Physics:Blasius theorem

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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

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

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

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


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


  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.