Abstract
The problem of steady axisymmetric deformations of a liquid sessile drop on a flat solid surface under an impinging gas jet is of interest for understanding the fundamental behavior of free surface flows as well as for establishing the theoretical basis in process design for the Aerosol \({{\rm Jet}^{\circledR}}\) direct-write technology. It is studied here numerically using a Galerkin finite-element method, by computing solutions of Navier–Stokes equations. For effective material deposition in Aerosol \({{\rm Jet}^{\circledR}}\) printing, the desired value of Reynolds number for the laminar gas jet is found to be greater than ~500. The sessile drop can be severely deformed by an impinging gas jet when the capillary number is approaching a critical value beyond which no steady axisymmetric free surface deformation can exist. Solution branches in a parameter space show turning points at the critical values of capillary number, which typically indicate the onset of free surface shape instability. By tracking solution branches around turning points with an arc-length continuation algorithm, critical values of capillary number can be accurately determined. Near turning points, all the free surface profiles in various parameter settings take a common shape with a dimple at the center and bulge near the contact line. An empirical formula for the critical capillary number for sessile drops with \({45^{\circ}}\) contact angle is derived for typical ranges of jet Reynolds number and relative drop sizes especially pertinent to Aerosol \({{\rm Jet}^{\circledR}}\) printing.
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References
Zeleny J.: Instability of electrified liquid surfaces. Phys. Rev. 10, 1–6 (1917)
Macky W.A.: Some investigations on the deformation and breaking of water drops in strong electric fields. Proc. R. Soc. Lond. A 133, 565–587 (1931)
Taylor G.I.: Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397 (1964)
Basaran O.A., Scriven L.E.: Axisymmetric shapes and stability of pendant and sessile drops in an electric field. J. Colloid Interface Sci. 140(1), 10–30 (1990)
Zollmer V., Muller M., Renn M., Busse M., Wirth I., Godlinski D., Kardos M.: Printing with aerosols: a maskless deposition technique allows high definition printing of a variety of functional materials. Eur. Coating J. 07(08), 46–55 (2006)
Kahn B.E.: The M3D aerosol jet system, an alternative to inkjet printing for printed electronics. Organ. Print. Electron. 1, 14–17 (2007)
Christenson, K.K., Paulsen, J.A., Renn, M.J., McDonald, K., Bourassa, J.: Direct printing of circuit boards using Aerosol \({{\rm Jet}^{\circledR}}\). In: Proceedings NIP 27 Digital Fabrication, pp. 433–436 (2011)
Tryggvason G., Bunner B., Esmaeeli A., Juric D., Al-Rawahi N., Tauber W., Han J., Nas S., Jan Y.-J.: A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759 (2001)
Bozzi L.A., Feng J.Q., Scott T.C., Pearlsein A.J.: Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number. J. Fluid Mech. 336, 1–32 (1997)
Feng J.Q.: A deformable liquid drop falling through a quiescent gas at terminal velocity. J. Fluid Mech. 658, 438–462 (2010)
Strang G., Fix G.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)
Christodoulou K.N., Scriven L.E.: Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99, 39–55 (1992)
Ortega J.M., Rheinboldt W.C.: Iterative Soluti on of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Hood, P.: Frontal solution program for unsymmetric matrices. Int. J. Numer. Methods Eng. 10, 379–399 (1976; see ibid., 11, 1055 (1977) for corrigendum)
Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinovich, P.H. (ed.) Applications of Bifurcation Theory, pp. 359–384. Academic (1977)
Boudouvis, A.G.: Mechanisms of surface instabilities and pattern formation in ferromagnetic liquids, Ph.D. thesis, University of Minnesota, Minneapolis (1987)
Feng J.Q.: Contact behavior of spherical elastic particles: a computational study of particle adhesion and deformation. Colloids Surf. A 172, 175–198 (2000)
Zuckerman N., Lior N.: Jet impingement heat transfer: physics, correlations, and numerical modeling. In: Greene, G.A. (ed.) Advances in Heat Transfer, vol. 39, pp. 565–631. Elsevier, Amsterdam (2006)
Bergthorson J.M. et al.: Impinging laminar jets at moderate Reynolds numbers and separation distances. Phys. Rev. E. 72, 0066307 (2005)
Minaki, H.: A multiscale analysis of dynamic wetting, Ph.D. thesis, University of California, Berkeley (2013)
Sprittles J.E., Shikhmurzaev Y.D.: Finite element simulation of dynamic wetting flows as an interface formation process. J. Comput. Phys. 233, 34–65 (2013)
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Communicated by S. Balachandar.
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Feng, J.Q. Sessile drop deformations under an impinging jet. Theor. Comput. Fluid Dyn. 29, 277–290 (2015). https://doi.org/10.1007/s00162-015-0353-x
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DOI: https://doi.org/10.1007/s00162-015-0353-x