基于Tolman长度的Lucas-Washburn渗吸模型改进及数值模拟

doi:10.19596/j.cnki.1001-246x.8347

摘要/Abstract

摘要：

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。

关键词: 毛细现象, Lucas-Washburn方程, 非线性微分方程, Tolman长度, 数值模拟

Abstract:

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.

Key words: capillarity, Lucas-Washburn equation, nonlinear differential equation, Tolman length, numerical simulation

中图分类号:

O357.3

王俊捷, 寇继生, 蔡建超, 潘益鑫, 钟振. 基于Tolman长度的Lucas-Washburn渗吸模型改进及数值模拟[J]. 计算物理, 2021, 38(5): 521-533.

Junjie WANG, Jisheng KOU, Jianchao CAI, Yixin PAN, Zhen ZHONG. Tolman Length-based Modified Lucas-Washburn Capillary-driven Model and Numerical Simulation[J]. Chinese Journal of Computational Physics, 2021, 38(5): 521-533.

图/表 21

图1 多种模型形式

Fig.1 Various model forms

图2 变半径模型水的高度-时间图像

Fig.2 Water height-time profile of variable radius model

图3 局部管径突变小后水的高度-时间图像

Fig.3 Water height-time profiles with sudden decrease of local tube radius

图4 局部管径突变小后水的高度-时间图像(0~0.008 s)

Fig.4 Water height-time profiles with sudden decrease of local tube radius(0~0.008 s)

图5 局部管径突变大后水的高度-时间图像

Fig.5 Water height-time profiles with sudden increase of local tube radius

图6 局部管径突变大后水的高度-时间图像(0~0.006 s)

Fig.6 Water height-time profiles with sudden increase of local tube radius(0~0.006 s)

图7 不同突变管径对应的到达平衡位置时间散点图

Fig.7 Scatter diagrams of equilibrium time with respect to variable tube radius

图8 不同突变管径对应的平衡位置高度散点图

Fig.8 Scatter diagram of equilibrium height with respect to variable tube radius

图9 不同突变管径对应的最高水位散点图

Fig.9 Scatter diagram of highest water level with respect to variable tube radius

图10 不同突变细位置对应的到达平衡位置时间散点图

Fig.10 Scatter diagram of equilibrium time with respect to position where the tube radius decreases

图11 不同突变细位置对应的平衡位置高度散点图及拟合

Fig.11 Scatter diagram of equilibrium height with respect to position where the tube radius decreases

图12 不同突变粗位置对应的到达平衡位置时间散点图

Fig.12 Scatter diagram of equilibrium time with respect to position where the tube radius increases

图13 不同突变粗位置对应的平衡位置高度散点图及拟合

Fig.13 Scatter diagram of equilibrium height with respect to position where the tube radius increases

表1 不同管径下的平衡时间及平衡位置高度

Table 1 Equilibrium time and height at various tube radii

管径/m	到达平衡位置时间/s	平衡位置高度/(10^-5 m)
0.000 07	18.13	1 323
0.000 08	13.35	1 158
0.000 09	7.85	1 029
0.000 1	6.534	926.1
0.000 5	0.043	232.6
0.001	0.031	133.9
0.002	0.023	69.03
0.003	0.019	46.22
0.004	0.016	34.71
0.005	0.014	27.78
0.006	0.013	23.16
0.007	0.012	19.85
0.008	0.011	17.37
0.009	0.011	15.44
0.01	0.01	13.9
0.05	0.005	2.78
0.1	0.004	1.39
0.5	0.001 4	0.277 7
1	0.001 1	0.138 4

图14 平衡位置高度关于半径倒数的散点图及拟合

Fig.14 Scatter diagram of equilibrium height with respect to radius and its fitting curve

图15 到达平衡位置时间关于半径倒数的散点图及拟合

Fig.15 Scatter diagram of equilibrium time with respect to radius and its fitting curves

表2 未经Tolman长度改进时不同管径下的平衡时间及平衡位置高度

Table 2 Equilibrium time and height at various tube radii without Tolman length

管径/m	到达平衡位置时间/s	平衡位置高度/(10^-5 m)
0.000 07	18.46	1 324
0.000 08	10.76	1 158
0.000 09	9.19	1 030
0.000 1	7.315	926.7
0.000 5	0.043	232.6
0.001	0.031	133.9
0.002	0.023	69.03
0.003	0.019	46.22
0.004	0.016	34.71
0.005	0.014	27.78
0.006	0.013	23.16
0.007	0.012	19.85
0.008	0.011	17.37
0.009	0.011	15.44
0.01	0.01	13.9
0.05	0.005	2.78
0.1	0.003 1	1.39
0.5	0.001 4	0.277 7
1	0.001 1	0.138 4

图16 未经Tolman长度改进的平衡位置高度关于半径倒数的散点图及拟合

Fig.16 Scatter diagram of equilibrium height with respect to radius without Tolman length and its fitting curve

图17 未经Tolman长度改进的到达平衡位置时间关于半径倒数的散点图及拟合

Fig.17 Scatter diagram of equilibrium time with respect to radius without Tolman length and its fitting curves

图18 改进前后到达平衡位置时间关于半径的散点图

Fig.18 Scatter diagrams of equilibrium time with respect to radius with and without Tolman length

图19 改进前后平衡位置高度关于半径的散点图

Fig.19 Scatter diagrams of equilibrium height with respect to radius with and without Tolman length

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