# Physics:Blasius theorem

In fluid dynamics, Blasius theorem states that [1][2][3] the force experienced by a two-dimensional fixed body in a steady irrotational flow is given by

$\displaystyle{ F_x-iF_y = \frac{i\rho}{2} \oint_C \left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^2\mathrm{d}z }$

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

$\displaystyle{ M=\Re\left\{-\frac{\rho}{2}\oint_C z \left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^2\mathrm{d}z\right\}. }$

Here,

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

The first formula is sometimes called Blasius–Chaplygin formula.[4]

The theorem is named after Paul Richard Heinrich Blasius, who derived it in 1911.[5] The Kutta–Joukowski theorem directly follows from this theorem.

## References

1. Lamb, H. (1993). Hydrodynamics. Cambridge university press. pp. 91
2. Milne-Thomson, L. M. (1949). Theoretical hydrodynamics (Vol. 8, No. 00). London: Macmillan.
3. Acheson, D. J. (1991). Elementary fluid dynamics.
4. Eremenko, Alexandre (2013). "Why airplanes fly, and ships sail". Purdue University.
5. Blasius, H. (1911). Mitteilung zur Abhandlung über: Funktionstheoretische Methoden in der Hydrodynamik. Zeitschrift für Mathematik und Physik, 59, 43-44.