Equilibrium fluctuations in fluid layers: Effects of elastic solid boundaries

Abstract

The hydrodynamic theory of the dynamics of equilibrium fluctuations in one-component fluid layers confined by two identical elastic solid boundaries is presented. The dynamic structure factor for a fluid layer and the interfacial modes are analyzed under assumptions of continuity of stresses (no interfacial stress) and velocities (stick boundary condition) across the fluid-solid interfaces. There are four interfacial modes in such a system with dispersion relations which depend on the component of the wave vector parallel to the interfaces, ${k}_{\ensuremath{\parallel}}$. In the limit of $\frac{\overline{\ensuremath{\rho}}}{{\ensuremath{\rho}}_{s}}\ensuremath{\rightarrow}0$, where $\overline{\ensuremath{\rho}}$ and ${\ensuremath{\rho}}_{s}$ are the mean densities of the fluid and solid, respectively, two of these modes are just the well-known Rayleigh waves and the remaining two are the interfacial fluid modes found previously in the case of rigid solid walls [D. Gutkowicz-Krusin and I. Procaccia, Phys. Rev. Lett. 48, 417 (1982); Phys. Rev. A 27, 2585 (1983)]. For finite small values of $\frac{\overline{\ensuremath{\rho}}}{{\ensuremath{\rho}}_{s}}$ the attenuation of the interfacial fluid modes is due to dissipative interfacial transport and varies as ${k}_{\ensuremath{\parallel}}^{\frac{3}{2}}\frac{\overline{\ensuremath{\rho}}}{{\ensuremath{\rho}}_{s}}$, provided that $cl{c}_{t}$, where $c$ is the speed of sound in the fluid and ${c}_{t}$ is the transverse sound speed in the solid. If, however, $cg{c}_{t}$, the energy of the interfacial fluid mode is radiated into the solid and the resulting attenuation along the interface varies as ${k}_{\ensuremath{\parallel}}{(\frac{\overline{\ensuremath{\rho}}}{{\ensuremath{\rho}}_{s}})}^{2}$, for small ${k}_{\ensuremath{\parallel}}$. Since the speed of the interfacial fluid modes is less than $c$, the corresponding peaks in the dynamic structure factor are better separated from the unbounded fluid Brillouin peaks and thus easier to observe experimentally than would have been the case with rigid solid walls.