R. Byron Bird

A nonlinear viscoelastic model for polymer solutions and melts—I

Abstract A rheological model for polymeric fluids is proposed. This model can describe non-Newtonian viscosity, shear-rate-dependent normal stresses, frequently-dependent complex viscosity, stress relaxation after large-deformation shear flow, recoil, and hysteresis loops. The model may be expanded in such a way as to get an approximate model valid for slowly-varying flows.

A kinetic theory for polymer melts. I. The equation for the single‐link orientational distribution function

In this series of papers we show how a kinetic theory of undiluted polymers can be developed using the Curtiss–Bird–Hassager phase‐space formulation. The polymer molecule is modeled as a Kramers freely jointed bead–rod chain. The objective is to obtain a molecular‐theory expression for the stress tensor from which the rheological properties of polymer melts can be obtained. This development is put forth as an alternative to the Doi–Edwards theory; using a very different approach, we have rederived some of their results and generalized or extended others. In this first paper we develop the partial differential equation for the chain configurational distribution function, and then proceed to get the equation for the orientational distribution function for a single link in the chain. A modification of Stokes’ law is introduced that includes a tensor drag coefficient, characterized by two scalar parameters ζ (the friction coefficient) and e (the link tension coefficient). In addition, to describe the increase of the drag force on a bead with chain length, at constant bead density, a ’’chain constraint exponent’’ β is used, which can vary from zero (the Doi–Edwards limit) to about 0.5. Solutions to the partial differential equation for the single‐link distribution function are given in several forms, including an explicit series solution to terms of third order in the velocity gradients.

Theory of Diffusion

Publisher Summary In chemical engineering, diffusion is responsible for mass transfer. Three different patterns of diffusion are responsible in this process: ordinary diffusion, thermal diffusion, and pressure diffusion. Theory of diffusion primarily focuses on the mass-flux vector and its relation to concentration gradients and diffusion coefficients. These diffusion coefficients have to be calculated or estimated to make calculations of practical interest. There are numerous ways of expressing concentration in diffusion problems, the most important for the purposes being mass density, molar density, mass fraction, and mole fraction. It is necessary to know values, for chemical engineering calculations, of the coefficients of diffusion and thermal diffusion for a wide range of chemical systems. The theory that forms the basis for discussions of the transport phenomena in dense gases is Enskogs kinetic theory for a pure gas made up of rigid spheres. Enskogs kinetic theory for dense gases takes into account the fact that the diameter of the molecules is not small with respect to the mean free path. This chapter discusses theories and experiments of diffusion and measuring different diffusion coefficients. Solutions of the diffusion equations of interest in chemical engineering are illustrated.

The Graetz-Nusselt problem for a power-law non-newtonian fluid☆

Abstract The authors show how the Graetz-Nusselt problem in heat transfer theory may be extended to non-Newtonian flow. A simple rheological model, “the power law,” is used to obtain a partial differential equation for the temperature profiles, to which a semi-analytical solution may be found. The temperature profiles are then used to calculate average outlet temperatures, as well as Nusselt numbers for several degrees of non-Newtonian behaviour. Tables are given for estimating the effect of non-Newtonian flow on the heating and cooling of fluids in tubes.

An Experimental Appraisal of Viscoelastic Models

Eleven differential and nine integral rheological models for viscoelastic fluids are tabulated in a uniform notation. Included among these are several very recently published models and three previously unpublished models. Representative experimental data are presented and used in qualitative and quantitative evaluations of the models. Comparisons between the models and the experimental data are summarized in a table where a six‐point rating scheme is used to assess the ability of several models to quantitatively describe seven material functions. The table indicates which models show promise for future work.

A kinetic theory for polymer melts. II. The stress tensor and the rheological equation of state

An expression for the stress tensor of an undiluted polymer is derived by starting with the phase‐space kinetic theory of Curtiss, Bird, and Hassager. The final formula contains two integrals, the first of which is the Doi–Edwards expression; the second integral, multiplied by the ’’link tension coefficient’’ e, is new, and consequently a rheological equation of state is obtained whose properties are different from those of the Doi–Edwards theory. As illustrations, the constants in the retarded‐motion expansion are determined to third order, and the relaxation modulus of linear viscoelasticity is obtained.

On coil–stretch transitions in dilute polymer solutions

In this paper we examine molecular stretching in the inception of uniaxial elongational flow of dilute polymer solutions. The polymer molecules are modeled as bead–spring chains with finitely extensible nonlinear elastic springs, and we use the Peterlin approximation. This work is distinguished from earlier work because we model the macromolecules with chains instead of dumbbells, and we examine the time dependence of three average quantities describing the chain conformation in unsteady flows: root‐mean‐square end‐to‐end distance, root‐mean‐square extensions of the individual links, and mean moment of inertia about the axis of elongation. We observe a gradual transition from the coiled equilibrium state of the chain to the stretched state after the inception of strong uniaxial elongational flow, and we describe the nature of this transition which takes place in roughly four stages: I equilibrium coil; II deformed coil; III spring stretched (‘‘locally unraveled’’); and IV unfolded chain. Inclusion of hydr...

A finitely extensible bead-spring chain model for dilute polymer solutions

Abstract The FENE-P dumbbell, that is, the finitely extensible non-linear elastic dumbbell with the Peterlin approximation, has been used extensively to describe the rheological behavior of dilute polymer solutions. The model is, however, severely limited, since it cannot describe the broad distribution of relaxation times that real polymer molecules possess. Hence, it is important to extend the model to a chain of finitely extensible springs. The FENE-P chain, however, has not been as widely used, presumably because of the large number of coupled equations that must be solved simultaneously in order to calculate the stress tensor. We present here a new model, the FENE-PM chain, as an alternative to the FENE-P chain. The stress tensor for the FENE-PM chain is calculated from a much smaller set of equations than the FENE-P and at the same time yields nearly the same results for the material properties in shear and elongational flow as the FENE-P. The reduced number of equations greatly expedites calculations for longer chains.

Interactions between two spheres falling along their line of centers in a viscoelastic fluid

Abstract An experimental investigation is presented in which the distance between two identical spheres falling along their line of centers in a viscoelastic fluid is determined as a function of time. It is found, for all five experimental fluids, that for small initial separations of the spheres, the two spheres eventually converge. However, for large initial separations, the two spheres eventually diverge. This leads to the definition of a critical initial separation distance. From this quantity a characteristic time can be derived for the two sphere fluid system which depends solely upon fluid properties. The implications of this study for suspension rheology are discussed.

New Variational Principle for Incompressible Non‐Newtonian Flow

A variational principle is given which describes the steady laminar motion of simple non‐Newtonian fluids, for which the viscosity is a function of the second invariant of the rate of deformation tensor. This principle simplifies to von Helmholtzs principle for the Newtonian fluid, and to Tomitas principle for the Ostwald‐de Waele fluid. For the latter two types of fluids, the equations of continuity and motion are equivalent to the statement that the rate of entropy production in the system is a minimum; for more general fluids, the variational principle does not admit this simple interpretation.