System Parameters

• np : (int) number of all particles (both mobile and fixed)
• nm : (int) number of mobile particles
• x : (list or vector of doubles) initial positions of all particles
• a : (list or vector of doubles) radius of particles (list or vector with length np) by default (if not given), monodisperse system
• slip : (list or vector of doubles) slip length of particles (list or vector with length np) by default (if not given), no-slip particles
• Ui : (list or vector of doubles) imposed translational velocity $\bm{u}^\infty$
• Oi : (list or vector of doubles) imposed angular velocity $\bm{\Omega}^\infty$
• Ei : (list or vector of doubles) imposed strain velocity $\bm{E}^\infty$
• shear-mode : (int)
  0 imposed flow is given by Ui, Oi, Ei (default)
  1 for simple shear (x = flow dir, y = grad dir)
  2 for simple shear (x = flow dir, z = grad dir)
  NOTE: (Ui,Oi,Ei) is overwritten for the case ${\tt shear-mode} \ne 0$
• shear-rate : (double) shear rate for the case ${\tt shear-mode} \ne 0$
• shear-shift : the initial cell-shift for ${\tt shear-mode} \ne 0$
• F0 : (list or vector of doubles) applied force $\bm{F}_{\text{ext}}$
• T0 : (list or vector of doubles) applied force $\bm{T}_{\text{ext}}$
• stokes : (doubles) effective Stokes number $\widetilde{\text{St}}$

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

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

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

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

The vector (or list) Oi has three elements

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

and the vector (or list) Ei has five elements

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

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

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

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

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

and zero for others. See the gradient

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

This is given by parameters Oi and Ei as

$${\tt Oi} = \left(0, \dot{\gamma}/2, 0\right),$$ (2.8)

$${\tt Ei} = \left( 0, 0, \dot{\gamma}/2, 0, 0 \right) .$$ (2.9)

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

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

is given by

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

and zero for others. Note that $E^{\infty}_{yy} = 1-E^{\infty}_{xx}-E^{\infty}_{zz} = 0$. Therefore, it is given by

$${\tt Oi} = \left(0, 0, 0\right),$$ (2.12)

$${\tt Ei} = \left( \dot{\epsilon}, 0, 0, 0, 0 \right) .$$ (2.13)