# Stokesian dynamics

Stokesian dynamics[1] is a solution technique for the Langevin equation, which is the relevant form of Newton's 2nd law for a Brownian particle

$m\frac{du}{dt} = F^{H} + F^{B} + F^{P}.$

In the above equation $F^{H}$ is the hydrodynamic force, i.e., force exerted by the fluid on the particle due to relative motion between them. $F^{B}$ is the stochastic Brownian force due to thermal motion of fluid particles. $F^{P}$ is the inter particle force,e.g. electrostatic repulsion between like charged particles. Brownian dynamics is one of the popular techniques of solving the Langevin equation, but the hydrodynamic interaction in Brownian dynamics is highly simplified and normally includes only the isolated body resistance. On the other hand, Stokesian dynamics includes the many body hydrodynamic interactions. Hydrodynamic interaction is very important for non-equilibrium suspensions, like a sheared suspension, where it plays a vital role in its microstructure and hence its properties. Stokesian dynamics is used primarily for non-equilibrium suspensions where it has been shown to provide results which agree with experiments.[citation needed]

## Hydrodynamic interaction

One of the key features of Stokesian dynamics is its handing of the hydrodynamic interactions, which is fairly accurate without being computationally inhibitive (like boundary integral methods) for a large number of particles. Classical Stokesian dynamics requires $O(N^{3})$ operations where N is the number of particles in the system (usually a periodic box). Recent advances have reduced the computational cost to about $O(N^{1.25} \, \log N).$[2][3]