Rheological models with microstructural constraints

Abstract.

Rheological models of complex fluids with a physically restricted microstructure are analyzed to obtain general classes of dynamical evolution equations for these materials. These classes insure that the appropriate mathematical constraints, associated with each type of physical restriction, are consistently incorporated into the corresponding model development. Describing the microstructure of the complex fluid with a second-rank tensor variable, a general class of dynamical evolution equations is derived for three physically meaningful constraints associated with constancy of the invariants of this microstructural tensor. The physical rationale for each of these constraints is discussed, and a corresponding set of constrained dynamical evolution equations is derived in general terms.

Appendix

Even though the formulation presented in the main body of the paper is complete, it is useful to regard it also in the light of some other types of formulations and arguments. First, note that the terms representing the dissipation in Eqs. (9) and (10), i.e., the last term on the right-hand side of Eq. (10) and the last two terms on the right-hand side of Eq. (9), can be recast in the forms of $- {{\delta \Psi } \over {\delta \left( {{{\delta H} \over {\delta {\bf u}}}} \right)}}$ and $- {{\delta \Psi } \over {\delta \left( {{{\delta H} \over {\delta {\bf C}}}} \right)}}$ , respectively, where the functional , called the dissipation potential, is given by

$$ \displaylines{ \Psi = \int {{1 \over 2}\Lambda _{\alpha \beta \gamma \varepsilon } {{\delta H} \over {\delta C_{\alpha \beta } }}} {{\delta H} \over {\delta C_{\gamma \varepsilon } }}d^3 r + \int {{1 \over 2}B_{\alpha \beta \gamma \varepsilon \eta \nu } \left( {\nabla _\gamma {{\delta H} \over {\delta C_{\alpha \beta } }}} \right)\left( {\nabla _\nu {{\delta H} \over {\delta C_{\varepsilon \eta } }}} \right)} d^3 r \cr + \int {{1 \over 2}Q_{\alpha \beta \gamma \varepsilon } \left( {\nabla _\alpha {{\delta H} \over {\delta u_\beta }}} \right)\left( {\nabla _\gamma {{\delta H} \over {\delta u_\varepsilon }}} \right)} d^3 r \cr} $$

(A1)

The most significant aspect of the concept of the dissipation potential is that it allows one to formulate the dissipative part of the time evolution equations that depends nonlinearly on ${{\delta H} \over {\delta {\bf C}}}$ and ${{\delta H} \over {\delta {\bf u}}}$ . The physical and geometrical significance of the dissipation potential has been recently investigated by Grmela (2002). From the historical point of view, the dissipation potential can be regarded as an appropriate generalization of the dissipation potential introduced by Rayleigh (1945). As in the main body of the paper, one can arrive at the constrained version of the dissipative part of the time evolution by a dissipation potential that depends, for the constant extension example, on ${{\delta H} \over {\delta C_{\alpha \beta } }}$ only through its dependence on ${{\delta H} \over {\delta C_{\alpha \beta } }} - {1 \over 3}{{\delta H} \over {\delta C_{\gamma \gamma } }}\delta _{\alpha \beta }$ .

The second observation concerns the first term on the right-hand side of the Eq. (10) for the extra stress tensor field, σ. Note that one can alternatively derive it by using an argument that is weaker than the one used above. It follows from Eq. (4) that in the limit of non-dissipative time evolution, i.e., if $\left[ {F,H} \right] = 0$ , then ${{dH} \over {dt}} = 0$ as a consequence of the antisymmetry property of the Poisson bracket. This now raises the following question: given Eqs. (7), (8), and (9) with Λ=0 and ${\bf B} = {\bf 0}$ , what is the expression for σ appearing on the right-hand side of Eq. (8) with which the given time evolution equations imply ${{dH} \over {dt}} = 0$ ? This question has been answered by Grmela (1985), and later in a more general setting by Jongschaap (1990, 2001). The answer is the following: $\sigma _{\alpha \gamma } = - {{\delta H} \over {\delta C_{\beta \delta } }}{{\delta \Xi _{\beta \delta } } \over {\delta (\nabla _\alpha \nu _\gamma )}}$ , where $\Xi _{\alpha \beta } ({\bf C},\nabla {\bf v})$ denotes the right-hand side of Eq. (9) with Λ=0 and ${\bf B} = {\bf 0}$ . One can directly verify that this expression leads to the first term on the right-hand side of Eq. (10). Thus, this provides an alternative derivation of the first term on the right-hand side of Eq. (10), one that is based on the conservation of the free energy during the reversible time evolution. This requirement is weaker than the requirement that the reversible time evolution possesses a Hamiltonian structure.