Flutter analysis of a flag of fractional viscoelastic material

We develop a two-dimensional model to study the effects of the material viscoelasticity on the dynamics of a flag in flow. Two periodic states of an elastic flag are firstly identified with different dimensionless bending stiffness: a lower frequency state and a higher frequency state. The Scott–Blair model and the fractional Kelvin–Voigt model are further used to represent the viscoelasticity of the flag material. When the Scott–Blair model is used, with the increase of the fractional derivative order α, the flag flapping frequency of the higher frequency state decreases abruptly, and that of the lower frequency state also shows a downward trend. When the system parameters are in a certain range, an interesting phenomenon is observed, where the time needed to achieve the periodic steady state initially increases and then decreases with increasing α. The phenomenon implies that the flag has a higher energy harvesting speed when α approaches 1. When the fractional Kelvin–Voigt model is used, the increasing α also causes the transition from the higher frequency state to the lower frequency state, and quasi-periodic states are observed during the transition. The fractional Kelvin–Voigt type viscoelasticity produces complex effects on the lower frequency state.