Perfect slipping

The more incisive calculation of Springett, et al., (1968) allows the trapped electron wave function to penetrate into the liquid a little, which results in a somewhat modified criterion often quoted as 47r/)y/V02< 0.047 for the stability of the trapped electron. It should be noted that this criterion is also approximate. It predicts correctly the stability of quasi-free electrons in LRGs and the stability of trapped electrons in liquid 3He, 4He, H2, and D2, but not so correctly the stability of delocalized electrons in liquid hydrocarbons (Jortner, 1970). The computed cavity radii are 1.7 nm in 4He at 3 K, 1.1 nm in H2 at 19 K, and 0.75 nm in Ne at 25 K (Davis and Brown, 1975). The calculated cavity radius in liquid He agrees well with the experimental value obtained from mobility measurements using the Stokes equation p = eMriRr], with perfect slip condition, where TJ is liquid viscosity (see Jortner, 1970). Stokes equation is based on fluid dynamics. It predicts the constancy of the product Jit rj, which apparently holds for liquid He but is not expected to be true in general. 

In compression there is not often perfect slip at the compressed ends and a general relationship for compression with bonded ends is ... 

This is identical to the drag force on a solid sphere at whose surface perfect slip occurs. 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

The similarities between gas-liquid and solid-liquid interfaces end here. Since it is only necessary to adopt the exact fluid dynamics solution for vg within the range 0 < y < Sc where 8c/R 1, if the interface is characterized by perfect-slip and zero-shear, then the first-order term in the polynomial expansion for the tangential velocity component should be identically zero. Hence,... 

The error function profile for P(ri) in the liquid phase adjacent to a perfect-slip zero-shear interface is... 

This allows one to calculate the dimensionless molar density profile RiX) for O2 transport in water at the particular values of r and B mentioned above. Since the gas-liquid interface is characterized by zero shear and perfect slip, P(X) is obtained from the incomplete gamma function when the argument n = and the variable A. = f The first three terms of the infinite series yield the following result ... 

Flow Regime Solid-Liquid, High Shear, No Slip Gas-Liquid, Zcto Shear, Perfect Slip... 

Apparent slip Boundary condition at interface Perfect slip... 

Boundary Slip of Liquids, Fig. 1 Schematic representation of the no-slip, partial slip, and perfect slip boundary conditions. Under no-slip boundary condition, the relative velocity. Vs, between the fluid and the solid wall is zero at... 

Three possible velocity profiles near a solid boundary are shown in Fig. 1. Figure la shows that the velocity of fluid particles near the stationary solid wall is equal to zero and represents the no-sUp boundary condition. Rgure lb shows that the velocity near the stationary solid wall is non-zero with a relative velocity between the two and represents the slip boundary condition. Rgure Ic shows the perfect slip condition, for which there is no influence of the boundary surface on the velocity profile. The velocity when extrapolated towards the wall matches that of the wall at some distance Lg away from it (Fig. lb), which is known as the slip length and is used as a measure of the slip. The slip length is a fictitious distance below the surface at which the velocity would be equal to zero if extrapolated linearly. The velocity difference between the boundary surface and the adjacent fluid particles is known as the slip velocity, Vs and is related to the velocity gradient (8V/Sy) near the solid boundary as [1]... 

Figure 3.7 Deviations of diffusion coefficients from the Stokes-Einstein value (2). Comparison of the diffusion coefficients of symmetrical solutes in five protic and aprotic solvents with the prediction of the Stokes-Einstein equation. The diffusion coefficient — viscosity product Drj is plotted against the reciprocal of the solute radius the predictions of Stokes law for perfect stick (kT/6n) and perfect slip kT/An) are shown by dotted lines. See text. From Ref. [13,b].

All of the investigations discussed thus far have assumed a no>slip boundary condition at the walls of the channel. The effect of wall slip on extrudate swell from three-dimensional dies appears also to have been unreported in the literature. However, the effect of slip on die swell for the two-dimensional planer jet has been considered by Silliman [21] where the effects of inertia have been neglected. His results show that die swell decreases with slip and approaches 1 in the limit of perfect slip. 

The results showing the effect of the slip coefficient on the final shape of the extrudate are shown in Fig. 10. As in the two-dimensional case, die swell is reduced by wall slip and approaches 1 in the limit of perfect slip (i.e., P(i/H = 00). These results demonstrate that slip will reduce the distortion of the extrudate from that of the channel from which it emerges. Dies with wall slip would also exhibit less pressure drop than those without slip. For these reasons, die manufacturers have attempted to promote slip where possible. Internal die surfaces are ground to mirror like finishes, and in some cases, slip-enhancing coatings are applied near the exit of the die. Even so, introducing perfect slip is not achievable in practice. Therefore, the effects of die swell should be considered in the design of extrusion dies. 

The effects of the Reynolds number on the extrusion of Newtonian fluid from square and rectangular dies has been considered. As with planar and axisymmetric jets, extrudates from three-dimensional dies swell at low Reynolds numbers but contract at high ones. Depending on its aspect ratio, limiting die swell from the rectangle varies that of the square (0.7255) and that of two-dimensional planar case (0.8333). Wall slip reduces die swell and in the cases of perfect slip, completely eliminates it. 

A similar asymptotic solution can be obtained for power-law fluids. One counterintuitive phenomenon with power-law behavior occurs when n = in that the flow is radial everywhere with partial slip the downstream asymptotic behavior changes from nearly perfect slip to no slip as the power-law index n decreases below... 

We conclude, therefore, that since H is fixed, the maximal Qy/Qx corresponds to the largest physically possible b, that is, the perfect slip at the gas sectors. 

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