in an unbounded three-dimensional Stokes fluid:

In a three-dimensional exterior Stokes flow with no body force, $\nabla \cdot \mathbf{\sigma }=0$. The divergence theorem on the fluid between a rigid body and a large sphere $S_R$ gives the displayed force on the body, where $\mathbf{n}$ points outward from the sphere. The body's outward normal points oppositely to the inner normal of this fluid domain. A nonzero resultant force produces a leading Stokeslet, so velocity is $O(R^{-1})$, gauge-fixed pressure is $O(R^{-2})$ and rate-of-strain tensor is $O(R^{-2})$. The finite force is therefore compatible with pointwise decay of the stress tensor. Force-free bodies may have faster decay.

The Oseen tensor is the unbounded three-dimensional incompressible Stokes flow Green tensor relating a point force to the velocity it produces: $u_i={\mathsf{G}}_{ij}{F}_j$. At separation $r$, its longitudinal mobility is twice its transverse mobility. It generates the leading far-field hydrodynamic interaction between externally forced particles. The singular value at $r=0$ is not a particle's self mobility; a no-slip sphere instead has self mobility $1/(6\pi \eta a)$. Walls and near-contact lubrication require different or corrected tensors.

A normal point force $F\mathbf{n}$ at height $d$ above a plane with a no-slip boundary condition is balanced by an image Stokeslet $-F\mathbf{n}$, a stresslet of tensor strength $S$, and a source dipole of moment $\mathbf{M}$ at the reflected point. Here the stresslet convention is $u_i=-S_{jk}{\partial }_k{G}_{ij}$, with $G$ the free-space Stokes tensor. Substitution on the plane cancels both normal and tangential velocity. The original and image forces cancel their $r^{-1}$ terms, while the image stresslet cancels the remaining $r^{-2}$ force-dipole term; the generic fixed-angle far field is $O(r^{-3})$.

Place a normal Stokeslet below a plane and an equal, oppositely directed image at the reflected point above it. The normal velocity of their sum is odd across the plane and the tangential velocity is even. The plane therefore has zero normal velocity and zero normal derivative of tangential velocity, giving zero tangential stress. For a sphere of radius $a$ translating towards that plane at speed $U$ from distance $d\gg a$, the leading radial surface flow is $3aUrd/[2(r^2+d^2{)}^{3/2}]$, directed outward. The image's incident flow at the sphere changes its required force by relative order $a/d$, so the first correction to surface velocity at fixed $r/d$ is $O(U(a/d{)}^2)$; the isolated sphere's potential-dipole correction occurs only at third order.

Differentiating the Stokeslet gives its rate-of-strain tensor and vorticity, displayed above for $r=|\mathbf{x}|>0$. The pressure is $p=\mathbf{F}\cdot \mathbf{x}/(4\pi r^3)$. Integrating the Newtonian fluid stress tensor over a sphere enclosing the origin gives traction resultant $-\mathbf{F}$, fixing the point-force normalization.

An impermeable stress-free plane is enforced for a normally oriented Stokeslet by an oppositely directed image Stokeslet at the reflected point. Two equal normal point forces at the same depth then attract laterally through their image flows.