纤维涂层的数学模拟
Mathematical modelling of fibre coating
摘要：
在这份报告中，我们制定和分析的数学模型描述的薄液膜涂层通过挤出过程的演变。我们考虑二维不可压缩牛顿流体的纳维-斯托克斯方程与流体表面张力与曲率的标准方程耦合。采用润滑理论近似，并假设稳态，该问题归结为一个单一的三阶微分方程的薄膜高度。最后的膜高度的近似解析解推导出，并与通过拍摄计划获得的数值解进行比较。两个解之间的一致性很好，导致约5\%的相对误差。近似解表明，该过程的关键控制参数是初始膜高度，流体的表面张力和粘度，线速度和挤出机的出口角。
英文摘要：
In this report we formulate and analyse a mathematical model describing the evolution of a thin liquid film coating a wire via an extrusion process. We consider the Navier-Stokes equations for a 2D incompressible Newtonian fluid coupled to the standard equation relating the fluid surface tension with the curvature. Taking the lubrication theory approximation and assuming steady state, the problem is reduced to a single third-order differential equation for the thin film height. An approximate analytical solution for the final film height is derived and compared with a numerical solution obtained by means of a shooting scheme. Good agreement between the two solutions is obtained, resulting in a relative error of around 5\%. The approximate solution reveals that the key control parameters for the process are the initial film height, the fluid surface tension and viscosity, the wire velocity and the angle of exit at the extruder.
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