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Effect of Compliant Boundaries on Weakly Nonlinear Shear Waves in Channel Flow


Rotenberry, James Michael (1989) Effect of Compliant Boundaries on Weakly Nonlinear Shear Waves in Channel Flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/gva0-r175.


There exists a critical Reynolds number (at which a linear instability first appears for an incompressible fluid flowing in a channel with compliant walls (Hains and Price, [1962]). It is proven that, for fixed non-dimensionalized wall parameters, to any unstable disturbance in three dimensions there corresponds an unstable disturbance in two dimensions at a lower Reynolds number. Consequently, the Ginzburg-Landau equation is used to study the weakly nonlinear two-dimensional evolution of a disturbance in a channel with compliant walls for Reynolds number near its critical value. The coefficients of this equation are found by numerically integrating solutions of the Orr-Sommerfeld equation and its adjoint as well as solutions of the perturbation equations.

For rigid walls the finite amplitude two-dimensional plane wave solution that bifurcates from laminar Poiseuille flow at the critical Reynolds number is itself unstable to two-dimensional disturbances. It is found that for compliant walls this solution is stable to disturbances of the same type.

The formalism developed by Landman [1987] is used to study a class of quasisteady solutions to the Ginzburg-Landau equation. This class includes solutions describing a transition from the laminar solution to finite amplitude states and nonperiodic, "chaotic" attracting sets. It is shown that for compliant walls the transition solutions persist while the "chaotic" ones do not.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Applied Mathematics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Saffman, Philip G.
Thesis Committee:
  • Saffman, Philip G. (chair)
  • Cohen, Donald S.
  • Keller, Herbert Bishop
  • Meiron, Daniel I.
  • Wu, Theodore Yao-tsu
Defense Date:12 August 1988
Funding AgencyGrant Number
Charles Lee Powell FellowshipUNSPECIFIED
Office of Naval Research (ONR)N00014-85-K-0205
Army Research Office (ARO)DAAG29-85- K-0092
Record Number:CaltechETD:etd-02152007-075746
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:640
Deposited By: Imported from ETD-db
Deposited On:15 Feb 2007
Last Modified:19 Oct 2021 23:22

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