Abstract

Basing on the rheological mesoscopic Pokrovskii–Vinogradov model, the equations of nonrelativistic magneto-hydrodynamics, and the heat conduction equation with dissipative terms, we obtain a closed coupled system of nonlinear partial differential equations that describes the flow of solutions and melts of linear polymers. We take into account the rheology and induced anisotropy of a polymeric fluid flow as well as mechanical, thermal, and electromagnetic impacts. The parameters of the equations are determined by mechanical tests with up-to-date materials and devices used in additive technologies (as \(3D\) printing). The statement is given of the problems concerning stationary polymeric fluid flows in channels with circular and elliptical cross-sections with thin inclusions (some heating elements). We show that, for certain values of parameters, the equations can have three stationary solutions of high order of smoothness. Just these smooth solutions provide the defect-free additive manufacturing. To search for them, some algorithm is used that bases on the approximations without saturation, the collocation method, and the relaxation method. Under study are the dependences of the distributions of the saturation fluid velocity and temperature on the pressure gradient in the channel.

中文翻译：

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带有内部加热元件的通道中聚合物流体的稳态非等温MHD流动的模拟

摘要

基于流变介观的Pokrovskii–Vinogradov模型，非相对论性磁流体动力学方程以及带有耗散项的热传导方程，我们获得了描述非线性聚合物微分方程解的封闭式耦合系统。我们考虑了聚合物流体的流变学和各向异性，以及机械，热和电磁的影响。方程的参数由机械测试确定，并使用最新的材料和添加剂技术中使用的设备（如\（3D \）印刷）。陈述了涉及固定聚合物流体在具有薄夹杂物（某些加热元件）的圆形和椭圆形横截面的通道中流动的问题。我们表明，对于某些参数值，这些方程式可以具有三个具有高阶平滑度的固定解。正是这些平滑的解决方案提供了无缺陷的增材制造。为了搜索它们，使用了一种基于不饱和的近似算法，并置方法和松弛方法的算法。正在研究的是饱和流体速度和温度的分布与通道中压力梯度的关系。