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

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

Abstract

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

CLC Number:

O357.3

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.

Figures/Tables 21

Fig.1 Various model forms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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