Analytical and numerical results on the positivity of steady state solutions of a thin film equation

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July 2013, 18(5): 1305-1321. doi: 10.3934/dcdsb.2013.18.1305

Analytical and numerical results on the positivity of steady state solutions of a thin film equation

Daniel Ginsberg ^1, and Gideon Simpson ^2,

1.	Department of Mathematics, University of Toronto, Toronto, Canada
2.	School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, United States

Received October 2011 Revised December 2012 Published March 2013

Abstract
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We consider an equation for a thin film of fluid on a rotating cylinder and present several new analytical and numerical results on steady state solutions. First, we provide an elementary proof that both weak and classical steady states must be strictly positive so long as the speed of rotation is nonzero. Next, we formulate an iterative spectral algorithm for computing these steady states. Finally, we explore a non-existence inequality for steady state solutions from the recent work of Chugunova, Pugh & Taranets.

Keywords: spectral methods, positivity., numerical methods, Thin films, steady state solutions.

Mathematics Subject Classification: Primary: 74K35, 65M70; Secondary: 35B0.

Citation: Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305

References:

[1]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, New York-London, 2003
[2]	J. Ashmore, A. E. Hosoi and H. A. Stone, The effect of surface tension on rimming flows in a partially filled rotating cylinder, Journal of Fluid Mechanics, 479 (2003), 65-98.
[3]	D. Badali, M. Chugunova, D. Pelinovsky and S. Pollack, Regularized shock solutions in coating flows with small surface tension, Physics of Fluids, 23 (2011), 093103.
[4]	B. Jürgen and G. Günther, The thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2005), S291-S306.
[5]	E. Benilov, M. Benilov and N. Kopteva, Steady rimming flows with surface tension, Journal of Fluid Mechanics, 597 (2008), 91-118. doi: 10.1017/S0022112007009585.
[6]	E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920.
[7]	F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Diff. Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.
[8]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: The moving contact line with a "porous media'' cut-off of van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564.
[9]	A. Burchard, M. Chugunova and B. Stephens, Convergence to equilibrium for a thin-film equation on a cylindrical surface, Comm. Partial Diff. Equations, 37 (2012), 585-609. doi: 10.1080/03605302.2011.648704.
[10]	M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. on Math. Anal., 42 (2010), 1826-1853. doi: 10.1137/090777062.
[11]	R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 1131-1198.
[12]	E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265-284.
[13]	L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010.
[14]	A. Oron and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931-980.
[15]	K. Pougatch and I. Frigaard, Thin film flow on the inside surface of a horizontally rotating cylinder: Steady state solutions and their stability, Physics of Fluids, 23 (2011), 022102.
[16]	V. V. Pukhnachev, Motion of a liquid film on the surface of a rotating cylinder in a gravitational field, J. of App. Mech. and Tech. Physics, 18 (1977), 244-351.
[17]	V. V. Pukhnachev, Asymptotic solution of the rotating film problem, Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauk, "Mathematics and Continuum Mechanics", (2004), 191-199.
[18]	V. V. Pukhnachev, On the equation of a rotating film, Siberian Math. J., 46 (2005), 913-924. doi: 10.1007/s11202-005-0088-9.
[19]	A. E. Shishkov and R. M. Taranets, On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Math. Bull., 1 (2004), 402-444.
[20]	R. M. Taranets and A. E. Shishkov, A singular Cauchy problem for the equation of the flow of thin viscous films with nonlinear convection, Ukr. Math. J., 58 (2006), 250-271. doi: 10.1007/s11253-006-0066-9.
[21]	L. N. Trefethen, "Spectral Methods in MATLAB," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.

show all references

References:

[1]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, New York-London, 2003
[2]	J. Ashmore, A. E. Hosoi and H. A. Stone, The effect of surface tension on rimming flows in a partially filled rotating cylinder, Journal of Fluid Mechanics, 479 (2003), 65-98.
[3]	D. Badali, M. Chugunova, D. Pelinovsky and S. Pollack, Regularized shock solutions in coating flows with small surface tension, Physics of Fluids, 23 (2011), 093103.
[4]	B. Jürgen and G. Günther, The thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2005), S291-S306.
[5]	E. Benilov, M. Benilov and N. Kopteva, Steady rimming flows with surface tension, Journal of Fluid Mechanics, 597 (2008), 91-118. doi: 10.1017/S0022112007009585.
[6]	E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920.
[7]	F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Diff. Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.
[8]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: The moving contact line with a "porous media'' cut-off of van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564.
[9]	A. Burchard, M. Chugunova and B. Stephens, Convergence to equilibrium for a thin-film equation on a cylindrical surface, Comm. Partial Diff. Equations, 37 (2012), 585-609. doi: 10.1080/03605302.2011.648704.
[10]	M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. on Math. Anal., 42 (2010), 1826-1853. doi: 10.1137/090777062.
[11]	R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 1131-1198.
[12]	E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265-284.
[13]	L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010.
[14]	A. Oron and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931-980.
[15]	K. Pougatch and I. Frigaard, Thin film flow on the inside surface of a horizontally rotating cylinder: Steady state solutions and their stability, Physics of Fluids, 23 (2011), 022102.
[16]	V. V. Pukhnachev, Motion of a liquid film on the surface of a rotating cylinder in a gravitational field, J. of App. Mech. and Tech. Physics, 18 (1977), 244-351.
[17]	V. V. Pukhnachev, Asymptotic solution of the rotating film problem, Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauk, "Mathematics and Continuum Mechanics", (2004), 191-199.
[18]	V. V. Pukhnachev, On the equation of a rotating film, Siberian Math. J., 46 (2005), 913-924. doi: 10.1007/s11202-005-0088-9.
[19]	A. E. Shishkov and R. M. Taranets, On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Math. Bull., 1 (2004), 402-444.
[20]	R. M. Taranets and A. E. Shishkov, A singular Cauchy problem for the equation of the flow of thin viscous films with nonlinear convection, Ukr. Math. J., 58 (2006), 250-271. doi: 10.1007/s11253-006-0066-9.
[21]	L. N. Trefethen, "Spectral Methods in MATLAB," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.

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