高分子材料近來廣泛被應用於各種工業以及消費性產品。其在加工融熔流動過程，由於兼具彈性與黏滯性，以玫流變現象迴異於一般之牛頓流體（ Newtonian fluid）。

如欲瞭解此類”黏彈性”流體在流動過程之局部（ local）應力、速度分佈，或探討所可能產生之次級流動（secondary flow）現象，則有必要對黏彈流動進行模擬與分析。本論文研究目的之一即在於發展一穩定之數值方法用來模擬高分子融熔流體之二維穩定態流動。所探用之流變模式為里歐諾夫（Leonov）黏彈流體模式。第二個目的為分析探討黏彈性流體在收縮流動（contraction flow）中，次級渦流（secondary vortex flow） 現象與流變參數間之關聯性。

本研究探用格勒京氏( Galerkin）有限元素分析法求包含連續方程、動量方程和里歐諾夫流變方程式之非線性偶合聯立微分方程式組。藉速度與應力互相疊代分別獨立求解之步驟，而得以將統御方程式（ggoverning equations）”去偶合”（decouple）。

以平板流動和擴散徑向流動等一維流動問題測試，有限元素模擬結果和常微分數值解得到定量上令人滿意的吻合。對於聚苯乙烯（PS）和低密度聚乙烯（LDPE）融熔流體於60°斜角和直角之收縮流動，模擬計算之應力分佈和實驗值得到定性上之一致。數值上之誤差可歸因於收縮入口附近有相當明顯的拉伸流動（extension flow）現象；而實際計算所用之流變參數係由剪切流動（shear flow）所得之流變數據再以流變模式去迴歸所得到。由於黏彈流體在拉伸流動和剪切流動中具有不同流變特性，故造成模擬計算上之誤差。以管徑4:1 之平面直角收縮流動問題而言，本模擬計算可收斂至We數（Weissenberg number）等於2左右。

經分析歸納PS和LDPE於直角收縮流動之模擬結果並配合文獻之實驗數據，發現收縮流動之渦流現象和We數有明顯之關聯性。渦流範圍和強度，在We數小於1 時都很小；反之，當We數超過1 以後，則有很明顯的增大趨勢。此關聯性適用於壁剪切率小於10秒-1以下之範圍。

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A finite element method with Galerkin weighted residual formulation has

been developed for the numerical modeling of vescoelastic flow with Leonov

rheological model. Several problems have been solved for one- and

two-dimensional steady creeping flows. These include plane Poiseuille

flow, diverging radial flow, 60°-tapered entry flow and planar 4:1 sudden

contraaction flow.

The nonlinear simultaneous system equations are decoupled and treated by

means of a two-stage cyclic iterative numerical schems, with the velocity

and elastic strain fields solved separately. The ''artificial viscosity''

and ''under-relaxation'' techniques have been employed to stabilize the

numerical computations and thus suppress oscillations in velocity and

elastic strain fields. Numerical divergence occurs over a critical value

of Deborah number, depending on the flow problem.

One-dimensional problems such as a plane Poiseuille flow and a diverging

radial flow were taken as the preliminary tests for the present numerical

method. The quantitative agreement between the two distinct numerical

solutions shows that this finite element scheme yields efficient and

accurate convective integrations for the Leonov equation for a given

velocity field.

In the 60°-tapered entry flow calculation, although the predicted

stresses only portray the experimental trends qualitatively, they agree

with the numerical results of Upadhyay quantitatively where the elastic

strain equations were solved by the method of streamwise integration.

Through the investigation of predicted elongational stress and velocity

gradient along the converging centerline, it can be found that the

elongational feature of the viscoelastic converging flow may induce a

large elastic stress relative to its viscous counterpart near the

re-entrant region.

In the sudden contraction flow simulation, a salient vortex growth is

found for LDPE but not for polystyrene, which is consistent with what has

been observed experimentally in the literature. The semiquantitative

agreement between the predicted and experimental results for the

streamline patterns and stress profiles at the centerline has been

obtained. The discrepancy between the numerical and experimental stress is

attributed to the fact that the rheological parameters, which were

obtained by fitting the viscometric data of tested fluids, may not do very

well in describing the planar elongational flow behavior along the

contraction centerline. Elaborate analysis of the vortex behavior

exhibited by LDPE and polystyrene with various flow rates leads to the

correlation between the secondary vortex flow and the Weissenberg number.

It has been observed that there exists a critical value of We (ca.,

unity), beyond which the vortex growth occurs along with more intense

recirculation. This correlation is valid up to a limit of wall shear rate

about 10 sec-1.

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