Miroslav Grmela

Dynamics and thermodynamics of complex fluids. I. Development of a general formalism

We recognize some universal features of macroscopic dynamics describing the approach of a well-established level of description (that is, successfully tested by experimental observations) to equilibrium. The universal features are collected in a general equation for the nonequilibrium reversible-irreversible coupling (abbreviated as GENERIC). In this paper we formulate a GENERIC, derive properties of its solutions, and discuss their physical interpretation. The relation of the GENERIC with thermodynamics is most clearly displayed in a formulation that uses contact structures. The GENERIC is also discussed in the presence of noise. In applications we either search for new governing equations expressing our insight into a particular complex fluid or take well-established governing equations and cast them into the form of a GENERIC. In the former case we obtain the governing equations as particular realizations of the GENERIC structure; in the latter case we justify the universality of the GENERIC and derive some properties of solutions. Both types of applications are discussed mainly in the following paper [Phys. Rev. E 56, 6633 (1997)].

Relationships between rheology and morphology for immiscible molten blends of polypropylene and ethylene copolymers under shear flow

The linear and nonlinear viscoelastic properties of immiscible polymer blends polypropylene/ethylenevinylacetate–ethylenemethylacrylate [PP/(EVA–EMA)] have been investigated. The transient shear flow experiments reflect the structural changes of the blends during the flow. Overshoots in stress growth experiments are observed when the dispersed phase is deformable. In this case, a good description of these transient rheological data is obtained using modified versions of the Lee and Park [J. Rheol. 38, 1405 (1994)] and the Grmela and Ait-Kadi [J. Non-Newtonian Fluid Mech. 55, 191 (1994)] models. Predictions of the morphological evolution of the blend under transient shear flows were calculated from the modified models which are shown to describe the breakup and coalescence phenomena under moderately large deformation shear flow. When the dispersed phase is undeformable, these models, which are either based on the original Doi and Ohta [J. Chem. Phys. 95, 1242 (1991)] theory or derived to retrieve an extens...

Bracket formulation of dissipative fluid mechanics equations

Abstract The bracket formulation of the Euler fluid mechanics equations is extended to the fluid mechanics equations corresponding to the Navier-Stokes-Fourier and the Edelen constitutive relations.

Quantitative relationship between rheology and morphology in emulsions

Calculations based on the Grmela et al. model [M. Grmela, M. Bousmina, and J. F. Palierne, Rheol. Acta 40, 560 (2001)] for the simple case of ellipsoidal droplet-type morphology express a direct quantitative relationship between flow and microstructure both in the steady and transient regimes. The results of calculations show that in shear, elongational, and planar hyperbolic flow it is possible to extract the morphology (the deformation and the orientation of the droplet) from rheological material functions and, conversely, rheological material functions can be obtained from in situ morphological observation.

Modeling of oscillatory shear flow of emulsions under small and large deformation fields

A model for a mixture of two Newtonian liquids that undergo oscillatory shear flow is presented. The model expresses qualitatively the relationship between the oscillating stress and the oscillating shape of the drops characterized by a second order symmetric tensor called a morphology tensor. The governing equations of the model are solved analytically in small-amplitude oscillatory shear (SAOS) flow and, to the second order of the capillary number, in large-amplitude oscillatory shear (LAOS) flow. Maxwell-type dynamic moduli under SAOS are found to give quite similar predictions as those of Palierne [J. F. Palierne, Rheol, Acta 29, 204 (1990)] and Bousmina [M. Bousima, Rheol, Acta 38, 73 (1999)] emulsion models. Nonlinear dependence of the shear stress and the difference in first normal stress on strain are predicted for LAOS. The predictions of the model are found to be in agreement with the experimental results of Cavallo et al. [R. Cavallo et al., Rheol. Acta (in press, 2002)].

Conformation tensor rheological models

Abstract A systematic study of conformation tensor rheological models is presented in this paper. The approach unifies and extends results obtained previously by Hand, Giesekus, and Maugin and Drouot. Rheological properties that are independent of details of the conformational dependence of the intrasegmental force and the mobility are identified. A closed form solution is found for rigid-in-average dumbbell models.

Bracket formulation of diffusion-convection equations

Abstract An abstract diffusion-convection equation is introduced. It is shown that many well-known and experimentally tested time evolution equations of physical systems (e.g. the Boltzmann and the Enskog kinetic equations, and the Navier-Stokes-Fourier fluid mechanics equations) are identical to or well approximated by realizations of the abstract diffusion-convection equation. The physical content of the abstract diffusion-convection equation is compatibility with equilibrium thermodynamics. It means that properties of solutions of the abstract diffusion-convection equation are mathematical formulations of the experience on which equilibrium thermodynamics is based. The requirement that all admissible time evolution equations are realizations of the abstract diffusion-convection equation is used to find admissible closures in the BBGKY and the Maxwell-Grad hierarchies, and admissible constitutive relations in generalized hydrodynamics.

Reciprocity relations in thermodynamics

Dissipative time evolution of macroscopic systems describing the experimentally observed approach to reduced descriptions is put into the context of contact geometry. This viewpoint of nonequilibrium thermodynamics unifies many well-established mesoscopic dynamical theories, as for example the Boltzmann kinetic theory and the Navier–Stokes–Fourier hydrodynamics, and leads to new reciprocity relations that extend the Maxwell and the Onsager–Casimir relations.

Hamiltonian dynamics of incompressible elastic fluids

Abstract Equations that govern the time evolution of incompressible elastic fluids are shown to possess the hamiltonian structure.

Generalized constitutive equation for polymeric liquid crystals Part 1. Model formulation using the Hamiltonian (poisson bracket) formulation

Abstract The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets, developed in Z.R. Iwinski and L.A. Turski, Lett. Appl. Eng. Sci., 4 (1976), 179–191, P.J. Morrison and J.M. Greene, Phys. Rev. Lett., 45 (1980) 790–794, I.E. Dzyaloshinskii and G.E. Volovick, Ann. Phys., 125 (1980) 67–97, D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep., 123 (1985) 1–116, M. Grmela, Phys. Lett. A, 130 (1988) 81–86, and A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55–78, for a class of incompressible fluids, is used here in order to generate a constitutive equation for the stress and the order parametr tensor for a polymeric liquid crystal. A free energy expression, of the type used by Doi in his theory for concentrated solutions of rigid rods, is used in addition to the Frank elasticity expression employed in the Leslie—Ericksen—Parodi (LEP) theory to model the effect of spatial gradients in the liquid crystalline microstructure. For homogeneous systems, the analysis leads to a model which is equivalent to a generalization of Doi theory out to fourth-order terms in S . Truncating this model at second-order terms gives the Doi equations exactly. To evaluate the expanded model, results for steady simple shear and extensional flows are compared against the Doi model predictions. The constitutive equation resulting from the expanded model is compared against the LEP constitutive equation and the parameters between the two are correlated. The additional stress terms for non-homogeneous systems reduce to a recently presented (B.J. Edwards and A.N. Beris, J. Rheol., 33 (1989) 1189–1193; M. Grmela, Phys. Lett. A, 137 (1989) 342–348) generalization of the Ericksen stress expression in terms of the second-order parameter tensor. The model presented is a generalization and extension of the order-parameter-based theory of Doi which allows a greater flexibility in describing the rheological properties of polymeric liquid crystalline systems