Kirchhoff equations explained

In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid.

$\begin & = \times \boldsymbol\omega + \times \mathbf v + \mathbf Q_h + \mathbf Q, \\[10pt] & = \times \boldsymbol\omega + \mathbf F_h + \mathbf F, \\[10pt]T & = \left(\boldsymbol\omega^T \tilde I \boldsymbol\omega + m v^2 \right) \\[10pt]\mathbf Q_h & = -\int p \mathbf x \times\hat\mathbf n \, d\sigma, \\[10pt]\mathbf F_h & = -\int p \hat\mathbf n \, d\sigma\end$

where

\boldsymbol\omega

and

are the angular and linear velocity vectors at the point

, respectively;

\tildeI

is the moment of inertia tensor,

is the body's mass;

\hatn

isa unit normal vector to the surface of the body at the point

;

is a pressure at this point;

Q_h

and

F_h

are the hydrodynamictorque and force acting on the body, respectively;

and

likewise denote all other torques and forces acting on thebody. The integration is performed over the fluid-exposed portion of thebody's surface.

If the body is completely submerged body in an infinitely large volume of irrotational, incompressible, inviscid fluid, that is at rest at infinity, then the vectors

Q_h

and

F_h

can be found via explicit integration, and the dynamics of the body is described by the Kirchhoff – Clebsch equations:

$= \times \boldsymbol\omega + \times \mathbf v, \quad = \times \boldsymbol\omega,$

$L(\boldsymbol\omega, \mathbf v) = (A \boldsymbol\omega,\boldsymbol\omega) + (B \boldsymbol\omega,\mathbf v) + (C \mathbf v,\mathbf v) + (\mathbf k,\boldsymbol\omega) + (\mathbf l,\mathbf v).$

Their first integrals read $J_0 = \left(\boldsymbol\omega \right) + \left(\mathbf v \right) - L, \quadJ_1 = \left(\right), \quad J_2 = \left(\right).$

Further integration produces explicit expressions for position and velocities.

References

Kirchhoff G. R. Vorlesungen ueber Mathematische Physik, Mechanik. Lecture 19. Leipzig: Teubner. 1877.
Lamb, H., Hydrodynamics. Sixth Edition Cambridge (UK): Cambridge University Press. 1932.