# American Institute of Mathematical Sciences

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

## On stability of a capillary liquid down an inclined plane

 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.
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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