Journal of Engineering Mathematics
, Volume 40, Issue 3, pp 227248
First online:
The unsteady motion of twodimensional flags with bending stiffness
 A.D. FittAffiliated withFaculty of Mathematical Studies, University of Southampton
 , M.P. PopeAffiliated withFaculty of Mathematical Studies, University of Southampton
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The motion of a twodimensional flag at a timedependent angle of incidence to an irrotational flow of an inviscid, incompressible fluid is examined. The flag is modelled as a thin, flexible, impermeable membrane of finite mass with bending stiffness. The flag is fixed at the leading edge where it is assumed to be either freely hinged or clamped with zero gradient. The angle of incidence to the outer flow is assumed to be small and thin aerofoil theory and simple beam theory are employed to obtain a partial singular integrodifferential equation for the flag shape. Steady solutions to the problem are calculated analytically for various limiting cases and numerically for order one values of a nondimensional parameter that measures the relative importance of outer flow momentum flux and flexural rigidity. For the unsteady problem, the stability of steady solutions depends only upon two nondimensional parameters. Stability analysis is performed in order to identify the regions of instability. The resulting quadratic eigenvalue problem is solved numerically and the marginal stability curves for both the hinged and the clamped flags are constructed. These curves show that both stable and unstable solutions may exist for various values of the mass and flexural rigidity of the membrane and for both methods of attachment at the leading edge. In order to confirm the results of the linear stability analysis, the full unsteady flag equation is solved numerically using an explicit method. The numerical solutions agree with the predictions of the linear stability analysis and also identify the shapes that the flag adopts according to the magnitude of the flexural rigidity and mass.
 Title
 The unsteady motion of twodimensional flags with bending stiffness
 Journal

Journal of Engineering Mathematics
Volume 40, Issue 3 , pp 227248
 Cover Date
 200107
 DOI
 10.1023/A:1017595632666
 Print ISSN
 00220833
 Online ISSN
 15732703
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 unsteady incompressible aerodynamics
 singular integrodifferential equations
 asymptotics
 numerics
 stability.
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