Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,[1] is a type of fluid flow where advective inertial forces are small compared with viscous forces.[2] The Reynolds number is low, i.e.
Re\ll1
The equations of motion for Stokes flow, called the Stokes equations, are a linearization of the Navier–Stokes equations, and thus can be solved by a number of well-known methods for linear differential equations.[4] 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.[5] The Stokeslet was first derived by Oseen in 1927, although it was not named as such until 1953 by Hancock.[6] 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[7] and micropolar[8] fluids.
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{\nabla} ⋅ \sigma+f=\boldsymbol{0}
where
\sigma
f
| \partial\rho | |
| \partialt |
+\nabla ⋅ (\rhou)=0
where
\rho
u
\rho
Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term
\rho
| \partialu | |
| \partialt |
Re\to0.
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.
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.[11]
A Taylor–Couette system can create laminar flows in which concentric cylinders of fluid move past each other in an apparent spiral.[12] 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, 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.[13] [14] [15]
In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form:
\begin{align}\mu\nabla2u-\boldsymbol{\nabla}p+f&=\boldsymbol{0}\\ \boldsymbol{\nabla} ⋅ u&=0\end{align}
where
u
\boldsymbol{\nabla}p
\mu
f
With the velocity vector expanded as
u=(u,v,w)
f=(fx,fy,fz)
\begin{align} \mu\left(
| \partial2u | |
| \partialx2 |
+
| \partial2u | |
| \partialy2 |
+
| \partial2u | |
| \partialz2 |
\right)-
| \partialp | |
| \partialx |
+fx&=0\\ \mu\left(
| \partial2v | |
| \partialx2 |
+
| \partial2v | |
| \partialy2 |
+
| \partial2v | |
| \partialz2 |
\right)-
| \partialp | |
| \partialy |
+fy&=0\\ \mu\left(
| \partial2w | |
| \partialx2 |
+
| \partial2w | |
| \partialy2 |
+
| \partial2w | |
| \partialz2 |
\right)-
| \partialp | |
| \partialz |
+fz&=0\\ {\partialu\over\partialx}+{\partialv\over\partialy}+{\partialw\over\partialz}&=0 \end{align}
We arrive at these equations by making the assumptions that
P=\mu\left(\boldsymbol{\nabla}u+(\boldsymbol{\nabla}u)T\right)-pI
\rho
The equation for an incompressible Newtonian Stokes flow can be solved by the stream function method in planar or in 3-D axisymmetric cases
| Type of function | Geometry | Equation | Comments | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Stream function, \psi | 2-D planar | \nabla4\psi=0 \Delta2\psi=0 | \Delta | ||||||||||||||||||
| Stokes stream function, \Psi | 3-D spherical | E2\Psi=0, E={\partial2\over\partialr2}+{\sin{\theta}\overr2}{\partial\over\partial\theta}\left({1\over\sin{\theta}}{\partial\over\partial\theta}\right) | For derivation of the E | ||||||||||||||||||
| 3-D cylindrical |
\Psi=0, L-1=
+
-
| For L-1 |
The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function,
J(r)
\begin{align} \mu\nabla2u-\boldsymbol{\nabla}p&=-F ⋅ \delta(r)\\ \boldsymbol{\nabla} ⋅ u&=0\ |u|,p&\to0 as r\toinfty \end{align}
where
\delta(r)
F ⋅ \delta(r)
u(r)=F ⋅ J(r), p(r)=
| F ⋅ r | |
| 4\pi|r|3 |
J(r)={1\over8\pi\mu}\left(
| I | |
| |r| |
+
| rr | |
| |r|3 |
\right)
F ⋅ (rr)=(F ⋅ r)r
The terms Stokeslet and point-force solution are used to describe
F ⋅ J(r)
F
For a continuous-force distribution (density)
f(r)
u(r)=\intf\left(r'\right) ⋅ J\left(r-r'\right)dr', p(r)=\int
| f\left(r'\right) ⋅ \left(r-r'\right) | |
| 4\pi\left|r-r'\right|3 |
dr'
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.
The Papkovich–Neuber solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic potentials.
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.
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 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.[9]
Lamb's general solution arises from the fact that the pressure
p
\begin{align}u&=
| n=infty | |
| \sum | |
| n=-infty,n ≠ -1 |
\left[
| (n+3)r2\nablapn | |
| 2\mu(n+1)(2n+3) |
-
| nxpn | |
| \mu(n+1)(2n+3) |
\right]
| n=infty | |
| +...\ \sum | |
| n=-infty |
[\nabla\Phin+\nabla x (x\chin)]\\ p&=
| n=infty | |
| \sum | |
| n=-infty |
pn\end{align}
where
pn,\Phin,
\chin
n
\begin{align}pn&=rn
| m=n | |
| \sum | |
| m=0 |
| m(\cos\theta)(a | |
| P | |
| mn |
\cosm\phi+\tilde{a}mn\sinm\phi)\\ \Phin&=rn
| m=n | |
| \sum | |
| m=0 |
| m(\cos\theta)(b | |
| P | |
| mn |
\cosm\phi+\tilde{b}mn\sinm\phi)\\ \chin&=rn
| m=n | |
| \sum | |
| m=0 |
| m(\cos\theta)(c | |
| P | |
| mn |
\cosm\phi+\tilde{c}mn\sinm\phi)\end{align}
and the
| m | |
| P | |
| n |
n<0
n>0
n\to-n-1
See also: Stokes' law. The drag resistance to a moving sphere, also known as Stokes' solution is here summarised. Given a sphere of radius
a
U
\mu
FD
FD=6\pi\muaU
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.[1]
The Lorentz reciprocal theorem states a relationship between two Stokes flows in the same region. Consider fluid filled region
V
S
u
u'
V
\sigma
\sigma'
\intSu ⋅ (\boldsymbol{\sigma}' ⋅ n)dS=\intSu' ⋅ (\boldsymbol{\sigma} ⋅ n)dS
Where
n
S
Faxén's laws are direct relations that express the multipole moments in terms of the ambient flow and its derivatives. First developed by Hilding Faxén to calculate the force,
F
T
\begin{align} F&=6\pi\mua\left(1+
| a2 | |
| 6 |
\nabla2\right)
| infty(x)| | |
| v | |
| x=0 |
-6\pi\muaU\ T&=8\pi\mua3(\Omegainfty(x)-\omega)|x=0\end{align}
where
\mu
a
vinfty
U
\Omegainfty
\omega
Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops.
. Leal, L. G. . L. Gary Leal . Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes . 2007.
. Batchelor, G. K.. George Batchelor . Introduction to Fluid Mechanics . 2000 . Cambridge University Press . 978-0-521-66396-0.