Hele-Shaw flow

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

Next, we consider a two-dimensional, isothermal, viscous flow analysis. Most mould cavities are much longer and wider than they are thick, so the flow is approximately two-dimensional. In the Hele-Shaw flow... 

Selection of Dentritic Growth and Viscous Fingerings in Hele-Shaw Flow By Jian-Jun Xu... 

Housiadas KD, Tanner RI (2009) On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid. J Non-Newtonian Huid Mech 2009 162 88-92 Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29 329-349 Huilgol RR (2006) On the derivation of the symmetric and asymmetric Hele-Shaw flow equations for viscous and viscoplastic fluids using the viscometric fluidity function. J Non-Newtonian Fluid Mech 138 209-213... 

Richardson SM (1972) Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J Fluid Mech 56 609-618 Rizzo FJ, Shippy DJ (1970) A method of solution for certain problems of transient heat conduction. AIAA J 8 2004-2009... 

Minor modifications have been made to the parallel-plate flow chamber geometry in order to produce a spatially dependent range of shear stresses across a single substrate. For example, a variable gap height device has been constructed in which /i is a linear function of the distance from the inlet, x (Figure 34.5b) [ 103 ]. In another device the width of the flow path varies inversely with distance from the inlet, w oc 1/x, to produce a Hele-Shaw flow pattern and a hydrodynamic shear stress that increases linearly with distance, t OCX [104,105]. 

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 ... 

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. 

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... 

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. 

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. 

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). 

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. 

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