Brownian motion in a fluid in simple shear flow

We study Brownian motion of a sphere suspended in a fluid in stationary homogeneous flow. Using hydrodynamic fluctuation theory, we derive a Langevin equation and show that the fluctuation-dissipation theorem is modified by the macroscopic homogeneous flow. It is found that the “strength” of the random force is no longer given in terms of the (shear-dependent) friction coefficient alone as it is when the unperturbed fluid is at rest. The modification originates from the tensile nature of the macroscopic flow. As a special case, we analyze the Langevin equation for the simple shear case in detail and give the velocity autocorrelation function and the mean square displacement of the sphere for some special time regimes. The possibility to defince a diffusion coefficient in these regimes is also discussed