Slip length

The results also show that the normalized slip length and the average friction factor-Reynolds number product exhibit Reynolds dependence. Furthermore, the predictions reveal that the impact of the vapor cavity depth on the overall frictional resistance is minimal provided the depth of the vapor cavity is greater than 25% of its width. 

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 ... 

When the fractions of molecules reflected specularly and diffusively are known, the slip length can be determined, as shovm by Maxwell. Maxwell introduced a tangential momentum accommodation coefficient defined as... 

Based on the accommodation coefficient, the slip length is given by... 

Apparently, the slip length diverges when 0. In practice, the shear strain in Eq. (5) will approach zero in such a case, thus leaving the velocity jump finite. Several experimental results on cf have been reported [6, 7], most of them indicating values between 0.8 and 1.0, compatible with rough walls. 

Reference slip length Thermal entrance length... 

Figure 8 Left Schematic graph of the setup for the simulation of rubbing surfaces. Upper and lower walls are separated by a fluid or a boundary lubricant of thickness D. The outermost layers of the walls, represented by a dark color, are often treated as a rigid unit. The bottom most layer is fixed in a laboratory system, and the upper most layer is driven externally, for instance, by a spring of stiffness k. Also shown is a typical, linear velocity profile for a confined fluid with finite velocities at the boundary. The length at which the fluid s drift velocity would extrapolate to the wall s velocity is called the slip length A. Right The top wail atoms in the rigid top layer are set onto their equilibrium sites or coupled elastically to them. The remaining top wall atoms interact through interatomic potentials, which certainly may be chosen to be elastic.

However, in the case of large Kn, the no-slip approximation cannot be applied. This implies that the mean free path of the liquid is on the same length scale as the dimension of the system itself. In such a case, stress and displacement are discontinuous at the interface, so an additional parameter is required to characterize the boundary condition. A simple technique to model this is the one-dimensional slip length, which is the extrapolation length into the wall required to recover the no-slip condition, as shown in Fig. 1. If we consider... 

Figure 1. No-slip condition and slip condition with slip length, for one-dimensional shear flow. The slip length b is the extrapolation distance into the solid, to obtain the no-slip point. The slope of the linear velocity profile near the wall is the shear rate y.

To include the slip in this model, we use the slip length. In this approach, if slip occurs, the slip velocity v, is proportional to the velocity gradient at the wall... 

This result is interesting, since it gives the slip length as a function of parameters that can be measured experimentally or a priori, for simple systems in a linear approximation. The bulk shear viscosity can be approximated from the literature, and the monolayer density can be determined from optical techniques. To a first approximation, for rigidly adsorbed layers, the sliptime is related to the autocorrelation function of random momentum fluctuations in the film, given by [40]... 

The authors noted that when their friction parameter M= (pG/,) 8/G is real, it is equivalent to the real slip parameter s = fe used by McHale et al. [14]. From this analysis, a real interfacial energy G /8 is related to the slip length b, for a purely viscous fluid, by... 

Typically, the authors found slip lengths over a very wide range, from a few nanometers to a few centimeters, but this could be due to different slip lengths on each cylinder. 

From this, the velocities of particles flowing near the wall can be characterized. However, the absorption parameter a must be determined empirically. Sokhan et al. [48, 63] used this model in nonequilibrium molecular dynamics simulations to describe boundary conditions for fluid flow in carbon nanopores and nanotubes under Poiseuille flow. The authors found slip length of 3nm for the nanopores [48] and 4-8 nm for the nanotubes [63]. However, in the first case, a single factor [4] was used to model fluid-solid interactions, whereas in the second, a many-body potential was used, which, while it may be more accurate, is significantly more computationally intensive. 

We have examined the many of the various factors that determine the proper boundary condition to use at the solid-liquid interface and considered many of the models associated with theses factors. The single-valued slip length model is the simplest and most convenient boundary condition, and it has been used successfully in many studies. However, it cannot describe coupling changes where there are changes in both the storage and dissipation properties. In this situation, a two-parameter complex value may be necessary. 

The experimental data obtained at low surface densities, for end grafted chains, are in very good agreement with these theoretical predictions, not only for the overall evolution of the slip velocity vs the shear rate or of the slip length vs the slip velocity as shown in Fig. 19, but also for the molecular weight dependence of the critical velocity V which do follow exactly the laws implied by... 

Fig. 6.19 The basis of the mesoscale model of Rouby and Reynaud. The axial stress, oy in the fiber decreases linearly over the slip length, from a peak value of oy° at the edge of the matrix crack, to a value of o(Ef/Ec) at the end of the slip-zone (z = ls). It is assumed that the crack spacing / is larger than twice the load transfer length /,. It is assumed that the fiber stress decreases linearly to zero over the load transfer length l,. After Rouby and Reynaud.46...

The brittle film cracking with plastic deformation of the ductile substrate at the interface has been described by using the shear lag model. " This model, which was proposed in the analysis of the fragmentation of fiber composites," " develops a relation for the critical stress producing the steady-state cracking of the film. It assumes that the interfacial shear stress, on the one hand, is activated at each crack tip along the characteristic slip length r, and, on the... 

Fig. 16 Modeling of the stress distribution in a cracked film with plasticity at the interface using the shear lag approximation, (a) linear evolution of the normal stress along the characteristic slip length r/2 and (b) sharp evolution of the interfacial shear stress x. along the characteristic slip length r/2.

Figure 6. Slip length b deduced from the data of Fig. 4 and 5 as a function of the slip velocity Vg. The threshold for the onset of strong slip appears as a kink in the b (Vg) curve, at the critical velocity V. Above V, b increases with Vg. following a power law with an exponent 0.8 0.04. At very high shear rates, b deviates from the power law and the Vg dependence tends to saturate, indicating that a new regime of linear strong slip is approached.

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