# American Institute of Mathematical Sciences

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

## On the Benilov-Vynnycky blow-up problem

 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.
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), 195-213.  Google Scholar [2] E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math., 87 (1992), 1-14.  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), 93-98.  Google Scholar [4] E. S. Benilov and M. Vynnycky, Contact lines with a $180^{\circ}$ contact angle, J. Fluid Mech., 718 (2013), 481-506. 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), 181-191. 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), 14-24. 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), 45-60. 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), 135-170. 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), 275-292. 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), 49-64. doi: 10.1016/j.cam.2014.01.003.  Google Scholar

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##### References:
 [1] M. J. Ablowitz and J. Villarroel, On the Kadomtsev-Petviashvili equation and associated constraints, Stud. Appl. Math., 85 (1991), 195-213.  Google Scholar [2] E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math., 87 (1992), 1-14.  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), 93-98.  Google Scholar [4] E. S. Benilov and M. Vynnycky, Contact lines with a $180^{\circ}$ contact angle, J. Fluid Mech., 718 (2013), 481-506. 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), 181-191. 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), 14-24. 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), 45-60. 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), 135-170. 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), 275-292. 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), 49-64. doi: 10.1016/j.cam.2014.01.003.  Google Scholar
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