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Motivated by a paper by Puglisi and Truskinovsky (P&T) on dissipation in plastic behavior of materials, I briefly revisit an article of mine on fluid foams, showing that the kind of dissipation by instability discussed by P&T applies.

Several methods in nonadiabatic molecular dynamics are based on Madelung’s hydrodynamic description of nuclear motion, while the electronic component is treated as a finite-dimensional quantum system. In this context, the quantum potential leads to severe computational challenges and one often seeks to neglect its contribution, thereby approximating nuclear motion as classical. The resulting model couples classical hydrodynamics for the nuclei to the quantum motion of the electronic component, leading to the structure of a complex fluid system. This type of mixed quantum–classical fluid models has also appeared in solvation dynamics to describe the coupling between liquid solvents and the quantum solute molecule. While these approaches represent a promising direction, their mathematical structure requires a certain care. In some cases, challenging higher-order gradients make these equations hardly tractable. In other cases, these models are based on phase-space formulations that suffer from well-known consistency issues. Here, we present a new complex fluid system that resolves these difficulties. Unlike common approaches, the current system is obtained by applying the fluid closure at the level of the action principle of the original phase-space model. As a result, the system inherits a Hamiltonian structure and retains energy/momentum balance. After discussing some of its structural properties and dynamical invariants, we illustrate the model in the case of pure-dephasing dynamics. We conclude by presenting some invariant planar models.

We provide a thermodynamic basis for the development of models that are usually referred to as “phase-field models” for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive “phase-field models” both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631–651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier–Stokes–Fourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier–Stokes–Fourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313–1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn–Hilliard–Navier–Stokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn–Hilliard–Navier–Stokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).

An experimental investigation is presented of an oscillating disk submerged in water. The system is adapted from one modeled by the Bagley-Torvik equation, being modified to more closely approximate idealizations in the derivation. The modified system eliminates alignment problems that would violate the assumption of fluid forces being due only to shear in the fluid. An implicit finite difference model based on the Riemann-Liouville fractional derivative was used to model the observed oscillatory free response resulting from a non-homogeneous initial condition. The numerical model accounted for motion well until the amplitude was small. MSC 2010 : 26A33, 34A08, 76A99, 34K11