Kirchhoff equations

In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid. In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid. where ω → {displaystyle {vec {omega }}} and v → {displaystyle {vec {v}}} are the angular and linear velocity vectors at the point x → {displaystyle {vec {x}}} , respectively; I ~ {displaystyle { ilde {I}}} is the moment of inertia tensor, m {displaystyle m} is the body's mass; n ^ {displaystyle {hat {n}}} isa unit normal to the surface of the body at the point x → {displaystyle {vec {x}}} ; p {displaystyle p} is a pressure at this point; Q → h {displaystyle {vec {Q}}_{h}} and F → h {displaystyle {vec {F}}_{h}} are the hydrodynamictorque and force acting on the body, respectively; Q → {displaystyle {vec {Q}}} and F → {displaystyle {vec {F}}} 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 {displaystyle {vec {Q}}_{h}} and F → h {displaystyle {vec {F}}_{h}} can be found via explicit integration, and the dynamics of the body is described by the Kirchhoff – Clebsch equations: