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),
• ${\displaystyle {\mathrm {d} w}/{\mathrm {d} z}=u_{x}-iu_{y}}$ 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.