Mathematical modelling of unsteady problems in thin aerofoil theory
The de-icing of aircraft wings by the injection of fluid through a slot in the leading edge of the wing is analysed. A review of current de-icing methods is presented and the semi-infinite slot-injection equation derived, which is a singular partial integro-differential equation. The Stefan condition is used to close the system. A discretisation of the equation is presented and the subsequent numerical results are analysed. The model is then revised to account for the retraction of the ice layer away from the slot. An asymptotic result for the thin ice layers is also presented. The problem of describing the motion of a thin, flexible membrane fixed at both ends (a 'sail') is then considered. The steady sail is analysed for the case of an inextensible sail and previous work on this topic is extended by using a discretisation of the singular integro-differential equation that is pertinent to the later analysis of the unsteady sail. An asymptotic expression for the eigenvalues of the system, defined as the values of the tension parameter for which the sail generates zero lift, is also presented. The problem is then extended to that of an extensible sail and numerical results are presented for both the sail with excess length and the membrane without slack. The case where the angle of incidence of the sail to the free stream is a prescribed function of time is then analysed. Previous work on this subject is extended to include the extensible sail and numerical results are presented. A linear stability analysis is then undertaken for both the extensible and elastic sails; the resulting quadratic eigenvalue problem is solved numerically and is in agreement with the numerical experiments. The trailing edge of the membrane is now permitted to move freely and thus the motion of a 'flag' is analysed. The inclusion of bending stiffness is found to be crucial to the stability properties of the flag. The steady equation of motion is numerically approximated for both a hinged flag and a flag that is clamped at the leading edge. The unsteady flag equation is then discretised and numerical results are presented. A linear stability analysis is performed, the conclusions of which are consistent with the numerical approximations of the unsteady flag equation.