Stokesian dynamics

When the motion on the particle scale is such that the particle Reynolds number is small, the hydrodynamic force exerted on the particles in a suspension undergoing a bulk linear shear flow is:

${\displaystyle \mathbf {F} ^{\mathrm {H} }=-\mathbf {R} _{\mathrm {FU} }(\mathbf {U} -\mathbf {U} ^{\infty })+\mathbf {R} ^{\mathrm {FE} }:\mathbf {E} ^{\infty }.}$

Here, ${\displaystyle \mathbf {U} ^{\infty }}$ is the velocity of the bulk shear flow evaluated at the particle center, ${\displaystyle \mathbf {E} ^{\infty }}$ is the symmetric part of the velocity-gradient tensor; ${\displaystyle \mathbf {R} _{\mathrm {FU} }}$ and ${\displaystyle \mathbf {R} _{\mathrm {FE} }}$ are the configuration-dependent resistance matrices that give the hydrodynamic force/torque on the particles due to their motion relative to the fluid (${\displaystyle \mathbf {R} _{\mathrm {FU} }}$) and due to the imposed shear flow (${\displaystyle \mathbf {R} _{\mathrm {FE} }}$). Note that the subscripts on the matrices indicate the coupling between kinematic (${\displaystyle \mathbf {U} }$) and dynamic (${\displaystyle \mathbf {F} }$) quantities.

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 ${\displaystyle O(N^{3})}$ operations where N is the number of particles in the system (usually a periodic box). Recent advances have reduced the computational cost to about ${\displaystyle O(N^{1.25}\,\log N).}$[3][4]

The stochastic or Brownian force ${\displaystyle \mathbf {F} ^{\mathrm {B} }}$ arises from the thermal fluctuations in the fluid and is characterized by:

${\displaystyle \left\langle \mathbf {F} ^{\mathrm {B} }\right\rangle =0}$

${\displaystyle \left\langle \mathbf {F} ^{\mathrm {B} }(0)\mathbf {F} ^{\mathrm {B} }(t)\right\rangle =2kT\mathbf {R} _{\mathrm {FU} }\delta (t)}$

The angle brackets denote an ensemble average, ${\displaystyle k}$ is the Boltzmann constant, ${\displaystyle T}$ is the absolute temperature and ${\displaystyle \delta (t)}$ is the delta function. The amplitude of the correlation between the Brownian forces at time ${\displaystyle 0}$ and at time ${\displaystyle t}$ results from the fluctuation-dissipation theorem for the N-body system.

• Immersed boundary method
• Stochastic Eulerian Lagrangian method