液膜流动Marangoni效应边界层解析理论研究

Marangoni effects induced by the asymmetrical temperature or concentration distribution directly influence a liquid film's heat and mass transfer. Being limited by strong non-linear of the partial differential equations, the flow models of a thin liquid film were only focus on Newtonian fluid under Marangoni effect caused by temperature gradient and fixed wall temperature function. In this proposal, wall temperature function will be popularized to general type with the double-parameter transformation perturbation expansion method created by our group, combined with similarity transformation group theory. Marangoni effects induced by temperature and concentration gradient will be investigated, and Newtonian fluid will be generalized to common power law non-Newtonian fluid. Three types of mathematical model for thin liquid film boundary layer flow will be built. The uniformly valid analytical approximate solutions of the boundary layer velocity, temperature and the liquid film's thickness are obtained. In addition, the numerical solutions also are discussed with the Runge-Kutta method. In this project, A thin liquid film boundary layer flow equations of Newtonian fluid under Marangoni effects created by temperature and concentration gradient are fristly solved, also are the flow equations of non-Newtonian fluid. It has very important theoretical significance and utility value to show the change law of the boundary layer's velocity, temperature and the liquid film's thickness, to understand and design all kinds of efficient heat-exchange apparatus and chemical equipment.

界面温度或浓度分布不均匀产生Marangoni效应直接影响液膜传热传质。受偏微分方程组高度非线性的限制，目前液膜流动模型仅考虑牛顿流体温度梯度引发的Marangoni效应，且壁面温度函数形式固定。本项目针对数理模型的局限性，利用我们创建的双参数变形摄动展开方法，结合相似变换群理论，将壁面温度函数推广到一般形式，考察由温度梯度和浓度梯度共同作用下的Marangoni效应，进一步将牛顿流体推广到更为普遍存在的非牛顿流体，建立三种薄液膜边界层流动数理模型。求出在整个边界层区域一致有效的速度、温度、液膜厚度的解析近似解，并用龙格-库塔法求其数值解。本项目首次求解由温度梯度和浓度梯度同时引发的Marangoni效应下牛顿流体液膜边界层方程，以及非牛顿流体流动方程。对于揭示薄液膜形成边界层温度、速度以及厚度的变化规律，理解和设计各种热交换设备和化工设备具有重要理论意义和实用价值。

界面温度或浓度分布不均匀产生Marangoni效应直接影响液膜传热传质。受偏微分方程组高度非线性的限制，目前液膜流动模型仅考虑牛顿流体由温度梯度引发的Marangoni效应。本项目针对数理模型的局限性，将壁面温度函数形式推广，利用求解偏微分方程和常微分方程的解析方法，结合相似变换群理论，考察由温度梯度作用下的Marangoni效应，建立非牛顿流体薄液膜边界层流动数理模型。求出在整个边界层区域一致有效的速度、温度、液膜厚度的解析近似解，揭示薄液膜形成边界层温度、速度以及厚度的变化规律，对于理解和设计各种热交换设备和化工设备具有重要理论意义和实用价值。.主要研究工作如下：.(1)采用双参数摄动展开方法，利用相似变换群理论，研究浮升力、外部压力梯度和温度梯度引起的界面张力梯度共同影响的牛顿流体Marangoni混合对流边界层，深入研究各参数对液膜流动传热的影响。.(2)研究磁流体Marangoni效应边界层问题，建立具有磁场作用的液膜流动模型，并考虑Marangoni效应边界层条件，将温度条件推广，采用同伦分析方法得到相似解，分析各参数包括温度指数对流动传热的影响。.(3)讨论分数阶微分方程模拟液膜一维热传导问题，利用集总参数法将偏微分方程转化为分数阶常微分方程非线性边值问题，运用Laplace变换得到解析解，获得毕渥数较大时一种新的拟合方法。.(4)建立一种新的非牛顿流体液膜本构方程，考虑磁场和滑移的影响，利用Laplace变换方法，研究各物性参数对非牛顿液膜流动速度的影响。.鉴于液膜流动Marangoni效应研究的重要科研价值，本项目所做的理论分析，将为深入研究Marangoni效应下复杂流体流动和传热研究打下基础。同时，通过对结果的验证，说明同伦分析方法、双参数摄动展开方法和Laplace变换方法解决科学与工程中非线性问题的有效性。