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[[File:Stokes sphere.svg|thumb|right|200px|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''' or '''creeping motion''',<ref name=kim2005>Kim, S. &amp; Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. ISBN 0-486-44219-5.</ref> is a type of [[fluid flow]] where [[advection|advective]] [[inertia]]l forces are small compared with [[Viscosity|viscous]] forces.<ref name=Kirby>{{cite book | author=Kirby, B.J. | title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices.| url=http://www.kirbyresearch.com/textbook| year=2010| publisher=Cambridge University Press| isbn=978-0-521-11903-0}}</ref> The [[Reynolds number]] is low, i.e. <math>\textit{Re} \ll 1</math>. 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 [[microorganism]]s and [[sperm]]<ref>Dusenbery, David B. (2009). ''Living at Micro Scale''. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.</ref> and the flow of [[lava]]. In technology, it occurs in [[paint]], [[Microelectromechanical Systems|MEMS]] devices, and in the flow of viscous [[polymer]]s generally.

The equations of motion for Stokes flow, called the Stokes Equations, are a [[linearization]] of the [[Navier-Stokes Equations]], and thus can be solved by a number of well-known methods for linear differential equations.<ref name=Leal>{{cite book | author=Leal, L.G. | authorlink=L. Gary Leal | title=Advanced Transport Phenomena:Fluid Mechanics and Convective Transport Processes | year=2007}}</ref>  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 solution]]s can be obtained.<ref name="Chwang">Chwang, A. and Wu, T. (1974). [http://www.nada.kth.se/~annak/Chwang&Wu.pdf "Hydromechanics of low-Reynolds-number flow. Part 2.  Singularity method for Stokes flows"]. ''J. Fluid Mech. 62''(6), part 4, 787–815.</ref>

== Stokes equations ==

The equation of motion for Stokes flow can be obtained by linearizing the [[steady state]] Navier-Stokes Equations. The inertial forces are assumed to be negligible in the momentum balance equation in the [[Navier–Stokes equations]] simplifies to give the momentum balance equation in the Stokes equations:<ref name=kim2005 />

:<math>\boldsymbol{\nabla} \cdot \mathbb{P} + \mathbf{f} = 0</math>

where <math>\scriptstyle \mathbb{P} </math> is the [[Cauchy stress tensor]] representing viscous and pressure stresses,<ref name=Batchelor>{{cite book | author=Batchelor, G. K.| authorlink=George Batchelor | title=Introduction to Fluid Mechanics | year=2000}}</ref><ref name=Happel>Happel, J. &amp; Brenner, H. (1981) ''Low Reynolds Number Hydrodynamics'', Springer. ISBN 90-01-37115-9.</ref> and <math>\scriptstyle \mathbf{f}</math> an applied body force. The full Stokes equations also include an equation for the [[conservation of mass]], commonly written in the form:

:<math> \frac{D\rho}{Dt} + \rho \nabla \cdot \mathbf{u} = 0  </math>

Where <math>\scriptstyle \rho</math> is the fluid density, <math>\scriptstyle \mathbf{u}</math> the fluid velocity, and <math>\scriptstyle \frac{D}{Dt}</math> the [[material derivative]].  To obtain the equations of motion for incompressible flow, it is assumed that the density, <math>\scriptstyle \rho</math>, is a constant.

Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term <math>\scriptstyle \rho \frac{\partial\mathbf{u}}{\partial t}</math> is added to the left hand side of the momentum balance equation.<ref name=kim2005/>

===Properties===
The Stokes equations represent a considerable simplification of the full [[Navier–Stokes equations]], especially in the incompressible Newtonian case.<ref name=Kirby /><ref name=Leal /><ref name=Batchelor/><ref name="Happel"/> They are the [[leading-order]] simplification of the full Navier–Stokes equations, valid in the distinguished limit <math>Re \to 0.</math>

; Instantaneity
:A Stokes flow has no dependence on time other than through time-dependent [[boundary condition]]s. 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.

While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of [[non-Newtonian fluid]]s 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 solution for the Stokes equations around an infinitely long cylinder.<ref>{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|year=1945|publisher=Dover Publications|location=New York|pages=602–604|edition=Sixth}}</ref>

===Demonstration of time-reversibility===

A [[Taylor–Couette flow|Taylor–Couette system]] can create spiral laminar flows.<ref>C. David Andereck, S. S. Liu and Harry L. Swinney (1986). Flow regimes in a circular Couette system with independently rotating cylinders. Journal of Fluid Mechanics, 164, pp 155–183 doi:10.1017/S0022112086002513</ref> 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.<ref>Dusenbery, David B. (2009). ''Living at Micro Scale'', pp.46. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.</ref><ref>http://www.youtube.com/watch?v=p08_KlTKP50&feature=related</ref><ref>http://panda.unm.edu/flash/viscosity.phtml</ref>

==Incompressible flow of Newtonian fluids==

In the common case of an incompressible [[Newtonian fluid]], the Stokes equations take the (vectorized) form:

:<math> \begin{align} \mu \nabla^2 \mathbf{u} -\boldsymbol{\nabla}p + \mathbf{f} &= 0 \\
 \boldsymbol{\nabla}\cdot\mathbf{u}&=0 \end{align}</math>

Where <math>\scriptstyle \mathbf{u}</math> is the [[velocity]] of the fluid, <math>\scriptstyle \boldsymbol{\nabla} p</math> is the gradient of the [[pressure]], <math>\scriptstyle \mu</math> is the dynamic viscosity, and <math>\scriptstyle \mathbf{f}</math> an applied body force.  The resulting equations are linear in velocity and pressure, and therefore can take advantage of a variety of linear differential equation solvers.<ref name=Leal />

===Cartesian coordinates===
With the velocity vector expanded as <math>\scriptstyle \mathbf{u}=(u,v,w)</math> and similarly the body force vector <math>\scriptstyle \mathbf{f} = (f_x, f_y, f_z) </math>, we may write the vector equation explicitly,

:<math>  \begin{align}  \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) - \frac{\partial p}{\partial x} + f_x &= 0 \\
  \mu \left(\frac{\partial^2 v}{\partial x^2}  + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) - \frac{\partial p}{\partial y} + f_y &= 0 \\
  \mu \left(\frac{\partial^2 w}{\partial x^2}  + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) - \frac{\partial p}{\partial z} + f_z &= 0 \\
  {\partial u \over \partial x} + {\partial v \over \partial y} + {\partial w \over \partial z} &= 0 \end{align}</math>

We arrive at theses equations by making the assumptions that <math>\scriptstyle \mathbb{P} =\frac{1}{2}(\boldsymbol{\nabla}\mathbf{u} + (\boldsymbol{\nabla}\mathbf{u})^T) - p\mathbb{I}</math> and the density <math>\rho</math> is a constant.<ref name=Batchelor />

===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
{| class=wikitable
!Type of function
!Geometry
!Equation
!Comments
|-
|[[Stream function]] <math>(\psi)</math>
|2-D planar
| <math>\nabla^4 \psi = 0</math> or <math>\Delta^2 \psi = 0</math> ([[biharmonic equation]])
| <math>\Delta</math> is the [[Laplacian]] operator in two dimensions
|-
|[[Stokes stream function]] <math>(\Psi)</math>
|3-D spherical
|<math>E^2 \Psi = 0,</math> where <math>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)</math>
|For derivation of the <math>E</math> operator see [[Stokes stream function#Vorticity]]
|-
|[[Stokes stream function]] <math>(\Psi)</math>
|3-D cylindrical
|<math>L_{-1}^2 \Psi = 0,</math> where <math>L_{-1} = \frac{\partial^2}{\partial z^2} + \frac{\partial^2}{\partial \rho^2} - \frac{1}{\rho}\frac{\partial}{\partial\rho}</math>
|For <math>L_{-1}</math> see <ref>{{Cite journal
| volume = 7
| issue = 04
| pages = 529–549
| last = Payne
| first = LE
| coauthors = WH Pell
| title = The Stokes flow problem for a class of axially symmetric bodies
| journal = Journal of Fluid Mechanics
| year = 1960
|bibcode = 1960JFM.....7..529P |doi = 10.1017/S002211206000027X }}</ref>
|}

====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]], <math>\mathbb{J}(\mathbf{r})</math>, exists.  The Green's function is found by solving the Stokes equations with the forcing term replaced by a point force acting at the origin, and boundary conditions vanishing at infinity:

:<math> \begin{align} \mu \nabla^2 \mathbf{u} -\boldsymbol{\nabla}p &= \mathbf{F}\cdot\mathbf{\delta}(\mathbf{r})\\
 \boldsymbol{\nabla}\cdot\mathbf{u}&=0 \\ 
|\mathbf{u}|, p &\to 0 \quad \mbox{as} \quad r\to\infty \end{align}</math>

where <math>\mathbf{\delta}(\mathbf{r})</math> is the [[Dirac delta function]], and <math>\mathbf{F}\cdot\delta(\mathbf{r})</math> represents a point force acting at the origin. The solution for the pressure ''p'' and velocity '''u''' with |'''u'''| and ''p'' vanishing at infinity is given by<ref name=kim2005/>

:<math> \mathbf{u}(\mathbf{r}) = \mathbf{F} \cdot \mathbb{J}(\mathbf{r}), \qquad
p(\mathbf{r}) = \frac{\mathbf{F}\cdot\mathbf{r}}{4 \pi |\mathbf{r}|^3} </math>

where

:<math>\mathbb{J}(\mathbf{r}) = {1 \over 8 \pi \mu} \left( \frac{\mathbb{I}}{|\mathbf{r}|} + \frac{\mathbf{r}\mathbf{r}}{|\mathbf{r}|^3} \right)</math>
is a second-rank [[tensor]] (or more accurately [[tensor field]]) known as the [[Oseen tensor]] (after [[Carl Wilhelm Oseen]]).

The terms Stokeslet and point-force solution are used to describe <math>\mathbf{F}\cdot\mathbb{J}(\mathbf{r})</math>. Analogous to the point charge in [[Electrostatics]], the Stokeslet is force-free everywhere except at the origin, where it contains a force of strength <math>\mathbf{F}</math>.

For a continuous-force distribution (density) <math>\mathbf{f}(\mathbf{r})</math> the solution (again vanishing at infinity) can then be constructed by superposition:
:<math>\mathbf{u}(\mathbf{r}) = \int \mathbf{f}(\mathbf{r'}) \cdot \mathbb{J}(\mathbf{r} - \mathbf{r'}) \mathrm{d}\mathbf{r'}, \qquad
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'} </math>

This integral representation of the velocity can be viewed as a reduction in dimensionality: from the three-dimensional partial differential equation to a two-dimensional integral equation for unknown densities.<ref name=kim2005 />

====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 function|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.

===Stokes flow in particular geometries===

====Hele-Shaw flow====
[[Hele-Shaw flow]] is an example of a geometry for which inertia forces are negligible.  It is defined by two parallel plates arranged very close together with the space between the plates occupied partly by fluid and partly by obstacles in the form of cylinders with generators normal to the plates.<ref name=Batchelor />

====Slender-body theory====
[[Slender-body theory]] in Stokes flow is a simple approximate method  of determining the irrotational flow field around bodies whose length is large compared with their width.  The basis of the method is to choose a distribution of flow singularities along a line (since the body is slender) so that their irrotational flow in combination with a uniform stream approximately satisfies the zero normal velocity condition.<ref name=Batchelor/>

====Lamb's solution in spherical coordinates====
[[Horace Lamb|Lamb]]'s general solution arises from the fact that the pressure <math>p</math> satisfies the [[Laplace equation]], and can be expanded in a series of solid [[spherical harmonics]] in spherical coordinates.  As a result the solution to the Stokes equations can be written:

:<math> \begin{align} \mathbf{u} &= \sum_{n=-\infty, n\neq1}^{n=\infty} \left[ \frac{(n+3)r^2\nabla p_n}{2\mu(n+1)(2n+3)} - \frac{n\mathbf{x}p_n}{\mu(n+1)(2n+3)}\right] + \sum_{n=-\infty}^{n=\infty} [\nabla\Phi_n + \nabla \times (\mathbf{x}\chi_n)] \\
p &= \sum_{n=-\infty}^{n=\infty}p_n  \end{align} </math>

where <math>p_n, \Phi_n,</math> and <math>\chi_n</math> are solid spherical harmonics of order <math>n</math>:

:<math>\begin{align} p_n &= r^n \sum_{m=0}^{m=n} P_n^m(\cos\theta)(a_{mn}\cos m\phi +\tilde{a}_{mn} \sin m\phi) \\
\Phi_n &= r^n \sum_{m=0}^{m=n} P_n^m(\cos\theta)(b_{mn}\cos m\phi +\tilde{b}_{mn} \sin m\phi) \\
\chi_n &= r^n \sum_{m=0}^{m=n} P_n^m(\cos\theta)(c_{mn}\cos m\phi +\tilde{c}_{mn} \sin m\phi) \end{align}</math>

and the <math>P_n^m</math> are the [[associated Legendre polynomials]]. The Lamb's solution can be used to describe the motion of fluid either inside or outside a sphere.  For example, it can be used to describe the motion of fluid around a solid spherical particle, or to describe the ow inside a spherical drop of fluid.  For interior flows, the terms with <math>n<0</math> are dropped, while for exterior flows the terms with <math>n>0</math> are dropped (often the convention <math>n\to -n-1</math> is assumed for exterior flows to avoid indexing by negative numbers).<ref name=kim2005 />

===Theorems about Stokes flow===

====Stokes' Law (Stokes' Drag)====
The [[Stokes' law]] for the resistance to a moving sphere, also known as Stokes' drag, is a relation describing the drag force on a sphere exerted by the surrounding fluid in a Stokes flow.  Given a sphere of radius <math>a</math>, travelling at velocity <math>U</math>, in a Stokes fluid with dynamic viscosity <math>\mu</math>, the drag force <math>F_D</math> is given by:<ref name=Batchelor />

:<math> F_D = 6 \pi \mu a U </math>

====Minimum Energy Dissipation Theorem====
According to the Minimum Energy Dissipation theorem, the Stokes solution dissipates less energy than any other [[solenoidal vector field]] with the same boundary velocities.<ref name=kim2005/>

====Lorentz Reciprocal Theorem====
The Lorentz Reciprocal Theorem states a relationship between two Stokes flows in the same region.  Consider  fluid filled region <math>V</math> bounded by surface <math>S</math>.  Let the velocity fields <math>\mathbf{u}</math> and <math>\mathbf{u}' </math> solve the Stokes equations in the domain <math>V</math>, each with corresponding stress fields <math>\mathbf{\sigma}</math> and <math>\mathbf{\sigma}'</math>. Then the following equality holds:

:<math> \int_S \mathbf{u}\cdot (\mathbf{\sigma}' \cdot \mathbf{n}) dS = \int_S \mathbf{u}' \cdot (\mathbf{\sigma} \cdot \mathbf{n}) dS </math>

Where <math>\mathbf{n} </math> is the unit normal on the surface <math>S</math>.  The Lorentz Reciprocal theorem can be used to show that Stokes flow "transmits" unchanged the total force and torque from an inner closed surface to an outer enclosing surface.<ref name=kim2005 />  The Lorentz Reciprocal theorem can also be used to relate the swimming speed of a microorganism, such as [[cyanobacterium]], to the surface velocity which is prescribed by deformations of the body shape via [[cilia]] or [[flagella]].<ref name=Stone>{{cite journal|last=Stone|first=Howard A.|coauthors=Samuel, Aravinthan D. T.|title=Propulsion of Microorganisms by Surface Distorsions|journal=Physical Review Letters|date=November 1996|volume=77|series=19|pages=4102–4104}}</ref>

====Faxén's Laws====
The [[Faxén's law]]s are direct relations that express the [[multipole expansion|multipole]] moments in terms of the ambient flow and its derivatives.  First developed by [[Hilding Faxén]] to calculate the force, <math>\mathbf{F}</math>, and torque, <math>\mathbf{T}</math> on a sphere, they took the following form:

:<math>\begin{align} \mathbf{F} &= 6\pi\mu a \left( 1 + \frac{a^2}{6}\nabla^2 \right) \mathbf{v}^{\infty}(\mathbf{x})|_{x=0} - 6\pi\mu a \mathbf{U} \\ 
\mathbf{T} &= 8\pi\mu a^3(\mathbf{\Omega}^{\infty}(\mathbf{x}) - \mathbf{\omega})|_{x=0}  \end{align}</math>

where <math>\mu</math> is the dynamic viscosity, <math>a</math> is thee particle radius, <math>\mathbf{v}^{\infty}</math> is the ambient flow, <math>\mathbf{U}</math> is the speed of the particle, <math>\mathbf{\Omega}^{\infty}</math> is the angular velocity of the background flow, and <math>\mathbf{\omega}</math> is the angular velocity of the particle.

The Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops.<ref name=kim2005 />

== See also ==
{{columns-list|colwidth=25em|
* [[Stokes' law]]
* [[Darcy's law]]
* [[Hele-Shaw flow]]
* [[Laminar flow]]
* [[Lubrication theory]]
* [[Oseen equations]]
* [[Poiseuille]]
* [[Slender-body theory]]
* [[Volumetric flow rate]]
* [[Faxén's law]]
}}

== References ==
{{reflist}}
* Ockendon, H. & Ockendon J. R. (1995) ''Viscous Flow'', Cambridge University Press. ISBN 0-521-45881-1.

==External links==
*[http://www.youtube.com/watch?v=p08_KlTKP50&feature=related Video demonstration of time-reversibility of Stokes flow] by UNM Physics and Astronomy

[[Category:Fluid dynamics]]
[[Category:Equations of fluid dynamics]]