Slip Model Derivation

Maxwell s derivation of velocity slip and temperature jump boundary condition is based on kinetic theory of gases. A similar boundary condition can be derived by an approximate analysis of the motion of gas in an isothermal condition, which has been presented in this section. 

We can write the tangential momentum flux on a surface s located near the wall as equal to Here, is the number density of molecules crossing surface Y m is the molecular mass is the tangential (slip) velocity on the surface and Vg is the mean thermal speed of the molecule. 

It may be noted that thermal speed Vg is a measure of temperature. It is strictly not velocity, that is, it is not a vector quantity. Rather it is simply a scalar speed and has been defined in 

Let us assume that approximately half of the molecules passing through control surface s are coming from a layer of gas at the distance proportional to one mean free path 

The tangential momentum flux of these incoming molecules can be written as 

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It is important to remind the reader that U is the velocity of the fluid phase seen by the particle, U - U is the slip velocity, dp is the particle diameter, and Vf is the kinematic viscosity of the fluid phase. Note that Eq. (5.33) depends on the particle velocity U and is valid in the zero-Stokes-number limit where U = U so that particles follow the fluid. The correlation in Eq. (5.31) is valid only for RCp < 1 and Sc > 200. For larger particle Reynolds numbers the following correlations can be used Sh = 2 -i- 0.724Rep Sc, which is valid for 100 < RCp < 2000, and Sh = 2 -i- 0.425RCp Sc, which is valid for 2000 < RCp <10. Among the other correlations available, it is important to cite the one proposed by Ranz Marshall (1952) for macroparticles Sh = 2.0 -i- O.bReJ Sc. These expressions assume that the fluid velocity U is known. For micron-sized (or smaller) particles moving in turbulent fluids for which only the ensemble-mean fluid velocity (Uf) is known, it is instead better to employ the mesoscale model derived by Armenante Kirwan (1989) Sh = 2.0 -i- 0.52(Re ) Sc, where Re = is the modi-... 

The affine and the phantom models derive the behavior of the network from the statistical properties of the individual molecules (single chain models). In the more advanced constrained junction fluctuation model the properties of these two classical models are bridged and interchain interactions are taken into account. We remark for completeness that other molecular models for rubber networks have been proposed [32,57,75-87], however, these are not nearly as widely used and remain the subject of much debate. Here we briefly summarize the basic concepts of the affine, phantom, constrained junction fluctuation, diffused constraint, tube and slip-tube models. 

Based on the nanoscale effect function, Peng et al. further derived the modihed Re5molds equations of the first-order slip, the second-order slip and the Fukui-Kaneko models [19]. The flow rate coefficients become the following expressions. 

A similar equation to that of Eq. (43) was proposed by Bankoff (B6) on the basis of a bubble-flow model for vertical flow. His derivations are discussed in the following section (Section V, B). Finally, it should be mentioned that the momentum exchange model of Levy (L4), and the slip-ratio model of Lottes and Flinn (L7) are more readily applied for the determination of void fractions than for pressure drops. In general, these methods seem to give rather poorer accuracy than those already discussed. 

It is possible to derive an expression for the pressnre profile in the x direction using a simple model. We assnme that the flow is steady, laminar, and isothermal the flnid is incompressible and Newtonian there is no slip at the walls gravity forces are neglected, and the polymer melt is uniformly distribnted on the rolls. With these assnmptions, there is only one component to the velocity, v dy), so the equations of continuity and motion, respectively, reduce to... 

An extension, by accounting for gas-liquid interfacial interactions via velocity-and shear-slip factors was then proposed by Al-Dahhan et al. [42] to lift the model disparities observed for conditions of high gas throughputs and elevated pressures. Later, Iliuta et al. [43] derived more general slip-corrective correlations. 

Example 3.4 Flow of a Power Law Fluid in Tubes For an isothermal, laminar, fully developed steady pressure flow of an incompressible Power Law model fluid in a horizontal tube without slip, we wish to derive (a) the velocity profile and (b) the flow rate. 

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... 

Various model parameters involved in the derivation of the stability criterion need to be specified in order to use the stability criterion for quantitative predictions. Model parameters essential for this purpose include the slip velocity, the virtual mass coefficient, and the dispersion coefficient. The procedure for estimation of these parameters is given for gas-solid (and solid-liquid) fluidized beds and bubble columns. 

For smooth roll compactors a formula can be derived that correlates roller diameter and gap. With the definitions of Figure 227 and the restrictions imposed by the modified strip model [i.e. beginning at the line (cxe) horizontal increments move with the peripheral speed of the rolls (no slip) and remain absolutely horizontal (no distortion)], the following equation for the porosity min at the narrowest point (a = 0) is obtained ... 

In this section we derive the algebraic-slip mixture model equations for cold flow studies starting out from the multi-fluid model equations derived applying the time- after volume averaging operator without mass-weighting [204, 205]. The momentum equations for the dispersed phases are determined in terms... 

The force-displacement relations in contact mechanics are often nonlinear. A prominent example is the transition from stick to slip. Even for nonlinear interactions, there is a strictly quantitative relationship between the shifts of frequency and bandwidth. A/ and AT, on the one hand, and the force acting on the crystal, T(t), on the other. A/ and Ar are proportional to the in-phase and the out-of-phase component of F(t), respectively. Evidently, F(t) cannot be explicitly derived from A/ and Ar. Still, any contact-mechanical model (like the Mindlin model of partial slip) can be tested by comparing the predicted... 

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