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Next: Hydrodynamic Interaction Parameters Up: stokes3: Stokesian Dynamics Simulator Previous: Output Parameters

System Parameters

Currently, the applied force $ \bm{F}_{\text{ext}}$ and torque $ \bm{T}_{\text{ext}}$ are given by the single vector, respectively. (That is, all particles have the same force and torque.) ** This should be extended soon. **

The imposed flow $ \bm{u}^{\infty}$ is given by three parameters $ \bm{U}^{\infty}$, $ \bm{\Omega}^{\infty}$, and $ \bm{E}^{\infty}$ as

$\displaystyle \bm{u}^{\infty}(\bm{x}) = \bm{U}^{\infty} + \bm{\Omega}^{\infty}\times\bm{x} + \bm{E}\cdot\bm{x} .$ (2.1)

Therefore, this is a linear flow and the gradient tensor $ \bm{\nabla}\bm{u}^{\infty}$ is given by

$\displaystyle \bm{\nabla}\bm{u}^{\infty} = \left[ \begin{array}{ccc} E^{\infty}...
...}-\Omega^{\infty}_{x} & 1-E^{\infty}_{xx}-E^{\infty}_{yy} \end{array} \right] .$ (2.2)

The vector (or list) Oi has three elements

$\displaystyle {\tt Oi} = \left(\Omega^{\infty}_{x}, \Omega^{\infty}_{y}, \Omega^{\infty}_{z}\right) ,$ (2.3)

and the vector (or list) Ei has five elements

$\displaystyle {\tt Ei} = \left( E^{\infty}_{xx}, E^{\infty}_{xy}, E^{\infty}_{xz}, E^{\infty}_{yz}, E^{\infty}_{yy} \right) .$ (2.4)

Note the order of the element. From these five elements, all nine elements of $ \bm{E}^{\infty}$ is recovered, using the properties

$\displaystyle E^{\infty}_{ij} = E^{\infty}_{ji},\quad E^{\infty}_{ii} = 0.$ (2.5)

For example, the simple shear flow in $ xz$-plane (velocity direction in $ x$ and vorticity direction in $ y$) is given by

$\displaystyle \Omega^{\infty}_{y} = \frac{1}{2}\dot{\gamma},\quad E^{\infty}_{xz} = \frac{1}{2}\dot{\gamma},$ (2.6)

and zero for others. See the gradient

$\displaystyle \bm{\nabla}\bm{u}^{\infty} = \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \dot{\gamma} & 0 & 0 \end{array} \right] .$ (2.7)

This is given by parameters Oi and Ei as
$\displaystyle {\tt Oi}$ $\displaystyle =$ $\displaystyle \left(0, \dot{\gamma}/2, 0\right),$ (2.8)
$\displaystyle {\tt Ei}$ $\displaystyle =$ $\displaystyle \left(
0,
0,
\dot{\gamma}/2,
0,
0
\right)
.$ (2.9)

The planar extension in $ xz$-plane with the gradient

$\displaystyle \bm{\nabla}\bm{u}^{\infty} = \left[ \begin{array}{ccc} \dot{\epsilon} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -\dot{\epsilon} \end{array} \right] ,$ (2.10)

is given by

$\displaystyle E^{\infty}_{xx} = -E^{\infty}_{zz} = \dot{\epsilon},$ (2.11)

and zero for others. Note that $ E^{\infty}_{yy} = 1-E^{\infty}_{xx}-E^{\infty}_{zz} = 0$. Therefore, it is given by
$\displaystyle {\tt Oi}$ $\displaystyle =$ $\displaystyle \left(0, 0, 0\right),$ (2.12)
$\displaystyle {\tt Ei}$ $\displaystyle =$ $\displaystyle \left(
\dot{\epsilon},
0,
0,
0,
0
\right)
.$ (2.13)


next up previous contents
Next: Hydrodynamic Interaction Parameters Up: stokes3: Stokesian Dynamics Simulator Previous: Output Parameters
Kengo Ichiki 2008-10-12