Stokes' law

 ( Fluid Dynamics ) 

$F\_e\; =\; (\; \backslash rho\_p\; -\; \backslash rho\_f)\backslash ,\; g\backslash ,\; \backslash frac\{4\}\{3\}\backslash pi\backslash ,\; R^3,$

where (in SI units):

• is the mass density of the sphere kg/m³
• is the mass density of the fluid kg/m³
• is the gravitational acceleration m/s

$$v\; =\; \backslash frac\{2\}\{9\}\backslash frac\{\; \backslash rho\_p\; -\; \backslash rho\_f\}\; \backslash mu\; g\backslash ,\; R^2\; \backslash quad$$

where (in SI units):

• is the gravitational field strength m/s²
• is the radius of the spherical particle m
• is the mass density of the particle kg/m³
• is the mass density of the fluid kg/m³
• is the dynamic viscosity kg/(m•s).

$\backslash begin\{align\}$

 &\nabla p = \mu\, \nabla^2 \mathbf{u} = - \mu\, \nabla \times \mathbf{ \boldsymbol{\omega} }, \\[2pt]
 &\nabla \cdot \mathbf{u} = 0,
     

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

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:

$\backslash nabla^2\; \backslash boldsymbol\{\backslash omega\}=0$   and   $\backslash nabla^2\; p\; =\; 0.$

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 this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream function , depending on and :Batchelor (1967), section 2.2, p. 78.Lamb (1994), §94, p. 126.

 u_z = \frac{1}{r}\frac{\partial\psi}{\partial r},
 \qquad
 u_r = -\frac{1}{r}\frac{\partial\psi}{\partial z},
     

with and the flow velocity components in the and direction, respectively. The azimuthal velocity component in the –direction is equal to zero, in this axisymmetric case. The volume flux, through a tube bounded by a surface of some constant value , is equal to and is constant.

For this case of an axisymmetric flow, the only non-zero component of the vorticity vector is the azimuthal –component Batchelor (1967), section 4.9, p. 230Batchelor (1967), appendix 2, p. 602.

 \omega_\varphi = \frac{\partial u_r}{\partial z} - \frac{\partial u_z}{\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 , becomes in this cylindrical coordinate system with axisymmetry:

$\backslash nabla^2\; \backslash omega\_\backslash varphi\; =\; \backslash frac\{1\}\{r\}\backslash frac\{\backslash partial\}\{\backslash partial\; r\}\backslash left(\; r\backslash ,\; \backslash frac\{\backslash partial\backslash omega\_\backslash varphi\}\{\backslash partial\; r\}\; \backslash right)\; +\; \backslash frac\{\backslash partial^2\; \backslash omega\_\backslash varphi\}\{\backslash partial\; z^2\}\; -\; \backslash frac\{\backslash omega\_\backslash varphi\}\{r^\{2\}\}\; =\; 0.$

From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity in the –direction and a sphere of radius , the solution is found to beLamb (1994), §337, p. 598.

 \psi(r,z) = - \frac{1}{2}\, u\, 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 solution of velocity in cylindrical coordinates and components follows as:

$\backslash begin\{align\}$

 u_r(r, z) &= \frac{3R r z u}{4 \sqrt{r^2 + z^2}} \left( \left( \frac{R}{r^2+z^2} \right)^2 - \frac{1}{r^2+z^2} \right) \\[4pt]
     

 u_z(r, z) &= u + \frac{3Ru}{4 \sqrt{r^2 + z^2}} \left(
 \frac{2 R^2 + 3 r^2}{3 (r^2 + z^2)}
 -\left(\frac{r R}{r^2 + z^2}\right)^2
 - 2
 \right)
     

\end{align}

The solution of vorticity in cylindrical coordinates follows as:

$\backslash omega\_\backslash varphi(r,\; z)\; =\; -\; \backslash frac\{3\; Ru\}\{2\}\; \backslash cdot\; \backslash frac\{r\}\{\backslash sqrt\{r^2+z^2\}^3\}$

The solution of pressure in cylindrical coordinates follows as:

$p(r,\; z)\; =\; -\; \backslash frac\{3\; \backslash mu\; R\; u\}\{2\}\; \backslash cdot\; \backslash frac\{z\}\{\backslash sqrt\{r^2+z^2\}^3\}$

The solution of pressure in spherical coordinates follows as:

 p(r, \theta) = - \frac{3\mu R u}{2} \cdot \frac{\cos\theta}{r^2}
     

The formula of pressure is also called dipole potential analogous to the concept in electrostatics.

A more general formulation, with arbitrary far-field velocity-vector $\backslash mathbf\{u\}\_\{\backslash infty\}$, in cartesian coordinates $\backslash mathbf\{x\}=\; (x,\; y,\; z)^T$ follows with:

$\backslash boldsymbol\{\backslash omega\}(\backslash mathbf\{x\})\; =\; -\; \backslash frac\{3R\}\{2\}\; \backslash cdot\; \backslash frac\{\backslash mathbf\{u\}\_\{\backslash infty\}\backslash times\; \backslash mathbf\{x\}\}\{\backslash |\backslash mathbf\{x\}\backslash |^3\}$

$p\backslash left(\backslash mathbf\{x\}\backslash right)=\; -\; \backslash frac\{3\backslash mu\; R\}\{2\}\; \backslash cdot\; \backslash frac\{\backslash mathbf\{u\}\_\{\backslash infty\}\; \backslash cdot\; \backslash mathbf\{x\}\}\{\backslash |\backslash mathbf\{x\}\backslash |^3\}$

In this formulation the non-conservative term represents a kind of so-called Stokes flow. The Stokeslet is the Green's function 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 electromagnetism.

Alternatively, in a more compact way, one can formulate the velocity field as follows:

$\backslash mathbf\{u\}(\backslash mathbf\{x\})$

where $\backslash mathrm\{H\}\; =\; \backslash nabla\backslash otimes\backslash nabla$ is the Hessian matrix differential operator and $\backslash mathrm\{S\}\; =\; \backslash mathbf\{I\}\; \backslash nabla^2\; -\; \backslash mathrm\{H\}$ is a differential operator composed as the difference of the Laplacian and the Hessian. In this way it becomes explicitly clear, that the solution is composed from derivatives of a Coulomb potential ($1/\backslash |\backslash mathbf\{x\}\backslash |$) and a Biharmonic potential ($\backslash |\backslash mathbf\{x\}\backslash |$). The differential operator $\backslash mathrm\{S\}$ applied to the vector norm $\backslash |\backslash mathbf\{x\}\backslash |$ generates the Stokeslet.

The following formula describes the viscous stress tensor 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 $\backslash nabla\; \backslash mathbf\{u\}$ is identical to the Jacobian matrix. The matrix represents the identity matrix.

$\backslash boldsymbol\{\backslash sigma\}\; =\; -\; p\; \backslash cdot\; \backslash mathbf\{I\}\; +\; \backslash mu\; \backslash cdot\; \backslash left((\backslash nabla\; \backslash mathbf\{u\})\; +\; (\backslash nabla\; \backslash mathbf\{u\})^T\; \backslash right)$

The force acting on the sphere can be calculated via the integral of the stress tensor over the surface of the sphere, where represents the radial unit-vector of spherical-coordinates:

$\backslash begin\{align\}$

          &=  \int_{0}^{\pi}\int_{0}^{2\pi} \boldsymbol{\sigma}\cdot \mathbf{e_r}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\[4pt]
          &=  \int_{0}^{\pi}\int_{0}^{2\pi}  \frac{3\mu \cdot \mathbf{u}_{\infty}}{2 R}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\[4pt]
          &= 6\pi\mu R \cdot \mathbf{u}_{\infty}
     

$\backslash begin\{align\}$

          &= \int_{0}^{\pi} \int_{0}^{2\pi}  (R \cdot \mathbf{e_r}) \times \left( \boldsymbol{\sigma} \cdot \mathbf{e_r} \cdot R^2 \sin\theta \text{d}\varphi \text{d}\theta \right) \\
          &= 8\pi\mu R^3  \cdot \boldsymbol{\omega}_{R}
     

• Einstein relation (kinetic theory)
• Scientific laws named after people
• Ballistics
• Drag equation
• Viscometry
• Equivalent spherical diameter
• Deposition (geology)
• Stokes number – A determinant of the additional effect of turbulence on terminal fall velocity for particles in fluids

• G.K. Batchelor (1967). 9780521663960, Cambridge University Press. ISBN 9780521663960
• H. Lamb (1994). 9780521458689, Cambridge University Press. ISBN 9780521458689
  Originally published in 1879, the 6th extended edition appeared first in 1932.