Wall slip theories

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

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. 

Three theories were proposed to explain wall-slip (a) adhesive failure at the wall, (b) cohesive failure within the material as a result of disentanglement of chains in the bulk and chains absorbed on the wall, and (c) the creation of a lubricating surface layer at the wall either by a stress-induced transition, or by a lubricating additive. If the polymer contains low molecular weight components or slip-additives, their diffusion to the wall will create a thin lubricating layer at the wall, generating apparent slip. 

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... 

Qiu and Rao (Qiu, C. G. and Rao, M. A. J. Texture Stud., submitted) determined slip coefficients and slip velocities for apple sauce in a concentric cylinder viscometer as well as the effect of insoluble solids content on them. Three concentric cylinder units specified in the theory of Mooney (42.) were employed. Rotational speeds were determined with the different concentric cylinder systems at the same magnitude of torque. Figure 2 shows, for one sample of apple sauce, the shear rates uncorrected and corrected for slip plotted against the shear stress. The magnitudes of the flow behavior index of the power law model (Equation 2) did not change significantly due to correction for wall slip however, the magnitudes of the consistency index increased due to wall slip corrections. 

While experimental evidence indicates that fluid flow in microdevices differs from flow in macroscale, existing experimental results are often inconsistent and contradictory because of the difficulties associated with such experiments and the lack of a guiding rational theory. Koo and Kleinstreuer [6] summarized experimental observations of liquid microchannel flows and computational results concerning chamiel entrance, wall slip, non-Newtonian fluid, surface roughness, and other effects. Those contradictory results suggest the need for applying molecular-based models to help establish a theoretical frame for the fluid mechanics in microscale and nanoscale. 

Wang SQ (1999) Molecular transitions and dynamics at melt/wall interfaces origins of flow instabilities and wall slip. Adv Polym Sci 138 227-275 Weissenberg K (1947) A continuum theory of rheological phenranena. Natiue 159 310-311... 

Plug flow generally does not happen in polymer melts, except in the case of wall slip (PVC). However, it does occur with granular polymeric solids. The solids conveying theory of single screw extruders is based on the assumption of plug flow of the solid polymer. 

Rosenbaum EE, Hatzikiriakos SG (1997) Wall slip in the capillary flow of molten polymers subject to viscous heating. AICKE J 43 598-608 Rubin II (1972) Injection molding-theory and practice. Wiley, New York Santhanam N, Chiang HH, Himasekhar K, Tuschak P, Wang KK (1991) Postmolding and load-induced deformation analysis of plastic parts in the injection molding process. Adv Polym Tech 11 77-89... 

The physical reason for the velocity slip is the fact that close to the wall the gas is not in thermal equilibrium. For the same reason, a temperature jump is induced, and a more detailed investigation based on the kinetic theory of gases shows that heat transfer and momentum transfer are coupled. Expressions for velocity slip and temperature jump valid in the case of non-isothermal conditions are given by... 

According to the lubrication approximation, we can quite accurately assume that locally the flow takes place between two parallel plates at H x,z) apart in relative motion. The assumptions on which the theory of lubrication rests are as follows (a) the flow is laminar, (b) the flow is steady in time, (c) the flow is isothermal, (d) the fluid is incompressible, (e) the fluid is Newtonian, (f) there is no slip at the wall, (g) the inertial forces due to fluid acceleration are negligible compared to the viscous shear forces, and (h) any motion of fluid in a direction normal to the surfaces can be neglected in comparison with motion parallel to them. 

Values of To [34] close to those reported in [5,33], are obtained for foams with expansion ratio in the range of 86 to 300, and consisting of bubbles with 0.3-1 mm diameter, i.e. To,si = 2-12 Pa. It was established also that the value of To., depends on the viscosity of the dispersion medium, which does not conform with the theory presented in [10]. Probably, this is related to the slip at the tube walls during foam flow. 

The basic law of viscosity was formulated before an understanding or acceptance of the atomic and molecular structure of matter although just like Hooke s law for the elastic properties of solids the basic equation can be derived from a simple model, where a flnid is assumed to consist of hypothetical spherical molecules. Also like Hooke s law, this theory predicts linear behavior at low rates of strain and deviations at high strain rates. But we digress. The concept of viscosity was first introduced by Newton, who considered what we now call laminar flow and the frictional forces exerted between layers within a fluid. If we have a fluid placed between a stationary wall and a moving wall and we assume there is no slip at the walls (believe it or not, a very good assumption), then the velocity profile illustrated in Figure... 

In spite of the experimental results mentioned above, there is at present no accepted theory for macroscopic slip with polymer fluids. Analytical modelling results have indeed already been published [9-10, 17-20]. Generally, the laws assume a linear relation between stress at the wall and slip velocity [9], which can then be extended by a non-linear relation to high stress values [10, 17-20]. These often have the advantage of being simple to use, but nevertheless have serious drawbacks. 

Relatively complete elaborations for the cylinder model have been given by for instance, Anderson and Koh and Levine et al. K In these two theories the solution Is assumed to contain (1-1) electrolytes with =u. Both theories fail to account for conduction behind the slip plane, and both solve the electrokinetic equations, taking double layer overlap into account. Anderson and Koh assume this overlap to take place at fixed surface charge (which, because of the implicit rigid particle model of the cylinder wall, comes down to fixed tr =cT ), whereas Levine et al. do so for constant surface potential (essentially fixed Anderson and Koh also considered capUlaries of other... 

The Doi-Edwards model has been extended to allow processes of primitive-path fluctuations, constraint release, and tube stretching. These extensions of the theory allow accurate prediction of many steady-state and time-dependent phenomena, including shear thinning, stress overshoots, and so on. Predictions of strain localization and slip at walls... 

In micro parallel plate channel the interactions between the fluid and the wall become significant, because the molecular mean free path is comparable to 2 y 1. The gas slip along the wall with a finite velocity in the axial direction as described by Maxwell in 1890 [13]. The kinetic theory of gases gives the following boundary condition at the surface of the channel [28] ... 

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