Stokes flow

File:Stokes sphere.svg

An object moving through a gas or liquid experiences a force in direction opposite to its motion. Terminal velocity is achieved when the drag force is equal in magnitude but opposite in direction to the force propelling the object. Shown is a sphere in Stokes flow, at very low Reynolds number.

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. \(\textit{Re} \ll 1\). 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 .

Stokes equations

For this type of flow, the inertial forces are assumed to be negligible and the Navier–Stokes equations simplify to give the Stokes equations:

\[\boldsymbol{\nabla} \cdot \mathbb{P} + \mathbf{f} = 0\]

where \(\mathbb{P}\) is the stress tensor, and \(\mathbf{f}\) 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:

\[\boldsymbol{\nabla}p = \mu \nabla^2 \mathbf{u} + \mathbf{f}\] \[\boldsymbol{\nabla}\cdot\mathbf{u}=0\]

Here \(\mathbf{u}\) is the velocity of the fluid, \(\boldsymbol{\nabla} p\) is the gradient of the pressure, and \(\mu\) is the dynamic viscosity.

Properties

The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case.

Instantaneity

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.

Time-reversibility

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; a dramatic demonstration is possible of apparently mixing two fluids and then unmixing them by reversing the direction of the mixer.

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.

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 \((\psi)\)	2-D planar	\(\nabla^4 \psi = 0\) or \(\Delta^2 \psi = 0\) (biharmonic equation)	\(\Delta\) is the Laplacian operator in two dimensions
Stokes stream function \((\Psi)\)	3-D spherical	\(E^2 \Psi = 0,\) where \(E = {\partial^2 \over \partial r^2} + {\sin{\theta} \over r^2} {\partial \over \partial \theta} \left({ 1 \over \sin{\theta}} {\partial \over \partial \theta}\right)\)	For derivation of the \(E\) operator see Stokes_stream_function#Vorticity
Stokes stream function \((\Psi)\)	3-D cylindrical	\(L_{-1}^2 \Psi = 0,\) where \(L_{-1} = \frac{\partial^2}{\partial z^2} + \frac{\partial^2}{\partial \rho^2} - \frac{1}{\rho}\frac{\partial}{\partial\rho}\)	For \(L_{-1}\) 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, \(\mathbb{J}(\mathbf{r})\) – 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 \(\mathbf{F}\delta(\mathbf{r})\) acting at the origin with |u| and p vanishing at infinity is given by

\[\begin{align} \mathbf{u}(\mathbf{r}) &= \mathbf{F} \cdot \mathbb{J}(\mathbf{r}) \\ p(\mathbf{r}) &= \frac{\mathbf{F}\cdot\mathbf{r}}{4 \pi |\mathbf{r}|^3} \end{align}\]

where

\[\mathbb{J}(\mathbf{r}) = {1 \over 8 \pi \mu} \left( \frac{\mathbb{I}}{|\mathbf{r}|} + \frac{\mathbf{r}\mathbf{r}^\mathrm{T}}{|\mathbf{r}|^3} \right)\]  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) \(\mathbf{f}(\mathbf{r})\) the solution (again vanishing at infinity) can then be constructed by superposition: \[\begin{align}\mathbf{u}(\mathbf{r}) &= \int \mathbf{f}(\mathbf{r'}) \cdot \mathbb{J}(\mathbf{r} - \mathbf{r'}) \mathrm{d}\mathbf{r'} \\ p(\mathbf{r}) &= \int \frac{\mathbf{f}(\mathbf{r'})\cdot(\mathbf{r}-\mathbf{r'})}{4 \pi |\mathbf{r}-\mathbf{r'}|^3} \, \mathrm{d}\mathbf{r'} \end{align}\]

By Papkovich–Neuber solution

The Papkovich–Neuber solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic potentials.

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.

See also

• Darcy's law
• Laminar flow
• Lubrication theory
• Hele-Shaw flow
• Oseen equations
• Poiseuille
• Slender-body theory

References

• Ockendon, H. & Ockendon J. R. (1995) Viscous Flow, Cambridge University Press. ISBN 0-521-45881-1.bs:Stokesovo strujanje