Blasius theorem

In fluid dynamics, Blasius theorem states that Lamb, H. (1993). Hydrodynamics. Cambridge university press. pp. 91Milne-Thomson, L. M. (1949). Theoretical hydrodynamics (Vol. 8, No. 00). London: Macmillan.Acheson, D. J. (1991). Elementary fluid dynamics. the force experienced by a two-dimensional fixed body in a steady irrotational flow is given by

:$F\_x-iF\_y\; =\; \backslash frac\{i\backslash rho\}\{2\}\; \backslash oint\_C\; \backslash left(\backslash frac\{\backslash mathrm\{d\}w\}\{\backslash mathrm\{d\}z\}\backslash right)^2\backslash mathrm\{d\}z$

and the moment about the origin experienced by the body is given by

:$M=\backslash Re\backslash left\backslash \{-\backslash frac\{\backslash rho\}\{2\}\backslash oint\_C\; z\; \backslash left(\backslash frac\{\backslash mathrm\{d\}w\}\{\backslash mathrm\{d\}z\}\backslash right)^2\backslash mathrm\{d\}z\backslash right\backslash \}.$

Here,

• $(F\_x,F\_y)$ is the force acting on the body,
• $\backslash rho$ is the density of the fluid,
• $C$ is the contour flush around the body,
• $w=\backslash phi+\; i\backslash psi$ is the complex potential ($\backslash phi$ is the velocity potential, $\backslash psi$ is the stream function),
• $\{\backslash mathrm\{d\}w\}/\{\backslash mathrm\{d\}z\}\; =\; u\_x-i\; u\_y$ is the complex velocity ($(u\_x,u\_y)$ is the velocity vector),
• $z=x+iy$ is the complex variable ($(x,y)$ is the position vector),
• $\backslash Re$ is the real part of the complex number, and
• $M$ is the moment about the coordinate origin acting on the body.

The first formula is sometimes called Blasius–Chaplygin formula.{{cite web|last1=Eremenko|first1=Alexandre (2013)|title=Why airplanes fly, and ships sail|url=https://www.math.purdue.edu/~eremenko/dvi/airplanes.pdf|publisher=Purdue University}}

The theorem is named after Paul Richard Heinrich Blasius, who derived it in 1911.Blasius, H. (1911). Mitteilung zur Abhandlung über: Funktionstheoretische Methoden in der Hydrodynamik. Zeitschrift für Mathematik und Physik, 59, 43-44. The Kutta–Joukowski theorem directly follows from this theorem.