Slip parameter

Figure 2. Mass-spring model for slip motion. The upper and lower masses are allowed to move horizontally in one dimension, based on the spring dynamics. The slip parameter is the ratio of the displacements a = nj(2)/ j(l)-...

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

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... 

Rh( 5-C9H7)( 74-COD)] (757)]. Calculation of the slip parameter from the l3C NMR spectral data of 73 gives a A<5(C3a,7a) = - 15 ppm 151), indicating that the hapticity of the ligand is best described as a distorted >/5-indenyl (759), in agreement with the crystallographic results. 

It can be concluded that significant differences in metal-carbon bond distances to coordinated olefins are not reflected in either the slip parameter or the barrier to rotation of the indenyl ring in this series of complexes. However, the relative strengths of the interactions between the [Rh( /5-indenyl)] fragment and OFCOT and COT are illustrated by some reactivity comparisons in Section VIII,F,2. 

A detailed derivation of Eq. (1.6a) using the no-slip boundary condition is provided in Section A.2 of the Appendix. If we were to generalize the analysis above with the partial-slip boundary condition, that is, Sv/Sz = pv (p = slip parameter) instead of the no-slip condition in Eq. (A.2) at the lubricant/solid boundary with q (z) = p, we would obtain... 

This is equivalent to the assumption that there is some slippage of network junctions with respect to the continuum during the deformation. The local velocity field is thus different from the macroscopic one. The adjustable parameter a materializes the slippage and is called the slip parameter. To avoid confusion, it should be noticed that equation (36) is often written as ... 

The slip parameter can be easily determined from various experiments in shear situations by some fit of the steady state shear viscosity and primary normal stress coefficient. Analytic expressions are easily derived in steady state and transient flows in the form ... 

Alternatively, in transient flows the slip parameter can also be determined using the time position of the maximum of the experimental fimctions (tT for tangential stress and tN for normal stress). However, this can only be performed accurately if the stress overshoot is large enough to avoid uncertainties in these values and this can only be achieved at high shear rates. In this case, according to equations (50c) and (50d) ... 

As can be seen from these results and from Fig. 10 (for HD), satisfactory agreement caumot be achieved by the use of a single value of the slip parameter for both tangential and normal stresses. 

It requires that the principal stress axes should coincide with the principal strain axes. This rrile has been experimentally checked hy many authors [24, 56] Actually, the use of the Gordon-Schowalter derivative involves the violation of the Lodge - Meissner rule, indeed when a equals 0 or 2, either the upper or the lower convected derivatives implies that the relationship is respected. In the general case, the double value of the slip parameter is a natural way to accommodate this rule. 

The connection between the double value of the slip parameter obtained from the viscometric functions and the violation of the Lodge-Meissner rule becomes more evident when the time-strain separability of the model is considered. For this purpose, the Johnson-Segalman model can be rewritten under the form of a single integral equation, cancelling the Cauchy term, which gives the following form in simple shear flows ... 

Nevertheless, a rather good fit of the experimental damping function (as determined by equation (45) of the preceeding section) can be obtained until y = 4 for all the materials with equations (50), provided that the corresponding slip parameters are used (ax for equation (50a) and aN for equation (50b)). 

Keeping in mind the previous remarks, it must be recognized that the lower the value of the slip parameter, the smaller the deviation to the Lodge-Meissner relation. This may be especially interesting, since in such a case, one can expect a single value of a that enables some kind of compromise for an acceptable depiction of the rheological shear functions. This can be expected,... 

Lodge Meissner rule unsatisfied (2 slip parameters). Oscillations in transient stress growth. Negative relaxation modulus in large step shear strain. 

If the slip parameter a is a non-zero constant, the requirement that the shear stress be a monotonically increasing function of the shear rate in simple shear flow imposes a constraint upon the viscosity ratio. Using a spectrum of relaxation times loosens this constraint and allows for more realistic fitting of the rheological data. 

Ad slip parameter for wall boundary condition in granular theory (m) Af t) Eulerian longitudinal integral length scale (m)... 

For the purpose of a qualitative evaluation of a complex extruder design it often may be sufficient to carry out the simulation with reduced physics, that is to pass on wall slip and to perform the simulation for example with a constant low viscosity. But the more exact the real conditions should be followed, the more activities to locate the material parameters have to be spent. A reasonable access to avoid disaccords in the interpretation of the simulations on complex geometries is to calibrate the physical model on a simple nozzle. On such a basic structure it is possible to compare the measured pressures with the calculates values and to customize the complex physics of Bingham or wall slip parameters. 

The slip parameter takes the value of the mean free path of the molecules. Xd is normally estimated by ... 

Slip parameter for wall boundary condition in granular theory (m)... 

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