Abstract The non-Newtonian fluids presenting viscoelastic flow behavior are found in many engineering applications. The development of a new numerical scheme for solution of this class of problems is the main goal of the present work. The proposed methodology adopts a second-order fully implicit finite difference approximation to discretize the convection and diffusion terms in the governing equations. Besides, the discretization is accomplished in a collocated mesh arrangement being used an Euler implicit pseudo-transient march in time aiming at steady-state solutions. Finally, it is worth mentioning that under-relaxation parameters are not needed, and the odd-even decoupling problem is avoided using artificial dissipations terms that are externally controlled by the user. The examples illustrating the application of the present method are: the non-Newtonian flows of viscoelastic materials in a plane channel and in a lid-driven cavity. The validation/verification performed indicates that the results are truly encouraging.

中文翻译：

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一种模拟非牛顿粘弹性流动的数值方法

摘要 在许多工程应用中发现了呈现粘弹性流动行为的非牛顿流体。开发解决此类问题的新数值方案是当前工作的主要目标。所提出的方法采用二阶完全隐式有限差分近似来离散控制方程中的对流和扩散项。此外，离散化是在使用欧拉隐式伪瞬态行进的并置网格排列中完成的，针对稳态解。最后，值得一提的是，不需要欠松弛参数，并且使用用户外部控制的人工耗散项避免了奇偶解耦问题。说明本方法应用的例子是：粘弹性材料在平面通道和盖驱动腔中的非牛顿流动。执行的验证/验证表明结果确实令人鼓舞。