Slip-flow

Foam rheology has been a challenging area of research of interest for the yield behavior and stick-slip flow behavior (see the review by Kraynik [229]). Recent studies by Durian and co-workers combine simulations [230] and a dynamic light scattering technique suited to turbid systems [231], diffusing wave spectroscopy (DWS), to characterize coarsening and shear-induced rearrangements in foams. The dynamics follow stick-slip behavior similar to that found in earthquake faults and friction (see Section XU-2D). 

The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = X/D. Molecular flow is characterized by Kn > 1.0 continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. 

Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802-813 [1946]), which uses the parameter... 

For slip flow through square channels, see Milhgan and Wilker-son (J. Eng. Jnd., 95, 370-372 [1973]). For slip flowmrough annuli, see Maegley and Berman (Phys. Fluids, 15, 780-785 [1972]). 

For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the Vacuum Flow subsec tion) may be applied when the pore is of the same or smaller order as the mean free path, as described by Monet and Vermeulen (Chem. E/ig. Pi og., 55, Symp. Sei , 25 [1959]). 

When the size of a particle approaches the same order of magnitude as the mean free path of the gas molecules, the setthng velocity is greater than predicted by Stokes law because of molecular shp. The slip-flow correc tion is appreciable for particles smaller than 1 [Lm and is allowed for by the Cunningham correc tion for Stokes law (Lapple, op. cit. Licht, op. cit.). The Cunningham correction is apphed in calculations of the aerodynamic diameters of particles that are in the appropriate size range. 

The flow entering the rotor of a radial-inflow turbine must have a eertain ineidenee angle eorresponding to the slip flow in a eentrifugal impeller and not to zero ineidenee. By relating this eoneept to the radial-inflow turbine, the following relationship ean be obtained for the ratio of whirl veloeity to blade tip speed ... 

Stick Slip Flow. The continuous sudden stoppage and resumption of catalyst flow in a standpipe. This is usually caused by underaeration. 

Stick-Slip Flow is erratic circulation caused when the catalyst packs and bridges across the standpipe. 

I I. Tayi.or, T.D. Physics of Fluids 6 (1963) 987. Heat transfer from single spheres in a low Reynolds number slip flow. 

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

Kn = 0.1-10 Transition flow between slip flow and free molecular flow, treated statistically, e.g., by the Boltzmann equation... 

Because fluid slip occurs at highly water-repellent walls when the contact angle is about 150°, Watanabe et al. (1999) analyzed the friction factor of slip flow in a circular pipe. For a fully developed steady flow in a pipe, the Navier-Stokes equation can be written as... 

In 1959, Burgdorfer [39] first introduced a concept of the kinetic theory to the field of gas film lubrication. This was to derive an approximation equation, called the modified Reynolds equation, using a slip flow velocity boundary con-... 

Mitsuya, Y, Modified Reynoids Equation for Uitra-Thin Fiim Gas Lubrication Using 1.5-Order Slip-Flow Model and Considering Surface Accommodation Coefficient," ASME J. Tri-bol, Voi. 115,1993, pp. 289-294. 

Since the middle of the 1990s, another computation method, direct simulation Monte Carlo (DSMC), has been employed in analysis of ultra-thin film gas lubrication problems [13-15]. DSMC is a particle-based simulation scheme suitable to treat rarefied gas flow problems. It was introduced by Bird [16] in the 1970s. It has been proven that a DSMC solution is an equivalent solution of the Boltzmann equation, and the method has been effectively used to solve gas flow problems in aerospace engineering. However, a disadvantageous feature of DSMC is heavy time consumption in computing, compared with the approach by solving the slip-flow or F-K models. This limits its application to two- or three-dimensional gas flow problems in microscale. In the... 

In most of the gas lubrication problems in nano-gaps, gas flow usually locates in the slip flow or the transient flow regime, depending on working conditions and local geometry. Therefore, both of the macroscopic and microscopic models are introduced to analyze the gas lubrication problems. 

XlylmnkT/ird ), or h = [TT[i 2RTI2p), as long as substituting the gap-dependent viscosity rather than the bulk viscosity. Because the effective viscosity decreases as the Knudsen number enters the slip flow and transition flow ranges, and thus the mean free path becomes smaller as discussed by Morris [20] on the dependence of slip length on the Knudsen number. 

For applications in the field of micro reaction engineering, the conclusion may be drawn that the Navier-Stokes equation and other continuum models are valid in many cases, as Knudsen numbers greater than 10 are rarely obtained. However, it might be necessary to use slip boimdaty conditions. The first theoretical investigations on slip flow of gases were carried out in the 19th century by Maxwell and von Smoluchowski. The basic concept relies on a so-called slip length L, which relates the local shear strain to the relative flow velocity at the wall ... 

In order to obtain a qualitative view of how the transition regime differs from the continuum flow or the slip flow regime, it is instructive to consider a system close to thermodynamic equilibrium. In such a system, small deviations from the equilibrium state, described by thermodynamic forces X, cause thermodynamic fluxes J- which are linear functions of the (see, e.g., [15]) ... 

As mentioned earlier, the drift-flux model is used in slip flows. Equation (3-63) can be written in the form... 

On the basis of extensive analysis of available data, Weisman et al. (1978) concluded that, for abrupt expansions, oq and a2 should be evaluated by assuming slip flow. They recommended Hughmark s (1962) relationship for obtaining a from x,... 

Later, Weisman et al. (1978) also found that assuming homogeneous flow everywhere provided nearly as good a correlation of the data as the slip flow model. The total pressure drop across a contraction can be approximated by... 

Average velocities of different magnitude exist for each phase (i.e., a slip flow exists). 

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