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A phase‐space kinetic theory of dilute polymer solutions is developed to account for the effects of nonhomogeneous velocity and stress fields. The theory allows the configuration distribution function to depend on spatial location and explicitly treats the polymer molecule as an extended object in space. Constitutive equations for the mass flux vector and stress tensor are derived that predict polymer migration induced by stress gradients and nonuniform velocity gradients. In addition, the constitutive equation for stress contains a diffusive term in stress, and hence the model does not fall within the class of simple fluids. Simple shear flow between parallel plates is solved to illustrate the features of the constitutive equations. Asymptotic analysis and numerical calculations show the formation of boundary layers in stress, velocity gradient, and polymer concentration that arise near solid walls as a result of preferential orientation of the polymer molecules there. The thickness of these layers scale...

We present a simple linear stability analysis of the diffusion equation for nematic polymers that delivers the equilibrium bifurcation diagram for rigid rod, excluded volume potentials. Symmetry properties of the diffusion equation yield insight into invariant subspaces of solutions and allow for interpretation of recently published simulation results.

The initial stage of spinodal decomposition is investigated for a rigid-rod system. Spinodal decomposition proceeds through either of two mechanisms: (1) The randomly aligned rods rotate toward a common director with no inherent length scale. (2) The rods diffuse axially and segregate into regions of common alignment with a selected length scale [script-l]. Previous studies on spinodal decomposition yielded radically different conclusions about which mechanism is dominant. A computational method is employed to analyze the growth rate and properties of the dominant fluctuation mode through an eigenvalue analysis of the linearized Doi diffusion equation in Fourier space. The linearized operator is discretized in Fourier mode and orientation space (k,theta,phi) space, and the maximum eigenvalue and corresponding eigenvector of the operator are calculated. Our analysis generalizes the results of previous studies and shows that the conflicts seen in those studies are due to differences in the diffusivities for rotational motion, motion perpendicular to the rod axis, and motion along the rod axis. For a given system density, a plot of the dominant perturbation wave number k(max) as a function of the diffusivity ratios shows two separate regions corresponding to mechanisms (1) and (2). High rotational diffusivity corresponds to mechanism (1), whereas high axial diffusivity corresponds to mechanism (2). The transition between the two mechanisms depends on the ratio of diffusivities and on system density. Also, the dominant wave number increases with increasing density indicating that a deeper quench into the spinodal regime leads to a smaller characteristic length scale.

Recent experiments have shown that porous filters preadsorbed with polymer molecules exhibit an anomalously high pressure drop at high rates of flow. We have modeled the adsorbed polymers as dynamical systems and have found that the introduction of hydrodynamic interaction between molecules destabilizes at a high applied shear. As a direct result this instability will cause the molecules to unravel and stretch far into the cross section of the pore, and thus by inference, cause the observed anomalously high pressure drop. Although much of this paper is devoted to the stability characteristics of the deterministic system, Brownian motion is also considered, and an account of the statistics of the Brownian system when the deterministic system becomes unstable is given. The examples revealed in this paper are not of sufficient complexity to calculate with any accuracy the magnitude of this anomalous pressure drop. We simply present a procedure by which a large variety of more complex models could be undertak...

For a dilute solution of macromolecules idealized as elastic dumbbells with Brownian motion, expressions are obtained giving the stress tensor in terms of rate‐of‐strain tensors for an arbitrary, steady, homogeneous flow. The rheological equations of state are fit into a retarded motion expansion. For dumbbells with arbitrary, nonlinear, elastic connectors, the result is given up through terms of second‐order. Stress tensor equations for two specific models, Hookean dumbbells and finitely extendible non‐linear elastic (FENE) dumbbells, are given up through terms of third order.

An investigation into the conformation of a polymer molecule exposed to a stochastic flow field is made utilizing a cumulant expansion. The study begins by considering the full Kirkwood–Riseman n‐bead chain with an arbitrary connector potential as a model for the polymer molecule, digressing to the two‐bead dumbbell molecule for detailed calculations. It is found that the effect of the stochastic velocity field on the molecule can be interpreted as a ‘‘renormalization’’ of the connector potential. The introduction of this renormalized potential greatly simplifies the mathematics and lends an intuitive understanding to the effect of the turbulence on the polymer. It is shown that this analysis will recover the anomalous tendency of the radius of gyration to become infinite for the special case of a Hookean potential which has been observed by previous investigators. Further, this study involves the Eulerian velocity–velocity correlation function rather than the Lagrangian correlation used in the previous studies.

For an arbitrary, time‐dependent, homogeneous flow, an expression is obtained for the stress tensor describing the behavior of a dilute solution of macromolecules idealized as elastic dumbbells with Brownian motion. The result is given in terms of the relaxation modulus from linear viscoelasticity. The general relaxation modulus found for an arbitrary, nonlinear, elastic dumbbell is specialized to that for a finitely extendible nonlinear elastic (FENE) dumbbell. Good agreement is found when the result is compared with Warner's numerical calculations for the FENE dumbbell.

Fano’s tetradic representation of Liouville–Neumann operators is used as the basis of a systematic procedure for solving an important class of rheological equations of state. This procedure is complementary to the techniques which Bird and his co‐workers have used in their studies of complicated rheological models. To illustrate how our method works it is applied to a rheological model of the Maxwell–Oldroyd type. Explicit formulas are derived for the stress tensors associated with a number of time‐independent, homogeneous flows.

A convincing prrof of the positive definite property of an altered form of the Kirkwood–Riseman diffusion tensor is provided by excluding the nonphysical singularities. (AIP)

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.

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.