Velocity fluctuations in laminar fluid flow

Fluctuating hydrodynamics, originally developed for fluctuations in fluids in equilibrium, can be extended to deal also with thermally excited hydrodynamic fluctuations in non-equilibrium states. After first reviewing some results earlier obtained for temperature fluctuations in fluids subjected to an externally imposed temperature gradient, we use in this paper fluctuating hydrodynamics to determine the enhancement of velocity fluctuations in laminar fluid flow. Adopting the case of planar Couette flow as a representative example, we show how the fluctuations of the wall-normal component of the velocity and of the wall-normal component of the vorticity can be obtained as solutions of a stochastic Orr-Sommerfeld equation and a stochastic Squire equation, respectively. By solving these fluctuating hydrodynamic equations we obtain quantitative estimates of the flow-induced non-equilibrium enhancements of the velocity and vorticity fluctuations as a function of the Reynolds number and of the wave number of the fluctuations.