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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

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 & 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  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 1131-1198. Google Scholar

[12]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265-284.  Google Scholar

[13]

L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010.  Google Scholar

[14]

A. Oron and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931-980. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

L. N. Trefethen, "Spectral Methods in MATLAB," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, New York-London, 2003  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 1131-1198. Google Scholar

[12]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265-284.  Google Scholar

[13]

L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010.  Google Scholar

[14]

A. Oron and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931-980. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

L. N. Trefethen, "Spectral Methods in MATLAB," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.  Google Scholar

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