Empirically motivated modifications of the Stokes-Einstein formula, in which the diffusion coefficient is inversely proportional to a fractional power of the viscosity, are discussed. We suggest that the results should be analyzed in terms of an effective hydrodynamic radius.An empirical modification of the Stokes-Einstein (SE) relation between diffusion coefficient D and fluid viscosity ' T] is proposed occasionally, most recently by Pollack and Enyeart. I (These authors refer to earlier work). This relation has the form

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SUMMARYLagrangian hydrodynamics of strength-free materials continues to present open issues, even in one dimension. We focus on the problem of closing a system of equations for a two-material cell under the assumption of a single velocity model. There are several existing models and approaches, each possessing different levels of fidelity to the underlying physics and each exhibiting unique features in the computed solutions. We consider three models that take different approaches to breaking the assumption of instantaneous pressure equilibration in the mixed-material cell. The first of these is the well-known method of Tipton, in which a viscosity-like pressure relaxation term is coupled with an otherwise isentropic pressure update to obtain closed-form expressions for the materials' volume fractions and corresponding sub-cell pressures. The second is the physics-inspired, geometry-based pressure relaxation model of Kamm and Shashkov, which is based on an optimization procedure that uses a local, exact Riemann problem. The third model is the unique aspect of this paper, inspired by the work of Delov and Sadchikov and Goncharov and Yanilkin. This sub-scale dynamics approach is motivated by the linearized Riemann problem to initialize volume fraction changes, which are then modified, via the materials' SIEs, to drive the mixed cell toward pressure equilibrium. Each of these approaches is packaged in the framework of a two-step time integration scheme. We compare these multi-material pressure relaxation models, together with corresponding pure-material calculations, on idealized, two-material problems with either ideal-gas or stiffened-gas equations of state.

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