Lubrication theory approximation

LAT lubrication approximation theory LIMM hght intensity modulation method... 

According to the lubrication approximation, we can quite accurately assume that locally the flow takes place between two parallel plates at H x,z) apart in relative motion. The assumptions on which the theory of lubrication rests are as follows (a) the flow is laminar, (b) the flow is steady in time, (c) the flow is isothermal, (d) the fluid is incompressible, (e) the fluid is Newtonian, (f) there is no slip at the wall, (g) the inertial forces due to fluid acceleration are negligible compared to the viscous shear forces, and (h) any motion of fluid in a direction normal to the surfaces can be neglected in comparison with motion parallel to them. 

The system of equations is closed by thermodynamic equations defiifing p, to be derived in Sections 2 and 3, which replace an equation of state used in standard hydrodynamic theory. Combined thermo-hydrodynanfic theory based on Eq. (1) and including nonlocal interactions is formidable, but the problem can simplified using a natural scale separation in thin films [13, 14], which is also used in the standard hydrodynamic lubrication approximation [7]. 

The above theory blends hydrodynamic and thermodynamic relations into a common formalism in the framework of lubrication approximation. This provides a common framework for discussion of different physical effects. It still... 

In 1959, Burgdorfer [39] first introduced a concept of the kinetic theory to the field of gas film lubrication. This was to derive an approximation equation, called the modified Reynolds equation, using a slip flow velocity boundary con-... 

In the past, various resin flow models have been proposed [2,15-19], Two main approaches to predicting resin flow behavior in laminates have been suggested in the literature thus far. In the first case, Kardos et al. [2], Loos and Springer [15], Williams et al. [16], and Gutowski [17] assume that a pressure gradient develops in the laminate both in the vertical and horizontal directions. These approaches describe the resin flow in the laminate in terms of Darcy s Law for flow in porous media, which requires knowledge of the fiber network permeability and resin viscosity. Fiber network permeability is a function of fiber diameter, the porosity or void ratio of the porous medium, and the shape factor of the fibers. Viscosity of the resin is essentially a function of the extent of reaction and temperature. The second major approach is that of Lindt et al. [18] who use lubrication theory approximations to calculate the components of squeezing flow created by compaction of the plies. The first approach predicts consolidation of the plies from the top (bleeder surface) down, but the second assumes a plane of symmetry at the horizontal midplane of the laminate. Experimental evidence thus far [19] seems to support the Darcy s Law approach. 

In this section we consider the detailed analysis for two applications of lubrication theory the classic slider-block problem that was depicted in the previous section and the motion of a sphere toward an infinite plane wall when the sphere is very close to the wall. It is the usual practice in lubrication theory to focus directly on the motion in the thin gap using (5-69)-(5-72), or their solutions (5-74) and (5-79), without any mention of the asymptotic nature of the problem or of the fact that these equations (and their solutions) represent only a first approximation to the full solution in the lubrication layer. We adopt the same approach here but with the formal justification of the preceding section. 

Fortunately, in most practical investigations, such complete information is unnecessary. Rather, it usually suffices to know only certain components of these dyadics, and then only in limiting cases. If ajl represents a characteristic particle-to-wall dimension ratio, these limiting cases correspond to the extreme cases where ajl is either very small or very near unity. In the former case the method of reflections (cf. H9) provides a useful technique for obtaining the wall correction. In the latter case, corresponding to the situation where the particle is extremely close to the wall, lubrication-theory type approximations (B7, B29, Cll, D7, G5d, H15, K8, M9, MIO, S8) normally suffice to obtain the required correction. Intermediate cases are rarely of interest. 

This result is identical to that obtained from a conventional lubrication-theory approximation (Ml), based on the flow field generated when two parallel plane approach one another (L6). The general formula (130) is in excellent agreement, over the entire a/h range, with the experimental results of MacKay et al. (Ml, M2). 

Matching the lubrication equation to thermodynamic theory requires some caution, since thermodynamic theory yielding an expression for pL should be applied to the entire system including dense (liquid) and dilute (vapor) phases in equilibrium, whereas only the dense phase may have a suitable aspect ratio. To make the approximation applicable, one has to assume that the interface dividing the dense and the dilute phase is only weakly inclined relative to the substrate and weakly curved, so that its position can be expressed by a function h x, t) with derivatives obeying the above lubrication scahng. Thermodynamic theory, either local or nonlocal, can be used to compute an equilibrium density profile across the interface (in the vertical direction), po z — h x, t)), which is weakly dependent on the horizontal 2D position and time only through its dependence on h, e.g. 

Consequently analytical methods are mostly confined to creeping flows. Roughly, there are two types of problems that can be solved. The first of these deals with interfaces that show small deviations from simple geometric forms, as for instance the case of a slightly deformed sphere settling in an infinite fluid. The second type constitutes cases where interfacial position changes, but only very slowly. Then its variation can be neglected to the first approximation and the lubrication theory approximation or the slender body approximation applied. It should be noted that both the above methods yield approximate solutions. 

The cases of a sphere and slightly deformed sphere in a uniform flow field are considered first in Sections 4 and 5. The mathematical method used conventionally in these problems is the regular asymptotic expansion. The reader is introduced to this method. In Section 6, the dip coating problem under the lubrication theory approximation is examined. (The closely related slender body approximation is outlined in Problem 7.5.) A more sophisticated method of matched asymptotic expansions is used to solve this problem and its main features... 

The lubrication theory approximation is used to formulate this problran. In this scheme it is assumed that the film is almost flat. Consequently, if x = h(y) is the profile shape, then dhldy 1. Thus the boundary eonditions apphcable on h(y) are used by treating h to be almost a constant. This procedure also allows the use of the assumption that, to a good approximation, the flow is fully developed, with 0 and a function of x alone. Then the equations of motion become... 

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