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Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,Kim, S. & Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. . is a type of
fluid flow In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
where advective inertial forces are small compared with
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
forces. The
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
is low, i.e. \mathrm \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 Lubrication is the process or technique of using a lubricant to reduce friction and wear and tear in a contact between two surfaces. The study of lubrication is a discipline in the field of tribology. Lubrication mechanisms such as fluid-lubric ...
. In nature this type of flow occurs in the swimming of
microorganism A microorganism, or microbe,, ''mikros'', "small") and ''organism'' from the el, ὀργανισμός, ''organismós'', "organism"). It is usually written as a single word but is sometimes hyphenated (''micro-organism''), especially in olde ...
s,
sperm Sperm is the male reproductive cell, or gamete, in anisogamous forms of sexual reproduction (forms in which there is a larger, female reproductive cell and a smaller, male one). Animals produce motile sperm with a tail known as a flagellum, whi ...
and the flow of
lava Lava is molten or partially molten rock (magma) that has been expelled from the interior of a terrestrial planet (such as Earth) or a moon onto its surface. Lava may be erupted at a volcano or through a fracture in the crust, on land or un ...
. In technology, it occurs in
paint Paint is any pigmented liquid, liquefiable, or solid mastic composition that, after application to a substrate in a thin layer, converts to a solid film. It is most commonly used to protect, color, or provide texture. Paint can be made in many ...
,
MEMS Microelectromechanical systems (MEMS), also written as micro-electro-mechanical systems (or microelectronic and microelectromechanical systems) and the related micromechatronics and microsystems constitute the technology of microscopic devices, ...
devices, and in the flow of viscous
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
s generally. The equations of motion for Stokes flow, called the Stokes equations, are a
linearization In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, lineariz ...
of the Navier–Stokes equations, and thus can be solved by a number of well-known methods for linear differential equations. 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.Chwang, A. and Wu, T. (1974)
"Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows"
. ''J. Fluid Mech. 62''(6), part 4, 787–815.
The Stokeslet was first derived by Oseen in 1927, although it was not named as such until 1953 by Hancock. The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian and micropolar fluids.


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 comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations: :\boldsymbol \cdot \sigma + \mathbf = \boldsymbol where \sigma is the stress (sum of viscous and pressure stresses),Happel, J. & Brenner, H. (1981) ''Low Reynolds Number Hydrodynamics'', Springer. . and \mathbf an applied body force. The full Stokes equations also include an equation for the conservation of mass, commonly written in the form: : \frac + \nabla\cdot(\rho\mathbf) = 0 where \rho is the fluid density and \mathbf the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, \rho, is a constant. Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term \rho \frac is added to the left hand side of the momentum balance equation.


Properties

The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case. They are the
leading-order The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.math.ucdavis.edu/~ ...
simplification of the full Navier–Stokes equations, valid in the
distinguished limit In mathematics, a distinguished limit is an appropriately chosen scale factor used in the method of matched asymptotic expansions. External links Singular perturbation theory Scholarpedia ''Scholarpedia'' is an English-language wiki-based onli ...
\mathrm \to 0. ; 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 that 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 In the science of fluid flow, Stokes' paradox is the phenomenon that there can be no creeping flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial steady-state solution for the Stokes equations arou ...
: 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.


Demonstration of time-reversibility

A Taylor–Couette system can create laminar flows in which concentric cylinders of fluid move past each other in an apparent spiral. A fluid such as corn syrup with high viscosity fills the gap between two cylinders, with colored regions of the fluid visible through the transparent outer cylinder. The cylinders are rotated relative to one another at a low speed, which together with the high viscosity of the fluid and thinness of the gap gives a low
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
, so that the apparent mixing of colors is actually
laminar Laminar means "flat". Laminar may refer to: Terms in science and engineering: * Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules * Laminar armour or "band ...
and can then be reversed to approximately the initial state. This creates a dramatic demonstration of seemingly mixing a fluid and then unmixing it by reversing the direction of the mixer.


Incompressible flow of Newtonian fluids

In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form: : \begin \mu \nabla^2 \mathbf -\boldsymbolp + \mathbf &= \boldsymbol \\ \boldsymbol\cdot\mathbf&= 0 \end where \mathbf is the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of the fluid, \boldsymbol p is the gradient of the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, \mu is the dynamic viscosity, and \mathbf 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.


Cartesian coordinates

With the velocity vector expanded as \mathbf=(u,v,w) and similarly the body force vector \mathbf = (f_x, f_y, f_z) , we may write the vector equation explicitly, :\begin \mu \left(\frac + \frac + \frac\right) - \frac + f_x &= 0 \\ \mu \left(\frac + \frac + \frac\right) - \frac + f_y &= 0 \\ \mu \left(\frac + \frac + \frac\right) - \frac + f_z &= 0 \\ + + &= 0 \end We arrive at these equations by making the assumptions that \mathbb = \mu\left(\boldsymbol\mathbf + (\boldsymbol\mathbf)^\mathsf\right) - p\mathbb and the density \rho is a constant.


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


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(\mathbf), 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 \mu \nabla^2 \mathbf -\boldsymbolp &= -\mathbf\cdot\mathbf(\mathbf)\\ \boldsymbol\cdot\mathbf &= 0 \\ , \mathbf, , p &\to 0 \quad \mbox \quad r\to\infty \end where \mathbf(\mathbf) is the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, and \mathbf \cdot \delta(\mathbf) 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 : \mathbf(\mathbf) = \mathbf \cdot \mathbb(\mathbf), \qquad p(\mathbf) = \frac where :\mathbb(\mathbf) = \left( \frac + \frac \right) is a second-rank
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
(or more accurately
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
) known as the Oseen tensor (after
Carl Wilhelm Oseen Carl Wilhelm Oseen (17 April 1879 in Lund – 7 November 1944 in Uppsala) was a theoretical physicist in Uppsala and Director of the Nobel Institute for Theoretical Physics in Stockholm. Life Oseen was born in Lund, and took a Fil. Kand. degre ...
). Here, r r is a quantity such that \mathbf \cdot (\mathbf \mathbf) = (\mathbf \cdot \mathbf) \mathbf. The terms Stokeslet and point-force solution are used to describe \mathbf\cdot\mathbb(\mathbf). Analogous to the point charge in
electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
, the Stokeslet is force-free everywhere except at the origin, where it contains a force of strength \mathbf. For a continuous-force distribution (density) \mathbf(\mathbf) the solution (again vanishing at infinity) can then be constructed by superposition: : \mathbf(\mathbf) = \int \mathbf\left(\mathbf\right) \cdot \mathbb\left(\mathbf - \mathbf\right) \mathrm\mathbf, \qquad p(\mathbf) = \int \frac \, \mathrm\mathbf 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.


By Papkovich–Neuber solution

The
Papkovich–Neuber solution The Papkovich–Neuber solution is a technique for generating analytic solutions to the Newtonian incompressible Stokes equations, though it was originally developed to solve the equations of linear elasticity. It can be shown that any Stokes ...
represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
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.


Some geometries


Hele-Shaw flow

Hele-Shaw flow Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap, named after Henry Selby Hele-Shaw, who studied the problem in 1898. Various problems in fluid mechanics can be approximated to Hele ...
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.


Slender-body theory

Slender-body theory In fluid dynamics and electrostatics, slender-body theory is a methodology that can be used to take advantage of the slenderness of a body to obtain an approximation to a field surrounding it and/or the net effect of the field on the body. Principal ...
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.


Spherical coordinates

Lamb's general solution arises from the fact that the pressure p satisfies the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
, and can be expanded in a series of solid
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...
in spherical coordinates. As a result, the solution to the Stokes equations can be written: : \begin \mathbf &= \sum_^ \left \frac - \frac\right+...\\ \sum_^ nabla\Phi_n + \nabla \times (\mathbf\chi_n)\\ p &= \sum_^p_n \end where p_n, \Phi_n, and \chi_n are solid spherical harmonics of order n: :\begin p_n &= r^n \sum_^ P_n^m(\cos\theta)(a_\cos m\phi +\tilde_ \sin m\phi) \\ \Phi_n &= r^n \sum_^ P_n^m(\cos\theta)(b_\cos m\phi +\tilde_ \sin m\phi) \\ \chi_n &= r^n \sum_^ P_n^m(\cos\theta)(c_\cos m\phi +\tilde_ \sin m\phi) \end and the P_n^m are the
associated Legendre polynomials In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently ...
. 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 spherical particle with prescribed surface flow, a so-called
squirmer The squirmer is a model for a spherical microswimmer swimming in Stokes flow. The squirmer model was introduced by James Lighthill in 1952 and refined and used to model Paramecium by John Blake in 1971. Blake used the squirmer model to describe ...
, 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 -n-1 is assumed for exterior flows to avoid indexing by negative numbers).


Theorems


Stokes solution and related Helmholtz theorem

The drag resistance to a moving sphere, also known as Stokes' solution is here summarised. 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: : F_D = 6 \pi \mu a U The Stokes solution dissipates less energy than any other solenoidal vector field with the same boundary velocities: this is known as the
Helmholtz minimum dissipation theorem In fluid mechanics, Helmholtz minimum dissipation theorem (named after Hermann von Helmholtz who published it in 1868) states that ''the steady Stokes flow motion of an incompressible fluid has the smallest rate of dissipation than any other incomp ...
.


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 and \mathbf' solve the Stokes equations in the domain V, each with corresponding stress fields \mathbf and \mathbf'. Then the following equality holds: : \int_S \mathbf\cdot (\boldsymbol' \cdot \mathbf) dS = \int_S \mathbf' \cdot (\boldsymbol \cdot \mathbf) dS Where \mathbf 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. 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 The cilium, plural cilia (), is a membrane-bound organelle found on most types of eukaryotic cell, and certain microorganisms known as ciliates. Cilia are absent in bacteria and archaea. The cilium has the shape of a slender threadlike projecti ...
or
flagella A flagellum (; ) is a hairlike appendage that protrudes from certain plant and animal sperm cells, and from a wide range of microorganisms to provide motility. Many protists with flagella are termed as flagellates. A microorganism may have f ...
.


Faxén's laws

Faxén's laws are direct relations that express the
multipole A multipole expansion is a Series (mathematics), mathematical series representing a Function (mathematics), function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and Azimuth, azimuthal angles) f ...
moments in terms of the ambient flow and its derivatives. First developed by
Hilding Faxén Olov Hilding Faxén (29 March 1892 – 1 June 1970) was a Swedish physicist who was primarily active within mechanics. Faxén received his doctorate in 1921 at Uppsala University with the thesis ''Einwirkung der Gefässwände auf den Widerstan ...
to calculate the force, \mathbf, and torque, \mathbf on a sphere, they take the following form: :\begin \mathbf &= 6\pi\mu a \left( 1 + \frac\nabla^2 \right) \mathbf^\infty(\mathbf), _ - 6\pi\mu a \mathbf \\ \mathbf &= 8\pi\mu a^3(\mathbf^\infty(\mathbf) - \mathbf), _ \end where \mu is the dynamic viscosity, a is the particle radius, \mathbf^ is the ambient flow, \mathbf is the speed of the particle, \mathbf^ is the angular velocity of the background flow, and \mathbf is the angular velocity of the particle. Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops.


See also


References

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


External links


Video demonstration of time-reversibility of Stokes flow
by UNM Physics and Astronomy Fluid dynamics Equations of fluid dynamics