Zero-shear interface

Generalized vector analysis is presented in this section for fluid flow adjacent to zero-shear interfaces in the laminar regime. The following adjectives have been used to characterize potential flow inviscid, irrotational, ideal, and isentropic. Ideal fluids experience no viscous stress because their viscosities are exceedingly small (i.e., ii 0). Hence, the V r term in the equation of motion is negligible... 

Mass transfer across a perfect-shp zero-shear interface at high Schmidt numbers is discussed in light of the previous results for no-slip interfaces. The following assumptions are invoked to develop the model ... 

Neither the r9 component of the rate-of-strain tensor nor the simple velocity gradient dvg/dr vanishes at the gas-liquid interface. This is expected for inviscid flow because viscous stress is not considered, even in the presence of a signiflcant velocity gradient. Once again, the leading term in the polynomial expansion for vg, given by (11-126), is used to approximate the tangential velocity component for flow of an incompressible fluid adjacent to a zero-shear interface ... 

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

Answer For boundary layer mass transfer in an incompressible liquid that contacts a zero-shear interface, a previous example problem on pages 311 and 312 reveals that the relative importance of the second term on the right side of the spherical coordinate expression for radial diffusion,... 

Now, calculate the normal component of the total local molar flux of species A at the nondeformable zero-shear interface. Since the radial component of the flnid velocity vector vanishes at r = R, species A is transported across the interface exclusively via concentration diffnsion (i.e.. Pick s law). Then, the diffusional flux of species A in the radial direction, evalnated at the interface, is equated to the product of a local mass transfer coefficient and the overall concentration driving force for mass transfer (i.e., Ca. equilibrium — CA.buik)- The... 

The intimate contact data shown in Figure 7.16 were obtained from three-ply, APC-2, [0°/90o/0o]7- cross-ply laminates that were compression molded in a 76.2 mm (3 in.) square steel mold. The degree of intimate contact of the ply interfaces was measured using scanning acoustic microscopy and image analysis software (Section 7.4). The surface characterization parameters for APC-2 Batch II prepreg in Table 7.2 and the zero-shear-rate viscosity for PEEK resin were input into the intimate contact model for the cross-ply interface. Additional details of the experimental procedures and the viscosity data for PEEK resin are given in Reference 22. 

From the correspondence between the calculated and experimental curves we can extract other information. For example the temperature (ca. 72 °C) at which x = 1/2 is shown on Fig. 14b. Above this temperature no more chains break at this temperature and higher, the craze growth is disentanglement dominated. We can use the fact that = 1/2 and Eq. (19) to extract a value for the corresponding to disentanglement of chains at the void interface under these conditions this value is 1.5 x 10" N-s/m, a value that is only reached for polystyrene melts (from zero shear viscosity or diffusion measurements) at a temperature of about 120 °C, or 20° above T. ... 

A more common source of Marangoni effects in systems of interest to chemical engineers is surfactants, as discussed in Chap. 2. This is particularly pertinent to the motion of gas bubbles (or drops) in water, or in any liquid that has a large surface tension (the surface tension of a pure air-water interface is approximately 70 dyn/cm). Experiments on the motion of gas bubbles in water at low Reynolds numbers show the perplexing result illustrated in Fig. 7-18. For bubbles larger than about 1 mm millimeter in diameter, the translation velocity is approximately equal to the predicted value for a spherical bubble with zero shear stress at the interface, that is,... 

The limit in this case can be seen to reduce the equations to the linearized stability equations for an inviscid fluid. As a consequence, not all of the interface boundary conditions can be satisfied. Our experience from Chap. 10 shows that we should not expect the solution to satisfy the zero-shear-stress condition, which will come into play only if we were to... 

The issue of slip at the solid-liquid interface has been a topic of much debate [103]. The influence of slip on the frequency of the QCM is discussed in detail in the chapter by M. Urbakh et al. 2006, in this volume. Shp can be very easily integrated into the framework of the multilayer formahsm and we briefly show this connection. We represent slip by a layer close to the solid surface (a film ) with a reduced viscosity. Inside this layer, the shear gradient is increased, leading to the flow profile indicated in Fig. 10. The slope of the profile dv(z)/dz is proportional to o (z)- The slip length, hs, is the difference between the location of the surface and the extrapolated plane of zero shear. One can show that the slip length, b, is given by ... 

If one applies equation (8-153) to gas-liquid interfaces, the total force exerted by the fluid on the bubble across the nondeformable zero-shear boundary is due... 

Obviously, integration constant A must be zero to satisfy the zero shear condition 3 at the gas-liquid interface. Now condition 2 is satisfied when 2B = —l Vbubbie-The final results for the stream function and the fluid velocity profile are... 

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,... 

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 ... 

If T terface and Tbuik replace Ca, equilibrium and Ca, bulks respectively, in the definition of the dimensionless profile P, and the thermal diffusiv-ity replaces a. mix. then the preceding equation represents the thermal energy balance from which temperature profiles can be obtained. The tangential velocity component within the mass transfer boundary layer is calculated from the potential flow solution for vg if the interface is characterized by zero shear and the Reynolds number is in the laminar flow regime. Since the concentration and thermal boundary layers are thin for large values of the Schmidt and Prandtl... 

If the interface between particle A and fluid B is better represented by zero-shear rather than no-slip, then the friction coefficient for creeping flow of an incompressible Newtonian fluid around a bubble is... 

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