Generalized Reynolds-Orr Energy Equation with Wall Slip

Abstract:

The Reynolds-Orr energy equation is generalized to include the velocity slip effect at the walls, for investigating the energy gain or loss in the disturbed slip flow. Taken the Poiseuille flow, typical prototype of parallel shear flows as an example, it is found that for the very weak effect of wall slip, the disturbance energy being transferred from the basic flow overcomes the viscous dissipation, resulting in the growth of disturbance energy and the destabilizing role of wall slip. Otherwise, the viscous dissipation overcomes the energy production resulting in the decay of the disturbance energy and the stabilizing role of wall slip.

• Periodical: Applied Mechanics and Materials (Volumes 117-119)
• Pages: 674-678
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.117-119.674
• Online since: October 2011
• Authors: Ling Ren, De Hong Xia
• Keywords: Instability, Poiseuille Flow, Reynolds-Orr Energy Equation, Wall Slip

References

[1] J.M. Gersting, Hydrodynamic stability of plane porous slip flow, Phys. Fluids 17 (1974) 2126-2127.

DOI: 10.1063/1.1694672

[2] A. Spille, A. Rauh, H. Bühring, Critical curves of plane Poiseuille flow with slip boundary conditions, Nonlinear Phenomena in Complex Systems 3 (2000) 171-173.

[3] E. Lauga, C. Cossu, A note on the stability of slip channel flows, Phys. Fluids 17 (2005) 088106.

DOI: 10.1063/1.2032267

[4] C.J. Gan, Z.N. Wu, Short-wave instability due to wall slip and numerical observation of wall-slip instability for microchannel flows, J. Fluid Mech. 550 (2006) 289-306.

DOI: 10.1017/s0022112005008086

[5] L. Ren, J.G. Chen, K.Q. Zhu, Dual role of wall slip on linear stability of plane Poiseuille flow, Chin. Phys. Lett. 25 (2008) 601-603.

DOI: 10.1088/0256-307x/25/2/067

[6] Y.X. Zhu, S. Granick, Limits of the hydrodynamic no-slip boundary condition, Phys. Rev. Lett. 88 (2002) 106102.

DOI: 10.1103/physrevlett.88.106102

[7] G. Ahmadi, Stability of a micropolar fluid layer heated from below, Int. J. Engng Sci. 14 (1976) 81-89.

DOI: 10.1016/0020-7225(76)90058-6

[8] G.P. Galdi, Nonlinear stability of the magnetic Bénard problem via a generalized energy method, Arch. Rat. Mech. Anal. 87 (1985) 167-186.

DOI: 10.1007/bf00280699

[9] G.P. Galdi, M. Padula, A new approach to energy theory in the stability of fluid motion, Arch. Rat. Mech. Anal. 110 (1990) 187-286.

DOI: 10.1007/bf00375129

[10] P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981.

[11] P.J. Schmid, D.S. Henningson, Stability and Transition in Shear Flows, Springer, New York, 2001.

[12] W.O. Criminale, T.L. Jackson, R.D. Joslin, Theory and Computation of Hydrodynamic Stability, Cambridge University Press, Cambridge, 2003.