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A summary is given of some recent attempts to relate the results of the kinetic theory of rigid and flexible macromolecules to continuum mechanics results.

For a suspension of rigid macromolecules with Brownian motion, expressions are obtained for the components of the stress tensor for any time‐dependent shearing flow. The results are expressed up to terms of the fourth order as a series of memory integrals. The first two terms in the series enable one to calculate the explicit expressions for the kernel functions for second‐order viscoelasticity. It is established that the Oldroyd six‐constant model does not give kernel functions which are of the same form as those given by the molecular theory of rigid macromolecules. The general time‐dependent elongational flow results are also given through terms of the third order.

Publisher Summary This chapter describes the two methods for evaluating constitutive equations for viscoelastic fluids: the use of material functions as determined by rheometry and the comparison of measured and computed velocity, and stress fields in complex flows. It also presents the way complex flows give useful insight into the proper constitutive description of materials and to the associated evaluation of constitutive equations. The study of complex flows in the context of rheometry can be useful for three reasons: (1) information may be provided as to which of the host of possible material functions are most important in determining flow fields in general classes of flows; (2) new methods for measuring material functions may be suggested; and (3) the ability of a constitutive equation to interpolate between kinematics such as simple shear and simple elongational flow can be assessed.

Because of their complex structure the mechanical behavior of polymeric materials is not well described by the classical constitutive equations: Hooke’s law (for elastic solids) or Newton’s law (for viscous liquids). Polymeric materials are said to be ‘viscoelastic’ inasmuch as they exhibit both viscous and elastic responses. This viscoelastic behavior has played a key role in the development of the understanding of polymer structure.1 Viscoelasticity is also important in the understanding of various measuring devices needed for rheometric measurements.2–4 In the fluid dynamics of polymeric liquids, viscoelasticity also plays a crucial role.5–13 Also in the polymer-processing industry it is necessary to include the role of viscoelastic behavior in careful analysis and design.14,15 Finally there are important connections between viscoelasticity and flow birefringence.16

The use of the retarded motion expansion as a constitutive equation has long been known to be valid only for small Deborah number flow. We demonstrate that ‘solution’ which cannot be obtained as a regular perturbation about the Newtonian solution is physically meaningless. This is discussed in connection with a recent paper in this journal.

Turbulent drag reduction by the addition of a small amount of polymer is studied using the bead‐string model for the polymer moleculses. (AIP)

An equation for the stress tensor in terms of the rate of strain tensor is obtained for a dilute suspension of rigid macromolecules with Brownian motion. The result is given as a series of memory integrals up through terms of third order; the kernel functions contain two constants: the number density and time constant for the macromolecules. In obtaining this series, the distribution function for arbitrary unsteady homogeneous flows is developed up through terms of second order. It is shown that one need consider only irrotational flow in determining the kernel functions for the memory integral expansion. Relations between the kernel functions and the coefficients in the retarded motion expansion are also given. In addition it is shown how the Eulerian components of the rate‐of‐deformation tensor are related to the fixed components by using the theory of matrizants.