Hele-Shaw equation

1 Flow in Thin Cavity of Arbitrary In-plane Dimensions 

Most injection-molded parts are thin waUed, i.e., they have a small thickness compared to other typical dimensions. Therefore, one can reduce the three-dimensional flow to a simpler two-dimensional problem, using the lubrication approximation (Richardson 1972). We consider a polymer flow through a thin cavity with a slowly varying gap-wise dimension and arbitrary in-plane dimensions. Assume that x, X2 are the planar coordinates, x is the gap-wise direction coordinate. The flow occurs between two walls at JC3 = h/2. Adjacent to each wall there is a frozen layer of the solidified polymer so that the polymer melt flows between two solid-liquid interfaces at xj, = s (xi,X2) and JC3 = s x, x2) (see Fig. 3.1). 

It is convenient to use a fluidity here (Huilgol 2006), which provides a way of defining the shear rate in terms of the shear stress, such that 

Zheng et al.. Injection Molding, DOI 10.1007/978-3-642-21263-5 3, Springer-Verlag Berlin Heidelberg 2011 

using the no-slip condition at X3 =, we obtain the velocity field 

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This resulting equation is known as the Hele-Shaw equation. 

The Hele-Shaw equation solves for the pressure problem, which is coupled with the temperature equation ... 

In the derivation of the Hele-Shaw equation, we have applied the so-called no-slip boundary condition at the liquid-solid interface, where the liquid is assumed to adhere, thus to have no velocity relative to the solid surface. 

The Hele-Shaw equation for the determination of pressure has been derived for a two-dimensional geometry. To solve the pressure problem for a thin eavity of general planar geometry in three-dimensional space, we use a finite-element (FE) representation on the midplane of the cavity (Fig. 8.1). Each element is assigned a thickness. The Hele-Shaw equation is diseretized on eaeh element using the local coordinate system associated with that element. The unknown node pressure and the volumetric flow rate are all scalar quantities and they are not linked to the coordinate system. In addition, we use a finite dififerenee (FD) method to discretize the time- and gap-wise coordinates to solve the energy equation for the temperature field. In the following derivation of the FE/FD equations, only the cavity planar flow is considered. Derivation of the axisymmetrie form of the equations for the runner flow can be done in the same manner. This approach deals with a 2-D pressure field, eoupled to a 3-D temperature field, and therefore it is called a 2.5D simulation. 

To describe the finite element formulation of the discretized Hele-Shaw equation, we write the Hele-Shaw equation in the following form... 

The limitations of the automatic midplane generation stimulated the innovation of the so-called dual domain approach (Yu and Thomas 2000 Yu et al. 2004). This approach uses the external mesh on a 3-D geometry. It still solves the Hele-Shaw equation but eliminates the need for a midplane mesh. 

The flow of a viscoelastic liquid between infinite parallel walls is a viscometric flow, or a flow with constant stretch history. The velocity profile for a fluid that is isotropic at rest is determined in such a flow only by the shear viscosity, although the stress distribution depends on the viscoelastic parameters. A nearly parallel flow for which the Deborah number is low, and stress growth and relaxation is not important, can be treated as though the local flow were that between infinite parallel walls in that case the viscoelasticity is not important for determining the flow field and the process can be analyzed with the lubrication or Hele-Shaw equations as though the poljmier were purely viscous. Effects attributable to the viscoelastic parameters (eg, interface movement in co-extrusion)... 

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... 

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... 

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... 

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. 

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 ... 

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 ... 

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