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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

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 & 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.   Google Scholar

[2]

Finn, R., On equations of capillarity,, J. Math. Fluid Mech., 3 (2001), 139.  doi: 10.1007/PL00000966.  Google Scholar

[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.  doi: 10.1007/s00021-004-0104-z.  Google Scholar

[4]

M. Padula, On nonlinear stability of MHD equilibrium figures,, Adv. Math. Fluid Mech., (2009), 301.   Google Scholar

[5]

M. Padula, On nonlinear stability of linear pinch,, Appl. Anal., 90 (2011), 159.  doi: 10.1080/00036811.2010.490527.  Google Scholar

[6]

M. Padula, On stability of a capillary liquid down an inclined plane,, preprint n. 341 of Math. Dept. of Ferrara., ().   Google Scholar

[7]

Chia-Shun Yih, "Dynamics of Nonhomogeneous Fluids,", The Macmillan Series in Advanced Mathematics and Theoretical Physics, (1965).   Google Scholar

[8]

Chia-Shun Yih, Stability of parallel laminar flow with a free surface,, Proc. 2nd U.S. Nat. Congr. Appl. Mech., (1954), 623.   Google Scholar

show all references

References:
[1]

T. B. Benjamin, Wave formation in laminar flow down an inclined plane,, J. Fluid Mech., 2 (1957), 554.   Google Scholar

[2]

Finn, R., On equations of capillarity,, J. Math. Fluid Mech., 3 (2001), 139.  doi: 10.1007/PL00000966.  Google Scholar

[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.  doi: 10.1007/s00021-004-0104-z.  Google Scholar

[4]

M. Padula, On nonlinear stability of MHD equilibrium figures,, Adv. Math. Fluid Mech., (2009), 301.   Google Scholar

[5]

M. Padula, On nonlinear stability of linear pinch,, Appl. Anal., 90 (2011), 159.  doi: 10.1080/00036811.2010.490527.  Google Scholar

[6]

M. Padula, On stability of a capillary liquid down an inclined plane,, preprint n. 341 of Math. Dept. of Ferrara., ().   Google Scholar

[7]

Chia-Shun Yih, "Dynamics of Nonhomogeneous Fluids,", The Macmillan Series in Advanced Mathematics and Theoretical Physics, (1965).   Google Scholar

[8]

Chia-Shun Yih, Stability of parallel laminar flow with a free surface,, Proc. 2nd U.S. Nat. Congr. Appl. Mech., (1954), 623.   Google Scholar

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