Transverse-velocity fluctuations in a liquid under steady shear

Abstract

We present an analysis of the transverse-velocity fluctuations in an isothermal liquid layer with a uniform shear rate between two parallel horizontal boundaries as a function of the wave number and the Reynolds number. The results were obtained by solving a stochastic version of the Orr-Sommerfeld equation subject to no-slip boundary conditions in a second-order Galerkin approximation. We find that the spatial Fourier transform of the transverse-velocity fluctuations exhibits a maximum as a function of the (horizontal) wave number q(parallel to). This maximum is associated with a crossover from a q(parallel to)(-4) dependence for larger q(parallel to) to a q(parallel to)(2) dependence for small q(parallel to). The q(parallel to)(-4) dependence at larger wave numbers is independent of the boundary conditions, but the small-q(parallel to) behavior is strongly affected by the boundary conditions. The nonequilibrium enhancement of the intensity of the transverse-velocity fluctuations remains finite for all values of the Reynolds number, but increases approximately with the square of the Reynolds number. The relation between our results and those obtained by previous authors in the absence of boundary conditions is elucidated.