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 sperm^{[3]} and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
The equations of motion for Stokes flow, called the Stokes Equations, are a linearization of the NavierStokes Equations, and thus can be solved by a number of wellknown methods for linear differential equations.^{[4]} 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.^{[5]}
The fundamental solution due to a point force in a steady Stokes flow was first derived by the Nobel Laureate, Lorentz, as far back as 1896. This solution is now known by the name Stokeslet, although Stokes never knew about it. The name Stokeslet was coined by Hancock in 1953. The closedform fundamental solutions for generalized unsteady Stokes and Oseen flows associated with arbitrary timedependent translational and rotational motions have been derived for Newtonian^{[6]} and micropolar^{[7]} fluids.
Stokes equations
The equation of motion for Stokes flow can be obtained by linearizing the steady state NavierStokes Equations. The inertial forces are assumed to be negligible compared with the viscous forces, which reduces the momentum balance in the Navier–Stokes equations to the momentum balance in the Stokes equations:^{[1]}

\boldsymbol{\nabla} \cdot \mathbb{P} + \mathbf{f} = 0
where \scriptstyle \mathbb{P} is the Cauchy stress tensor representing viscous and pressure stresses,^{[8]}^{[9]} and \scriptstyle \mathbf{f} an applied body force. The full Stokes equations also include an equation for the conservation of mass, commonly written in the form:

\frac{d\rho}{dt} + \rho\nabla\cdot\mathbf{u} = 0
Where \scriptstyle \rho is the fluid density and \scriptstyle \mathbf{u} the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, \scriptstyle \rho, is a constant.
Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term \scriptstyle \rho \frac{\partial\mathbf{u}}{\partial t} is added to the left hand side of the momentum balance equation.^{[1]}
Properties
The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case.^{[2]}^{[4]}^{[8]}^{[9]} They are the leadingorder simplification of the full Navier–Stokes equations, valid in the distinguished limit Re \to 0.

Instantaneity

A Stokes flow has no dependence on time other than through timedependent 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.

Timereversibility

An immediate consequence of instantaneity, timereversibility means that a timereversed 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 nonlinear and sometimes timedependent nature of nonNewtonian 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 nontrivial solution for the Stokes equations around an infinitely long cylinder.^{[10]}
Demonstration of timereversibility
A Taylor–Couette system can create spiral laminar flows.^{[11]} 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.^{[12]}^{[13]}^{[14]}
Incompressible flow of Newtonian fluids
In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form:

\begin{align} \mu \nabla^2 \mathbf{u} \boldsymbol{\nabla}p + \mathbf{f} &= 0 \\ \boldsymbol{\nabla}\cdot\mathbf{u}&=0 \end{align}
Where \scriptstyle \mathbf{u} is the velocity of the fluid, \scriptstyle \boldsymbol{\nabla} p is the gradient of the pressure, \scriptstyle \mu is the dynamic viscosity, and \scriptstyle \mathbf{f} 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.^{[4]}
Cartesian coordinates
With the velocity vector expanded as \scriptstyle \mathbf{u}=(u,v,w) and similarly the body force vector \scriptstyle \mathbf{f} = (f_x, f_y, f_z) , we may write the vector equation explicitly,

\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}
We arrive at theses equations by making the assumptions that \scriptstyle \mathbb{P} =\frac{1}{2}(\boldsymbol{\nabla}\mathbf{u} + (\boldsymbol{\nabla}\mathbf{u})^T)  p\mathbb{I} and the density \rho is a constant.^{[8]}
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 3D axisymmetric cases
Type of function

Geometry

Equation

Comments

Stream function (\psi)

2D planar

\nabla^4 \psi = 0 or \Delta^2 \psi = 0 (biharmonic equation)

\Delta is the Laplacian operator in two dimensions

Stokes stream function (\Psi)

3D 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)

3D 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 ^{[15]}

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}), 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:

\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}
where \mathbf{\delta}(\mathbf{r}) is the Dirac delta function, and \mathbf{F}\cdot\delta(\mathbf{r}) 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^{[1]}

\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}
where

\mathbb{J}(\mathbf{r}) = {1 \over 8 \pi \mu} \left( \frac{\mathbb{I}}{\mathbf{r}} + \frac{\mathbf{r}\mathbf{r}}{\mathbf{r}^3} \right)
is a secondrank tensor (or more accurately tensor field) known as the Oseen tensor (after Carl Wilhelm Oseen).
The terms Stokeslet and pointforce solution are used to describe \mathbf{F}\cdot\mathbb{J}(\mathbf{r}). Analogous to the point charge in Electrostatics, the Stokeslet is forcefree everywhere except at the origin, where it contains a force of strength \mathbf{F}.
For a continuousforce distribution (density) \mathbf{f}(\mathbf{r}) the solution (again vanishing at infinity) can then be constructed by superposition:

\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'}
This integral representation of the velocity can be viewed as a reduction in dimensionality: from the threedimensional partial differential equation to a twodimensional integral equation for unknown densities.^{[1]}
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 3dimensional flows.
Stokes flow in particular geometries
HeleShaw flow
HeleShaw 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.^{[8]}
Slenderbody theory
Slenderbody 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.^{[8]}
Lamb's solution in spherical coordinates
Lamb's general solution arises from the fact that the pressure p 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:

\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}
where p_n, \Phi_n, and \chi_n are solid spherical harmonics of order n:

\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}
and the P_n^m 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 flow inside a spherical drop of fluid. For interior flows, the terms with n<0 are dropped, while for exterior flows the terms with n>0 are dropped (often the convention n\to n1 is assumed for exterior flows to avoid indexing by negative numbers).^{[1]}
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 a, travelling at velocity U, in a Stokes fluid with dynamic viscosity \mu, the drag force F_D is given by:^{[8]}

F_D = 6 \pi \mu a U
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.^{[1]}
Lorentz Reciprocal Theorem
The Lorentz Reciprocal Theorem states a relationship between two Stokes flows in the same region. Consider fluid filled region V bounded by surface S. Let the velocity fields \mathbf{u} and \mathbf{u}' solve the Stokes equations in the domain V, each with corresponding stress fields \mathbf{\sigma} and \mathbf{\sigma}'. Then the following equality holds:

\int_S \mathbf{u}\cdot (\mathbf{\sigma}' \cdot \mathbf{n}) dS = \int_S \mathbf{u}' \cdot (\mathbf{\sigma} \cdot \mathbf{n}) dS
Where \mathbf{n} is the unit normal on the surface S. 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.^{[1]} 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.^{[16]}
Faxén's Laws
The Faxén's laws are direct relations that express the multipole moments in terms of the ambient flow and its derivatives. First developed by Hilding Faxén to calculate the force, \mathbf{F}, and torque, \mathbf{T} on a sphere, they took the following form:

\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}
where \mu is the dynamic viscosity, a is thee particle radius, \mathbf{v}^{\infty} is the ambient flow, \mathbf{U} is the speed of the particle, \mathbf{\Omega}^{\infty} is the angular velocity of the background flow, and \mathbf{\omega} 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.^{[1]}
See also
References

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Kim, S. & Karrila, S. J. (2005) Microhydrodynamics: Principles and Selected Applications, Dover. ISBN 0486442195.

^ ^{a} ^{b} Kirby, B.J. (2010). Micro and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices.. Cambridge University Press.

^ Dusenbery, David B. (2009). Living at Micro Scale. Harvard University Press, Cambridge, Mass. ISBN 9780674031166.

^ ^{a} ^{b} ^{c}

^ Chwang, A. and Wu, T. (1974). "Hydromechanics of lowReynoldsnumber flow. Part 2. Singularity method for Stokes flows". J. Fluid Mech. 62(6), part 4, 787–815.

^ Shu, JianJun; Chwang, A.T. (2001). "Generalized fundamental solutions for unsteady viscous flows". Physical Review E 63 (5): 051201.

^ Shu, JianJun; Lee, J.S. (2008). "Fundamental solutions for micropolar fluids". Journal of Engineering Mathematics 61 (1): 69–79.

^ ^{}a ^{b} ^{c} ^{d} ^{e} ^{f}

^ ^{}a ^{b} Happel, J. & Brenner, H. (1981) Low Reynolds Number Hydrodynamics, Springer. ISBN 9001371159.

^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 602–604.

^ 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

^ Dusenbery, David B. (2009). Living at Micro Scale, pp.46. Harvard University Press, Cambridge, Mass. ISBN 9780674031166.

^ http://www.youtube.com/watch?v=p08_KlTKP50&feature=related

^ http://panda.unm.edu/flash/viscosity.phtml

^ Payne, LE; WH Pell (1960). "The Stokes flow problem for a class of axially symmetric bodies". Journal of Fluid Mechanics 7 (04): 529–549.

^ Stone, Howard A.; Samuel, Aravinthan D. T. (November 1996). "Propulsion of Microorganisms by Surface Distorsions". Physical Review Letters. 19 77: 4102–4104.

Ockendon, H. & Ockendon J. R. (1995) Viscous Flow, Cambridge University Press. ISBN 0521458811.
External links

Video demonstration of timereversibility of Stokes flow by UNM Physics and Astronomy
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