# Radial Imbibition in Paper under Temperature Differences

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Temperature on Circular Plates

## 3. Imbibition into a Porous Medium

## 4. Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Muller, R.H.; Clegg, D.L. Physical and geometric factors. Ann. Chem.
**1951**, 23, 403–408. [Google Scholar] [CrossRef] - Gillespie, T. The capillary rise of a liquid in a vertical strip of filter paper. J. Colloid Sci.
**1959**, 14, 123–130. [Google Scholar] [CrossRef] - Ridgeway, C.J.; Gane, P.A.C. Controlling the absorption dynamic of water-based ink into porous pigmented coating structures to enhance print performance. Nord. Pulp Pap. Res. J.
**2002**, 17, 119–129. [Google Scholar] [CrossRef] - Marmur, A. The radial capillary. J. Colloid Interface Sci.
**1988**, 124, 301–308. [Google Scholar] [CrossRef] - Danino, D.; Marmur, A. Radial capillary penetration into paper: limited and unlimited liquid reservoirs. J. Colloid Interface Sci.
**1994**, 166, 245–250. [Google Scholar] [CrossRef] - Medina, A.; Pérez-Rosales, C.; Pineda, A.; Higuera, F.J. Imbibition in pieces of paper with different shapes. Rev. Mex. Fis.
**2001**, 47, 537–541. [Google Scholar] - Starov, V.M.; Kostvintsev, S.R.; Sobolev, V.D.; Velarde, M.G.; Zhdanov, S.A. Spreading of liquid drops over dry porous layers: Complete wetting case. J. Colloid. Interface Sci.
**2002**, 252, 397–408. [Google Scholar] [CrossRef] [PubMed] - Das, S.; Milacic, E.; Deen, N.G.; Kuipers, J.A.M. Droplet spreading and capillary imbibition in a porous medium: A coupled IB-VOF method based numerical study. Phys. Fluids
**2018**, 30, 012112. [Google Scholar] [CrossRef] - Chen, Y.-J.; Watanabe, S.; Yoshikawa, K. Roughening dynamics of radial imbibition in a porous medium. Phys. Chem. C
**2015**, 119, 12508. [Google Scholar] [CrossRef] - Middleman, S. Modeling Axisymmetric Flows; Academic Press: San Diego, CA, USA, 1995. [Google Scholar]
- Babadagli, T. Temperature effect on heavy-oil recovery by imbibition in fractured reservoirs. J. Pet. Sci. Eng.
**1996**, 14, 197–208. [Google Scholar] [CrossRef] - Morrow, N.R.; Mason, G. Recovery of oil by spontaneous imbibitions. Curr. Opin. Colloid Interface Sci.
**2001**, 6, 321–337. [Google Scholar] [CrossRef] - Amadu, M.; Pegg, M.J. Analytical solution to spontaneous imbibition under vertical temperature gradient based on the theory of spontaneous imbibition dynamics. J. Pet. Sci. Eng.
**2019**, 172, 627–635. [Google Scholar] [CrossRef] - Eastathopoulos, N.; Nicholas, M.G.; Devret, B. Wettability at High Temperatures; Pergamon Materials Series; Elsevier: Oxford, UK, 1999; Volume 3. [Google Scholar]
- Medina, A.; Pineda, A.; Treviño, C. Imbibition driven by a temperature gradient. J. Phys. Soc. Jpn.
**2003**, 72, 979–982. [Google Scholar] [CrossRef] - Washburn, E.W. The dynamics of capillary flow. Phys. Rev.
**1921**, 17, 273–283. [Google Scholar] [CrossRef] - Bear, J. Dynamics of Fluids in Porous Media; Dover: New York, NY, USA, 1988. [Google Scholar]
- Dees, P.J.; Tjan, T.G.; Polderman, J. Determination of pore diameter by permeability measurements. Powder Techol.
**1980**, 27, 29–36. [Google Scholar] [CrossRef] - Chang, S.; Seo, J.; Hong, S.; Lee, D.-G.; Kim, W. Dynamics of liquid imbibition through paper with intra-fiber pores. J. Fluid Mech.
**2018**, 845, 36–50. [Google Scholar] [CrossRef] - Gomba, J.M.; Homsy, G.M. Regimes of thermocapillary migration of droplets under partial wetting conditions. J. Fluid Mech.
**2010**, 647, 125–142. [Google Scholar] [CrossRef] - Incropera, F.P.; Dewitt, D.P. Fundamentals of Heat and Mass Transfer; Wiley: Hoboken, NJ, USA, 2002. [Google Scholar]
- Lavrykov, S.A.; Ramarao, B.V. Thermal properties of copy paper sheets. Dry. Technol.
**2012**, 30, 297–311. [Google Scholar] [CrossRef] - Dortmund Data Ban. Available online: http://ddbonline.ddbst.de/VogelCalculation/VogelCalculationCGI.exe?component=Water (accessed on 1 May 2018).
- Pallas, N.R.; Harrison, Y. An automated drop shape apparatus and the surface tension of pure water. Colloids Surf.
**1990**, 43, 169–194. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the imbibition process in a thin paper on a circular copper plate. The paper sample has an inner radius ${R}_{0}$ and an outer radius ${R}_{1}$ and a thickness e. Temperatures at $r={R}_{0}$ and $r={R}_{1}$ are ${T}_{0}$ and ${T}_{1}$, respectively. The green sector indicates the imbibed region, and the circular profile $r=R\left(t\right)$ indicates the instantaneous position of the imbibition front.

**Figure 2.**Temperature distribution on dry blotting paper for several cases: (

**a**) (top) positive mean gradient, (

**b**) (middle) negative mean gradient, and (

**c**) (bottom) isothermal case. Thermographies are on the left-hand side, while the measured temperature profiles are on the right-hand side. The respective profiles fit approximately Equation (1), and fluctuations are related to the measurement error, which in these cases was around $\pm 0.1$${}^{\circ}$C.

**Figure 3.**Dimensional plot of the time evolution of the experimental imbibition fronts (symbols) for positive and negative gradients and for the isotherm case where ${T}_{0}={T}_{1}=301.2$ K (28 ${}^{\circ}$C). Dashed curves correspond to the respective numerical solutions: red dashed line for case $\overline{G}>0$, green dashed line for $\overline{G}<0$, and blue dashed line for $\overline{G}=0$. Symbol sizes correspond to the standard deviation of 5%.

**Figure 5.**Plots of the averaged of velocity front as a function of time for several mean gradients. The same data as in Figure 3 were used.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

López-Villa, A.; Medina, A.; Higuera, F.J.; Mac Intyre, J.R.; Perazzo, C.A.; Gomba, J.M.
Radial Imbibition in Paper under Temperature Differences. *Fluids* **2019**, *4*, 86.
https://doi.org/10.3390/fluids4020086

**AMA Style**

López-Villa A, Medina A, Higuera FJ, Mac Intyre JR, Perazzo CA, Gomba JM.
Radial Imbibition in Paper under Temperature Differences. *Fluids*. 2019; 4(2):86.
https://doi.org/10.3390/fluids4020086

**Chicago/Turabian Style**

López-Villa, Abel, Abraham Medina, F. J. Higuera, Jonatan R. Mac Intyre, Carlos Alberto Perazzo, and Juan Manuel Gomba.
2019. "Radial Imbibition in Paper under Temperature Differences" *Fluids* 4, no. 2: 86.
https://doi.org/10.3390/fluids4020086