Wall slip

In the derivation of the Hele-Shaw equation, we have applied the so-called no-slip boundary condition at the liquid-solid interface, where the liquid is assumed to adhere, thus to have no velocity relative to the solid surface. 

Wall slipping has also been proposed as a possible mechanism for some surface defects of injection-molded parts (Denn 2001). In a study of several blends of BPA polycarbonate and ABS resins, Hobbs (1996) found that periodic surface irregularities can be made at high injection rates. The periodicity is produced due to the oscillation of the flow front between the two mold walls, as the wall slip first occurs on one surface and then the other. 

Early microstructural discussions suggested that slip occurs because the polymer molecules at the wall align themselves more strongly with the flow than those 

Some phenomenological approaches for handling the slip boundary condition have been proposed, where the slip velocity is takes as an empirical function of the wall shear stress Rosenbaum and Hatzikiriakos (1997) introduced a power-law relation for slip velocity  

Phan-Thien (1988) and Tanner (1994) have considered the situation of non-uniform stresses at outlet and inlet. Thus the idea and methods regarding the partial slip were discussed. 

In addition to these impediments to rheological measurements, some complex fluids exhibit wall slip, yield, or a material instability, so that the actual fluid deformation fails to comply with the intended one. A material instability is distinguished from a hydrodynamic instability in that the former can in principle be predicted from the constitutive relationship for the material alone, while prediction of a flow instability requires a mathematical analysis that involves not only the constitutive equation, but also the equations of motion (i.e., momentum and mass conservation). 

In some cases the presence of slip is fairly obvious, as are its causes. For example, when an aqueous foam is sheared between smooth surfaces, the water in the foam can easily form a lubricating layer at the wall, leaving the bulk of the foam less sheared than intended (Yoshimura and Prud homme 1988 Khan et al. 1988). Gelled colloidal suspensions are elastic materials that contain solvents capable of lubricating rheometer tool surfaces, and slip is a problem (Buscall et al. 1993 Persello et al. 1994). In these and other cases, slip can be counteracted in a number of ways, for example by using roughened rheometer surfaces (Khan et al. 1988 Buscall et al. 1993), 

Other complex fluids, such as polymer melts, contain no solvent that can serve as a lubricant, and mechanisms for shp at or near a solid surface—and even the existence of wall slip-—are less obvious (Denn 1990). Suspicion that slip may be occurring is aroused by observations of jumps, or abrupt slope changes, in curves of shear stress versus shear rate, or by oscillations in stress or pressure at fixed apparent flow rate, suggesting stick-slip — that is, alternating periods of stick and slip (Benbow and Lamb 1963 Blyler and Hart 1970 Vinogradov et al. 1972 Kalika and Denn 1987 Lim and Schowalter 1989 Piau et al. 1990 Hatzikiriakos and Dealy 1992). But molecular theories of slip for complex fluids such as 

As mentioned in Chapter 20, one very important point that must be considered in any rheological measurement is the possibility of slip during the measurements. This is particularly the case with highly concentrated dispersions, whereby the flocculated system may form a plug in the gap of the platens so as to leave a thin liquid film at the walls of the concentric cylinder or cone-and-plate geometry. This behaviour is caused by some syneresis of the formulation in the gap of the concentric cylinder or cone and plate. In order to reduce shp, roughened walls should be used for the platens, and a vane rheometer may also be used. 

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Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

In the following sections examples of the application of this procedure to the analysis of specific phenomena such as wall slip and stress overshoot which affect polymeric flow processes are illustrated. 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

Predicted velocity fields in a segment adjacent to the tip of the blade in the single-blade mixer, described in a previous sub-section, before and after imposition of the wall slip are shown in Figures 5.14a and 5.14b, respectively. As expected, momentum transfer from the rotating wall to the fluid is significantly affected by the imposition of the wall slip. In Figures 5.15a and 5.15b temperature contours corresponding to these velocity fields are shown. 

The fact that the appearance of a wall slip at sufficiently high shear rates is a property inwardly inherent in filled polymers or an external manifestation of these properties may be discussed, but obviously, the role of this effect during the flow of compositions with a disperse filler is great. The wall slip, beginning in the region of high shear rates, was marked many times as the effect that must be taken into account in the analysis of rheological properties of filled polymer melts [24, 25], and the appearance of a slip is initiated in the entry (transitional) zone of the channel [26]. It is quite possible that in reality not a true wall slip takes place, but the formation of a low-viscosity wall layer depleted of a filler. This is most characteristic for the systems with low-viscosity binders. From the point of view of hydrodynamics, an exact mechanism of motion of a material near the wall is immaterial, since in any case it appears as a wall slip. 

In this sense for calculating a bulk output Q during the flow with wall slip as a first approximation the following formula can be used ... 

It seems that indeed the answers to many fundamental questions are obtained, at least in qualitative form. Perhaps, the most important exception are thixotropic phenomena. There are many of them and the necessary systematization and mathematical generalization are absent here. Thus, it is not clear how to describe the effect of an amplitude on nonlinear dynamic properties. It is not clear what is the depth and kinetics of the processes of fracture-reduction of structure, formed by a filler during deformation. Further, there is no strict description of wall effects and a friction law for a wall slip is unknown in particular. 

At the walls of the pipe, that is where s — r, the velocity ux must be zero in order to satisfy the condition of zero wall slip. Substituting the value ux = 0, when s — r. then ... 

If it is known that a particular form of relation, such as the power-law model, is applicable, it is not necessary to maintain a constant shear rate. Thus, for instance, a capillary tube viscometer can be used for determination of the values of the two parameters in the model. In this case it is usually possible to allow for the effects of wall-slip by making measurements with tubes covering a range of bores and extrapolating the results to a tube of infinite diameter. Details of the method are given by Farooqi and Richardson. 21 ... 

Kn = 0.01-0.1 Slip flow rarefaction effects that can be modeled with a modified continuum theory with wall slip taken into consideration... 

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

Over the twentieth century, the mbber industry has developed special rheometers, essentially factory floor instmments either for checking process regularity or for quality control purposes, for instance, the well-known Mooney rheometer (1931), the oscillating disk rheometer (1962), and the rotorless rheometer (1976). All those instmments basically perform simple drag flow measurements but they share a common feature During the test, the sample is maintained in a closed cavity, under pressure, a practice intuitively considered essential for avoiding any wall slip effects. Indeed it has... 

Wang, S.-Q. Molecular Transitions and Dynamics at Polymer/Wall Interfaces Origins of Flow Instabilities and Wall Slip. VoL 138, pp. 227-276. 

The discussion above that led to Eqs. (4.2.6 and 4.2.7) assumes that the no-slip condition at the wall of the pipe holds. There is no such assumption in the theory for the spatially resolved measurements. We have recently used a different technique for spatially resolved measurements, ultrasonic pulsed Doppler velocimetry, to determine both the viscosity and wall slip velocity in a food suspension [2]. From a rheological standpoint, the theoretical underpinnings of the ultrasonic technique are the same as for the MRI technique. Flence, there is no reason in principle why MRI can not be used for similar measurements. 

The velocity profile during slip flow in a cylindrical tube is shown in Figure 21. As in conventional fluid flow, the flow velocity in the z direction, u(r), is parabolic, but rather than reach zero at the tube wall, slip occurs, and the velocity at the wall is greater than zero. The velocity does not reach zero until distance h from the wall surface. The derivation of the mass flux equation proceeds along the same lines as the derivation of Poiseuille s law in conventional hydrodynamics, but in slip flow, u(r) = 0 at r = a + h instead of reaching zero at r = a. 

Yoshimura, A.S. Prud homme, R.K. "Viscosity Measurements in the Presence of Wall Slip in Capillary, Couette, and Parallel-Disk Geometries," SPE Reservoir Engineering, May 1988, 735-742. 

When trying to determine the flow behaviour of a material suspected of exhibiting wall slip, the procedure is first to establish whether slip occurs and how significant it is. The magnitude of slip is then determined and by subtracting the flow due to slip from the measured flow rate, the genuine flow rate can be determined. The standard Rabinowitsch-Mooney equation can then be used with the corrected flow rates to determine the tw-jw curve. Alternatively, the results can be presented as a plot of tw against the corrected flow characteristic, where the latter is calculated from the corrected value of the flow rate. 

In order to determine whether slip occurs with a particular material, it is essential to make measurements with tubes of various diameters. In equation 3.66, the value of the integral term is a function of the wall shear stress only. Thus, in the absence of wall slip, the flow characteristic 8 u/dt is a unique function of tw. However, if slip occurs, the term 8vjd will be different for different values of d, at the same value of tu., as shown in Figure 3.11. It is clear from equation 3.66 that for a given value of the slip velocity vs, the effect of slip is greater in tubes of smaller diameter. If the effect of slip is dominant, that is the bulk of the material experiences negligible shearing, then it can be seen from equation 3.66 that on a plot of... 

The varying effect of wall slip in tubes of different diameters... 

Having established that wall slip occurs but is not dominant, the procedure is to estimate the value of v, and hence calculate a corrected flow rate by subtracting the slip flow from the measured flow rate. In general it is found that the slip velocity increases with tw and decreases with d, although in some cases vs is independent of dt. Consequently, it can be seen from equation 3.66 that the effect of slip decreases as d, increases, and becomes negligible at very large diameters. 

The other approach is to scale up the genuine flow, then add the slip flow for the appropriate pipe diameter. Scale up of the genuine flow can be done as described in Section 3.3 or Section 3.4. In order to assess the flow due to wall slip in the pipe, it is necessary to have information about the variation of vs with tw and dt unless it is assumed that the pipe is large enough for the effect of slip to be negligible. If slip velocity data are available, implying that the apparent fluidity plots are also available, then it would be easier to use these plots directly. 

It is implicit in these methods that the wall slip behaviour in the pipe is similar to that in the tubes. There is evidence [Fitzgerald (1990)] that the wall roughness can have a dramatic effect on the flow of some suspensions, making the assumption of similar slip behaviour very dangerous. 

Not all suspensions will exhibit wall slip. Concentrated suspensions of finely ground coal in water have been found to exhibit wall slip [Fitzgerald (1990)]. This is to be expected because the coal suspension has a much higher apparent viscosity than the water. In contrast, when the liquid is a very viscous gum, the addition of solids may have a relatively small effect. In this case, the layer at the wall will behave only marginally differently from the material in the bulk. 

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