Lubrication Approximation Solution

The solution will proceed as follows (1) the dimensions of the die will be determined (2) an expression for AP versus Q will be determined using the lubrication approximation (3) from this relation one can calculate Q for the given AP and the average velocity v and (4) the time required to extrude the parison is calculated. 

From this information and the equation for the mass of the parison (wjp), 

ISOTHERMAL FLOW OF PURELY VISCOUS NON-NEWTONIAN FLUIDS 

FIGURE 2.15 Conical region of the blow molding die showing the dimensions required to produce the parison as requested in Design Problem I. 

AP/L is replaced by -dp/dz. H is the gap which is constant andisPo(l -Ko)otRi( -/Ci), where Po andPz. are the outer radii of the tapered annulus at z = 0 and z = L, respectively. It is also noted that although the gap is constant, the ratio of the inner to outer radii, k, varies slightly with the distance 

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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.

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]. 

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. 

FEM versus Analytical Solution of Flow in a Tapered Gap Consider isothermal pressure flow of a constant viscosity Newtonian fluid, between infinite plates, 10 cm long with a linearly decreasing gap size of 1.5 cm at the entrance and 1 cm at the exit. The distance between the entrance and the exit is 10 cm. The pressure at the inlet and outlet are 2 atmospheres and zero, respectively, (a) Calculate the pressure distribution invoking the lubrication approximation, (b) Calculate the pressure profile using the FEM formulation with six equal-sized elements, and compare the results to (a). 

If two particles are interacting across an electrolyte solution, the equations of continuity and the momentum balance. Equation 5.249, in lubrication approximation reads ... 

In any case, the very important conclusion from (5-81) is that the leading-order approximation to the solution in the thin-film region always can be determined completely without the need to determine anything about the velocity field in the rest of the domain where the geometry is much more complicated. This constitutes a very considerable simplification When the lubrication approximation can be made, we need focus our attention on only the lubrication equations within the thin gap, and in this region the general solutions (5-74) and (5-79) have been worked out already. We shall see in the next section that the dominant contributions to the forces or torques acting on a body near a second boundary always occur in the lubrication layer when e <[Pg.315]

The lubrication approximation can be applied to the case when the Reynolds number is small and when the distances between the particle surfaces are much smaller than their radii of curvature (Reynolds 1886). When the role of surfactants is investigated an additional assumption is made - the Peclet number in the gap is small. Below the equations of lubrication approximation are formulated in a cylindrical coordinate system, Orz, where the droplet interface, 5, is defined as z = H(t,r)/2 and H is the local film thickness (see Fig. 1). In additional only axial symmetric flows are considered when all parameters do not depend on the meridian angle. The middle plane is z = 0 and the unit normal at the surface S pointed to the drop phase is n. The solution in the film (continuous phase) is assumed to be a mixture of nonionic and ionic surfactants and background electrolyte with relative dielectric permittivity 8f. The general formulation can be found in Kralchevsky et al. (2002). 

The integrated mass balance equation (13) expresses the fact that the local change of the mass of molecules across the film is compensated by the bulk and surface convection and diffusion fluxes. In the case of lubrication approximation for small Peclet numbers the solution of the leading order of the diffusion equations (4) and (6) gives the Boltzmann type of the nonequilibrium concentration distribution in the bulk phase ... 

In Eq. (15) q is the vacuum dielectric constant. The condition for electroneutrality of the solution as a whole is equivalent to the Gauss law, which determines the surface charge density, q. In the lubrication approximation it reads (Kralchevsky et al. 1999) ... 

For ionic surfactant solution the body force tensor, Pb, is not isotropic - it is the Maxwell electric stress tensor, i.e. Pb = f6bEE - i6jE l2, where E = -V is the electric field (Landau and Lifshitz 1960). The density of the electric force plays the role of a spatial body force, f, in the Navier-Stokes equation of motion (3). In the lubrication approximation the pressure in the continuous phase depends on the vertical coordinate, z, only through its osmotic part generated from the electric potential and the pressure in the middle plane (or the pressure, pn, corresponding to the case of zero potential) ... 

Knowing the pressure distribution, Eq. (17), and the radial component of the surface velocity, Vs,r, the distribution of the radial component of the bulk velocity is derived from the solution of the Navier-Stokes equation (3) in the lubrication approximation to be (Valkovska and Danov 2001) ... 

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 computer simulation involves a f 1 nite-d1fference solution of the field equations which describe this problem which Involves coupling of fluid mechanics with pressure and temperature fields. The lubrication approximation is utilized to eliminate the momentum equation in the radial direction. This simplification is reasonable for the flow fields considered in this study since the following criteria cire met (21) ... 

The examples in this chapter were selected to illustrate the way in which numerical simulation complements and enhances the understanding of polymer processing operations that is gained from analytical solutions based on idealizations (infinite geometries and the lubrication approximation, for example). The analytical solutions are invaluable for providing insight, but detailed information and complete... 

The major practical use of Equation 11.1 lies in testing whether a film of uniform initial thickness ho remains stable or eventually become unstable with time. The solution of linearized equations of motion incorporating the effect of intermolecular forces, can be simplified by the lubrication approximation (non-inertial laminar flow in thin films) and admit space periodic solutions for the film thickness h(x, t) = ho + E sin(kx) exp(cof) with ... 

The radial flow example illustrates how rapidly problems involving the flow of non-Newtonian fluids become mathematically complicated. There are a number of times when the analysis can be simplified. There are two useful approximations that simpUly the differential equations that arise out of the equations of motion. These neglect the effect of curvature and the lubrication approximation. The solutions of several problems, which have already been dealt with, are used to illustrate these approximations. First, we examine how neglecting curvature can lead to a simplification of the differential equations. This is followed by the handling of geometries in which a variation in the dimension transverse to the flow direction occurs, such as the case of a tapered tube. 

The solution to Design Problem I is presented in this section. The lubrication approximation is used first to obtain a solution. This is followed by a numerical approach in which the die is broken into a series of annuli. 

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