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According to the previous investigations [4–6], fillinggoverning equations for the flow and temperature field can be written as:
Fig.2-1 a–d. Schematic procedure of the simulation with middle-plane model. a The 3-D surface model b The middle-plane model c The meshed middle-plane model d The display of the simulation result
In addition, boundary conditions in the gap-wise direction can be defined as:
where TW is the constant wall temperature (shown in Fig. 2a).
Combining Eqs. 1–4 with Eqs. 5–6, it follows that the distributions of the u, v, T, P at z coordinates should be symmetrical, with the mirror axis being z = 0, and consequently the u, v averaged in half-gap thickness is equal to that averaged in wholegap thickness. Based on this characteristic, we can pide the whole cavity into two equal parts in the gap-wise direction, as described by Part I and Part II in Fig. 2b. At the same time, triangular finite elements are generated in the surface(s) of the cavity (at z = 0 in Fig. 2b), instead of the middle-plane (at z = 0 in Fig. 2a). Accordingly, finite-difference increments in the gapwise direction are employed only in the inside of the surface(s) (wall to middle/center-line), which, in Fig. 2b, means from z = 0 to z = b. This is single-sided instead of two-sided with respect to the middle-plane (i.e. from the middle-line to two walls). In addition, the coordinate system is changed from Fig. 2a to Fig. 2b to alter the finite-element/finite-difference scheme, as shown in Fig. 2b. With the above adjustment, governing equations are still Eqs. 1–4. However, the original boundary conditions in the gapwise direction are rewritten as:
Meanwhile, additional boundary conditions must be employed at z = b in order to keep the flows at the juncture of the two parts at the same section coordinate [7]:
where subscripts I, II represent the parameters of Part I and Part II, respectively, and Cm-I and Cm-II indicate the moving free melt-fronts of the surfaces of the pided two parts in the filling stage.
It should be noted that, unlike conditions Eqs. 7 and 8, ensuring conditions Eqs. 9 and 10 are upheld in numerical implementations becomes more difficult due to the following reasons:
1. The surfaces at the same section have been meshed respectively, which leads to a distinctive pattern of finite elements at the same section. Thus, an interpolation operation should be employed for u, v, T, P during the comparison between the two parts at the juncture.
2. Because the two parts have respective flow fields with respect to the nodes at point A and point C (as shown in Fig. 2b) at the same section, it is possible to have either both filled or one filled (and one empty). These two cases should be handled separately, averaging the operation for the former, whereas assigning operation for the latter.
3. It follows that a small difference between the melt-fronts is permissible. That allowance can be implemented by time allowance control or preferable location allowance control of the melt-front nodes.
4. The boundaries of the flow field expand by each melt-front advancement, so it is necessary to check the condition Eq. 10 after each change in the melt-front.
5. In view of above-mentioned analysis, the physical parameters at the nodes of the same section should be compared and adjusted, so the information describing finite elements of the same section should be prepared before simulation, that is, the matching operation among the elements should be preformed.
Fig. 2a,b. Illustrative of boundary conditions in the gap-wise direction a of the middle-plane model b of the surface model
2.2.2 Numerical implementation
Pressure field. In modeling viscosity η, which is a function of shear rate, temperature and pressure of melt, the shear-thinning behavior can be well represented by a cross-type model such as:
where n corresponds to the power-law index, and τ∗ characterizes the shear stress level of the transition region between the Newtonian and power-law asymptotic limits. In terms of an