Stoke's Law
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In 1851,
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish migration to Great Britain, Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University ...
derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
objects with very small
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 ...
s in a
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 ...
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers 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 ...
.Batchelor (1967), p. 233.


Statement of the law

The force of viscosity on a small sphere moving through a viscous fluid is given by: :F_ = 6 \pi \mu R v where: * ''F''d is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle * ''μ'' is the dynamic
viscosity 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 inte ...
(some authors use the symbol ''η'') * ''R'' is the radius of the spherical object * ''v'' is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
relative to the object. In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, ''F''d is given in
newtons The newton (symbol: N) is the unit of force in the International System of Units (SI). It is defined as 1 kg⋅m/s, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It is named after Isaac Newton in r ...
(= kg m s−2), ''μ'' in Pa·s (= kg m−1 s−1), ''R'' in meters, and ''v'' in m/s. Stokes' law makes the following assumptions for the behavior of a particle in a fluid: *
Laminar flow In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
*
Spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
particles *Homogeneous (uniform in composition) material *Smooth surfaces *Particles do not interfere with each other. For
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s Stokes' law is used to define their Stokes radius and diameter. The CGS unit of kinematic viscosity was named "stokes" after his work.


Applications

Stokes' law is the basis of the falling-sphere
viscometer A viscometer (also called viscosimeter) is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a spe ...
, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches terminal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid, Stokes' law can be used to calculate the
viscosity 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 inte ...
of the fluid. A series of steel ball bearings of different diameters are normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses
glycerine Glycerol (), also called glycerine in British English and glycerin in American English, is a simple triol compound. It is a colorless, odorless, viscous liquid that is sweet-tasting and non-toxic. The glycerol backbone is found in lipids known ...
or golden syrup as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. Several school experiments often involve varying the temperature and/or concentration of the substances used in order to demonstrate the effects this has on the viscosity. Industrial methods include many different
oil An oil is any nonpolar chemical substance that is composed primarily of hydrocarbons and is hydrophobic (does not mix with water) & lipophilic (mixes with other oils). Oils are usually flammable and surface active. Most oils are unsaturated ...
s, and
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 ...
liquids such as solutions. The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least three Nobel Prizes. Stokes' law is important for understanding 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 and
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 ...
; also, the
sedimentation Sedimentation is the deposition of sediments. It takes place when particles in suspension settle out of the fluid in which they are entrained and come to rest against a barrier. This is due to their motion through the fluid in response to the ...
of small particles and organisms in water, under the force of gravity. In air, the same theory can be used to explain why small water droplets (or ice crystals) can remain suspended in air (as clouds) until they grow to a critical size and start falling as rain (or snow and hail). Similar use of the equation can be made in the settling of fine particles in water or other fluids.


Terminal velocity of sphere falling in a fluid

At terminal (or settling) velocity, the excess force ''Fg'' due to the difference between the
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
and
buoyancy Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the p ...
of the sphere (both caused by
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
) is given by: :F_g = \left( \rho_p - \rho_ \right)\, g\, \frac\pi\, R^3, with ''ρp'' and ''ρf'' the mass densities of the sphere and fluid, respectively, and ''g'' the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
. Requiring the force balance ''F''d = ''Fg'' and solving for the velocity ''v'' gives the terminal velocity ''vs''. Note that since the excess force increases as ''R3'' and Stokes' drag increases as ''R'', the terminal velocity increases as ''R2'' and thus varies greatly with particle size as shown below. If a particle only experiences its own weight while falling in a viscous fluid, then a terminal velocity is reached when the sum of the frictional and the buoyant forces on the particle due to the fluid exactly balances the
gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
. This velocity ''v'' (m/s) is given by:Lamb (1994), §337, p. 599. :v = \frac\frac g\, R^2 (vertically downwards if ''ρp'' > ''ρ''f'','' upwards if ''ρp'' < ''ρ''f ), where: * ''g'' is the gravitational field strength (m/s2) * R is the radius of the spherical particle (m) * ''ρp'' is the mass density of the particle (kg/m3) * ''ρ''f is the mass density of the fluid (kg/m3) * ''μ'' is the
dynamic viscosity 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 ...
(kg/(m*s)).


Derivation


Steady Stokes flow

In Stokes flow, at very 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 ...
, the
convective acceleration Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convecti ...
terms in 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 ...
are neglected. Then the flow equations become, for an
incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
steady 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) and ...
:Batchelor (1967), section 4.9, p. 229. : \begin &\nabla p = \mu\, \nabla^2 \mathbf = - \mu\, \nabla \times \mathbf, \\ &\nabla \cdot \mathbf = 0, \end where: * ''p'' is the
fluid 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 ...
(in Pa), * u is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
(in m/s), and * ''ω'' is the
vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...
(in s−1), defined as  \boldsymbol=\nabla\times\mathbf. By using some
vector calculus identities The following are important identities involving derivatives and integrals in vector calculus. Operator notation Gradient For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: \o ...
, these equations can be shown to result in
Laplace's 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 ...
s for the pressure and each of the components of the vorticity vector: :\nabla^2 \boldsymbol=0   and   \nabla^2 p = 0. Additional forces like those by gravity and buoyancy have not been taken into account, but can easily be added since the above equations are linear, so
linear superposition The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
of solutions and associated forces can be applied.


Transversal flow around a sphere

For the case of a sphere in a uniform
far field The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ant ...
flow, it is advantageous to use a
cylindrical coordinate system A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
(''r'', ''φ'', ''z''). The ''z''–axis is through the centre of the sphere and aligned with the mean flow direction, while ''r'' is the radius as measured perpendicular to the ''z''–axis. The
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
is at the sphere centre. Because the flow is axisymmetric around the ''z''–axis, it is independent of the
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...
''φ''. In this cylindrical coordinate system, the incompressible flow can be described with a
Stokes stream function In fluid dynamics, the Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry. A surface with a constant value of the Stokes stream function encloses a streamtube, ...
''ψ'', depending on ''r'' and ''z'':Batchelor (1967), section 2.2, p. 78. : u_z = \frac\frac, \qquad u_r = -\frac\frac, with ''ur'' and ''uz'' the flow velocity components in the ''r'' and ''z'' direction, respectively. The azimuthal velocity component in the ''φ''–direction is equal to zero, in this axisymmetric case. The volume flux, through a tube bounded by a surface of some constant value ''ψ'', is equal to ''2π ψ'' and is constant. For this case of an axisymmetric flow, the only non-zero component of the vorticity vector ''ω'' is the azimuthal ''φ''–component ''ωφ''Batchelor (1967), section 4.9, p. 230Batchelor (1967), appendix 2, p. 602. : \omega_\varphi = \frac - \frac = - \frac \left( \frac\frac \right) - \frac\, \frac. The
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, applied to the vorticity ''ωφ'', becomes in this cylindrical coordinate system with axisymmetry: :\nabla^2 \omega_\varphi = \frac\frac\left( r\, \frac \right) + \frac - \frac = 0. From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity ''u'' in the ''z''–direction and a sphere of radius ''R'', the solution is found to beLamb (1994), §337, p. 598. : \psi(r,z) = - \frac\, u\, r^2\, \left 1 - \frac \frac + \frac \left( \frac \right)^3\; \right The solution of velocity in
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
and components follows as: : u_r(r, z) = \frac \cdot \frac - \frac \cdot\frac : u_z(r, z) = \frac\cdot \left(\frac - \frac \right) + u - \frac\cdot \left( \frac + \frac \right) The solution of vorticity in cylindrical coordinates follows as: :\omega_\varphi(r, z) = - \frac \cdot \frac The solution of pressure in cylindrical coordinates follows as: :p(r, z) = - \frac \cdot \frac The solution of pressure in
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
follows as: : p(r, \theta) = - \frac \cdot \frac The formula of pressure is also called ''
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
potential'' analogous to the concept in electrostatics. A more general formulation, with arbitrary far-field velocity-vector \mathbf_, in
cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
\mathbf= (x, y, z)^T follows with: \begin \mathbf(\mathbf) &= \underbrace_ \; \underbrace_ \\ pt&= \left \frac \frac - \frac \frac - \frac \frac - \frac \frac + \mathbb \rightcdot \mathbf_ \end :\boldsymbol(\mathbf) = - \frac \cdot \frac :p\left(\mathbf\right)= - \frac \cdot \frac In this formulation the non-conservative term represents a kind of so-called Stokeslet. The Stokeslet is the
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 differential ...
of the Stokes-Flow-Equations. The conservative term is equal to the dipole gradient field. The formula of vorticity is analogous to the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
. The following formula describes the
viscous stress tensor The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress ...
for the special case of Stokes flow. It is needed in the calculation of the force acting on the particle. In Cartesian coordinates the vector-gradient \nabla \mathbf is identical to the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
. The matrix \mathbf represents the identity-matrix. :\boldsymbol = - p \cdot \mathbf + \mu \cdot \left((\nabla \mathbf) + (\nabla \mathbf)^T \right) The force acting on the sphere is to calculate by surface-integral, where \mathbf represents the radial unit-vector of spherical-coordinates: :\mathbf = \iint_\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \;\boldsymbol\cdot \text\mathbf = \int_^\int_^ \boldsymbol\cdot \mathbf\cdot R^2 \sin\theta \text\varphi\text\theta = \int_^\int_^ \frac\cdot R^2 \sin\theta \text\varphi\text\theta = 6\pi\mu R \cdot \mathbf_


Rotational flow around a sphere

:\mathbf(\mathbf) = - \;R^3 \cdot \frac :\boldsymbol(\mathbf) = \frac - \frac :p(\mathbf) = 0 :\boldsymbol = - p \cdot \mathbf + \mu \cdot \left( (\nabla \mathbf) + (\nabla \mathbf)^T \right) :\mathbf = \iint_\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \mathbf \times \left( \boldsymbol \cdot \text\boldsymbol \right) = \int_^ \int_^ (R \cdot \mathbf) \times \left( \boldsymbol \cdot \mathbf \cdot R^2 \sin\theta \text\varphi \text\theta \right) = 8\pi\mu R^3 \cdot \boldsymbol_


Other types of Stokes flow

Although the liquid is static and the sphere is moving with a certain velocity, with respect to the frame of sphere, the sphere is at rest and liquid is flowing in the opposite direction to the motion of the sphere.


See also

*
Einstein relation (kinetic theory) In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their works on ...
*
Scientific laws named after people This is a list of scientific laws named after people ( eponymous laws). For other lists of eponyms, see eponym. See also * Eponym * Fields of science * List of eponymous laws (overlaps with this list but includes non-scientific laws such as ...
*
Drag equation In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is: F_\, =\, \tfrac12\, \rho\, u^2\, c_\, A where *F_ is the drag force ...
*
Viscometry A viscometer (also called viscosimeter) is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a spe ...
*
Equivalent spherical diameter The equivalent spherical diameter of an irregularly shaped object is the diameter of a sphere of equivalent geometric, optical, electrical, aerodynamic or hydrodynamic behavior to that of the particle under investigation. The particle size of a pe ...
*
Deposition (geology) Deposition is the geological process in which sediments, soil and rocks are added to a landform or landmass. Wind, ice, water, and gravity transport previously weathered surface material, which, at the loss of enough kinetic energy in the fluid, ...


Sources

* * Originally published in 1879, the 6th extended edition appeared first in 1932.


References

{{Reflist Fluid dynamics