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| {{about|the expression for frictional force|the acoustics theorem|Stokes' law (sound attenuation)}}
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| {{distinguish|Stokes' theorem}}
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| {{lead too long|date=January 2013}}
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| In 1851, [[George Gabriel Stokes]] derived an expression, now known as '''Stokes' law''', for the frictional force – also called [[drag force]] – exerted on [[sphere|spherical]] objects with very small [[Reynolds number]]s (e.g., very small particles) in a continuous [[viscosity|viscous]] [[fluid]]. Stokes' law is derived by solving the [[Stokes flow]] limit for small Reynolds numbers of the [[Navier–Stokes equations]]:<ref name=Batch233>Batchelor (1967), p. 233.</ref>
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| :<math>F_d = 6 \pi\,\mu\,R\,v\,</math>
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| where ''F<sub>d</sub>'' is the frictional force – known as '''Stokes' drag''' – acting on the interface between the fluid and the particle (in [[Newton (unit)|N]]),''μ'' is the [[dynamic viscosity]] (kg /m*s),''R'' is the radius of the spherical object (in m), and ''v'' is the particle's velocity (in m/s).
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| Stokes' law makes the following assumptions for the behavior of a particle in a fluid:
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| :*[[Laminar Flow]]
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| :*[[sphere|Spherical]] particles
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| :*Homogeneous (uniform in composition) material
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| :*Smooth surfaces
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| :*Particles do not interfere with each other.
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| Note that for [[molecule]]s Stokes' law is used to define their [[Stokes radius]].
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| It is interesting to note the CGS unit of kinematic viscosity was named "stokes" after his work.
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| ==Applications==
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| Stokes's law is the basis of the falling-sphere [[viscometer]], 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]] 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]] 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]]s, and [[polymer]] liquids such as solutions.
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| The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least 3 Nobel Prizes.<ref>Dusenbery, David B. (2009). ''Living at Micro Scale'', p.49. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.</ref>
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| Stokes' law is important to understanding the swimming of [[microorganism]]s and [[sperm]]; also, the [[sedimentation]], under the force of gravity, of small particles and organisms, in water.<ref>Dusenbery, David B. (2009). ''Living at Micro Scale''. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.</ref>
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| 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 settlement of fine particles in water or other fluids.
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| ===Terminal Velocity of Sphere Falling in a Fluid===
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| [[Image:Stokes sphere.svg|thumb|right|175px|Creeping flow past a falling sphere in a fluid: [[Streamlines, streaklines, and pathlines|streamlines]], drag force ''F''<sub>d</sub> and force by gravity ''F''<sub>g</sub>.]] At terminal (or settling) velocity, the excess force ''F<sub>g</sub>'' due to the difference of the [[weight]] of the sphere and the [[buoyancy]] on the sphere, (both caused by [[Earth's gravity|gravity]]:<ref name=Lamb599/>)
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| :<math>F_g = \left( \rho_p - \rho_f \right)\, g\, \frac{4}{3}\pi\, R^3,</math>
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| with ''ρ<sub>p</sub>'' and ''ρ<sub>f</sub>'' the [[mass density]] of the sphere and the fluid, respectively, and ''g'' the [[Earth's gravity|gravitational acceleration]]. Demanding force balance: ''F<sub>d</sub>'' = ''F<sub>g</sub>'' and solving for the velocity ''V'' gives the terminal velocity ''V<sub>s</sub>''. Note that since buoyant force increases as ''R<sup>3</sup>'' and Stokes drag increases as ''R'', the terminal velocity increases as ''R<sup>2</sup>'' and thus varies greatly with particle size as shown below. If the particle is falling in the viscous fluid under its own weight due to [[Earth's gravity|gravity]], then a [[terminal velocity]], or settling velocity, is reached when this frictional force combined with the [[buoyant force]] exactly balances the [[gravitational force]]. The resulting terminal velocity (or settling velocity) is given by:<ref name=Lamb599>Lamb (1994), §337, p. 599.</ref> | |
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| :<math>v_s = \frac{2}{9}\frac{\left(\rho_p - \rho_f\right)}{\mu} g\, R^2</math>
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| where ''v<sub>s</sub>'' is the particle's settling velocity (m/s) (vertically downwards if ''ρ<sub>p</sub>'' > ''ρ<sub>f</sub>'', upwards if ''ρ<sub>p</sub>'' < ''ρ<sub>f</sub>'' ), ''g'' is the [[Earth's gravity|gravitational acceleration]] (m/s<sup>2</sup>), ''ρ<sub>p</sub>'' is the [[mass density]] of the particles (kg/m<sup>3</sup>), and ''ρ<sub>f</sub>'' is the mass density of the fluid (kg/m<sup>3</sup>).
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| ===Steady Stokes flow===
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| In [[Stokes flow]], at very low [[Reynold number|Reynolds number]], the [[advection|convective acceleration]] terms in the [[Navier–Stokes equations]] are neglected. Then the flow equations become, for an [[incompressible flow|incompressible]] [[steady flow]]:<ref name=Batch229>Batchelor (1967), section 4.9, p. 229.</ref>
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| :<math>
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| \begin{align}
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| &\nabla p = \eta\, \nabla^2 \mathbf{u} = - \eta\, \nabla \times \mathbf{\boldsymbol{\omega}},
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| \\
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| &\nabla \cdot \mathbf{u} = 0,
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| \end{align}
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| </math>
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| where:
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| * ''p'' is the [[fluid pressure]] (in Pa),
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| * '''u''' is the [[flow velocity]] (in m/s), and
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| * '''''ω''''' is the [[vorticity]] (in s<sup>-1</sup>), defined as <math>\boldsymbol{\omega}=\nabla\times\mathbf{u}.</math>
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| By using some [[vector calculus identities]], these equations can be shown to result in [[Laplace's equation]]s for the pressure and each of the components of the vorticity vector:<ref name=Batch229/>
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| :<math>\nabla^2 \boldsymbol{\omega}=0</math> and <math>\nabla^2 p = 0.</math>
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| 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]] of solutions and associated forces can be applied.
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| ===Flow around a sphere===
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| For the case of a sphere in a uniform [[far field]] flow, it is advantageous to use a [[cylindrical coordinate system]] ( ''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 (mathematics)|origin]] is at the sphere centre. Because the flow is [[axisymmetric]] around the ''z''–axis, it is independent of the [[azimuth]] ''φ''.
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| In this cylindrical coordinate system, the incompressible flow can be described with a [[Stokes stream function]] ''ψ'', depending on ''r'' and ''z'':<ref name=Batch78>Batchelor (1967), section 2.2, p. 78.</ref><ref>Lamb (1994), §94, p. 126.</ref>
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| :<math>
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| v = -\frac{1}{r}\frac{\partial\psi}{\partial z},
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| \qquad
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| w = \frac{1}{r}\frac{\partial\psi}{\partial r},
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| </math> | |
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| with ''v'' and ''w'' 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.<ref name=Batch78/>
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| For this case of an axisymmetric flow, the only non-zero component of the vorticity vector '''''ω''''' is the azimuthal ''φ''–component ''ω<sub>φ</sub>''<ref name=Batch230>Batchelor (1967), section 4.9, p. 230</ref><ref name=Batch602>Batchelor (1967), appendix 2, p. 602.</ref> | |
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| :<math>
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| \omega_\varphi = \frac{\partial v}{\partial z} - \frac{\partial w}{\partial r}
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| = - \frac{\partial}{\partial r} \left( \frac{1}{r}\frac{\partial\psi}{\partial r} \right) - \frac{1}{r}\, \frac{\partial^2\psi}{\partial z^2}.
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| </math>
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| The [[Laplace operator]], applied to the vorticity ''ω<sub>φ</sub>'', becomes in this cylindrical coordinate system with axisymmetry:<ref name=Batch602/>
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| :<math>\nabla^2 \omega_\varphi = \frac{1}{r}\frac{\partial}{\partial r}\left( r\, \frac{\partial\omega_\varphi}{\partial r} \right) + \frac{\partial^2 \omega_\varphi}{\partial z^2} - \frac{\omega_\varphi}{r^{2}} = 0.</math>
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| From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity ''V'' in the ''z''–direction and a sphere of radius ''R'', the solution is found to be<ref>Lamb (1994), §337, p. 598.</ref>
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| :<math>
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| \psi = - \frac{1}{2}\, V\, r^2\, \left[
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| 1
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| - \frac{3}{2} \frac{R}{\sqrt{r^2+z^2}}
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| + \frac{1}{2} \left( \frac{R}{\sqrt{r^2+z^2}} \right)^3\;
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| \right].
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| </math>
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| The viscous force per unit area '''σ''', exerted by the flow on the surface on the sphere, is in the ''z''–direction everywhere. More strikingly, it has also the same value everywhere on the sphere:<ref name=Batch233/>
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| :<math>\boldsymbol{\sigma} = \frac{3\, \eta\, V}{2\, R}\, \mathbf{e}_z</math>
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| with '''e'''<sub>''z''</sub> the [[unit vector]] in the ''z''–direction. For other shapes than spherical, '''σ''' is not constant along the body surface. Integration of the viscous force per unit area '''σ''' over the sphere surface gives the frictional force ''F<sub>d</sub>'' according to Stokes' law.
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| ==Other types of Stokes flow== | |
| {{Other uses-section|steady Stokes flow around a sphere|the forces on a sphere in unsteady Stokes flow|Basset–Boussinesq–Oseen equation}}
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| ==See also==
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| * [[Stokes flow]]
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| * [[Einstein relation (kinetic theory)]]
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| * [[Scientific laws named after people]]
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| * [[Drag equation]]
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| * [[Viscometry]]
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| * [[Equivalent spherical diameter]]
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| * [[Deposition (geology)]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book | first=G.K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 }}
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| *{{cite book | first=H. | last=Lamb | authorlink=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th edition| isbn=978-0-521-45868-9 }} Originally published in 1879, the 6th extended edition appeared first in 1932.
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| [[Category:Fluid dynamics]]
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