| 180 |
|
and oblate : |
| 181 |
|
\[ |
| 182 |
|
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 183 |
< |
}}{a} |
| 184 |
< |
\], |
| 183 |
> |
}}{a}. |
| 184 |
> |
\] |
| 185 |
|
one can write down the translational and rotational resistance |
| 186 |
|
tensors |
| 187 |
|
\[ |
| 231 |
|
\label{introEquation:tensorExpression} |
| 232 |
|
\end{equation} |
| 233 |
|
This equation is the basis for deriving the hydrodynamic tensor. In |
| 234 |
< |
1930, Oseen and Burgers gave a simple solution to Equation |
| 235 |
< |
\ref{introEquation:tensorExpression} |
| 234 |
> |
1930, Oseen and Burgers gave a simple solution to |
| 235 |
> |
Eq.~\ref{introEquation:tensorExpression} |
| 236 |
|
\begin{equation} |
| 237 |
|
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 238 |
|
R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} |
| 248 |
|
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 249 |
|
\label{introEquation:RPTensorNonOverlapped} |
| 250 |
|
\end{equation} |
| 251 |
< |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 252 |
< |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 253 |
< |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 251 |
> |
Both of the Eq.~\ref{introEquation:oseenTensor} and |
| 252 |
> |
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption |
| 253 |
> |
$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for |
| 254 |
|
overlapping beads with the same radius, $\sigma$, is given by |
| 255 |
|
\begin{equation} |
| 256 |
|
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 258 |
|
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 259 |
|
\label{introEquation:RPTensorOverlapped} |
| 260 |
|
\end{equation} |
| 261 |
– |
|
| 261 |
|
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 262 |
|
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 263 |
|
$B_{ij}$ blocks |
| 275 |
|
\] |
| 276 |
|
where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
| 277 |
|
matrix, we obtain |
| 279 |
– |
|
| 278 |
|
\[ |
| 279 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 280 |
|
{C_{11} } & \ldots & {C_{1N} } \\ |
| 281 |
|
\vdots & \ddots & \vdots \\ |
| 282 |
|
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 283 |
< |
\end{array}} \right) |
| 283 |
> |
\end{array}} \right), |
| 284 |
|
\] |
| 285 |
< |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 285 |
> |
which can be partitioned into $N \times N$ $3 \times 3$ block |
| 286 |
|
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 287 |
|
\[ |
| 288 |
|
U_i = \left( {\begin{array}{*{20}c} |
| 294 |
|
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 295 |
|
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 296 |
|
arbitrary origin $O$ can be written as |
| 297 |
< |
\begin{equation} |
| 298 |
< |
\begin{array}{l} |
| 301 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 297 |
> |
\begin{eqnarray} |
| 298 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 299 |
|
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 300 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 304 |
< |
\end{array} |
| 300 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 301 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 302 |
< |
\end{equation} |
| 302 |
> |
\end{eqnarray} |
| 303 |
|
|
| 304 |
|
The resistance tensor depends on the origin to which they refer. The |
| 305 |
|
proper location for applying friction force is the center of |
| 335 |
|
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 336 |
|
\end{array}} \right) |
| 337 |
|
\] |
| 338 |
< |
Using Equations \ref{introEquation:definitionCR} and |
| 339 |
< |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 340 |
< |
the position of center of resistance, |
| 338 |
> |
Using Eq.~\ref{introEquation:definitionCR} and |
| 339 |
> |
Eq.~\ref{introEquation:resistanceTensorTransformation}, one can |
| 340 |
> |
locate the position of center of resistance, |
| 341 |
|
\begin{eqnarray*} |
| 342 |
|
\left( \begin{array}{l} |
| 343 |
|
x_{OR} \\ |
| 354 |
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 355 |
|
\end{array} \right) \\ |
| 356 |
|
\end{eqnarray*} |
| 361 |
– |
|
| 357 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 358 |
|
joining center of resistance $R$ and origin $O$. |
| 359 |
|
|
| 366 |
|
\end{equation} |
| 367 |
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
| 368 |
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
| 369 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
| 370 |
< |
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
| 369 |
> |
$V_i = V_i(v_i,\omega _i)$. The right side of |
| 370 |
> |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in |
| 371 |
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
| 372 |
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
| 373 |
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
| 377 |
|
\[ |
| 378 |
|
\begin{array}{l} |
| 379 |
|
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
| 380 |
< |
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
| 381 |
< |
\end{array}. |
| 380 |
> |
F_{r,i}^l (t) = A^T F_{r,i}^b (t). \\ |
| 381 |
> |
\end{array} |
| 382 |
|
\] |
| 383 |
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
| 384 |
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
| 385 |
< |
angular velocity $\omega _i$, |
| 385 |
> |
angular velocity $\omega _i$ |
| 386 |
|
\begin{equation} |
| 387 |
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
| 388 |
|
f_{r,i}^b (t) \\ |
| 402 |
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
| 403 |
|
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
| 404 |
|
\end{equation} |
| 410 |
– |
|
| 405 |
|
The equation of motion for $v_i$ can be written as |
| 406 |
|
\begin{equation} |
| 407 |
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
| 423 |
|
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
| 424 |
|
+ \tau _{r,i}^b(t) |
| 425 |
|
\end{equation} |
| 432 |
– |
|
| 426 |
|
Embedding the friction terms into force and torque, one can |
| 427 |
|
integrate the langevin equations of motion for rigid body of |
| 428 |
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
| 442 |
|
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 443 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 444 |
|
\end{align*} |
| 452 |
– |
|
| 445 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
| 446 |
|
product of the three body-fixed rotations, |
| 447 |
|
\begin{equation} |
| 476 |
|
\end{array} |
| 477 |
|
\right). |
| 478 |
|
\end{equation} |
| 479 |
< |
All other rotations follow in a straightforward manner. |
| 479 |
> |
All other rotations follow in a straightforward manner. After the |
| 480 |
> |
first part of the propagation, the forces and body-fixed torques are |
| 481 |
> |
calculated at the new positions and orientations |
| 482 |
|
|
| 489 |
– |
After the first part of the propagation, the forces and body-fixed |
| 490 |
– |
torques are calculated at the new positions and orientations |
| 491 |
– |
|
| 483 |
|
{\tt doForces:} |
| 484 |
|
\begin{align*} |
| 485 |
|
{\bf f}(t + h) &\leftarrow |
| 491 |
|
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
| 492 |
|
\cdot {\bf \tau}^s(t + h). |
| 493 |
|
\end{align*} |
| 494 |
+ |
Once the forces and torques have been obtained at the new time step, |
| 495 |
+ |
the velocities can be advanced to the same time value. |
| 496 |
|
|
| 504 |
– |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
| 505 |
– |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 506 |
– |
torques have been obtained at the new time step, the velocities can |
| 507 |
– |
be advanced to the same time value. |
| 508 |
– |
|
| 497 |
|
{\tt moveB:} |
| 498 |
|
\begin{align*} |
| 499 |
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
