Hele-Shaw

Generalization of the Hele-Shaw approach to flow in thin curved layers... 

The issues of selection of the spatial wavelength and the deterministic character of the fine scale features of the microstructure are closely related to similar questions in nonlinear transitions in a host of other physical systems, such as macroscopic models of immiscible displacement in porous media - - the Hele Shaw Problem (15) - and flow transitions in fluid mechanical systems (16). 

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... 

A schematic diagram of a typical flow described by the Hele-Shaw model is presented in Fig. 5.18. 

Figure 5.20 Typical velocity profile for a Hele-Shaw flow.

There are various special forms and simplifications of the above equation and they are given below. In subsequent chapters of this book we will illustrate how the various forms of the Hele-Shaw model are implemented to solve realistic mold filling problems. 

Newtonian-isothermal Hele-Shaw model. A special form of the Hele-Shaw type flow governing equations is the isothermal Newtonian case where r/(z) = //,. This simplification leads to flow a conductance given by... 

Generalized Newtonian Hele-Shaw model using a power law viscosity. For a... 

Hele-Shaw model for compression molding of a Newtonian fluid. A special case, where the z-velocity component plays a significant role and must be included, is the compression molding process. The process is schematically depicted in Fig. 5.21. 

Using the Hele-Shaw model, analyze a compression molding problem where the melt is allowed to move only in the x-direction. Use a dimension in the x-direction of 2L, and in the y-direction of W. The gapwise thickness is h. The two flow fronts are located at x = L. 

For the solution of this problem we will assume a Newtonian and isothermal flow. The Hele-Shaw approximation gives an expression for the mean velocity profile as a function of the pressure as follows (see Chapter 5)... 

The Hele-Shaw model was used to describe the flow in the FAN formulation. For a Newtonian case, the flow is described by... 

The second integral on the right hand side of eqn. (9.67) can be evaluated for problems with a prescribed Neumann boundary condition, such as heat flow when solving conduction problems. For the Hele-Shaw approximation used to model some die flow and mold filling problems, where 8p/8n = 0, this term is dropped from the equation. 

As discussed in Chapter 8 of this book, the momentum balance and the continuity equation lead to the Hele-Shaw approximation given by... 

Write a two-dimensional finite element program, using constant strain triangles and ID tube elements, to predict the flow and pressure distribution in a variable thickness die. Use the Hele-Shaw model. Compare the FEM results with the analytical solution for an end-fed sheeting die. 

This structure is generated via the modified diffusion-limited aggregation (DLA) algorithm of [205] using the law p = a (m/N). Here, N = 2, 000 (the number of particles of the DLA clusters), a = 10 and ft = 0.5 are constants that determine the shape of the cluster, p is the radius of the circle in which the cluster is embedded, pc = 0.1 is the lower limit of p (always pc < p), and to is the number of particles sticking to the downstream portion of the cluster. This example corresponds to a radial Hele-Shaw cell where water has been injected radially from the central hole. Due to heterogeneity a sample cannot be used to calculate the dissolved amount at any time, i.e., an average value for the percent dissolved amount at any time does not exist. This property is characteristic of fractal objects and processes. 

Diffusing wave spectroscopy, used to measure statistics of fluctuations in relatively thin, Hele-Shaw configurations... 

Figure 5.10 is an example of the results obtained. While most investigators of VF used Hele-Shaw cells which approximate the three-dimensional bed with a pseudo-two-dimensional one, these last two groups have visualized the VF phenomenon in actual columns. 

Problem 8-7. The Hele-Shaw Cell. The Hele Shaw cell is perhaps one of the simplest constructions of a nearly unidirectional flow however, as we shall see in this problem the flow inside the cell has some remarkable properties. A Hele Shaw cell consists of two flat parallel rigid walls of vertical separation h and horizontal extent / with h/l 1. The gap between the fluids is occupied with Newtonian fluid and contains obstacles in the form of cylinders with generators normal to the walls The fluid is being driven through the cell by a steady horizontal pressure gradient Vp = G applied between the ends of the cell. 

This result implies that (at any fixed z) the flow past a cylindrical obstacle will correspond to the 2D potential flow past that obstacle. For the Hele Shaw cell the pressure p plays the role of the velocity potential velocity potential outside a circular cylinder of radius a. There is, however, one important caveat. Let us consider the circulation T defined by... 

Some excellent photgraphs of flow past obstacles in a Hele-Shaw cell are contained in An Album of Fluid Motion by M. Van Dyke (Parabolic Press, Stanford, CA, 1982). 

P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid, Proc. R. Soc. London Ser A 245, 312-29 (1958). 

Researchers have used physical models of porous media to study flow problems for many years. For example, the Hele-Shaw cell appeared in the late 1800s (Sahimi, 1993). The first reported use of such models for two-phase systems is attributed to Chatenever and Calhoun (1952), who used Lucite and glass bead packs to view immiscible displacement of brine and crude oil (Buckley, 1991). Subsequently, etched and photo-etched glass were used to construct physical models. The use of molded resins for model construction was introduced in the 1970s (Buck-ley, 1991). 

Wooding, R.A. I960. Instability of a viscous liquid of variable density in a vertical Hele-Shaw cell. J. Fluid Mech. 7 501-515. 

R. A. Wooding, Instability of a Viscous Liquid of Variable Density in a Vertical Hele-Shaw Cell, J. Fluid Mech. (7) 501-515,1960. 

In very large Hele-Shaw cells, at a distance from the walls much larger than the cell thickness, a hexagonal order is obtained only locally, and defects destroy the long-range order. This is expected to arise from the randomness of the interactions between columns and the bottom of the cell. Once a column spans the narrow height of the cell, the solid friction with the floor and ceiling freezes the... 

---