Search

Local similarity solutions are presented for the stress field of a fluid described by the Oldroyd-B viscoelastic constitutive equation near the singularity caused by the intersection of a planar free surface and a solid surface along which Navier’s slip law holds, the partial-slip/slip problem. For the case where the velocity field is given by Newtonian kinematics, the elastic stress field is predicted to have a logarithmic singularity as the point of attachment of the free surface is approached. Asymptotic analysis for the fully-coupled flow, where the stress and flow fields are determined simultaneously, results in a local form for the flow and elastic stress fields that is similar in form to that for the decoupled case. For both the coupled and decoupled flow problems, the strength of the singularity depends on the dimensionless solvent viscosity and the slip coefficient, but not upon the Deborah number. The asymptotic results for the coupled flow differ from the predictions with Newtonian kinematics i...

The local solution behavior near corners formed by the intersection of a slip surface with either a no-slip or a shear-free boundary is analyzed by finite element calculations of the two-dimensional flow of an inertialess Newtonian fluid in several model flow geometries; these flows are the flow in a tapered contraction, a sudden expansion and the extrudate swell from a planar die. Local finite element mesh refinement based on irregular, embedded elements is used to obtain extremely fine resolution of the velocity and pressure fields near the region where there is a sudden change in boundary condition. The calculations accurately reproduce the expected asymptotic behavior for a shear-free surface intersecting a no-slip boundary, where the solution is given by a self-similar form for the velocity and pressure fields. Replacing the shear-free condition with a slip condition yields a similar form for the local velocity and pressure fields and indicates that the slip boundary behaves, to leading order, as a shear-free surface. Calculations for a slip boundary intersecting a shear-free surface yield similar results, with the local behavior being given by asymptotic analysis for two shear-free surfaces intersecting to form a wedge. These results suggest that replacing the no-slip boundary condition in planar Newtonian die swell with a slip boundary condition can give rise to local behavior of velocity gradients and pressure which is more singular than the flow created with no-slip boundary conditions. This prediction is confirmed by calculations of Newtonian die swell with slip. These calculations also demonstrate that the local solution in Newtonian die swell is sensitive to the details of the numerical method.

The two‐dimensional, free‐surface flow of a Newtonian fluid exiting from a planar die is computed by finite element analysis using quasiorthogonal mesh generation and local mesh refinement with irregular, embedded elements to obtain extreme resolution of the velocity and pressure fields near the die edge, where the fluid sheet attaches to the solid boundary. Calculations for the limit of large surface tension, the stick‐slip problem, reproduce the singular behavior near the die edge expected from asymptotic analysis using a self‐similar form for the velocity field. Results for finite capillary number (Ca) predict that the meniscus separates from the die at a finite contact angle and suggest that the capillary force enters the dominant normal stress balance at the die edge through an infinite curvature, as previously suggested by Schultz and Gervasio. The size of this region with large positive curvature increases with increasing Ca, and the strength of the singularity is in good agreement with theoretical predictions for a straight meniscus attached to the die at the appropriate contact angle predicted by the simulations. The contact angle appears to be determined from matching of the inner solution structure valid near the singularity with the bulk flow, in agreement with arguments made by Ramalingam; increasing the Reynolds number decreases the contact angle, corroborating this effect. Introducing fluid slip along the surface of the die changes the structure of the singularity in the pressure and stresses, but does not alleviate the singular behavior. In fact, the calculations with slip coefficients small enough not to change the bulk solution are more difficult than calculations with the no‐slip boundary condition.

Finite‐amplitude waves propagating at constant speed down an inclined fluid layer are computed by finite element analysis of the Navier–Stokes equations written in a reference frame translating at the wave speed. The velocity and pressure fields, free‐surface shape and wave speed are computed simultaneously as functions of the Reynolds number Re and the wave number μ. The finite element results are compared with predictions of long‐wave, asymptotic theories and boundary‐layer approximations for the form and nonlinear transitions of finite‐amplitude waves that evolve from the flat film state. Comparisons between the finite element calculations and the long‐wave predictions for fixed μ and increasing Re agree well for small‐amplitude waves. However, for larger‐amplitude waves the long‐wave results diverge qualitatively from the finite element predictions; the long‐wave theories predict limit points in the solution families that do not exist in the finite element solutions. Comparisons between the finite ele...

A constitutive equation for computing particle concentration and velocity fields in concentrated monomodal suspensions is proposed that consists of two parts: a Newtonian constitutive equation in which the viscosity depends on the local particle volume fraction and a diffusion equation that accounts for shear‐induced particle migration. Particle flux expressions used to obtain the diffusion equation are derived by simple scaling arguments. Predictions are made for the particle volume fraction and velocity fields for steady Couette and Poiseuille flow, and for transient start‐up of steady shear flow in a Couette apparatus. Particle concentrations for a monomodal suspension of polymethyl methacrylate spheres in a Newtonian solvent are measured by nuclear magnetic resonance (NMR) imaging in the Couette geometry for two particle sizes and volume fractions. The predictions agree remarkably well with the measurements for both transient and steady‐state experiments as well as for different particle sizes.

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...

The dynamics of concentrated solutions of rodlike molecules in nonhomogeneous shear flow are explored using a consistent numerical simulation of the Doi diffusion equation and the nonhomogeneous Onsager model of excluded-volume rod interactions. Simulations of planar, wall-driven shear flow show that out-of-plane structure instabilities occur when nematic anchoring constraints at the boundaries are removed. A new composite state with misaligned logrolling and flow-aligning domains is observed for pressure-driven flow in a planar channel. These results mark the first use of the Doi diffusion equation to show how a nonhomogeneous flow field generates sharp inter-domain interfaces analogous to those observed in rheological experiments.

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.

Isotropic-Nematic and Nematic-Nematic transitions from a homogeneous base state of a suspension of high aspect ratio, rod-like magnetic particles are studied for both Maier-Saupe and the Onsager excluded volume potentials. A combination of classical linear stability and asymptotic analyses provides insight into possible nematic states emanating from both the isotropic and nematic non-polarized equilibrium states. Local analytical results close to critical points in conjunction with global numerical results (Bhandar, 2002) yields a unified picture of the bifurcation diagram and provides a convenient base state to study effects of external orienting fields., Comment: 3 Figures

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.