The thermal fluctuations of diluted emulsion

The collective excitations of the compressible diluted emulsion are studied. The diluted emulsion is consisted of molecules with intrinsic momentum. The motion equation for fluctuation fields of the velocity, density, intrinsic momentum and pressure were chosen here in the form of linearized Navier-Stokes equations with constant shear and volume viscosity, and diffusion coefficient of momentum, spontaneous stresses. For simplicity in the calculations, temperature fluctuations were not taken into account. We assume that the emulsion possesses constant surface tension. For hydrodynamic fields excited by spontaneous sources, continuity conditions on the surface tangents of the velocity and stress tensor components should be satisfied at the interface. The boundary conditions for the normal force components comprise the excess Laplace pressure and the random surface force determined by the difference of the radial values of the spontaneous volume stress densities. Because of the smallness of the surface displacements, all the boundary conditions should be applied on sphere. To construct the spectral densities of the thermal fields caused by random surface forces, we used the fluctuation-dissipate theorem. The spectral densities of the thermal amplitude fluctuations of the hydrodynamic currents are expressed in terms of susceptibilities. The spectrum of fluctuations contains all the collective excitations of the emulsion associated with fluctuating motions of the surface and with the properties of near-surface currents. Their analysis, based on strict inequalities between the drop's emulsion size, the penetration depth of a viscous wave, and the wavelength of sound, essentially depends on the order of magnitude of the boundary transitions.