View source for Zhukovskii function

The rational function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992801.png" /></td> </tr></table>

of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992802.png" />. It is important for its applications in fluid mechanics, which were discovered by N.E. Zhukovskii (see [[#References|[1]]], [[#References|[2]]]), particularly in constructing and studying the Zhukovskii profile (Zhukovskii wing). Suppose that a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992803.png" /> is given in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992804.png" />-plane passing through the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992805.png" /> (Fig. a), together with a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992806.png" /> touching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992807.png" /> on the outside at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992808.png" />, with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z0992809.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928010.png" />. Under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928011.png" />, the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928012.png" /> is a closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928013.png" /> with a cusp at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928014.png" />, touching an arc of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928015.png" /> (the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928016.png" />) at that point; this image is represented in Fig. band is the Zhukovskii profile.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/z099280a.gif" />

Figure: z099280a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/z099280b.gif" />

Figure: z099280b

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928017.png" /> maps the exterior of the unit circle in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928018.png" />-plane to the exterior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928019.png" />. To obtain a Zhukovskii profile of a more general shape and disposition, the generalized Zhukovskii function is applied (see [[#References|[3]]], [[#References|[4]]], [[#References|[5]]]):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099280/z09928020.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Zhukovskii,   "Collected works" , '''2. Hydrodynamics''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Zhukovskii,   "Collected works" , '''6. The theoretical foundations of flying''' , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1–2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.J. [L.I. Sedov] Sedov,   "Two-dimensional problems in hydrodynamics and aerodynamics" , Acad. Press  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Kochin,   I.A. Kibel',   N.V. Roze,   "Theoretical hydrodynamics" , '''1''' , Interscience  (1964)  (Translated from Russian)</TD></TR></table>



====Comments====
The Zhukovskii profiles (or Zhukovskii aerofoils) suffer the drawback that, as mentioned above, they have a cusp at the trailing edge. This implies that if one had to build wings with such a profile, one should obtain a very thin, and hence fragile, rear part of the wing. For this reason more general profiles, having a singularity with distinct tangents at the trailing edge, have been introduced (von Kármán–Trefftz profiles). Another generalization of the Zhukovskii profile goes in the direction of enlarging the number of parameters (von Mises profiles).

The Zhukovskii aerofoils are usually called the Kutta–Zhukovskii aerofoils in the Western literature.  "Zhukovskii"  is often spelled  "Joukowski"  in the Western literature.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Birhoff,   "Hydrodynamics, a study in logic, fact and similitude" , Princeton Univ. Press  (1960)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lighthill,   "An informal introduction to theoretical fluid mechanics" , Clarendon Press  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. von Mises,   "Theory of flight" , Dover, reprint  (1959)</TD></TR></table>