Hele-Shaw approximations

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 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... 

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. 

All technology discussed here is based on the assumption that the plastic part is thin walled and makes use of the Hele-Shaw approximation. Three-dimensional finite element analysis (3D/FEA) eliminates this requirement. In so doing, it introduces a new class of components to simulation. With 3D/FEA it is possible to. simulate the molding of parts for which a midplane is not available. Typically, such parts are chunky— some examples are given in Fig. 7.64. Many parts... 

In injection molding the pressure level can be very high, so that the assumption of pressure-independent viscosity can lead to incorrect simulation results. Some authors (e.g., Sherbelis and Friedl 1996) have considered pressure-dependent viscosity in their simulations. However, caution should be made because the pressure-dependent viscosity might invalidate the Hele-Shaw approximation. When the viscosity varies with pressure, the pressure gradient in the flow direction (say, xi direction) and that in the thickness direction have the following relation 0F/0X3 = 7i3(0j7/0P)(0F/0xi) (Huilgol and You 2006). Only if yi3(0f//0P) 1,... 

Then appeared finite element simulations, in 2D [145] or within the frame of the Hele-Shaw approximations [68,144,148]. Until now, only few studies concerned the implementation of a crystallization kinetics into 3D codes [149], which do not require any geometrical or kinematic assumption. 

The free surface flow was calculated by using the control volume finite element method (CVFEM). Since the depth of the microchannel is small compared with the length and width, we used the Hele-Shaw approximation for the momentum equation, which is given as follows ... 

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... 

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. 

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... 

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). 

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