Lubrication approximation

Let s use the steady, two-dimensional flow in a thin channel or a narrow gap between solid objects as schematically represented in Fig. 5.11. The channel height or gap width 

For this type of flow, the momentum equations (for a Newtonian fluid) are reduced to the steady Navier-Stokes equations, i.e. 

The lubrication approximation depends on two basic conditions, one geometric and one dynamic. The geometric requirement is revealed by the continuity equation. If Lx and Ly represents the length scales for the velocity variations in the x- and (/-directions, respectively, and let U and V be the respective scales for uz and uy. From the continuity equation we obtain 

In order to neglect pressure variation in the //-direction all the terms in the //—momentum equation must be small, in other words V/U 1. From the continuity scale analysis we get that the geometric requirement is, 

The velocity gradient components duldxi and dufdx2 are negligible comparing to duldx.  

At least for isothermal and pressure-independent Newtonian fluids, the pressure may be regarded as a function of Xi and X2 only, and thus dPIdx = 0. 

Assuming that stress components Th and T13 are comparable in magnitude, then one has dxn/dxi C 0x13/6x3. 

The Reynolds number is very small and the flow remains laminar. 

A typical example of the lubrication flow is a Newtonian flow in a slider bearing where a moving wall drags the fluid with a velocity U. Taking the above-mentioned assumptions into account, the equation of motion governing the bearing flow becomes simply 

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In the United States approximately 50% of the 40,000 t of chloriaated paraffins consumed domestically are used in metal-working lubricants. Approximately 20% are consumed as plastic additives, mainly fire retardants, and similarly 12% in mbber. The remainder as plasticizers in paint (9%) and caulks, adhesives, and sealants at 6%. 

II provides a transition between the two asymptotic limits. Viscous stresses now scale by the local thickness of the film, h, and the bubble shape varies from the constant thickness film to the spherical segment. Here the surfactant distribution along the interface may be important. Fortunately, for small capillary numbers, dh/dx < 1 and the lubrication approximation may be used throughout. Region II is quantified below. 

Based on the lubrication approximation, the momentum equations to solve the flow through a slightly tapered tube are the same equations that we use to solve for the equations that pertain to the straight circular tube, i.e.,... 

Today, the most widely used model simplification in polymer processing simulation is the Hele-Shaw model [5], It applies to flows in "narrow" gaps such as injection mold filling, compression molding, some extrusion dies, extruders, bearings, etc. The major assumptions for the lubrication approximation are that the gap is small, such that h < . L, and that the gaps vary slowly such that... 

For the flow of polymer melts, the Reynolds number, Re = pUch/r], is usually 10 5, with the exception of the reaction injection molding process, RIM, where Re —> 1 — 100 at the gate. The geometric and dynamic conditions of the lubrication approximation, applied... 

In Gaskell s treatment, aNewtonian flow was assumed with a very small gap-to-radiusratio, h -C R. This assumption allows us to assume the well known lubrication approximation with only velocity components ux y). In addition, Gaskell s model assumes that a very large bank of melt exists in the feed side of the calender. The continuity equation and momentum balance reduce to... 

As with the Newtonian model, we assume a lubrication approximation, where the momentum balance reduces to... 

Figure 11.19 presents the pressure distribution along the x-axis for a Newtonian solution using several bank-to-nip ratios. The solutions are presented with the analytical predictions using McKelvey s lubrication approximation model presented in Chapter 6. The graph shows that the two solutions are in good agreement. Fig. 11.20 presents a sample velocity field for the Newtonian case with a bank-to-nip ratio of 10. As can be seen, the velocities look plausible and present the recirculation pattern predicted by McKelvey s lubrication approximation model and seen in experimental work done in the past [18]. 

Figure 11.19 Comparison of lubrication approximation solution and RFM solution of the pressure profiles between the rolls for several values of bank, or fed sheet, to nip ratio for Newtonian viscosity model.

Mathematical Modeling, Common Boundary Conditions, Common Simplifying Assumptions and the Lubrication Approximation, 60... 

In polymer processing, we frequently encounter creeping viscous flow in slowly tapering, relatively narrow, gaps as did the ancient Egyptians so depicted in Fig. 2.5. These flows are usually solved by the well-known lubrication approximation, which originates with the famous work by Osborne Reynolds, in which he laid the foundations of hydrodynamic lubrication.14 The theoretical analysis of lubrication deals with the hydrodynamic behavior of thin films from a fraction of a mil (10 in) to a few mils thick. High pressures of the... 

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 lubrication approximation facilitates solutions to flow problems in complex geometries, where analytical solutions either cannot be obtained or are lengthy and difficult. The utility of this approximation can well be appreciated by comparing the almost exact solution of pressure flow in slightly tapered channels to that obtained by the lubrication approximation. 

The lubrication approximation as previously derived is valid for purely viscous Newtonian fluids. But polymer melts are viscoelastic and also exhibit normal stresses in shearing flows, as is discussed in Chapter 3 nevertheless, for many engineering calculations in processing machines, the approximation does provide useful models. 

Consider an incompressible Newtonian fluid in isothermal flow between two non-parallel plates in relative motion, as shown in Fig. E2.8, where the upper plate is moving at constant velocity V0 in the z direction. The gap varies linearly from an initial value of H0 to H over length L, and the pressure at the entrance is P0 and at the exit P. Using the lubrication approximation, derive the pressure profile. 

Invoking the lubrication approximation, the local velocity profile (at a given angle 0) in rectangular coordinates X, Y, with boundary conditions vx(0) = Ylr and vx(B) — 0 (see Example 2.5) is given by... 

Plot the ratio of pressure drops obtained by Eqs. (h) and (f) to show that for a < 10°, the error involved using the lubrication approximation is very small. 

The lubrication approximation was discussed in terms of Newtonian fluids. Considering a nearly parallel plate pressure flow (H = Ho — Az), where A is the taper, what additional considerations would have to be made to consider using the lubrication approximation for (a) a shear-thinning fluid flow, and (b) a CEF fluid ... 

These, together with the small Reynolds number in the film, justifies the use of the lubrication approximation. Moreover, the same considerations lead us to neglect exit effects (at x = W), and precise entrance conditions (at x = 0) need not be specified. 

We have substituted ordinary differentials for the partial differentials in the equation of motion because the left-hand side is only a function of r, whereas we assume the right-hand side is only a function of v (lubrication approximation). Therefore, they simply equal a constant. Equation 5.8-4 can now be integrated over y, with boundary conditions vr(0) = 0 and vr(<5) = 0, to give... 

The flow configuration of building block 3, of two non-parallel plates in relative motion, shown in Fig. 6.19, was analyzed in detail in Example 2.8 using the lubrication approximation and the Reynolds equation. This flow configuration is not only relevant to knife coating and calendering, but to SSEs as well, because the screw channel normally has constant-tapered sections. As shown in Fig. 6.19, the gap between the plates of length L is Ho and II at the entrance and exit, respectively, and the upper plate moves at constant velocity Vo-... 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

For incompressible isothermal flow of a Newtonian fluid, making the lubrication approximation and the no-slip condition, the equations of continuity and motion become... 

The vx(y) velocity profiles for Regions III and IVare shown in Fig. 10.45. Using Eq. 10.2-43, we can compute the whole velocity field and plot the velocity vector field. However, we must recall that the model assumed the lubrication approximation and neglected all acceleration and inertia effects. 

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