Stress correlations and stress memory kernels in viscoelastic fluids

We discuss the spatio-temporal correlations of stress fluctuations in viscoelastic fluids, including colloidal dispersions, polymer melts and glass-forming liquids. First, we relate the tensor of stress auto-correlation functions to its corresponding tensor of generalized Onsager transport kernels. The relation is valid at finite wavelengths and frequencies, and serves as basis for continuum mechanics approximations, where it leads to the viscosities in a fluid and to the elastic constants in a solid. Second, we consider the long wavelength limit and re-derive the far-field power-law stress correlations in fluid states. Two theoretical approaches are studied and give equivalent results. One approach is based on the familiar Zwanzig-Mori decomposition of the stress tensor, while the other approach decomposes the stress into a deterministic one due to the history of prior internal flow and a contribution due to "stress noise" that persists even if internal flow is suppressed. The general results are used to predict the distance dependence of the shear stress correlation function both for 2D and 3D systems. An encouraging agreement of these predictions with simulation data on 2D and 3D binary hard-sphere mixtures is observed.