A constitutive equation is derived from a Lodge—Yamamoto type of network theory for polymeric fluids. The network junctions are not assumed to move strictly as points of the continuum but allowed a certain “effective slip”. The rates of creation and destruction of junctions are assumed to depend on the instantaneous elastic energy of the network, or equivalently, the average extension of the network strand, in a simple manner. Agreement between model predictions and the I.U.P.A.C. data on L.D.P.E. is good.

A new constitutive equation derived from network theory

A new constitutive equation for whole human blood is derived using ideas drawn from temporary polymer network theory to model the aggregation and disaggregation of erythrocytes in normal human blood at different shear rates. Each erythrocyte is represented by a dumbbell. The use of a linear spring law in the dumbbells leads to a multi-mode generalized Maxwell equation for the elastic stress and both the relaxation times and viscosities are functions of a time-dependent structure variable. An approximate constitutive equation is derived by choosing a single mode corresponding to the cell aggregate size where the largest number of cells are to be found. This size is identified in the case of steady flows. The model exhibits shear-thinning, viscoelasticity and thixotropy and these are clearly related to the microstructural properties of the fluid. Agreement with the experimental data of Bureau et al. [M. Bureau, J.C. Healy, D. Bourgoin, M. Joly, Rheological hysteresis of blood at low shear rate, Biorheology 17 (1980) 191–203] in the case of a simple triangular step shear rate flow is convincing.

A new microstructure-based constitutive model for human blood

A consitutive theory for polymetric liquids based on the transient network concept is developed, following Wiegel and Jongschaap. The Phan-Thien-Tanner equation is shown to follow from the general theory with two critical assumption, one of these is quasi-equilibrium of the internal structure, which preludes consistency of application in ”fast flows“. The Marrucci model can be made consistent with the general format with a small change in the kinetic equation that can be deduced from asymptotic behaviour and leaves the steady viscometric behavior unchanged. The simplest genaral formulation requires the linear viscoelastic spectrum and two parameters; the latter cannot be determined uniquely from steady viscometric flow data.

Constitutive equations based on the transient network concept

A new numerical approach is presented to exactly solve the convection equation arising in network theories. The method is based on a direct stochastic interpretation of the convection equation. We show that with this approach models can be studied extensively which are not solvable analytically. It turns out that a conceptually simple approach to network theories predicts a qualitatively satisfying rheological behavior.

A numerical stochastic approach to network theories of polymeric fluids

The understanding of the flow behaviour of polymeric liquids is of great interest from a practical as well as a theoretical point of view. An important part of the research in this field consists of the development of suitable models, describing the rheological properties of the materials. Depending upon its purpose, such a model may be based upon empirical knowledge of the macroscopic flow behaviour or on information about the microstructure of the materials. Moreover, for a given system, different types of modelling may be possible. In order to provide an overview of the various approaches in this area the basic principles of some important models are discussed: continuum, bead-rod-spring, transient network, reptation and configuration tensor models. Emphasis has been put on a consistent treatment of the fundamentals of the various models and their interrelationship, rather than considering any of them in much detail.

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F.Th. De Bats

Papers

Rheological properties of a network model for macromolecular fluids