Slip occurs in the flow of two-phase systems because of the displacement of the disperse phase away from solid boundaries. This arises from steric, hydrodynamic, viscoelastic and chemical forces and constraints acting on the disperse phase immediately adjacent to the walls. The enrichment of the boundary near the wall with the continuous (and usually low-viscosity) phase means that any flow of the fluid over the boundary is easier because of the lubrication effect. Because this effect is usually confined to a very narrow layer — with typical thickness of 0.1–10 μm—it so resembles the slip of solids over surfaces that it has historically been given the same terminology. The restoring force for all the forces that cause an increase in concentration is usually osmotic, and this will always limit the effective slip. In dilute systems, concentration gradients can be present over relatively large distances out from walls, giving what might be interpreted on an overall basis as a thick solvent-only layer. However, as the concentration of the system increases, the layer gets thinner and thinner because it is more difficult to create with the large reverse osmotic force present. However, the enormous increase in the bulk viscosity with increase in concentration means that although thinner, the layer becomes, paradoxically, even more important. Slip manifests itself in such a way that viscosity measured in different size geometries gives different answers if calculated the normal way — in particular the apparent viscosity decreases with decrease in geometry size (e.g. tube radius). Also, in single flow curves unexpected lower Newtonian plateaus are sometimes seen, with an apparent yield stress at even lower stresses. Sudden breaks in the flow curve can also be seen. Large particles as the disperse phase (remember flocs are large particles), with a large dependence of viscosity on the concentration of the dispersed phase are the circumstances which can give slip, especially if coupled with smooth walls and small flow dimensions. The effect is usually greatest at low speeds/flow rates. When the viscometer walls and particles carry like electrostatic charges and the continuous phase is electrically conducted, slip can be assumed. In many cases we need to characterise the slip effects seen in viscometers because they will also be seen in flow in smooth pipes and condults in manufacturing plants. This is usually done by relating the wall shear stress to a slip velocity using a power-law relationship. When the bulk flow has also been characterized, the flow in real situations can be calculated. To characterise slip, it is necessary to change the size of the geometry, and the results extrapolated to very large size to extract unambigouos bulk-flow and slip data respectively. A number of mathematical manipulations are necessary to retrieve these data. We can make attempts to eliminate slip by altering the physical or chemical character of the walls. This is usually done physically by roughening or profiling, but in the extreme, a vane can be used. This latter geometry has the advantage of being easy to make and clean. In either case—by extrapolation or elimination—we end up with the bulk flow properties. This is important in situations where we are trying to understand the microstructure/flow interactions.

A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure

Nonlinear dynamics of the Frenkel-Kontorova model

We investigate the steady and transient shear and extensional rheological properties of a series of model hydrophobically modified ethoxylate−urethane (HEUR) polymers with varying degrees of hydrophobicity. A new nonlinear two-species network model for these telechelic polymers is described which incorporates appropriate molecular mechanisms for the creation and destruction of elastically active chains. Like other recent models we incorporate the contributions of both the bridging chains (those between micelles) and the dangling chains to the final stress tensor. This gives rise to two distinct relaxation time scales:  a short Rouse time for the relaxing chains and a longer network time scale that depends on the aggregation number and strength of the micellar junctions. The evolution equations for the fraction of elastically active chains and for the conformation tensors of each species are solved to obtain the total stress arising from imposed deformations. The model contains a single adjustable nonlinea...

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Rheology and Dynamics of Associative Polymers in Shear and Extension: Theory and Experiments

The stationary solution for the transient network model of reversibly crosslinked gels is found under arbitrary macrodeformation. For shear flow with constant shear rate γ, the number of active chains, stationary viscosity, first and second normal stress differences are calculated as functions of γ. Elongational flow with constant elongational rate ϵ is also studied. It is found that these stationary properties depend rather sensitively on the chain breakage function β(r) and the recombination probability p of the sticky dangling ends. On the basis of the polymer statistics, a specific form of β(r) = βo exp kr is proposed, where βo and k are functions of the temperature T and the molecular weight M of the polymer chain. Stationary viscoelastic properties are shown to exhibit an exponential dependence on T due to the activation process for the junction dissociation, differing markedly from an uncrosslinked polymer melt whose viscosity varies as a power of the temperature. Thickening and thinning conditions for both types of flow are examined. It is shown that limiting behaviour of the shear viscosity under high shear rate obeys the scaling law η(γ) ≈ γ −2 n ( n +1) if β(r) is proportional to rn at high stretching.

Viscoelastic properties of physically crosslinked networks

The rheological behaviour of dilute solutions of finitely extensible non-linear elastic (FENE) dumbbells in both steady state and transient shear and simple elongational flow is investigated. Three dumbbell models are compared: the original FENE model with the Warner spring force, which is treated by brownian dynamics simulations, and the FENE-P model based on the Peterlin approximation and the FENE-CR model as suggested by Chilcott and Rallison, which are treated by standard numerical techniques. It is shown that in the linear viscoelastic limit and in steady state flows the behaviour is similar, except for the FENE-CR dumbbell in shear flow, modelling a Boger fluid. In transient flows larger differences appear.

A detailed comparison of various FENE dumbbell models

A new numerical approach is presented to exactly solve the convection equation arising in network theories. The method is based on a direct stochastic interpretation of the convection equation. We show that with this approach models can be studied extensively which are not solvable analytically. It turns out that a conceptually simple approach to network theories predicts a qualitatively satisfying rheological behavior.

A numerical stochastic approach to network theories of polymeric fluids

A numerical stochastic approach allows the exact solution of the convection equation arising in network theories. We now want to show the flexibility and the limits of this approach by studying the rheological properties of different kinds of models.

Rheological properties of network models with configuration-dependent creation and loss rates

A dilute polymer solution is modeled by linear and nonlinear dumbbells suspended in a Newtonian solvent. The Langevin equations governing the motion of the dumbbells in the tube are solved with the help of Brownian dynamics simulations consistent with the momentum balance equation. To this purpose a previously presented consistent numerical approach has to be specialized to tube flow. The models show typical features as the slip effect, the flow enhancement, and the reduction in viscosity with decreasing tube radius.

A Consistent Numerical Analysis of the Tube Flow of Dilute Polymer Solutions

With the help of Brownian dynamics simulations we study the rheological properties of polymer Hookean dumbbell models with anisotropic friction. The friction tensor depends on the actual configuration of the dumbbell. The resulting equation of motion is a Langevin equation with multiplicative noise. The correlations of the ‘‘Langevin’’ Brownian forces appearing in this equation turn out to be anisotropic, too. They should be carefully distinguished from the ‘‘smoothed out’’ Brownian forces which appear in the standard derivations of the Fokker–Planck equation in kinetic theory. We will discuss the rheological properties in steady shear flow for two different friction tensors.

Rheological properties of polymer dumbbell models with configuration‐dependent anisotropic friction

Starting from rigorous expressions derived from phase space kinetic theory for dumbbell models of polymer solutions, a new numerical approach is presented. It enables one to solve the Langevin equations governing the motion of the dumbbells in a confined geometry consistently with the momentum balance equation. As an example, we discuss the flow of a polymer solution between two parallel shearing planes. For this purpose, we consider linear and nonlinear dumbbell models and investigate typical phenomena such as, for example, the slip effect.

Peter Biller

Papers

A numerical stochastic approach to network theories of polymeric fluids

Rheological properties of network models with configuration-dependent creation and loss rates

A Consistent Numerical Analysis of the Tube Flow of Dilute Polymer Solutions

Rheological properties of polymer dumbbell models with configuration‐dependent anisotropic friction

The flow of dilute polymer solutions in confined geometries: a consistent numerical approach