Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, 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 length-scales 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 and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
The primary Green's function of Stokes flow is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental solutions can be obtained.
For this type of flow, the inertial forces are assumed to be negligible and the Navier–Stokes equations simplify to give the Stokes equations:
Here is the velocity of the fluid, is the gradient of the pressure, and is the dynamic viscosity.
The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case. They are the leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit
- A Stokes flow has no dependence on time other than through time-dependent boundary conditions. This means that, given the boundary conditions of a Stokes flow, the flow can be found without knowledge of the flow at any other time.
- An immediate consequence of instantaneity, time-reversibility means then a time-reversed Stokes flow solves the same equations as the original Stokes flow. This property can sometimes be used (in conjunction with linearity and symmetry in the boundary conditions) to derive results about a flow without solving it fully. Time reversibility means that it is difficult to mix two fluids using creeping flow.
While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of non-Newtonian fluids means that they do not hold in the more general case.
- Stokes paradox
An interesting property of Stokes flow is known as the Stokes' paradox: that there can be no Stokes flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial, for the Stokes equations around an infinitely long cylinder.
Demonstration of time-reversibility
A Taylor–Couette system can create spiral laminar flows. Two fluids with very different viscosities (and therefore a very low Reynolds number) create spiral laminar flows which can then be reversed to approximately the initial state. This creates a dramatic demonstration of seemingly mixing two fluids and then unmixing them by reversing the direction of the mixer.
Methods of solution
By stream function
The equation for an incompressible Newtonian Stokes flow can be solved by the stream function method in planar or in 3-D axisymmetric cases
|Type of function||Geometry||Equation||Comments|
|Stream function||2-D planar||or (biharmonic equation)||is the Laplacian operator in two dimensions|
|Stokes stream function||3-D spherical||where||For derivation of the operator see Stokes stream function#Vorticity|
|Stokes stream function||3-D cylindrical||where||For see |
By Green's function: the Stokeslet
The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function, – known as a Stokeslet – for the equations can be found, where r is the position vector. The solution for the pressure p and velocity u due to a point force acting at the origin with |u| and p vanishing at infinity is given by
- is a second-rank tensor (or more accurately tensor field) known as the Oseen tensor (after Carl Wilhelm Oseen).
For a continuous-force distribution (density) the solution (again vanishing at infinity) can then be constructed by superposition:
By Papkovich–Neuber solution
By boundary element method
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 to both 2- and 3-dimensional flows.
- Stokes' law
- Darcy's law
- Hele-Shaw flow
- Laminar flow
- Lubrication theory
- Oseen equations
- Slender-body theory
- Volumetric flow rate
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