Stokes flow

 ( Fluid Dynamics ) 

Instantaneity

A Stokes flow has no dependence on time other than through time-dependent boundary conditions. This means that, given the boundary conditions of a Stokes flow, the flow can be found without knowledge of the flow at any other time.

Time-reversibility

An immediate consequence of instantaneity, time-reversibility means that a time-reversed Stokes flow solves the same equations as the original Stokes flow. This property can sometimes be used (in conjunction with linearity and symmetry in the boundary conditions) to derive results about a flow without solving it fully. Time reversibility means that it is difficult to mix two fluids using creeping flow.

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.

\boldsymbol{\nabla}\cdot\mathbf{u}&= 0 \end{align}
     

where $\backslash mathbf\{u\}$ is the velocity of the fluid, $\backslash boldsymbol\{\backslash nabla\}\; p$ is the gradient of the pressure, $\backslash mu$ is the dynamic viscosity, and $\backslash mathbf\{f\}$ an applied body force. The resulting equations are linear in velocity and pressure, and therefore can take advantage of a variety of linear differential equation solvers.

$\backslash begin\{align\}$

 \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) - \frac{\partial p}{\partial x} + f_x &= 0 \\
 \mu \left(\frac{\partial^2 v}{\partial x^2}  + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) - \frac{\partial p}{\partial y} + f_y &= 0 \\
 \mu \left(\frac{\partial^2 w}{\partial x^2}  + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) - \frac{\partial p}{\partial z} + f_z &= 0 \\
 {\partial u \over \partial x} + {\partial v \over \partial y} + {\partial w \over \partial z} &= 0
     

We arrive at these equations by making the assumptions that $\backslash mathbb\{P\}\; =\; \backslash mu\backslash left(\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\}\; +\; (\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\})^\backslash mathsf\{T\}\backslash right)\; -\; p\backslash mathbb\{I\}$ and the density $\backslash rho$ is a constant.

$\backslash begin\{align\}$

\mu \nabla^2 \mathbf{u} -\boldsymbol{\nabla}p &= -\mathbf{F}\cdot\mathbf{\delta}(\mathbf{r})\\
           \boldsymbol{\nabla}\cdot\mathbf{u} &= 0 \\
                              |\mathbf{u}|, p &\to 0 \quad \mbox{as} \quad r\to\infty
     

where $\backslash mathbf\{\backslash delta\}(\backslash mathbf\{r\})$ is the Dirac delta function, and $\backslash mathbf\{F\}\; \backslash cdot\; \backslash delta(\backslash mathbf\{r\})$ represents a point force acting at the origin. The solution for the pressure p and velocity u with | u| and p vanishing at infinity is given by

 \mathbf{u}(\mathbf{r}) = \mathbf{F} \cdot \mathbb{J}(\mathbf{r}), \qquad
          p(\mathbf{r}) = \frac{\mathbf{F}\cdot\mathbf{r}}{4 \pi |\mathbf{r}|^3}
     

$\backslash mathbb\{J\}(\backslash mathbf\{r\})\; =\; \{1\; \backslash over\; 8\backslash pi\backslash mu\}\; \backslash left(\; \backslash frac\{\backslash mathbb\{I\}\}$ + \frac{\mathbf{r}\mathbf{r}}{|\mathbf{r}|^3} \right)

The terms Stokeslet and point-force solution are used to describe $\backslash mathbf\{F\}\backslash cdot\backslash mathbb\{J\}(\backslash mathbf\{r\})$. Analogous to the point charge in electrostatics, the Stokeslet is force-free everywhere except at the origin, where it contains a force of strength $\backslash mathbf\{F\}$.

For a continuous-force distribution (density) $\backslash mathbf\{f\}(\backslash mathbf\{r\})$ the solution (again vanishing at infinity) can then be constructed by superposition:

 \mathbf{u}(\mathbf{r}) = \int \mathbf{f}\left(\mathbf{r'}\right) \cdot \mathbb{J}\left(\mathbf{r} - \mathbf{r'}\right) \mathrm{d}\mathbf{r'}, \qquad
 p(\mathbf{r}) = \int \frac{\mathbf{f}\left(\mathbf{r'}\right)\cdot\left(\mathbf{r} - \mathbf{r'}\right)}{4 \pi \left|\mathbf{r} - \mathbf{r'}\right|^3} \, \mathrm{d}\mathbf{r'}
     

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.

 \mathbf{F} &= 6\pi\mu a \left( 1 + \frac{a^2}{6}\nabla^2 \right) \mathbf{v}^\infty(\mathbf{x})|_{x=0} - 6\pi\mu a \mathbf{U} \\
 \mathbf{T} &= 8\pi\mu a^3(\mathbf{\Omega}^\infty(\mathbf{x}) - \mathbf{\omega})|_{x=0}