Universality of capillary rising in corners

TL;DR

The paper addresses viscous capillary rise in corner geometries and derives a PDE for the evolving meniscus profile $G(z,t)$ using the Onsager variational principle. It reveals a self-similar structure with a universal $t^{1/3}$ front-advance, predicting $Z_m/a_c=\chi_0 (n^2\cos^2\theta/3)^{1/3}(\gamma t/(\eta a_c))^{1/3}$ and a corner-dependent but nearly universal front factor $C$, consistent with experimental observations. The approach combines lubrication-based dissipation with a variational formulation to yield a clear, testable description of corner capillarity across power-law geometries. These results unify disparate corner shapes under a common scaling, providing predictive insight for designing systems where capillary rise in corners is relevant.

Abstract

We study the dynamics of capillary rising in corners. Using Onsager principle, we derive a partial differential equation that describes the time evolution of meniscus profile. We obtain both numerical solutions and self-similar solutions to this partial differential equation. Our results show that the advance of the meniscus front follows a time-scaling of $t^{1/3}$, in agreement with the experimental results and theoretical conjecture of Ponomarenko et al.

Figures (4)

• Figure 1: Schematic picture of the capillary rising in a power-law corner.
• Figure 2: Linear corner: (a) Meniscus shape at different times ($\tilde{t}=$200, 400, 600, 800, 1000 from bottom to top). The solution are obtained by solving the time evolution equation (\ref{['eq:te_1']}). (b) The position of the tip as a function of time. (c) Self-similar solution of equation (\ref{['eq:ss_1']}). Also shown are the meniscus shapes at different times (shown as symbols) and boundary conditions (\ref{['eq:ssbc0_1']}, red) and (\ref{['eq:ssbc1_1']}, green). (d) Comparison with experiments. Here the tip position is scaled by the capillary length $a_c$ and the time by $\eta a_c/\gamma$. $\blacksquare$ data from Higuera2008 ($a=2\tan 0.75^{\circ} \simeq 0.026$ and silicon oil V460); $\square$ data from Ponomarenko2011 ($a=2\tan 2.5^{\circ} \simeq 0.087$ and silicon oil V20); $\Diamond$ data from Ponomarenko2011 ($a=2\tan 6.5^{\circ} \simeq 0.228$ and silicon oil V20).
• Figure 3: Quadratic corner: (a) Meniscus shape at different times ($\tilde{t}=$1000, 2000, 3000, 4000, 50000 from bottom to top). The solution are obtained by solving the time evolution equation (\ref{['eq:te_2']}). (b) The position of the tip as a function of time. (c) Self-similar solution of equation (\ref{['eq:ss_2']}). Also shown are the meniscus shapes at different times (shown as symbols) and boundary conditions (\ref{['eq:ssbc0_2']}, red) and (\ref{['eq:ssbc1_2']}, green). (d) Comparison with experiments from Ponomarenko2011. Here the tip position is scaled by the capillary length $a_c$ and the time by $\eta a_c/\gamma$. The corner is given by $b=15$ mm$^{-1}$. $\blacksquare$ silicon oil V10; $\square$ silicon oil V20; $+$ silicon oil V170; $\triangle$ silicon oil V1000.
• Figure 4: Comparison of numerical calculation and experimental results for different corners. Here the tip position is scaled by the capillary length $a_c$ and the $x$-axis is the scaled time to the power of $1/3$, $[\gamma t/(\eta a_c)]^{1/3}$. Experimental data are from Higuera2008 and Ponomarenko2011. (i) linear corners ($E=ax$): $\blacksquare$$a=2\tan 0.75^{\circ} \simeq 0.026$ and silicon oil V460; $\square$$a=2\tan 2.5^{\circ} \simeq 0.087$ and silicon oil V20; $\Diamond$$a=2\tan 6.5^{\circ} \simeq 0.228$ and silicon oil V20. (ii) quadratic corner ($E=bx^2$): $\times$$b=15 \textrm{ mm}^{-1}$ and silicon oil V20. (iii) cubic corner ($E=cx^3$): $\boxplus$$c=18 \textrm{ cm}^{-2}$ and silicon oil V20.