Hydrodynamic Interaction in Stokes Flows

In the following, physical background is explained.

Fluid is described by Navier-Stokes equation

$$\frac{\partial}{\partial t} \Bigl[ \rho \bm{u} + \left( \bm{u} \cdot \bm{\nabla} \right) \bm{u} \Bigr] = - \bm{\nabla} p + \mu \Delta \bm{u} ,$$ (3.2)

where $\bm{u}$ is the fluid velocity, $p$ is the pressure, $\rho$ is the density, and $\mu$ is the viscosity. The differential operator

$$\bm{\nabla} = \left[ \begin{array}{c} \partial/\partial x\\ \partial/\partial y\\ \partial/\partial z \end{array} \right] ,$$ (3.3)

and $\Delta$ is the Laplacian

$$\Delta = \bm{\nabla} \cdot \bm{\nabla} = \partial^2/\partial x^2 + \partial^2/\partial y^2 + \partial^2/\partial z^2 .$$ (3.4)

This is the fluid equation of motion, in other words, the momentum balance equation and the counterpart of Newton's equation of motion $\bm{F} = m\bm{a}$. Actually, writing the viscous fluid stress by

$$\bm{\sigma} = p \bm{I} + \bm{\nabla}\bm{u} + \Bigl( \bm{\nabla}\bm{u} \Bigr)^\dagger ,$$ (3.5)

Navier-Stokes equation (3.2) can be written by the similar equation of motion

$$\frac{D}{D t} \Bigl[ \rho \bm{u} \Bigr] = - \bm{\nabla} \cdot \bm{\sigma} .$$ (3.6)

The Stokes flows, which is the target of libstokes, is where the viscous term in Eq. (3.2) is dominating to the inertial term. The ratio of the magnitude of these two terms are characterized by a dimensionless number called ``Reynolds number'' $Re$

$$Re := \frac{LU}{\mu} ,$$ (3.7)

where $L$ and $U$ are characteristic length and velocity. The Stokes flows are the flow of small Reynolds number.

In the limit $Re\rightarrow 0$, the governing equation of fluids becomes

$$\bm{0} = - \bm{\nabla} p + \mu \Delta \bm{u} ,$$ (3.8)

which is a linear partial differential equation. In the following, we are studying fluid motions governed by Eq. (3.8) for incompressible fluid

$$\bm{\nabla} \cdot \bm{u} = 0 .$$ (3.9)