基于Tolman长度的Lucas-Washburn渗吸模型改进及数值模拟

Tolman Length-based Modified Lucas-Washburn Capillary-driven Model and Numerical Simulation

经典的Lucas-Washburn(L-W)渗吸模型用Young-Laplace方程计算毛管压力,但该方程在管径细小情形得出的毛管压力值与真实值存在较大偏差。本文运用Tolman长度改进Young-Laplace方程,提出一种改进的L-W渗吸模型,并将等截面圆管扩展至任意变化截面圆管,得到变截面圆管中润湿流体注入长度随时间变化的数学模型。该模型为二阶非线性常微分方程,无法求出解析解,为此提出一种数值解法。选取截面变化的毛细管道,通过数值模拟计算出润湿液体注入长度与时间的对应关系,对Tolman长度的改进效果进行检验和分析。结果表明:在研究范围内Tolman长度对L-W渗吸模型的改进效果表现出毛细管道半径越小,效果越明显的规律。圆管局部缩小能改变渗吸水运动状态,依次呈现出三种运动模式;圆管局部扩大会缓慢改变渗吸水运动状态,只呈现单一运动模式。

The classical Lucas-Washburn(L-W)capillary-driven model calculates capillary pressure using the Young-Laplace equation,which leads to a deviation from the real values for a pipe with small diameter.In this work we use Tolman length to improve the Young-Laplace equation,and propose an improved L-W model.Moreover,circular pipes with rough and variable sections are considered rather than the circular pipe with a constant diameter.Relation between time and wetting fluid injection length in the circular pipe with variable sections is established.The mathematical model is described as a second-order nonlinear ordinary differential equation,which cannot be solved analytically,and thus an efficient numerical method is developed.A concrete pipe with variable cross-section is selected,and the relation between water length and time is calulated with numerical simulation.Numerical results are analyzed,and the effectiveness of Tolman length is verified.It shows that the local shrinkage of circular pipe changes the states of water motion significantly,and there are three kinds of movement modes.The local expansion of circular pipe changes slowly the state of seepage and water absorption,and only one single movement mode takes place.