On stability of a capillary liquid down an inclined plane

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October 2013, 6(5): 1343-1353. doi: 10.3934/dcdss.2013.6.1343

On stability of a capillary liquid down an inclined plane

Mariarosaria Padula ^1,

1.	Dipartimento di Matematica, University of Ferrara, Via Macchiavelli, 35, 44121 Ferrara, Italy

Received December 2011 Revised February 2012 Published March 2013

Abstract
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We consider capillary laminar fluid motions on an inclined plane and study spatially periodic surface waves with fixed periodicity on the line of maximum slope $\alpha_1$ and in the horizontal direction $\alpha_2$. Actually, we provide a sufficient condition on Reynolds and Weber numbers, and on the inclination angle, named condition (C), in order that the Poiseuille flow $(v_b,p_b,\Gamma_b)$ with upper flat free boundary $\Gamma_b$ and with periodicity conditions on the plane, is nonlinearly stable. Under condition (C), the perturbed surface $\Gamma_t$ is bounded for all time, and the free boundary Poiseuille flow is stable.

Keywords: free boundary problem, direct methods in nonlinear stability., Incompressible fluids, capillary laminar flow.

Mathematics Subject Classification: Primary: 76E99; Secondary: 76D0.

Citation: Mariarosaria Padula. On stability of a capillary liquid down an inclined plane. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1343-1353. doi: 10.3934/dcdss.2013.6.1343

References:

[1]	T. B. Benjamin, Wave formation in laminar flow down an inclined plane, J. Fluid Mech., 2 (1957), 554-574.
[2]	Finn, R., On equations of capillarity, J. Math. Fluid Mech., 3 (2001), 139-151. doi: 10.1007/PL00000966.
[3]	T. Nishida, Y. Teramoto, H. Yoshihara, Global in time behavior of viscous surface waves: horizontally periodic motion, J. Math. Fluid Mech., 7 (2005), 29-71. doi: 10.1007/s00021-004-0104-z.
[4]	M. Padula, On nonlinear stability of MHD equilibrium figures, Adv. Math. Fluid Mech., 2009, 301-331.
[5]	M. Padula, On nonlinear stability of linear pinch, Appl. Anal., 90 (2011), 159-192. doi: 10.1080/00036811.2010.490527.
[6]	M. Padula, On stability of a capillary liquid down an inclined plane, preprint n. 341 of Math. Dept. of Ferrara.
[7]	Chia-Shun Yih, "Dynamics of Nonhomogeneous Fluids," The Macmillan Series in Advanced Mathematics and Theoretical Physics, New York, 1965.
[8]	Chia-Shun Yih, Stability of parallel laminar flow with a free surface, Proc. 2nd U.S. Nat. Congr. Appl. Mech., 1954, 623-628.

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References:

[1]	T. B. Benjamin, Wave formation in laminar flow down an inclined plane, J. Fluid Mech., 2 (1957), 554-574.
[2]	Finn, R., On equations of capillarity, J. Math. Fluid Mech., 3 (2001), 139-151. doi: 10.1007/PL00000966.
[3]	T. Nishida, Y. Teramoto, H. Yoshihara, Global in time behavior of viscous surface waves: horizontally periodic motion, J. Math. Fluid Mech., 7 (2005), 29-71. doi: 10.1007/s00021-004-0104-z.
[4]	M. Padula, On nonlinear stability of MHD equilibrium figures, Adv. Math. Fluid Mech., 2009, 301-331.
[5]	M. Padula, On nonlinear stability of linear pinch, Appl. Anal., 90 (2011), 159-192. doi: 10.1080/00036811.2010.490527.
[6]	M. Padula, On stability of a capillary liquid down an inclined plane, preprint n. 341 of Math. Dept. of Ferrara.
[7]	Chia-Shun Yih, "Dynamics of Nonhomogeneous Fluids," The Macmillan Series in Advanced Mathematics and Theoretical Physics, New York, 1965.
[8]	Chia-Shun Yih, Stability of parallel laminar flow with a free surface, Proc. 2nd U.S. Nat. Congr. Appl. Mech., 1954, 623-628.

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