Stokes\' law - Citizendia

For a theorem in differential geometry, see Stokes' theorem. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from

Creeping flow past a sphere: streamlines, drag force F_d and force by gravity F_g.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force — also called drag force — exerted on spherical objects with very small Reynolds numbers (e. 1851 ( MDCCCLI) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian Calendar (or a Common year Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio g. , very small particles) in a continuous viscous fluid. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the generally unsolvable Navier–Stokes equations:^[1]

$F_d = 6 \pi\, \mu\, R\, V \,$

where:

F_d is the frictional force (in N),
μ is the fluid's dynamic viscosity (in Pa s),
R is the radius of the spherical object (in m), and
V is the particle's velocity (in m/s). Stokes flow (named after George Gabriel Stokes) is a type of Fluid flow where advective inertial forces are small compared with viscous The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International

If the particles are falling in the viscous fluid by their own weight due to gravity, then a terminal velocity, also known as the settling velocity, is reached when this frictional force combined with the buoyant force exactly balance the gravitational force. Earth's gravity, denoted by g, refers to the Gravitational attraction that the Earth exerts on objects on or near its surface A free falling object achieves its terminal velocity when the downward force of gravity ( Fg)equals the upward force of drag ( Fd) In Physics, buoyancy ( BrE IPA: /ˈbɔɪənsi/ is the upward Force on an object produced by the surrounding liquid or gas in which it is Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass The resulting settling velocity (or terminal velocity) is given by:^[2]

$V_s = \frac{2}{9}\frac{\left(\rho_p - \rho_f\right)}{\mu} g\, R^2$

where:

V_s is the particles' settling velocity (m/s) (vertically downwards if ρ_p > ρ_f, upwards if ρ_p < ρ_f ),
g is the gravitational acceleration (m/s²),
ρ_p is the mass density of the particles (kg/m³), and
ρ_f is the mass density of the fluid (kg/m³). Earth's gravity, denoted by g, refers to the Gravitational attraction that the Earth exerts on objects on or near its surface The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different

Note that for molecules Stokes' law is used to define their Stokes radius. In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by The Stokes radius, Stokes-Einstein radius, or Hydrodynamic radius R H, named after George Gabriel Stokes, is

1 Applications
2 Stokes flow around a sphere
3 See also
4 References
5 Notes

Applications

Stokes's law is the basis of the falling sphere viscometer,in which the fluid is stationary in a vertical glass tube. A viscometer (also called viscosimeter) is an instrument used to measure the Viscosity of a Fluid. 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. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. A series of steel ball bearings of different diameter is normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses glycerine as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. It includes many different oils, and polymer liquids such as solutions. An oil is a substance that is in a viscous Liquid state ( "oily") at ambient temperatures or slightly warmer and is A polymer is a large Molecule ( Macromolecule) composed of repeating Structural units typically connected by Covalent Chemical bonds

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.

The CGS unit of kinematic viscosity was named "stokes" after his work.

Stokes flow around a sphere

Steady Stokes flow

In Stokes flow, at very low Reynolds number, the convective acceleration terms in the Navier–Stokes equations are neglected. Stokes flow (named after George Gabriel Stokes) is a type of Fluid flow where advective inertial forces are small compared with viscous Advection, in mechanical and chemical engineering is a transport mechanism of a substance or a conserved property with a moving Fluid. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such Then the flow equations become, for an incompressible steady flow:^[3]

$\begin{align} &\nabla p = \mu\, \nabla^2 \mathbf{u} = - \mu\, \nabla \times \mathbf \boldsymbol{\omega}, \\ &\nabla \cdot \mathbf{u} = 0, \end{align}$

where:

p is the fluid pressure (in Pa),
u is the flow velocity (in m/s), and
ω is the vorticity (in s^-1), defined as $\boldsymbol{\omega}=\nabla\times\mathbf{u}.$

By using some vector calculus identities, these equations can be shown to result in Laplace's equations for the pressure and each of the components of the vorticity vector:^[3]

$\nabla^2 \boldsymbol{\omega}=0$ 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 of solutions and associated forces can be applied. In Fluid mechanics or more generally Continuum mechanics, an incompressible flow is Solid or Fluid flow in which the Divergence of Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Fluid pressure is the Pressure at some point within a Fluid, such as water or air In Fluid dynamics the flow velocity, or velocity field, of a fluid is a Vector field which is used to mathematically describe the motion of a fluid Vorticity is a mathematical concept used in Fluid dynamics. It can be related to the amount of " circulation " or "rotation" (or more strictly the The following identities are important in Vector calculus: Single operators (summary This section explicitly lists what some symbols mean for clarity In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems

Flow around a sphere

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 near field and far field of an antenna or other isolated source of Electromagnetic radiation are regions around the source where different parts of the field The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually 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 is at the sphere centre. In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference Because the flow is axisymmetric around the z–axis, it is independent of the azimuth φ. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Azimuth ( is a mathematical concept defined as the angle usually measured in degrees (° between a reference plane and a point.

In this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream function ψ, depending on r and z:^[4]^[5]

$v = +\frac{1}{r}\frac{\partial\psi}{\partial z}, \qquad w = -\frac{1}{r}\frac{\partial\psi}{\partial r},$

with v and w the flow velocity components in the r and z direction, respectively. In Fluid dynamics, the Stokes stream function is used to describe the Streamlines and Flow velocity in a three-dimensional Incompressible flow 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. ^[4]

For this case of an axisymmetric flow, the only non-zero of the vorticity vector ω is the azimuthal φ–component ω_φ^[6]^[7]

$\omega_\varphi = \frac{\partial v}{\partial z} - \frac{\partial w}{\partial r} = \frac{\partial}{\partial r} \left( \frac{1}{r}\frac{\partial\psi}{\partial r} \right) + \frac{1}{r}\, \frac{\partial^2\psi}{\partial z^2}.$

The Laplace operator, applied to the vorticity ω_φ, becomes in this cylindrical coordinate system with axisymmetry:^[7]

$\nabla^2 \omega_\varphi = \frac{1}{r}\frac{\partial}{\partial r}\left( r\, \frac{\partial\omega_\varphi}{\partial r} \right) + \frac{\partial^2 \omega_\varphi}{\partial z^2} = 0.$

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^[8]

$\psi = - \frac{1}{2}\, V\, r^2\, \left[ 1 - \frac{3}{2} \frac{R}{\sqrt{r^2+z^2}} + \frac{1}{2} \left( \frac{R}{\sqrt{r^2+z^2}} \right)^3\; \right].$

The viscous force per unit area σ, exerted by the flow on the surface on the sphere, is in the z–direction everywhere. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after More strikingly, it has also the same value everywhere on the sphere:^[1]

$\boldsymbol{\sigma} = \frac{3\, \mu\, V}{2\, R}\, \mathbf{e}_z$

with e_z the unit vector in the z–direction. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length 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_d according to Stokes' law.

Terminal velocity

At terminal velocity — or settling velocity — the frictional force F_d on the sphere is balanced by the excess force F_g due to the difference of the weight of the sphere and its buoyancy, both caused by gravity:^[2]

$F_g = \left( \rho_p - \rho_f \right)\, g\, \frac{4}{3}\pi\, R^3,$

with ρ_p and ρ_f the mass density of the sphere and the fluid, respectively, and g the gravitational acceleration. In the Physical sciences weight is a Measurement of the gravitational Force acting on an object In Physics, buoyancy ( BrE IPA: /ˈbɔɪənsi/ is the upward Force on an object produced by the surrounding liquid or gas in which it is Earth's gravity, denoted by g, refers to the Gravitational attraction that the Earth exerts on objects on or near its surface The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Earth's gravity, denoted by g, refers to the Gravitational attraction that the Earth exerts on objects on or near its surface Demanding force balance: F_d = F_g and solving for the velocity V gives the terminal velocity V_s.

References

Batchelor, G.K. (1967). Stokes flow (named after George Gabriel Stokes) is a type of Fluid flow where advective inertial forces are small compared with viscous The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such In Physics (namely in Kinetic theory) the Einstein relation (also known as Einstein–Smoluchowski relation) is a previously unexpected connection revealed This is a list of scientific laws named after people ( Eponymous laws) In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a The basis for determination of Molecular weight according to the Staudinger method (since replaced by the more general Mark -Houwink equation is the fact that George Keith Batchelor ( March 8 1920 - March 30 2000) was an Australian Applied mathematician and Fluid dynamicist An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0521663962.
Lamb, H. (1994). Sir Horace Lamb FRS ( 29 November 1849 – 4 December 1934) was a British Applied mathematician and author of several influential Hydrodynamics, 6^th edition, Cambridge University Press. ISBN 9780521458689. Originally published in 1879, the 6^th extended edition appeared first in 1932.

Notes

^ ^a ^b Batchelor (1967), p. 233.
^ ^a ^b Lamb (1994), §337, p. 599.
^ ^a ^b Batchelor (1967), section 4. 9, p. 229.
^ ^a ^b Batchelor (1967), section 2. 2, p. 78.
^ Lamb (1994), §94, p. 126.
^ Batchelor (1967), section 4. 9, p. 230
^ ^a ^b Batchelor (1967), appendix 2, p. 602.
^ Lamb (1994), §337, p. 598.

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Contents

Applications

Stokes flow around a sphere

Steady Stokes flow

Flow around a sphere

Terminal velocity

See also

References

Notes