Velocity slip-, methods

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

There is no analytic nor numerical model available, which provides the particle drag coefficient for particles over all the regimes of rarefied flows. The earlier methods to correct for rarefied flow effects were based on a correction to Stokes drag, derived by Basset to account for velocity slip at the surface. In that case, the drag coefficient can be expressed as... 

Kamiadakis and Beskok [6] developed a code H Flow with implementation of spectral element methods. They employed both the Navier-Stokes (incompressible and compressible) and energy equations in order to compute the relative effects of compressibility and rarefaction in gas microflow simulations. In addition, they also considered the velocity slip, temperature jump, and thermal creeping boundary conditions in the code Flow. The spatial discretization of fi Flow was based on spectral element methods, which are similar to the hp version of finite-element methods. A typical mesh for simulation of flow in a rough micro-channel with different types of roughness is shown in Fig. 1. The two-dimensional domain is broken up into elements, similar to finite elements, but each element employs high-order interpolants based on... 

Although the Navier-Stokes equations are not valid in the Knudsen layer, due to a nonlinear stress/strain-rate behavior in this small layer [4], their use with appropriate boundary velocity slip and temperature jump conditions proved to be accurate for predicting mass flow rates [5] ( methods for flow rate measurements) and velocity profiles out of the Knudsen layer. Classically, the real flow is not simulated within the JCnudsen layer, but the influence of the Knudsen layer on the flow outside this non-equilibrium layer is taken into account, replacing the no-slip condition at the wall with a slip-flow condition. For that purpose, a fictitious slip velocity MsUp is introduced (Fig. 2). Real slip at the wall, gas — wall. is due to the fact that gas molecules very close to the wall have actually a mean... 

Sutherland (1975). Orifice flow rates are underpredicted by about the same factor with the energy balance method and with the NEM. Discharge predictions for short (0.2-m) pipes are overpredicted by the energy balance method. In this region, the assumption of homogeneous equilibrium is not justified. A model that takes slip velocity into account may improve predictions for short pipes. 

The fractional dispersed phase holdup h is normally correlated on the basis of a characteristic velocity equation, which is based on the concept of a slip velocity for the drops vsiip, which then can be related to the free rise velocity of single drops, using some correctional functional dependence on holdup f(h). The normal method of correlating dispersed phase holdup is normally of the form... 

Plot of apparent fluidity against 1/di to determine the slip velocity (Mooney s method)... 

Mooney s method has been modified in various ways to allow for the observation that, with many suspensions, the slip velocity depends on the tube diameter as well as the wall shear stress. Jastrzebski (1967) deduced that, for certain kaolinite-water suspensions, vs was inversely proportional to d Thus a modified slip coefficient Cj may be defined by... 

As mentioned earlier, Reynolds numbers determined for the bulk flow have to be discerned from Reynolds numbers characterizing a particle-liquid dissolution system. The latter were calculated for drug particles of different sizes using the Reynolds term according to the combination model. The kinematic viscosity of the dissolution medium at 37°C is about 7 x 10-03 cm2/sec. The fluid velocities (Ua) employing the paddle method at stirring rates of 50-150 rpm can be taken from the literature and may arbitrarily be used as the slip velocities at the particle surfaces. 

The definition of friction factor using mean fluid properties has been most widely used because it reduces to the correct single-phase value for both pure liquid and pure gas flow. This technique is very similar to the so-called homogeneous model, because it has a clear physical significance only if the gas and liquid have equal velocities, i.e., without slip. Variations of this approach have also been used, particularly the plotting of a ratio of a two-phase friction factor to a single-phase factor against other variables. This approach is then very similar to the Lockhart-Martinelli method, since it can be seen that (G4)... 

The design procedure used by Kosters, of Shell Oil Co., who developed this equipment, requires pilot plant measurements on the particular system of HTU and slip velocity as functions of power input. The procedure for scaleup is summarized in Table 14.5, and results of a typical design worked out by Kosters (in Lo et al., 1983, pp. 391-405) are summarized in Example 14.11. Scaleup by this method is said to be reliable in going from 64 mm dia to 4-4.5 m dia. The data of Figure 14.18 are used in this study. 

With the supposition that the slip layer is thin and the slip velocity is constant, various analyses have been developed in the search for the ideal experimental method to define slip. The Mooney analysis (20) for both tube flow and concentric cylinder flow has been applied to a wide range of materials including polymer solutions (21), filled suspensions (22), semisolid foods (23), fruit purees (24), and ketchups (25). Alternate estimates of slip velocity have been determined experimentally from, parallel plate torsion flow (26), from flow data in channels and inclined planes, and from porous medium geometries (8). 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

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