Finite-size effects on fluctuations in a fluid out of thermal equilibrium

In this paper, we consider a horizontal liquid layer in the presence of a stationary temperature gradient. Specifically, we calculate the structure factor neglecting gravity, but taking into account the finite height of the liquid layer. For this purpose, we consider the linearized Boussinesq equations, in the limit of negligible Rayleigh number, supplemented with Langevin noise terms and assuming free-slip boundary conditions. The nonequilibrium temperature fluctuations are obtained by expanding the solution in a complete set of orthogonal functions satisfying the boundary conditions. It is shown that the finite height of the system restricts the spatial range of the temperature fluctuations not only in the direction of the temperature gradient, but also in the horizontal direction away from the boundaries. It is demonstrated that the q−4 dependence of the structure factor in the absence of finite-size effects now crosses over to a q2 dependence for very small values of the wave number q. Estimates of the wave numbers where light-scattering experiments will be affected by these finite-size effects are presented.