×

A numerical method to study the dynamics of capillary fluid systems. (English) Zbl 1351.76322

Summary: We propose a numerical approach to study both the nonlinear dynamics and linear stability of capillary fluid systems. In the nonlinear analysis, the time-dependent fluid region is mapped onto a fixed numerical domain through a coordinate transformation. The hydrodynamic equations are spatially discretized with the Chebyshev spectral collocation technique, while an implicit time advancement is performed using second-order backward finite differences. The resulting algebraic equations are solved with the iterative Newton-Raphson technique. The most novel aspect of the method is the fact that the elements of the Jacobian of the discretized system of equations are symbolic functions calculated before running the simulation. These functions are evaluated numerically in the Newton-Raphson iterations to find the solution at each time step, which reduces considerably the computing time. Besides, this numerical procedure can be easily adapted to solve the eigenvalue problem which determines the linear global modes of the capillary system. Therefore, both the nonlinear dynamics and the linear stability analysis can be conducted with essentially the same algorithm. We validate this numerical approach by studying the dynamics of a liquid bridge close to its minimum volume stability limit. The results are virtually the same as those obtained with other methods. The proposed approach proves to be much more computationally efficient than those other methods. Finally, we show the versatility of the method by calculating the linear global modes of a gravitational jet.

MSC:

76Z05 Physiological flows
92C35 Physiological flow
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pomeau, Y.; Villermaux, E., Two hundred years of capillary research, Phys. Today, 59, 39-44 (2006)
[2] Eggers, J.; Villermaux, E., Physics of liquid jets, Rep. Prog. Phys., 71, Article 036601 pp. (2008)
[3] Stone, H. A., Dynamics of drop deformation and breakup in viscous fluids, Annu. Rev. Fluid Mech., 26, 65-102 (1994) · Zbl 0802.76020
[4] Schatz, M. F.; Neitzel, G. P., Experiments on thermocapillary instabilities, Annu. Rev. Fluid Mech., 33, 93-127 (2001) · Zbl 1106.76320
[5] Stone, H. A., A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface, Phys. Fluids A, 2, 111 (1990)
[6] Rodd, L. E.; Scott, T. P.; Cooper-White, J. J.; Mckinley, G. H., Capillary break-up rheometry of low-viscosity elastic fluids, Appl. Rheol., 15, 12-27 (2005)
[7] Eggers, J., Nonlinear dynamics and breakup of free-surface flows, Rev. Mod. Phys., 69, 865-929 (1997) · Zbl 1205.37092
[8] Yarin, A., Drop impact dynamics: splashing, spreading, receding, bouncing…, Annu. Rev. Fluid Mech., 38, 159-192 (2006) · Zbl 1097.76012
[9] Rider, W. J.; Kothey, D. B., Reconstructing volume tracking, J. Comput. Phys., 141, 112-152 (1998) · Zbl 0933.76069
[10] Hirt, C. W.; Nichols, B. D., Volume of Fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, 201-225 (1981) · Zbl 0462.76020
[11] Chang, Y. C.; Hou, T. Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys., 124, 449-464 (1996) · Zbl 0847.76048
[12] Brackbill, J. U.; Kothe, D. B.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354 (1992) · Zbl 0775.76110
[13] Verfurth, R., A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math., 50, 67-83 (1994) · Zbl 0811.65089
[14] Worner, M., Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications, Microfluid. Nanofluid., 12, 841-886 (2012)
[15] Thompson, J. F.; Warsi, Z. U.A., Boundary-fitted coordinate systems for numerical solution of partial differential equations—a review, J. Comput. Phys., 47, 1-108 (1982) · Zbl 0492.65011
[16] Thompson, J. F.; Warsi, Z. U.A.; Mastin, C. W., Numerical Grid Generation. Foundations and Applications (1985), Elsevier Science Ltd. · Zbl 0598.65086
[17] Ferziger, J. H.; Peric, M., Computational Methods for Fluid Dynamics (2013), Springer · Zbl 0869.76003
[18] Karniadakis, G.; Sherwin, S. J., Spectral/hp Element Methods for CFD (1999), Oxford University Press · Zbl 0954.76001
[19] Canuto, C.; Hussaini, M. Y.; Quarteroni, A., Spectral Methods: Fundamentals in Single Domains (2007), Springer
[20] Boyd, J. P., Chebyshev and Fourier Spectral Methods (1989), Springer-Verlag
[21] Khorrami, M. R., Application of spectral collocation techniques to the stability of swirling flows, J. Comput. Phys., 81, 206-229 (1989) · Zbl 0662.76057
[22] Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability (1961), Dover: Dover New York, USA · Zbl 0142.44103
[23] Herrada, M. A.; Montanero, J. M.; Vega, J. M., The effect of surface shear viscosity on the damping of oscillations in millimetric liquid bridges, Phys. Fluids, 23, Article 082102 pp. (2011)
[24] Vega, E. J.; Montanero, J. M.; Herrada, M. A.; Ferrera, C., Dynamics of an axisymmetric liquid bridge close to the minimum-volume stability limit, Phys. Rev. E, 90, Article 013015 pp. (2014)
[25] Montanero, J. M.; Herrada, M. A.; Ferrera, C.; Vega, E. J.; Gañán-Calvo, A. M., On the validity of a universal solution for viscous capillary jets, Phys. Fluids, 23, 122103 (2011)
[26] Herrada, M. A.; López-Herrera, J. M.; Gañán-Calvo, A. M.; Vega, E. J.; Montanero, J. M.; Popinet, S., Numerical simulation of electrospray in the cone-jet mode, Phys. Rev. E, 86, Article 026305 pp. (2012)
[27] Ferrera, C.; López-Herrera, J. M.; Herrada, M. A.; Montanero, J. M.; Acero, A. J., Dynamical behavior of electrified pendant drops, Phys. Fluids, 25, Article 012104 pp. (2013)
[28] Herrada, M. A.; Gañán-Calvo, A. M.; Montanero, J. M., Theoretical investigation of a technique to produce microbubbles by a microfluidic T-junction, Phys. Rev. E, 88, Article 033027 pp. (2013)
[29] Herrada, M. A.; Ferrera, C.; Montanero, J. M.; Gañán-Calvo, A. M., Absolute lateral instability in capillary coflowing jets, Phys. Fluids, 22, Article 064104 pp. (2010) · Zbl 1190.76048
[30] Gañán-Calvo, A. M.; Herrada, M. A.; Montanero, J. M., How does a shear boundary layer affect the stability of a capillary jet?, Phys. Fluids, 26, Article 061701 pp. (2014)
[31] Herrada, M. A.; Montanero, J. M.; Ferrera, C.; Gañán-Calvo, A. M., Analysis of the dripping-jetting transition in compound capillary jets, J. Fluid Mech., 649, 523-536 (2010) · Zbl 1189.76233
[32] Rubio-Rubio, M.; Sevilla, A.; Gordillo, J. M., On the thinnest steady threads obtained by gravitational stretching of capillary jets, J. Fluid Mech., 729, 471-483 (2013) · Zbl 1291.76147
[33] Theofilis, V., Global linear instability, Annu. Rev. Fluid Mech., 43, 319-352 (2011) · Zbl 1299.76074
[34] Lister, R.; Stone, H. A., Capillary breakup of a viscous thread surrounded by another viscous fluid, Phys. Fluids, 10, 2758-2764 (1998) · Zbl 1185.76548
[35] Slobozhanin, L. A.; Perales, J. M., Stability of liquid bridges between equal disks in an axial gravity field, Phys. Fluids, 5, 1305-1314 (1993) · Zbl 0800.76152
[36] Khorrami, M. R.; Malik, M. R.; Ash, R. L., Application of spectral collocation techniques to the stability of swirling flows, J. Comput. Phys., 81, 206-229 (1989) · Zbl 0662.76057
[37] Dizes, S. L., Global modes in falling capillary jets, Eur. J. Mech. B, Fluids, 16, 761-778 (1997) · Zbl 0974.76029
[38] Clanet, C.; Lasheras, J. C., Transition from dripping to jetting, J. Fluid Mech., 383, 307-326 (1999) · Zbl 0933.76021
[39] Sauter, U. S.; Buggisch, H. W., Stability of initially slow viscous jets driven by gravity, J. Fluid Mech., 533, 237-257 (2005) · Zbl 1074.76023
[40] Javadi, A.; Eggers, J.; Bonn, D.; Habibi, M.; Ribe, N. M., Delayed capillary breakup of falling viscous jets, Phys. Rev. Lett., 110, Article 144501 pp. (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.