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[1] Shaughnessy, E.J., Introduction to Fluid Mechanics, Oxford University Press: New York & Oxford, 2010.
[2] Quere, D., Wetting and roughness. Annual Review of Materials Research, 38, pp. 71–99, 2008. [Crossref]
[3] Bashforth, F. & Adams, J.C., An Attempt to Test the Theory of Capillary Action, Cambridge University Press: Warehouse, London, 1883.
[4] Bullard, J.W. & Garboczi, E.J., Capillary rise between planar surfaces. Physical Review E, 79(011604), pp. 1–8, 2009. [Crossref]
[5] Roura, P., Contact angle in thick capillaries: a derivation based on energy balance. European Journal of Physics, 28(4), pp. 27–32, 2007. [Crossref]
[6] Zhmud, B.V., Tiberg, F. & Hallstensson, K., Dynamics of capillary rise. Journal of Colloid and Interface Science, 228(2), pp. 263–269, 2000. [Crossref]
[7] Taroni, M. & Vella, D., Multiple equilibria in a simple elastocapillary system. Journal of Fluid Mechanics, 712, pp. 273–294, 2012. [Crossref]
[8] Weber, M.W. & Shandas, R., Computational fluid dynamics analysis of microbubble formation in microfluidic flow-focusing devices. Microfluidics and Nanofluidics, 3(2), pp. 195–206, 2007. [Crossref]
[9] Dadvand, A., Khoo, B.C., Shervani-Tabar, M.T. & Khalilpourazary, S., Boundary ele- ment analysis of the droplet dynamics induced by spark-generated bubble. Engineering Analysis with Boundary Elements, 36(11), pp. 1595–1603, 2012. [Crossref]
[10] Lim, L.K., Hua, J.S., Wang, C.H. & Smith, K.A., Numerical simulation of cone-jet for- mation in electrohydrodynamic atomization. AIChE Journal, 57(1), pp. 57–78, 2011. [Crossref]
[11] Sprittles, J.E. & Shikhmurzaev, Y.D., Finite element simulation of dynamic wetting flows as an interface formation process. Journal of Computational Physics, 233, pp. 34–65, 2013. [Crossref]
[12] Sprittles, J.E. & Shikhmurzaev, Y.D., Finite element framework for describing dynamic wetting phenomena. International Journal for Numerical Methods in Fluids, 68(10), pp. 1257–1298, 2012. [Crossref]
[13] Rayleigh, L., On the instability of jets. Proceedings of the London Mathematical Society, 1(1), pp. 4–13, 1878. [Crossref]
[14] Batchelor, G.K., The stability of a large gas bubble rising through liquid. Journal of Fluid Mechanics, 184, pp. 399–422, 1987. [Crossref]
[15] Sussman, M. & Smereka, P., Axisymmetric free boundary problems. Journal of Fluid Mechanics, 341, pp. 269–294, 1997. [Crossref]
[16] Berry, J.D., Neeson, M.J., Dagastine, R.R., Chan, D.Y.C. & Tabor, R.F., Measurement of surface and interfacial tension using pendant drop tensiometry. Journal of Colloid Interface Science, 454, pp. 226–237, 2015. [Crossref]
[17] Rio, O.I. & Neumann, A.W., Axisymmetric drop shape analysis: computational meth- ods for the measurement of interfacial properties from the shape and dimensions of pen- dant and sessile drops. Journal of Colloid Interface Science, 196(2), pp. 136–147, 1997. [Crossref]
[18] Cabezas, M.G., Bateni, A., Montanero, J.M. & Neumann, A.W., Determination of sur- face tension and contact angle from the shapes of axisymmetric fluid interfaces without use of apex coordinates. Langmuir, 22(24), pp. 10053–10060, 2006. [Crossref]
[19] Dingle, N.M., Tjiptowidjojo, K., Basaran, O.A. & Harris, M.T., A finite element based algorithm for determining interfacial tension (gamma) from pendant drop profiles. Journal of Colloid Interface Science, 286(2), pp. 647–660, 2005. [Crossref]
[20] Gille, M., Gorbacheva, Y., Hahn, A., Polevikov, V. & Tobiska, L., Simulation of a pend- ing drop at a capillary tip. Communications in Nonlinear Science and Numerical Simu- lation, 26(1–3), pp. 137–151, 2015. [Crossref]
[21] Danov, K.D., Dimova, S.N., Ivanov, T.B. & Novev, J.K., Shape analysis of a rotat- ing axisymmetric drop in gravitational field: Comparison of numerical schemes for real-time data processing. Colloids and Surfaces a-Physicochemical and Engineering Aspects, 489, pp. 75–85, 2016.
[22] Versteeg, H.K. & Malalasekera, W., An Introduction to Computational Fluid Dynamics: the Finite Volume Method, Pearson Education, Harlow, 2007.
[23] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, CRC Press, New York, 1980.
[24] Leonard, B.P., Order of accuracy of quick and related convection-diffusion Schemes. Applied Mathematical Modelling, 19(11), pp. 640–653, 1995. [Crossref]
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Open Access
Research article

Solution of Young–Laplace Equation with Finite-Volume Method and Overlapped Grid

Zeyi Zhang1,2,
Liqiu Wang1,2
1
Department of Mechanical Engineering, the University of Hong Kong, Hong Kong
2
HKU-Zhejiang Institute of Research and Innovation (HKU-ZIRI), China
International Journal of Computational Methods and Experimental Measurements
|
Volume 6, Issue 1, 2018
|
Pages 11-22
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
View Full Article|Download PDF

Abstract:

The Young–Laplace equation describes the stress balance between the interfacial tension and the gravi- tational body force. Its modified form can be applied to model the dynamics of the interface of the two-phase flow. It is analytically difficult to solve the Young–Laplace equation due to its strong non- linearity, in the form of the surface curvature and the implicit body force on the interface. This work aims to numerically solve the Young–Laplace equation with a finite-volume method (FVM) and an overlapped grid. The overlapped grid allocates the variable and its derivative together at every grid point, and is compared with the traditional staggered grid. The proposed overlapped grid can achieve fourth-order numerical accuracy, which is higher than the second-order accuracy of the staggered grid. Also, the overlapped grid, with full states defined at every grid point, offers convenience to the imple- mentation of the boundary condition as well as the coupling of multi-physics.

Keywords: finite-volume method, overlapped grid, young–laplace equation
References
[1] Shaughnessy, E.J., Introduction to Fluid Mechanics, Oxford University Press: New York & Oxford, 2010.
[2] Quere, D., Wetting and roughness. Annual Review of Materials Research, 38, pp. 71–99, 2008. [Crossref]
[3] Bashforth, F. & Adams, J.C., An Attempt to Test the Theory of Capillary Action, Cambridge University Press: Warehouse, London, 1883.
[4] Bullard, J.W. & Garboczi, E.J., Capillary rise between planar surfaces. Physical Review E, 79(011604), pp. 1–8, 2009. [Crossref]
[5] Roura, P., Contact angle in thick capillaries: a derivation based on energy balance. European Journal of Physics, 28(4), pp. 27–32, 2007. [Crossref]
[6] Zhmud, B.V., Tiberg, F. & Hallstensson, K., Dynamics of capillary rise. Journal of Colloid and Interface Science, 228(2), pp. 263–269, 2000. [Crossref]
[7] Taroni, M. & Vella, D., Multiple equilibria in a simple elastocapillary system. Journal of Fluid Mechanics, 712, pp. 273–294, 2012. [Crossref]
[8] Weber, M.W. & Shandas, R., Computational fluid dynamics analysis of microbubble formation in microfluidic flow-focusing devices. Microfluidics and Nanofluidics, 3(2), pp. 195–206, 2007. [Crossref]
[9] Dadvand, A., Khoo, B.C., Shervani-Tabar, M.T. & Khalilpourazary, S., Boundary ele- ment analysis of the droplet dynamics induced by spark-generated bubble. Engineering Analysis with Boundary Elements, 36(11), pp. 1595–1603, 2012. [Crossref]
[10] Lim, L.K., Hua, J.S., Wang, C.H. & Smith, K.A., Numerical simulation of cone-jet for- mation in electrohydrodynamic atomization. AIChE Journal, 57(1), pp. 57–78, 2011. [Crossref]
[11] Sprittles, J.E. & Shikhmurzaev, Y.D., Finite element simulation of dynamic wetting flows as an interface formation process. Journal of Computational Physics, 233, pp. 34–65, 2013. [Crossref]
[12] Sprittles, J.E. & Shikhmurzaev, Y.D., Finite element framework for describing dynamic wetting phenomena. International Journal for Numerical Methods in Fluids, 68(10), pp. 1257–1298, 2012. [Crossref]
[13] Rayleigh, L., On the instability of jets. Proceedings of the London Mathematical Society, 1(1), pp. 4–13, 1878. [Crossref]
[14] Batchelor, G.K., The stability of a large gas bubble rising through liquid. Journal of Fluid Mechanics, 184, pp. 399–422, 1987. [Crossref]
[15] Sussman, M. & Smereka, P., Axisymmetric free boundary problems. Journal of Fluid Mechanics, 341, pp. 269–294, 1997. [Crossref]
[16] Berry, J.D., Neeson, M.J., Dagastine, R.R., Chan, D.Y.C. & Tabor, R.F., Measurement of surface and interfacial tension using pendant drop tensiometry. Journal of Colloid Interface Science, 454, pp. 226–237, 2015. [Crossref]
[17] Rio, O.I. & Neumann, A.W., Axisymmetric drop shape analysis: computational meth- ods for the measurement of interfacial properties from the shape and dimensions of pen- dant and sessile drops. Journal of Colloid Interface Science, 196(2), pp. 136–147, 1997. [Crossref]
[18] Cabezas, M.G., Bateni, A., Montanero, J.M. & Neumann, A.W., Determination of sur- face tension and contact angle from the shapes of axisymmetric fluid interfaces without use of apex coordinates. Langmuir, 22(24), pp. 10053–10060, 2006. [Crossref]
[19] Dingle, N.M., Tjiptowidjojo, K., Basaran, O.A. & Harris, M.T., A finite element based algorithm for determining interfacial tension (gamma) from pendant drop profiles. Journal of Colloid Interface Science, 286(2), pp. 647–660, 2005. [Crossref]
[20] Gille, M., Gorbacheva, Y., Hahn, A., Polevikov, V. & Tobiska, L., Simulation of a pend- ing drop at a capillary tip. Communications in Nonlinear Science and Numerical Simu- lation, 26(1–3), pp. 137–151, 2015. [Crossref]
[21] Danov, K.D., Dimova, S.N., Ivanov, T.B. & Novev, J.K., Shape analysis of a rotat- ing axisymmetric drop in gravitational field: Comparison of numerical schemes for real-time data processing. Colloids and Surfaces a-Physicochemical and Engineering Aspects, 489, pp. 75–85, 2016.
[22] Versteeg, H.K. & Malalasekera, W., An Introduction to Computational Fluid Dynamics: the Finite Volume Method, Pearson Education, Harlow, 2007.
[23] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, CRC Press, New York, 1980.
[24] Leonard, B.P., Order of accuracy of quick and related convection-diffusion Schemes. Applied Mathematical Modelling, 19(11), pp. 640–653, 1995. [Crossref]
Cite this:
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Zhang, Z. Y. & Wang, L. Q. (2018). Solution of Young–Laplace Equation with Finite-Volume Method and Overlapped Grid. Int. J. Comput. Methods Exp. Meas., 6(1), 11-22. https://doi.org/10.2495/CMEM-V6-N1-11-22
Z. Y. Zhang and L. Q. Wang, "Solution of Young–Laplace Equation with Finite-Volume Method and Overlapped Grid," Int. J. Comput. Methods Exp. Meas., vol. 6, no. 1, pp. 11-22, 2018. https://doi.org/10.2495/CMEM-V6-N1-11-22
@research-article{Zhang2018SolutionOY,
title={Solution of Young–Laplace Equation with Finite-Volume Method and Overlapped Grid},
author={Zeyi Zhang and Liqiu Wang},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2018},
page={11-22},
doi={https://doi.org/10.2495/CMEM-V6-N1-11-22}
}
Zeyi Zhang, et al. "Solution of Young–Laplace Equation with Finite-Volume Method and Overlapped Grid." International Journal of Computational Methods and Experimental Measurements, v 6, pp 11-22. doi: https://doi.org/10.2495/CMEM-V6-N1-11-22
Zeyi Zhang and Liqiu Wang. "Solution of Young–Laplace Equation with Finite-Volume Method and Overlapped Grid." International Journal of Computational Methods and Experimental Measurements, 6, (2018): 11-22. doi: https://doi.org/10.2495/CMEM-V6-N1-11-22
ZHANG Z Y， WANG L Q. Solution of Young–Laplace Equation with Finite-Volume Method and Overlapped Grid[J]. International Journal of Computational Methods and Experimental Measurements， 2018， 6(1)： 11-22. https://doi.org/10.2495/CMEM-V6-N1-11-22