American Institute of Mathematical Sciences

Advanced
July  2015, 20(5): 1443-1460. doi: 10.3934/dcdsb.2015.20.1443

On the Benilov-Vynnycky blow-up problem

Marina Chugunova 1, , Chiu-Yen Kao 2, and  Sarun Seepun 3,

1. 

Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, United States
2. 

Department of Mathematical Sciences, Claremont McKenna College, Claremont, CA 91711, United States
3. 

Department of Mathematics, Pitzer College, Claremont, CA 91711, United States

Received  July 2013 Revised  June 2014 Published  May 2015

We study an initial-boundary value problem for a fourth-order parabolic partial differential equation with an unknown velocity. The equation originated from the linearization of a two-dimensional Couette flow model, that was recently proposed by Benilov and Vynnycky. In the case of a $180^{\circ}$-- contact angle between liquid and a moving plate Benilov and Vynnycky conjectured that the speed of the contact line blows up to infinity in finite time. In this paper we present numerical simulations and qualitative analysis of the model. We show that depending on the initial data and parameter values different long time behaviors of velocity can be observed. The speed of the contact line may blow up to infinity or converge to a constant.
Keywords: contact line singularity, contact angle, fourth-order parabolic equation., blow-up, Couette flow.
Mathematics Subject Classification: Primary: 76D08, 35K30; Secondary: 65M0.
Citation: Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443
References:
[1]

M. J. Ablowitz and J. Villarroel, On the Kadomtsev-Petviashvili equation and associated constraints,, Stud. Appl. Math., 85 (1991), 195. Google Scholar

[2]

E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom,, Stud. Appl. Math., 87 (1992), 1. Google Scholar

[3]

D. J. Benney and W. J. Timson, The rolling motion of a viscous fluid on and off a rigid surface,, Stud. Appl. Math, 63 (1980), 93. Google Scholar

[4]

E. S. Benilov and M. Vynnycky, Contact lines with a $180^{\circ}$ contact angle,, J. Fluid Mech., 718 (2013), 481. Google Scholar

[5]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (): 539. Google Scholar

[6]

L. A. Ostrovskii, Nonlinear internal waves in the rotating ocean,, Okeanologiia, 18 (1978), 181. Google Scholar

[7]

D. E. Pelinovsky and A. R. Giniyatullin, Finite-time singularities in the dynamical evolution of contact lines,, Bulletin of the Moscow State Regional University (Physics and Mathematics), 3 (2013), 14. Google Scholar

[8]

D. E. Pelinovsky, A. R. Giniyatullin and Y. A. Panfilova, On solutions of the reduced model for the dynamical evolution of contact lines,, Transactions of Nizhni Novgorod State Technical University n.a. Alexeev N.4, 94 (2012), 45. Google Scholar

[9]

D. E. Pelinovsky and C. Xu, On numerical modelling and the blow-up behavior of contact lines with a $180^{\circ}$ contact angle,, J. Engineer. Math., (2015). doi: 10.1007/s10665-014-9763-9. Google Scholar

[10]

J. Le Sommer, G. M. Reznik and V. Zeitlin, Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane,, Journal of Fluid Mechanics, 515 (2004), 135. doi: 10.1017/S0022112004000229. Google Scholar

[11]

M. Vynnycky and S. L. Mitchell, On the accuracy of a finite-difference method for parabolic partial differential equations with discontinuous boundary conditions,, Num. Heat Trans B, 64 (2013), 275. doi: 10.1080/10407790.2013.797312. Google Scholar

[12]

S. L. Mitchell and M. Vynnycky, On the numerical solution of two-phase Stefan problems with heat-flux boundary conditions,, J. Comp. Appl. Maths, 264 (2014), 49. doi: 10.1016/j.cam.2014.01.003. Google Scholar

show all references

References:
[1]

M. J. Ablowitz and J. Villarroel, On the Kadomtsev-Petviashvili equation and associated constraints,, Stud. Appl. Math., 85 (1991), 195. Google Scholar

[2]

E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom,, Stud. Appl. Math., 87 (1992), 1. Google Scholar

[3]

D. J. Benney and W. J. Timson, The rolling motion of a viscous fluid on and off a rigid surface,, Stud. Appl. Math, 63 (1980), 93. Google Scholar

[4]

E. S. Benilov and M. Vynnycky, Contact lines with a $180^{\circ}$ contact angle,, J. Fluid Mech., 718 (2013), 481. Google Scholar

[5]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (): 539. Google Scholar

[6]

L. A. Ostrovskii, Nonlinear internal waves in the rotating ocean,, Okeanologiia, 18 (1978), 181. Google Scholar

[7]

D. E. Pelinovsky and A. R. Giniyatullin, Finite-time singularities in the dynamical evolution of contact lines,, Bulletin of the Moscow State Regional University (Physics and Mathematics), 3 (2013), 14. Google Scholar

[8]

D. E. Pelinovsky, A. R. Giniyatullin and Y. A. Panfilova, On solutions of the reduced model for the dynamical evolution of contact lines,, Transactions of Nizhni Novgorod State Technical University n.a. Alexeev N.4, 94 (2012), 45. Google Scholar

[9]

D. E. Pelinovsky and C. Xu, On numerical modelling and the blow-up behavior of contact lines with a $180^{\circ}$ contact angle,, J. Engineer. Math., (2015). doi: 10.1007/s10665-014-9763-9. Google Scholar

[10]

J. Le Sommer, G. M. Reznik and V. Zeitlin, Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane,, Journal of Fluid Mechanics, 515 (2004), 135. doi: 10.1017/S0022112004000229. Google Scholar

[11]

M. Vynnycky and S. L. Mitchell, On the accuracy of a finite-difference method for parabolic partial differential equations with discontinuous boundary conditions,, Num. Heat Trans B, 64 (2013), 275. doi: 10.1080/10407790.2013.797312. Google Scholar

[12]

S. L. Mitchell and M. Vynnycky, On the numerical solution of two-phase Stefan problems with heat-flux boundary conditions,, J. Comp. Appl. Maths, 264 (2014), 49. doi: 10.1016/j.cam.2014.01.003. Google Scholar

[1]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170
[2]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
[3]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1
[4]

Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
[5]

Antonio DeSimone, Natalie Grunewald, Felix Otto. A new model for contact angle hysteresis. Networks & Heterogeneous Media, 2007, 2 (2) : 211-225. doi: 10.3934/nhm.2007.2.211
[6]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[7]

Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189
[8]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225
[9]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[10]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
[11]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[12]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[13]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[14]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
[15]

Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117
[16]

Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025
[17]

Xiao-Ping Wang, Xianmin Xu. A dynamic theory for contact angle hysteresis on chemically rough boundary. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1061-1073. doi: 10.3934/dcds.2017044
[18]

Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667
[19]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617
[20]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences