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Examining capillary dynamics in rectangular and circular conduits subject to unsteady surface tension

Published online by Cambridge University Press:  23 September 2022

Martin N. Azese Open the ORCID record for Martin N. Azese [Opens in a new window] ,
Jaures J. Engola ,
Jacques Hona Open the ORCID record for Jacques Hona [Opens in a new window] ,
Emile Jean Yap  and
Yves C. Mbono Samba Open the ORCID record for Yves C. Mbono Samba [Opens in a new window]
Martin N. Azese*
Affiliation:
Department of Mechanical Engineering, College of Engineering, Otterbein University, Westerville, OH 43081, USA Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA Applied Mechanics Laboratory, Faculty of Science, University of Yaounde I, PO Box 812, Yaounde, Cameroon
Jaures J. Engola
Affiliation:
Applied Mechanics Laboratory, Faculty of Science, University of Yaounde I, PO Box 812, Yaounde, Cameroon
Jacques Hona
Affiliation:
Applied Mechanics Laboratory, Faculty of Science, University of Yaounde I, PO Box 812, Yaounde, Cameroon
Emile Jean Yap
Affiliation:
Applied Mechanics Laboratory, Faculty of Science, University of Yaounde I, PO Box 812, Yaounde, Cameroon
Yves C. Mbono Samba
Affiliation:
Applied Mechanics Laboratory, Faculty of Science, University of Yaounde I, PO Box 812, Yaounde, Cameroon
*
†Email address for correspondence: martin.azese@ttu.edu
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Abstract

The unsteady effects of transitioning surface tension, $\gamma (t)$, on the dynamics of capillary imbibition in channels of arbitrary shape are analytically investigated with a focus on rectangular and circular channels. With proper scaling, two unsteady models for $\gamma (t)$ are defined and used to highlight this transient behaviour. The convoluted dynamics at the flow front are correctly captured in the governing equations, which are rigorously analysed using unsteady eigenfunction expansion. Then, the final solution and data are obtained by employing the Runge–Kutta fourth-order scheme elegantly applied simultaneously to two derived nonlinear ordinary differential equations. Ultimately, the results are more accurate compared with previous studies. Dynamics and kinematic similarity between rectangular and circular channels are investigated and discussed and the conditions for equivalence in both channels are highlighted. Using a small parameter ($\epsilon$) that stretches the time scale, we successfully use a robust asymptotic analysis to develop and capture the long-time dynamics. Ultimately, we recover the Lucas–Washburn regime analysed in Washburn (Phys. Rev., vol. 17, 1921, pp. 273–283), Lucas (Kolloidn. Z., vol. 23, 1918, pp. 15–22) for steady surface tension where the variations of depth and rate with time result in $h\thicksim t^{1/2}$ and $v \thicksim t^{-1/2}$. In the end, the three forces, namely the inertia, $F_{v}$, the viscous, $F_{\mu }$, and the surface tension, $F_{\gamma }$, are briefly analysed and used to highlight three main distinct regimes. We show that at early times, $F_{v}/F_{\gamma } \thicksim 1$, whereas at a long time, $F_{\mu }/F_{\gamma } \thicksim -1$.

JFM classification

Interfacial Flows (free surface): Capillary flows Waves/Free-surface Flows: Channel flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 949 , 25 October 2022 , A6
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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