FTS formulation

Equation (3.18) is the famous FTS formulation derived in the paper of Stokesian dynamics method[6].

As shown above, it is natural form in the mathematical sense that $\bm{U}$, $\bm{\Omega}$ and $\bm{E}$ are in the left-hand side and $\bm{F}$, $\bm{T}$ and $\bm{S}$ are in the right-hand side. However, from physical point of view, especially for the rigid particles, it is not always the case. For instance, in the resistance problem, $\bm{U}$ and $\bm{\Omega}$ are given and $\bm{F}$ and $\bm{T}$ are unknown, while in the mobility problem, $\bm{F}$ and $\bm{T}$ are given and $\bm{U}$ and $\bm{\Omega}$ are unknown. But from the rigidity of the particles, $\bm{E}$ is always a given parameter and $\bm{S}$ is unknown. That is, for the mobility problem, it would be the natural form that

$$\left[ \begin{array}{c} \bm{U}\\ \bm{\Omega}\\ \bm{S} \end{array}... ...}' \cdot \left[ \begin{array}{c} \bm{F}\\ \bm{T}\\ \bm{E} \end{array} \right] .$$ (3.22)

Actually, Kim et al.[21] take this form rather than Eq. (3.18). But, here, we stick to the FTS form of Eq. (3.18). Because the equation is linear, we can construct the FTE form from FTS as

$${\bm{M}'}_{UF} = \bm{M}_{UF} - \bm{M}_{US} \cdot \Bigl( \bm{M}_{ES} \Bigr)^{-1} \cdot \bm{M}_{EF} ,$$ (3.23)

$${\bm{M}'}_{UE} = \bm{M}_{US} \cdot \Bigl( \bm{M}_{ES} \Bigr)^{-1} ,$$ (3.24)

$${\bm{M}'}_{SF} = - \Bigl( \bm{M}_{ES} \Bigr)^{-1} \cdot \bm{M}_{EF} ,$$ (3.25)

$${\bm{M}'}_{SE} = \Bigl( \bm{M}_{ES} \Bigr)^{-1} ,$$ (3.26)

for

$$\left[ \begin{array}{c} \bm{U}\\ \bm{S} \end{array} \right] = \le... ...ay} \right] \cdot \left[ \begin{array}{c} \bm{F}\\ \bm{E} \end{array} \right] ,$$ (3.27)

where

$$\left[ \begin{array}{c} \bm{U}\\ \bm{E} \end{array} \right] = \le... ...ay} \right] \cdot \left[ \begin{array}{c} \bm{F}\\ \bm{S} \end{array} \right] .$$ (3.28)

Note that, for simplicity, we combine $\bm{\Omega}$ and $\bm{T}$ into $\bm{U}$ and $\bm{F}$ respectively.

This procedure is exactly applied for the higher orders of expansions for rigid particles, because not only $\bm{E}$ but the higher order velocity moments should be vanished (that is, these are always given parameters).