On the influence of boundary condition on stability of Hagen-Poiseuille flow

We analyze the influence of choice of boundary condition (no-slip and Navier’s slip boundary conditions) on linear stability of Hagen–Poiseuille flow. Several heuristic arguments based on detailed analysis of spectrum of the Stokes operator are given, and it is concluded that Navier’s slip boundary condition should have a destabilizing effect on the flow. Finally the linear stability problem is solved by numerical means, and quantitative results confirming the heuristic prediction are obtained. It is shown that the destabilization is not strong enough to maintain an unstable disturbance, and that the significant destabilization effects of Navier’s slip boundary condition are restricted to small values of the Reynolds number. As a byproduct we obtain explicit formulas for eigenfunctions of the Stokes operater subject to Navier’s slip boundary condition.

[1]  A. Davey,et al.  On the stability of flow in an elliptic pipe which is nearly circular , 1978, Journal of Fluid Mechanics.

[2]  H. Wedin,et al.  Exact coherent structures in pipe flow: travelling wave solutions , 2003, Journal of Fluid Mechanics.

[3]  O. Reynolds.  III. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels , 1883, Proceedings of the Royal Society of London.

[4]  S. Orszag.  Accurate solution of the Orr–Sommerfeld stability equation , 1971, Journal of Fluid Mechanics.

[5]  L. Trefethen,et al.  Linearized pipe flow to Reynolds number 10 7 , 2003 .

[6]  Sandeep Prabhakara,et al.  The no-slip boundary condition in fluid mechanics , 2004 .

[7]  Nikolai V Priezjev.  Rate-dependent slip boundary conditions for simple fluids. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Doo-sung Lee,et al.  The Eigenfunctions of the Stokes Operator in Special Domains. III , 1997 .

[9]  H. Schlichting.  Zur Enstehung der Turbulenz bei der Plattenströmung , 1933 .

[10]  B. Eckhardt,et al.  Traveling waves in pipe flow. , 2003, Physical review letters.

[11]  C. Grosch,et al.  The stability of Poiseuille flow in a pipe of circular cross-section , 1972, Journal of Fluid Mechanics.

[12]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[13]  F. Waleffe.  Exact coherent structures in channel flow , 2001, Journal of Fluid Mechanics.

[14]  W. Tollmien,et al.  Über die Entstehung der Turbulenz , 1930 .

[15]  Vít Prusa,et al.  Sufficient Conditions for Monotone Linear Stability of Steady and Oscillatory Hagen-Poiseuille Flow , 2006, SIAM J. Appl. Math..

[16]  J. T. Tozzi,et al.  The Stability of Oscillatory Hagen-Poiseuille Flow , 1986 .

[17]  D. Bucur,et al.  Influence of wall roughness on the slip behaviour of viscous fluids , 2008, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[18]  A. A. Darhuber,et al.  Slip behavior in liquid films on surfaces of patterned wettability: comparison between continuum and molecular dynamics simulations. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  D. J. Tritton,et al.  The Theory of Hydrodynamic Stability , 1977 .

[20]  Kumbakonam R. Rajagopal,et al.  Navier's slip and evolutionary Navier-Stokes like systems with pressure and shear-rate dependent viscosity , 2007 .

[21]  P. Schmid,et al.  Stability and Transition in Shear Flows. By P. J. SCHMID & D. S. HENNINGSON. Springer, 2001. 556 pp. ISBN 0-387-98985-4. £ 59.50 or $79.95 , 2000, Journal of Fluid Mechanics.

[22]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[23]  I. Herron.  Observations on the Role of Vorticity in the Stability Theory of Wall Bounded Flows , 1991 .

[24]  C. Grosch,et al.  Linear stability of Poiseuille flow in a circular pipe , 1980, Journal of Fluid Mechanics.

[25]  M. E. Galassi,et al.  GNU SCIENTI C LIBRARY REFERENCE MANUAL , 2005 .

[26]  K. Rajagopal,et al.  The effect of the slip boundary condition on the flow of fluids in a channel , 1999 .