Stokes flow (named after George Gabriel Stokes), also named creeping flow, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the lengthscales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm^{[1]} and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
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For this type of flow, the inertial forces are assumed to be negligible and the Navier–Stokes equations simplify to give the Stokes equations:
where is the comoving stress tensor, and an applied body force. There is also an equation for conservation of mass. In the common case of an incompressible Newtonian fluid, the Stokes equations are:
Here is the velocity of the fluid, is the gradient of the pressure, and μ is the dynamic viscosity.
The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case.^{[2]}^{[3]}^{[4]}^{[5]}
While these properties are true for incompressible Newtonian Stokes flows, the nonlinear and sometimes timedependent nature of nonNewtonian fluids means that they do not hold in the more general case.
It can be shown that in planar 2D systems, the stream function for an incompressible Newtonian Stokes flow satisfies the biharmonic equation .
In the 3D axisymmetric case, the Stokes stream function Ψ solves the equation E^{2}Ψ = 0, where
The Papkovich–Neuber Solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic potentials.
Certain problems, such as the evolution of the shape of a bubble in a Stokes flow, are conducive to numerical solution by the Boundary Element method. This technique can be applied in both 2 and 3dimensional flows.
The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function for the equations can be found. The solution for the pressure p and velocity due to a point force acting at the origin with and p vanishing at infinity is given by^{[7]}
where
is a secondrank tensor known as the Oseen tensor (after Carl Wilhelm Oseen).
The solution for a continuous force distribution (density) (again vanishing at infinity) can then be constructed by superposition:
