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Stokes flow (named after
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University of Cambridge, where he was the Luc ...
), 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
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
l 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 in ...
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 dom ...
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 old ...
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, ...
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 ...
. 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, linea ...
of the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
, and thus can be solved by a number of well-known methods for linear differential equations. The primary
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
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 In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
s 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 solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
s 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 In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. 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 Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
(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 In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
, 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 In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
, especially in the incompressible Newtonian case. They are the leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit \mathrm \to 0. ; Instantaneity :A Stokes flow has no dependence on time other than through time-dependent
boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
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 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 fluid A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for ex ...
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.


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 dom ...
, so that the apparent mixing of colors is actually laminar 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 A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
, 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 i ...
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 a ...
, \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 The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. T ...
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 In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
, \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 enti ...
, 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 tensor ...
(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. degree ( ...
). 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 a ...
, 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 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'', t ...
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 The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, el ...
. This technique can be applied to both 2- and 3-dimensional flows.


Some 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.


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.


Spherical coordinates

Lamb Lamb or The Lamb may refer to: * A young sheep * Lamb and mutton, the meat of sheep Arts and media Film, television, and theatre * ''The Lamb'' (1915 film), a silent film starring Douglas Fairbanks Sr. in his screen debut * ''The Lamb'' (1918 ...
'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 = \n ...
, 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 ...
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, 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 In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf ...
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 Cyanobacteria (), also known as Cyanophyta, are a phylum of gram-negative bacteria that obtain energy via photosynthesis. The name ''cyanobacteria'' refers to their color (), which similarly forms the basis of cyanobacteria's common name, blu ...
, 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 proje ...
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 fro ...
.


Faxén's laws

Faxén's law In fluid dynamics, Faxén's laws relate a sphere's velocity \mathbf and angular velocity \mathbf to the forces, torque, stresslet and flow it experiences under low Reynolds number (creeping flow) conditions. First law Faxen's first law was introduc ...
s are direct relations that express the
multipole A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
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 Widerst ...
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