Lubrication Forces

We noted at the beginning ofthis section that the practical significance of the journal-bearing geometry with e y I is that large pressure variations are generated in the gap, which can be used to support a load that is attached to the inner cylinder. Now that the pressure and the velocity components u(r0) and uf are known, we can actually calculate the total force on the inner cylinder. 

The discussion of the preceding paragraph was mainly qualitative. However, explicit results for the total force acting in the horizontal and vertical directions (see Fig. 5 1) can 

Thus the total force per unit length on the inner cylinder is 

Let us first consider the pressure contributions to the force F p), corresponding to the lubrication approximation to the pressure distribution, p,(( ). The horizontal component Fpp can be evaluated immediately. In particular, because p,(() is an odd function of 0, 

On the other hand, the contribution in the vertical direction is not zero  

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Note that for liquid solid systems, Eq. (20) should also include the short-range lubrication forces and the effects of other forces such as the virtual mass force. But this is beyond the scope of this chapter. 

For two polymer coated surfaces, a simple extension of Reynolds formula for the lubrication force gives [4,60]... 

In order to account for hydrodynamic interactions among the suspended particles, Bossis and Brady (1984) used both pairwise additivity of velocities (mobilities) and forces (resistances), discussing the advantages and disadvantages of each method. While their original work did not take explicit account of three- (or more) body effects, the recent formulation of Durlofsky, Brady, and Bossis (1988) does provide a useful procedure for incorporating both the far-field, many-body interactions and near-field, lubrication forces into the grand resistance and mobility matrices. 

A quahtative understanding of shear thickening is that hydro-dynamic forces (the Magnus force) drive particles close to physical contact such that hydrodynamic lubrication forces and frictional... 

The coefficient k(h/a) has been introduced by H. Brenner 2[ and represents the modification of the Stokes friction due to the bottom wall located at a distance /( from the particle. If hja — oc, then k h/a) = I and we recover the usual Stokes law for an isolated particle. On the other hand, if the particle comes very close to the plane (h/a — I), then k(h/a) — oc and the particle will never reach the plane, owing to the presence of lubrication forces. The sedimentation velocity follows from Hq. (2) ... 

Lionberger and Russel (1994) suggested that the stabilizing layers on the spheres of van der Werff et al. produce different lubrication forces than those between bare particles, such as those of Shikata and Pearson, and that this accounts for the differences between the high-frequency moduli of these two systems. Thus, it would appear, perhaps not surprisingly, that the high-frequency behavior of concentrated suspensions is sensitive to the details of interactions between spheres in near contact. 

Finally, a steric repulsive force = E i must be included to keep particles from overlapping. This steric force would not be required if the hydrodynamic forces were treated more realistically, because as particles approach each other closely, strong lubrication forces are produced by the solvent that must be squeezed from between the particles (Bonnecaze and Brady 1992a). However, these lubrication forces are omitted from one-particle Stokes hydrodynamics, and a steric force must be introduced. 

FIG. 21-114 Interparticle forces and granule deformability. Interparticle forces include capillary forces, viscous lubrication forces, and friction forces. Reprinted from Design and Optimization of Granulation and Compaction Processes for Enhanced Product Performance, Ennis, 2006, with permission of E[Pg.2335]

The naive picture sketched here is not correct for a number of reasons. It is well known that for hard spheres without HI the quiescent shear modulus diverges for short times, irHSnoHij-j Lubrication forces, which keep the particles... 

Figure 5-3. The dependence of the magnitude of the hydrodynamic lubrication force on the degree of eccentricity of the journal bearing, a = F /(12n fiQa/e2).

Hint The net torque on the plate will be zero - the torque that is due to lubrication forces will balance the torque that is due to gravity. 

The radial hydrodynamic component (y component) of the force is denoted by fj, and represents the net externally applied hydrodynamic force on the particle resulting from the particle being driven toward (or away from) the collector by the external flow (undisturbed or disturbed) plus any negative resistive lubrication force arising from a close approach of the particle to the collector. The attractive molecular London force acting along the line of centers is denoted by (Ad denotes adhesion). Because of the linearity of the Stokes-Oseen equation, the velocity fields and associated forces may be superposed. 

Another semi-empirical expression derives from the widely used work of Frankel Acrivos (1967) who assumed a dense suspension of monomodal spherical particles, where the particle separation is small compared to the particle radius. In this limit the dissipation is taken to be governed by lubrication forces associated with the relative velocity of the particles along their line of centers. In the low volume fraction or dilute limit, this effect vanishes and the dissipation is close to that associated with the externally applied shear rate. Sengun Probstein (1989a) calculated the dissipation in a unit cell assuming the total dissipation at any volume fraction to be given by the linear sum of the two dissipations. 

The two experiments are exemplified by the sketches in figure 3.34. In the first, one surface is moved normally to the other in a sinusoidal manner, whereas in the second one surface is moved parallel to the second surface. For the first case, solvent is either squeezed from or drawn into the space between the surfaces until the load applied is balanced by steric repulsion fi-om the increase (decrease) in osmotic pressure due to increase (decrease) in segment concentration in the intersurface volume. Hydrodynamic interactions between the two sinfaces may become dominant and interactions associated with the flow of solvent between the two surfaces are termed lubrication forces. For the geometry of the SFA the hydrodynamic force, Ffi, is related to the velocity of motion of the surfaces normal to each other by... 

An extended brush-like layer is formed for the zwitterionically terminated polystyrene and comparison of force-distance curves before and after oscillatory motion showed that they were not displaced by the lubrication forces. These force-distance curves showed that the two brush-like layers interact at distances of about 2500 A. Above this separation, values of G scale with D in precisely the same manner both in the presence and in the absence of tethered polymer. The slopes of these lines, moreover, gave a viscosity (j/o) that agreed with the bulk viscosity of toluene. For the tethered pol)nner the line through these large separation data has an x axis intercept that corresponds closely to twice the polymer layer s thickness. Hence, in this region the shear plane in the system has shifted by a distance of 2Lh (Lr is hydrod5mamic layer thickness equilibrium layer thickness) and thus equation (3.4.13) becomes... 

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