The Hele-Shaw model

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. 

We start the derivation with an order of magnitude analysis of the continuity equation 

With the continuity equation and the scales for the x- and y-velocities, we can solve for the 2-velocity scale as 

uz -C ux, uy and uz can be ignored. We must point out that this velocity plays a significant role in heat transfer and orientation in the flow front region, because the free flow front is dominated by what is usually referred to as a fountain flow effect. 

---

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. 

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. 

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

The material flow in a thin cavity can be modeled by the Hele-Shaw model. Also a lubrication approximation can be used with assumptions such as a negligible inertial force, dominant shear stress, a thin cavity (compared with the planar dimensions), incompressible fluid, a small capillary number and a generalized Newtonian fluid (Hieber and Shen, 1980 Ilicker and Folgar,1983). 

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). 

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... 

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. 

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. 

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). 

The approximation arises in the study of flow between parallel plates and is sometimes known as the Hele-Shaw approximation. As most injection-molded parts are thin walled, it turns out that this model is also applicable to the injection-molding process. 

The solution to the full set of conservation equations for mass, momentum and energy requires major computational resources and may become very costly. Instead, some assumptions are introduced to obtain simplified flow models. The most popular models are the generalized Hele-Shaw (GHS) model and the Barone-Caulk model. 

Utilised for this purpose. The simplest means of modelling the phenomenon is to assume a Hele-Shaw creeping flow, that is, flow in an open channel between two parallel plates with small separation H. The gap can in this case be treated like a virtual porous medium with an equivalent... 

Because many SMC parts have a thickness much less than the length in the direction of flow, an adequate model may be constructed based on classical Hele-Shaw flow the flow field is essentially a flat sheet of a constant thickness. Tucker and Folgar... 

We consider the flow of polymer melt into thin rectangular cavities as a means of illustrating the approach taken. Hieber and Shen (1980) model the flow of polymer melt in a thin cavity using classical Hele-Shaw flow. In this approach the velocity field is considered to consist of two components, Vx and Vy, which depend primarily on z but not on x or y (i.e., dvxidx dVx/dz). The components of the equation of motion are then taken as... 

The generalized Hele-Shaw flow model is employed to describe flow behavior. Governing equations for flow in the plane direction are as follows ... 

---