Equation of a bearing line with elliptical edges

V. E. Petrov

St. Petersburg

Abstract: We consider the Prandtl carrier line equation,
$$\Gamma(x)-{p(x)\over \pi } \int\limits _{-1}^1{\Gamma'(t)\over t-x}\, dt = p(x) H_0(x),\quad \Gamma(\pm1)=0.$$
describing the circulation of $\Gamma(x)$ on a thin wing with chord $p(x)$ in a uniform incident flow $H_0(x)=1$. At present, only one case of an exact solution is known — an elliptical wing, when $p(x)=p_0\sqrt{1-x^2}\,.$ We consider a generalization of this case, when the wing edges remain elliptical, but the geometry can be quite general, namely
$$p(x)={\alpha x + b\over \gamma x + d}\,\sqrt{1-x^2}\geqslant 0, \ \ -1 \leqslant x \leqslant 1.$$
The equation in this case is reduced to an infinite recurrent system with linear coefficients. It is solved by some modification of the Laplace method. As a result, an integral representation for the solution of the system is obtained and, using it, a representation of the solution of equation itself. The solution is also presented as the Appell hypergeometric function $F_1.$ The limiting cases $b=\alpha\ne0$ and $\alpha=0$ are considered separately.

Keywords: Prandtl equation for an asymmetric wing with a chord

UDC: 517.968.23

MSC: 45E05, 45A05