Velocity slip coefficient

The velocity slip coefficient has been studied by several investigators using... 

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... 

In some foods, a thin layer of low-viscosity fluid forms at the solid-fluid interface that in turn contributes to lower viscosity values. The boundary condition that at the solid-fluid interface the fluid velocity is that of the wall is not satisfied. This phenomenon is known as slip effect. Mooney (1931) outlined the procedures for the quantitative determination of slip coefficients in capillary flow and in a Couette system. The development for the concentric cylinder system will be outlined here for the case of the bob rotating and details of the derivation can be found in Mooney (1931). 

Velocity slip and temperature jump affect the heat transfer in opposite ways a large slip on the wall increases the convection along the surface. On the other hand, a large temperature jump decreases the heat transfer by reducing the temperature gradient at the wall. Therefore, neglecting temperature jump will result in the overestimation of the heat transfer coefficient. 

Sokhan and Quirke434 devise a method of computing interfacial friction and the Maxwell slip coefficients (a) by equilibrium MD, in which a is computed from the relaxation time, which itself can be estimated by an exponential fit to the collective velocity autocorrelation function. Their equilibrium method is compared to an NEMD method devised by the authors previously and excellent agreement is found. They study the density dependence of the slip... 

Here n is the unit normal to the boundary, u and T are the (continuum) velocity and stress, and P is an empirical parameter known as the slip coefficient. The Navier-slip condition says, simply, that there is a degree of slip at a solid boundary that depends on the magnitude of the tangential stress. We note, however, that it is generally accepted that the slip coefficient is usually very small, and then the no-slip condition (2 123) appears as an excellent approximation to (2-124) for all except regions of very high tangential stress. 

Qiu and Rao (Qiu, C. G. and Rao, M. A. J. Texture Stud., submitted) determined slip coefficients and slip velocities for apple sauce in a concentric cylinder viscometer as well as the effect of insoluble solids content on them. Three concentric cylinder units specified in the theory of Mooney (42.) were employed. Rotational speeds were determined with the different concentric cylinder systems at the same magnitude of torque. Figure 2 shows, for one sample of apple sauce, the shear rates uncorrected and corrected for slip plotted against the shear stress. The magnitudes of the flow behavior index of the power law model (Equation 2) did not change significantly due to correction for wall slip however, the magnitudes of the consistency index increased due to wall slip corrections. 

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. 

Recently, Hynes et al. [221, 222] have pointed out that continuum models of rotational relaxation become unreliable when the molecule of interest rotates in a solvent comprising molecules of similar size. To improve on the model, they considered a sphere to be surrounded by a first co-ordination shell of solvent molecules. All these were taken as rough spheres, that is hard spheres which reverse their relative velocity (normal and tangential components) on impulsive collision. Of specific interest are CCI4 and SF. The test sphere and its boundary layer is surrounded by a hydrodynamic continuum. To model this, Hynes et al. used linearised hydrodynamic equations for the solvent with a modified boundary condition between solvent and test molecule, which relates the rotational stress on the test sphere to the angular velocity of the sphere. A coefficient of proportionality, 3, is introduced as a slip coefficient (j3 0... 

The mass transfer coefficient, k, and slip velocity, us, for > JS- The terminal velocity-slip velocity theory gives no indication from first principles what value the exponent on Ns should be in the relationship... 

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... 

The curves show that the velocity at the membrane surface (A = 1) is 0 when is 0 as expected. As the slip velocity increases with increasing (f), the wall shear decreases, and the velocity profiles become flatter, approaching those for plug flow [12-14]. The effect of slip coefficient on axial pressure gradient (P) is as follows an increase in (j) results in a decrease in wall shear stress so that P also decreases. In addition, the transition from laminar to turbulent flow at a porous surface occurs at a Re of less than 2000, which is also the case with membrane systems. 

The effect of slip coefficient on concentration polarisation (CP) was mathematically modeled for flat membrane and tubular membrane systems [12,13,15,16]. Lowering of CP due to slip coefficient as a function of product water recovery ( ) for different normalised diffusion coefficients (a) is shown in Figure 6.8. The data show that CP decreases both with and a. Since a is a measure of particle diffusion from the membrane surface to the bulk solution, slip-flow possibly augments diffusive back-transport of particles from the membrane surface to the bulk solution. Thus, the slip-flow velocity model possibly accounts for higher or actual UF/MF flux, which is under-predicted by the gel polarisation model discussed in Chapter 1. 

Although the above equation seems quite simple, it is derived from a slip velocity profile which is correct for either mass flow rate calculation or velocity profile prediction. In fact, it does not include any viscosity coefficient correction. If the viscosity correction is incorporated in this expression, we obtain a more comprehensive formula similar to Eq. 15. Considering this point, Kamiadakis et al. [5] derived a relation for the slip coefficient using a uiufied velocity profile and their empirical relation for the viscosity coefficient. Their attempts resulted in... 

To calculate the Poiseuille coefficient Gp in the slip flow regime (5 < 8 < 100) the Navier-Stokes equation is solved with the velocity slip boundary conditions (8). Then for the channel the flow rate reads... 

Sharipov F, Kalempa D (2003) Velocity slip and temperature jump coefficients for gaseous mixtures. I. Viscous slip coefficient. Phys Fluid 15(6) 1800-1806... 

Thermal slip coefficient determines the tangential velocity of gas near a solid surface due to a longitudinal gradient of the surface temperature. 

Yang and Kwok [8] presented the analytical solution of fully developed electrokinetic flow subjected to sinusoidal pressure gradient or sinusoidal external electric field. The combined effect of slip flow and electrokinetics was demonstrated on the velocity profile in confined geometries. The velocity profile was observed to be a function of both slip coefficient and external electric field. They observed that both these effects play important roles for flow inside microchannels. 

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