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Colloids with short range attractions self-assemble into sample-spanning structures, whose dynamic nature results in a thermokinematic memory of the deformation history, also referred to as "thixotropy." Here, we study the origins of the thixotropic effect in these time- and rate-dependent materials by investigating hysteresis across different length scales: from particle-level local measurements of coordination number (microscale), to the appearance of density and velocity fluctuations (mesoscale), and up to the shear stress response to an imposed deformation (macroscale). The characteristic time constants at each scale become progressively shorter, and hysteretic effects become more significant as we increase the strength of the interparticle attraction. There are also strong correlations between the thixotropic effects we observe at each scale.

We identify the sequence of microstructural changes that characterize the evolution of an attractive particulate gel under flow and discuss their implications on macroscopic rheology. Dissipative Particle Dynamics (DPD) is used to monitor shear-driven evolution of a fabric tensor constructed from the ensemble spatial configuration of individual attractive constituents within the gel. By decomposing this tensor into isotropic and non-isotropic components we show that the average coordination number correlates directly with the flow curve of the shear stress vs. shear rate, consistent with theoretical predictions for attractive systems. We show that the evolution in non-isotropic local particle rearrangements are primarily responsible for stress overshoots (strain-hardening) at the inception of steady shear flow and also lead, at larger times and longer scales, to microstructural localization phenomena such as shear banding flow-induced structure formation in the vorticity direction., 15 pages, 4 figures

This study compares the predictions of several “suspension temperature” models of particle migration to laser Doppler velocimetry measurements in a concentrated suspension of noncolloidal spheres. We compare the shear rate, concentration, and suspension temperature profiles in narrow-gap Couette flow. The models predict the observed macroscopic shear rate and concentration profiles well at moderate bulk particle concentration but diverge from one another and from the data at high concentrations. In addition, the predictions of the models compare poorly with suspension temperature measurements. Most of the models greatly underpredict the magnitude of the scalar temperature, capturing instead only the magnitude of the smaller two diagonal components of the temperature tensor. Also, the models do not predict the observed variation of the suspension temperature with particle concentration. Our investigation shows that both neglect of suspension temperature anisotropy and qualitative choices of model coefficie...

High-resolution laser Doppler velocimetry and flow-induced birefringence measurements were used to determine velocity and stress fields and to calculate a transient planar elongation viscosity profile for viscoelastic flow on the shear-free centerplane of a planar contraction geometry. A novel, two-component Boger fluid was developed that enabled Weissenberg numbers, We, as high as 2.9 to be attained and allowed the stress-optical coefficients of the fluid components to be measured. The experimental apparatus was designed to subject the fluid traveling on the centerplane to Hencky strains, e, as great as 3.5. Despite the high We and e, the flow induced only a weakly nonlinear viscoelastic stress response in the fluid. A spectral decomposition of predictions of linear and nonlinear constitutive equations was used to demonstrate that the weakly nonlinear response was a result of the nonhomogeneous form of the centerplane strain-rate profile in conjunction with the finite response time of the individual relaxation modes. The degree of nonlinear stress response in this experiment cannot be increased by increasing the contraction ratio to increase the total Hencky strain experienced by a fluid element traveling along the centerplane, increasing the flow rate to increase the strain rate at a given point along the centerplane, or introducing a taper to the contraction geometry to change the strain rate profile. Thus, the experimental configuration of flow through a contraction appears to be of limited value in evaluating the accuracy of nonlinear constitutive models in predicting the stress response of a viscoelastic polymer solution to a generalized elongational flow.

The linear stability of the flow of an Oldroyd-B fluid through a linear array of cylinders confined in a channel is analyzed by computing both the steady-state flows and the linear stability of these states by finite element analysis. The linear stability of two-dimensional base flows to three-dimensional perturbations is computed both by time integration of the linearized evolution equations for infinitesimal perturbations and by an iterative eigenvalue analysis based on the Arnoldi method. These flows are unstable to three-dimensional perturbations at a critical value of the Weissenberg number that is in very good agreement with the experimental observations by Liu [Viscoelastic Flow of Polymer Solutions around Arrays of Cylinders: Comparison of Experiment and Theory, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1997] for flow of a polyisobutylene Boger fluid through a linear periodic array of cylinders. The wave number of the disturbance in the neutral or spanwise direction also is in good agreement with experimental measurements. For closely spaced cylinders, the mechanism for the calculated instability is similar to the mechanism proposed by Joo and Shaqfeh [J. Fluid Mech. 262 (1994) 27] for the non-axisymmetric instability observed in viscoelastic Couette flow, where perturbations to the shearing component of the velocity gradient interact with the polymer stresses in the base flow. However, when the cylinder spacing is increased, we discover a new mechanism for instability that involves the coupling between the elongational component of the velocity gradient and the streamwise normal stress in the base state. In addition, the most unstable disturbance in the Couette geometry is over-stable (leading to time-periodic states) whereas the instability calculated for closely spaced cylinders grows monotonically with time. This apparent inconsistency is resolved by the observation that in a complex flow, the evolution of the perturbations to the base flow can be transient in a Lagrangian frame of reference, while steady in an Eulerian frame.

Recent statistical constitutive models of suspensions of neutrally buoyant, noncolloidal, solid spheres in Newtonian fluids suggest that the particles migrate in response to gradients in “suspension temperature,” defined as the average kinetic energy contained in the particle velocity fluctuations. These models have not yet been compared systematically with experimental data. In addition the “temperature'’ models assume isotropic particle velocity fluctuations, since the suspension temperature is given as a scalar. However, highly anisotropic particle velocity fluctuations have been observed experimentally, which suggests that a suspension temperature tensor is more realistic. We use laser Doppler velocimetry to measure particle velocity fluctuations arising from interparticle collisions in a concentrated suspension under nearly homogeneous shear flow in a narrow-gap concentric cylinder Couette device. We compare the relative sizes of the fluctuating velocity components and determine the variation of each...

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.

Laser Doppler velocimetry and flow‐induced birefringence are used to measure the rate of deformation and the principal components of the refractive index tensor in a 5% polyisobutylene (PIB) solution in tetradecane (C14) flowing along the centerplane of an abrupt 3.97:1 planar contraction. The stress optical law is used to interpret the birefringence data in terms of the normal stress difference, which is used to calculate a transient elongational viscosity defined along the centerplane. These measurements are compared directly to predictions of six multimode, differential constitutive models (Oldroyd‐B, White–Metzner, Acierno et al., Giesekus, Bird–DeAguiar, and Phan‐Thien–Tanner) that are fit to steady and small amplitude oscillatory shear flow data for the PIB/C14 solution. The fluid exhibits slight elongational thickening followed by apparent extensional thinning at higher elongation rates. We believe that this ‘‘thinning’’ behavior is due to the decreased residence time of the polymer molecules in the high‐strain‐rate region as the flow rate (and maximum elongation rate) is increased. The nonlinear constitutive equations, except for the White–Metzner model, are virtually indistinguishable in their description of the dynamical response of the fluid in this experiment; however, the Phan‐Thien–Tanner model gives the best quantitative fit to the data. These results point to the need for experiments in which the fluid flowing along the centerline is subjected to a greater total elongational strain.

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...

Laser Doppler velocimetry (LDV) and video flow visualization are used to investigate the creeping motion of a highly elastic, constant-viscosity fluid flowing past a cylinder mounted centrally in a rectangular channel. A sequence of viscoelastic flow transitions are documented as the volumetric flow rate past the cylinder is increased and elastic effects in the fluid become increasingly important. Velocity profiles clearly show that elasticity has almost no effect on the kinematics upstream of the cylinder, but that the streamlines in the wake of the cylinder are gradually shifted further downstream . Finite element calculations with a nonlinear constitutive model closely reproduce the evolution of the steady two-dimensional velocity field. However, at a well defined set of flow conditions the steady planar stagnation ow in the downstream wake is experimentally observed to become unstable to a steady, three-dimensional cellular structure. The Reynolds number at the onset of the flow instability is less than 0.05 and inertia plays little role in the flow transition, LDV measurements in the wake close to the cylinder reveal large spatially periodic fluctuations of the streamwise velocity that extend along the length of the cylinder and more than five cylinder radii downstream of the cylinder. Fourier analysis shows that the characteristic spatial wavelength of these flow perturbations scales closely with the cylinder radius R . Flow visualization combined with LDV measurements also indicates that the perturbations in the velocity field are confined to the narrow region of strongly extensional flow near the downstream stagnation point. A second flow transition is observed at higher flow rates that leads to steady translation of the cellular structure along the length of the cylinder and time-dependent velocity oscillations in the wake. Measurements of the flow instability are presented for a range of cylinder sizes, and a stability diagram is constructed which shows that the onset point of the wake instability depends on both the extensional deformation of the fluid in the stagnation flow and the shearing flow between the cylinder and the channel.

Finite element analysis is used to compute the inertialess flow of a viscoelastic fluid around a sphere falling in a cylindrical tube. Calculations are reported for three differential constitutive models with constant viscosities: the Upper-Convected Maxwell model (UCM), the Oldroyd-B model (OLDB) and the dumbbell model of Chilcott and Rallison (CR). Calculations are based on the Explicitly Elliptic Momentum Equation (EEME) and the Elastic Viscous Split Stress (EVSS) methods for calculations with models without and with a Newtonian solvent contribution, respectively. The calculations converged with mesh refinement for all three models for values of De below 1.6. Calculations with the UCM and OLDB models are limited below De = 2 by loss of convergence of the Newton iteration. Mesh refinement does not seem to alleviate these limits. Calculations with the CR model and moderate values of the maximum extensibility of the dumbbell L converge to higher values of De ; however, computations for large values of L require fine meshes in the stagnation region behind the sphere.

Finite‐element predictions of steady, two‐dimensional viscoelastic flow based on the elastic‐viscous split stress formulation of Rajagopalan et al. [Rajagopalan, D., R. C. Armstrong, and R. A. Brown, ‘‘Finite Element Methods for Calculation of Steady, Viscoelastic Flow Using Constitutive Equations with a Newtonian Viscosity,’’ J. Non‐Newt. Fluid Mech. 36, 159–192 (1990)] are compared to birefringence measurements of stress for flow between eccentric rotating cylinders of a solution of polystyrene in tricresyl phosphate. Steady and small‐amplitude oscillatory shearing measurements are used to characterize the shear flow rheology of the fluid, and a four‐mode Giesekus model with a Newtonian solvent viscosity is fit to the rheological data. Comparison of the stress fields measured by flow‐induced birefringence in the one‐dimensional flow between concentric cylinders to the mechanically measured viscometric data show that the optical data compare reasonably well with the mechanical data. Comparison of optical...

The purpose of this work is to extend a theoretical model of layered (laminar) media for SHS combustion presented in an earlier article [1] to explore possible mechanisms for after-burning in SHS (i.e., gasless) combustion. As before, our particular interest is how the microscopic geometry of the solid reactants is reflected in the combustion wave and in the reaction product. The model is constructed from alternating lamina of two pure reactants that interdiffuse exothermically to form a product. Here, the laminar model is extended to contain layers of differing thicknesses. Using asymptotic theory, it was found that under certain conditions, the combustion wave can become “detached,” and an initial thin flame propagates through the media, leaving a slower, thicker flame following behind (i.e., afterburning). Thin laminae are consumed in the initial flame and are thick in the secondary. The thin flame has a width determined by the inverse of the activation energy of diffusion, as found previously. The width of the afterburning zone, however, is determined by the absolute time of diffusion for the thicker laminae. Naturally, when the laminae are all the same thickness, there is only one thin flame. The condition for the appearance of afterburning is found to be contingent on the square of the ratio of smallestto-largest thicknesses being considerably less than unity.

The EEME/finite-element method is used to compute steady flows of fluids with constitutive behavior described by the Modified Upper Convected Maxwell Model of Apelian and a modified form of the dumbbell model of Chilcott and Rallison through abrupt, axisymmetric contractions. Both constitutive models predict constant viscosity and shear thinning first normal stress coefficient, but differ qualitatively in the behavior of the elongational viscosity. Asymptotic analysis for both models predicts that the solution has Newtonian-like spatial structure near the reentrant corner and integrable stresses and velocity gradients there. With the Newtonian-like asymptotics, the stress field can be approximated by conventional Lagrangian finite elements and computed by the streamline upwind Petrov Galerkin (SUPG) method. The finite element calculations are stable and convergent: higher values of Deborah number are reached with increasing mesh refinement. Moreover, the predicted asymptotic structure of the stress and velocity fields is recovered near the corner in the calculations. Calculations with both constitutive equations show the stretching of the Newtonian corner vortex toward the reentrant corner and its growth upstream with increasing Deborah number for 4:1 and 8:1 contraction ratios. The characteristics of the computed vortex are in semi-quantitative agreement with experiments for Boger fluids for which the flow is axisymmetric and steady.

Time-dependent simulations using the EEME finite-element method are reported for calculation of the linear stability and nonlinear dynamics of the transition to time-periodic, viscoelastic flow in axisymmetric TaylorCouette flow between parallel cylinders. The linear stability analysis is based on time integration of the linearized finite-element equations and reproduces the oscillatory linear instability recently analyzed by Larson et al. past a critical Deborah number Dec. Linear analysis presented here shows that the viscoelastic instability corresponds to a secondary flow structure composed of multiple toroidal flow cells nested radially, and that these cells travel across the gap between the cylinders. Nonlinear simulations demonstrate the existence of supercritical, i.e. for De - Dec > 0, time-periodic flow states. For only small changes in De >Dec, the flatness of the neutral stability curve with variation in the flow cell height leads to nonlinear interactions between flows that are closely spaced in De. These interactions will make the observation of simple oscillations near onset extremely difficult.

Experimental measurements of the shear stress and normal stresses in the viscometric flow of a well-characterized polyisobutylene/polybutene solution between a cone-and-plate and between parallel plates show the onset of an “anti-thixotropic” flow transition, which results in a time-dependent, apparent shear-thickening of the viscosity and first normal stress difference. Measurements of the critical conditions for the onset of the flow transition, made by systematically varying the plate separation in the parallel-plate geometry, demonstrate that this rotational instability is a function of the angular velocity in the flow and initially grows exponentially in time, in agreement with theoretical predictions for an Oldroyd-B model. However, experiments show that the disturbance which causes this flow transition is nonaxisymmetric, over-stable in time and subcritical in γ. Spectral analysis also shows that the nonlinear flow which ultimately develops is temporally aperiodic. These features of the flow transition do not agree with the existing theoretical analysis. The experimentally constructed flow-stability diagram shows that, for a given fluid, the instability occurs at a critical Deborah number which is insensitive to the specific geometrical parameters, temperature, and shear-rate of the flow. This interpretation of the elastic instability clarifies recent measurements of the flow stability for the Ml viscoelastic test fluid.

It is generally assumed in this work, as elsewhere, that the mass diffusion coefficient has an Arrhenius temperature dependence. This allows a combustion wave propagated by thermal conduction to develop similar to more conventional combustion systems. It is found that for realistic systems the problem must involve a three length scale analysis. Similar to the classic theory on premixed, gas-phase flames, the largest scale is identified with thermal conduction and a “reaction zone” scale proportional to the inverse of the mass diffusion activation energy is associated with a small length over which significant mass diffusion is possible. Unlike classic flame theory a still smaller scale is identified with the length over which mass diffusion takes place (i.e., the size of a typical domain of alloyable constituent). Flame speeds are derived for three different geometric configurations of a binary system using a singular perturbation analysis. First a system where the constituents are arrayed in alt...

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 steady, two-dimensional flows and interface shapes of the rimming flow of Newtonian and viscoelastic liquid films are studied by finite-element analysis. The viscoelastic flow calculations are based on the elastic-viscous split stress (EVSS) formulation for differential constitutive models. The EVSS formulation is derived by taking into account the mathematical type of the momentum, continuity and constitutive equations and is extended in this paper to calculation of free-surface flows. Calculations for a viscous Newtonian fluid demonstrate the balance between viscous forces and gravity which sets the shape of the interface of the liquid film coating the inside of the rotating cylinder. The liquid shape evolves from a concentric and circular film at high rotation rates to become thicker on the rising surface as the rotation rate is lowered. No steady flows with continuous films are found to exist below a minimum rotation rate, Ω = Ωc, where the family of flows evolves back toward higher values of Ω. Multiple solutions are predicted for a range of rotation rates, Ω > Ωc, and unstable flows develop a pronounced bulge on the rising side of the film. Asymptotic analysis for a thin film predicts this limiting rotation rate. Adding viscoelasticity to the liquid, as modelled by the Giesekus constitutive equation, leads to the existence of steady solutions at lower rotation rates and causes the bulge to appear on stable films. The minimum rotation rate for steady, viscoelastic flow shifts to lower values as the time constant of the fluid is increased.

The steady-state and time-dependent flow transitions observed in a well-characterized viscoelastic fluid flowing through an abrupt axisymmetric contraction are characterized in terms of the Deborah number and contraction ratio by laser-Doppler velocimetry and flow visualization measurements. A sequence of flow transitions are identified that lead to time-periodic, quasi-periodic and aperiodic dynamics near the lip of the contraction and to the formation of an elastic vortex at the lip entrance. This lip vortex increases in intensity and expands outwards into the upstream tube as the Deborah number is increased, until a further flow instability leads to unsteady oscillations of the large elastic vortex. The values of the critical Deborah number for the onset of each of these transitions depends on the contraction ratio β, defined as the ratio of the radii of the large and small tubes. Time-dependent, three-dimensional flow near the contraction lip is observed only for contraction ratios 2 [les ] β [les ] 5, and the flow remains steady for higher contraction ratios. Rounding the corner of the 4:1 abrupt contraction leads to increased values of Deborah number for the onset of these flow transitions, but does not change the general structure of the transitions.

Time-dependent processes of solid propellant deflagration are modeled and a numerical simulation of ignition and extinction is undertaken. Since the analytic theory of ignition is well developed we rely on it to verify the ability of the numerical algorithm to track stiff, time-dependent features of an ignition event. The verification of our numerical method provides a degree of confidence for the subsequent study of deradiation, which is less well understood. An activation energy asymptotic theory for deradiation is developed, which predicts that an influx of radiation on the order of the steady heat release of the exothermic reaction will, at near term, extinguish the propellant. Beyond this, conduction processes inherited from the thermal profile at the instant of deradiation may, or may not, cause resumption of pyrolysis, and, presumably, the deflagration as a whole. The analytic theory ignores the subsequent impact of a possible gas-phase flame that is detached many conductive-diffusive lengths away from the gas solid surface. The assumption is based on the observation that for most physical systems the gas density—and hence its thermal capacitance—is many times smaller than that of the solid and thus lacks the potential to alter the surface temperature after deradiation. The simulation is used to verify more complicated and more physical situations than those accessible via the theory, including deradiation with a detached flame and in the presence of an instability.

The steady-state, similarity solutions of the flow of an upper-convected Maxwell fluid through a tube with a porous wall are constructed by asymptotic and numerical analyses as functions of the direction of flow through the tube, the amount of elasticity in the fluid, as measured by the Deborah number De, and the degree of fluid slip along the tube wall. Fluid slip is assumed to be proportional to the local shear stress and is measured by a slip parameter β that ranges between no-slip (β = 1) and perfect slip (β = 0). The most interesting results are for fluid injection into the tube. For β = 1, the family of flows emanating from the Newtonian limit (De = 0) has a limit point where it turns back to lower values of De. These solutions become asymptotic to De = 0) and develop an O(De) boundary layer near the tube wall with singularly high stresses matched to homogeneous elongational flow in the core. This solution structure persists for all nonzero values of the slip parameter. For β ≠ 1, a family of exact solutions is found with extensional kinematics, but nonzero shear stress convected into the tube through the wall. These flows differ for low De from the Newtonian asymptote only by the absence of the boundary layer at the tube wall. Finite difference calculations evolve smoothly between the Newtonian-like and extensional solutions because of approximation error due to under-resolution of the boundary layer. The radial gradient of the axial normal stress of the extensional flow is infinite at the centerline of the tube for De > 1; this singularity causes failure of the finite difference approximations for these Deborah numbers unless the variables are rescaled to take the asymptotic behavior into account.

Approximate analytical solutions of the local temperature variations, due to viscous heating, in channel flows of non-Newtonian fluids with small Nahme-Griffith numbers were obtained by the WKB-J method. The results are general and easy to use so that extensive numerical computations are not required. Comparisons between the approximate analytical solution and complete numerical solution showed that the relative differences were very small.

A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stressYgbelow which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only whenYgis below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value ofYgthere is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

The circumferential and radial profiles of velocity, pressure and stress are derived for the flow of model viscoelastic liquids between two slightly eccentric cylinders with the inner one rotating. Singular perturbation methods are used to derive expansions valid for small gaps between the cylinders, but for all Deborah numbers. Results for Newtonian, second-order, Criminale-Ericksen-Filbey, upper-convected Maxwell, and White-Metzner constitutive equation separate the effects of elasticity, memory, and shear thinning on the development of the large stress gradients that hinder numerical solutions with these models in more complicated geometries. The effect of the constitutive equation on the critical Deborah number for flow separation is presented.

A nonlinear response theory is employed to obtain the modification of the mean burning rate of a nonhomogeneous solid propellant. Although the methods used here are quite general, attention is focused on the interaction between the three-dimensional structure of the deflagration and the constitutive model. Based on a calculation of the mean burning rate as a function of the correlation length of a random nonhomogeneity, it is shown that the mean burning rate is greater than that of homogeneous burning for small correlation lengths, and is less for larger correlation lengths. Thus, this study indicates that a solid composed of particles of two different propellants would burn more slowly than their mean rate if the size of the particles were large, and more quickly if their size were small. As many of the methods used here are new to this application, some comment is also made on the utility of this approach in computing mean burning rate statistics for random media.

Under the influence of a uniform and parallel magnetic field, a ferromagnetic fiber suspended in a Newtonian fluid rotates to align with the field direction. This study examines the field-induced rotation process for an individual non-Brownian axisymmetric ellipsoid suspended in a stagnant Newtonian fluid. Theoretical predictions are derived by a perturbation analysis for the limiting case where the strength of the applied magnetic field far exceeds the saturation magnetization of the ellipsoid. Numerical calculations are performed for the more general problem of an ellipsoid with known isotropic, non-hysteretic magnetic properties, using nickel and a stainless steel as examples. The analysis encompasses materials with field-induced, nonlinear magnetic properties, distinguishing these results from the simpler cases where the particle magnetization is either independent of, or linearly dependent on, the strength of the applied external field. In this study, predictions indicate that when the ellipsoid is magnetically saturated, the particle rotation is governed by the magnetoviscous time constant,τ MV = ηs/e0 M s 2 . It is found that the rotation rate depends strongly on the aspect ratio,a/b, of the ellipsoid, but only weakly on the dimensionless magnetization,M s/H 0.

Laser Doppler velocimetry (LDV) measurements of the axial and radial velocity components are reported for the axisymmetric contraction flow of a viscoelastic solution of polyisobutylene dissolved in tetradecane. This fluid exhibits shear-thinning of both the viscosity and first normal stress difference; the viscosity is well described by the Carreau-Yasuda model. Shear-rate-dependent Reynolds and Deborah numbers are used to described the flow conditions. Measurements at low flow rates for contraction ratios of 2:1, 4:1, and 8:1 indicate that the flow near the contraction plane scales with the small tube radius. A viscoelastic corner vortex is detected at moderate flow rates, yet vortex growth is not observed with increased flow rate, because inertial forces press the vortex into the outer corner. At high flow rates, where both the Reynolds number and Deborah number are large, the flow near the contraction plane is divided into an accelerating core and an outer region where the flow retains its upstream profile. The magnitude of the local rate-of-strain is calculated from the axial and radial velocity profiles. In the core, the magnitude of >50s −1 and its value is dominated by the axial gradients ∂ v /∂ z and ∂ v r /∂ z . Large extension rates along the centerline suggest that extensional thinning occurs in this core region. Finite element calculations for a Carreau-Yasuda fluid with the same viscosity as the PIB/C14 solution and under the same flow conditions do not predict the two-regions of the flow.

The two-dimensional flows between two slightly eccentric cylinders with the inner one rotating are calculated by finite-element methods for five viscoelastic constitutive equations each derived as a limit of the Giesekus model. Comparisons with exact results for Newtonian, second-order, and corotational Maxwell-like fluids set the accuracy of the calculations as a function of eccentricity and Deborah number (De). Computer-implemented perturbation methods are used to demonstrate bifurcation and turning points in De for an upper-convected Maxwell fluid. The locations of the limit points are moderately stable to extensive mesh refinement and, therefore, seem to be an intrinsic property of this constitutive equation. Similar solution pathology is demonstrated for the three-constant Oldroyd-B model, but no limiting value of De is found for calculations with the Leonov-like version of the Giesekus fluid.

Finite element calculations of two-dimensional flows of viscoelastic fluids in a journal bearing geometry reported in an earlier paper (J. Non-Newt. Fluid Mech. 16 (1984) 141-172) are extended to higher eccentricity (ρ = 0.4); at this higher eccentricity flow separation occurs in the wide part of the gap for a Newtonian fluid. Calculations for the second-order fluid (SOF), upper-convected Maxwell (UCM), and the Giesekus models are continued in increasing Deborah number for each model until either a limit point is reached or oscillations in the solution make the numerical accuracy too poor to warrant proceeding. No steady solutions to the UCM model were found beyond a limit point De c , as was the case for results at low eccentricities. The value of De c was moderately stabel to mesh refinement. A limit point also terminated the calculations with a SOF model, in contradiction to the theorems for uniqueness and existence for this model. The critical value of De increased drastically with increasing refinement of the mesh, as expected for solution pathology caused by approximation error. Calculations for the Giesekus fluid with the mobility parameter α ≠ O showed no limit points, but failed when irregular oscillations destroyed the quality of the solution. The behavior of the recirculation region of the flow and the load on the inner cylinder were very sensitive to the value of α used in the Giesekus model. The recirculation disappeared at low values of De except when the mobility parameter α was so small that the viscosity was almost constant over the range of shear rates in the calculations. The recirculation persisted over the entire range of accessible De for the UCM fluid, the limit of α = O of the Giesekus model. The behavior of the recirculation is coupled directly to the viscosity by calculations with an inelastic fluid with the same viscosity predicted by the Giesekus model.

Numerical calculations of complex, two-dimensional flows of viscoelastic fluids fail when the elastic contribution to the stresses, measured by a Deborah number, becomes comparable to the viscous contribution. Reasons for the limit on Deborah number are explored by a sequence of finite element calculations with contravariant convected Maxwell and second-order fluid models. The possibilities of bifurcation or loss of a steady, two-dimensional flow field are ruled out by employing continuation methods for flow through a planar contraction and by existence and uniqueness proofs for a second-order fluid in a driven cavity. Error in the finite element approximation to steep gradients of stress causes the breakdown of all calculations. This is most clearly seen in the calculations for a second-order fluid, because the exact flow field for any Deborah number is known.

A new, automated, two-color laser Doppler velocimetry apparatus is described which has been constructed especially for mapping flow fields for viscoelastic fluids. This system is capable of measuring both slow, secondary flows and time-dependent motions that are characteristic of viscoelastic flow in complex geometries. Application of this apparatus to flow between eccentric cylinders and through an axisymmetric sudden contraction for a Newtonian fluid and for polyisobutylene (PIB) and polyacrylamide (PAC) solutions is described. In both geometries excellent agreement is found between calculations and experiments for Newtonian fluids, as expected. The Newtonian recirculation region which is present for flow between cylinders with large eccentricity disappears for the PAC solution and shrinks with increasing De for the PIB solution. These changes are attributed to the substantial shear thinning of the PAC solution and the slight shear thinning of the PIB mixture under the conditions studied. New flow transitions were discovered in the sudden contraction flow of the polyisobutylene solution. At a critical value of Deborah number De 1 ≈ 0.80 the flow changed from a steady, two-dimensional motion, nearly identical with the Newtonian result, to a time-periodic flow with a tangential velocity component that fluctuated about zero. Multiple, time-periodic motions and flow hysteresis were found for De 1 De De 2 ≈ 1.20. At De 2 the flow suddenly reverts to a two-dimensional, time-independent motion. A vortex emanating from the lip of the contraction was observed for De > De 2 .

We present a model for nonhomogeneous solid propellant combustion that treats the nonhomogeneity as a small random perturbation of a homogeneous deflagration. Our model is based on our previous work (Margolis and Armstrong, 1986) and, unlike most previous models which are quasi-steady and one dimensional, allows for a fully nonsteady, multidimensional response of the deflagration to both spatial and temporal nonhomogeneities. In one calculation, we assume a random spatial distribution for the solid pyrolysis rate coefficient and obtain fluctuation statistics for the burning propellant in terms of the statistics of the random propellant medium. In another calculation, we obtain the burning rate response to a deterministic external pressure fluctuation. Our results are particularly applicable to nitrocellulose/nitroglycerin propellants, which are typically “almost” homogeneous.

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