Lubrication theory, assumptions

To use the analytical (lubrication theory) results of Huppert (1982), we represent the plume material as an equivalent layer of constant viscosity rj and constant specific weight contrast Apg. We ignore viscous forces associated with downward flow of the normal mantle at the edges of the blob. The appropriateness of this assumption is discussed in the Appendix. The thickness of the material is h(R) where R is the radial distance from the centre of the blob. Outward flow of plume material is driven by the local slope at its base. The flux (in m s per circumferential length of the flow front) is analytically... 

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... 

The lubrication theory approximation and the closely related slend body approximation (see Problem 7.5) have been applied successfully in many cases. The critical assumption is that the variation of the velocity profile in the direction of flow is ignored in the first approximation. The flow under this approximation becomes one-dimensional in most cases, or at least retains a simple form. 

The governing equations are derived under the following assumptions usually adopted in lubrication theory ... 

One of the great strength of lubrication theory is that it can rely on a very powerful boundary condition between first and third bodies which states that there is no slip at the wall" between the oil film and the bordering surfaces. While that condition has occasionally been questioned (22), it has proved correct even under high shear conditions, and is one of the basic assumptions of Reynolds theory. 

After the dimple disappears, the resulting almost plane-parallel film thins at constant radius R. The theoretical description of thin-film drainage is based on the lubrication theory [59,466], in which the following assumptions are made (1) The thickness of the film, h. 

Several theories have been developed to account for the observed characteristics of the plasticization process Daniels has recently published a review of plasticization mechanisms and theories [8]. Although most mechanistic studies of plasticization have focused on PVC, much of this information can be adapted to other polymer systems. The lubricating theory of plasticization holds that plasticizers act as molecular lubricants to facilitate polymer chain movement when a force is applied to the plastic. It starts with the assumption that the unplasticized polymer chains do not move freely because of surface irregularities and van der Waals attractive forces. As the system is heated and mixed, the plasticizer molecules diffuse into the polymer and weaken the polymer-polymer interactions. Portions of the plasticizer molecule are strongly attracted to the polymer while other parts of the plasticizer molecule can shield the polymer chain and act as a lubricant. This reduction in intermolecular or van der Waals forces among the polymer chains increases the flexibility, softness, and elongation of the polymer. 

Due to limited computational power and relatively simple numerical algorithms in the past, early studies on the surface contact and lubrication were based either on an assumption of ideally smooth surfaces, or on stochastic models with limited numbers of surface roughness parameters. Those studies, such as presented in [ 1 -7], have laid a foundation for the conventional contact and lubrication theories that have been widely used in industry. Predictions of contact and lubrication characteristics by those theories have so far been useful references for designers. However, since most engineering surfaces are quite rough, predictions based on the smooth surface assumption may not be satisfactory. The stochastic models with only a few roughness parameters, on the other hand, may not be sufficient to give the detailed results on the contact and lubrication characteristics for the real surfaces. 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

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. 

This work is a continuation of the Tichy and Bou-Said s theory, which considers convective inertia terms and surface accelerations, and in which shear thinning is also taken into account. The lubricant is described as a power law fluid. One of the main assumptions concerns the use of the velocity... 

---