Coupled Massless Body - Fluid System

For a massless body in 2D the force and momentum balance are:

\[F_{x}=0\quad F_{y}=0\quad M_{z}=0\]

where \(F_{x}\) ,\(F_{y}\) and \(M_{z}\) are the force and momentum exerted on the body by the surrounding fluid.

The configuration of the body is described by the following variables:

\[\left\{ x(t),y(t),\theta(t),s_{1}(t),s_{2}(t),\dots\,\right\}\]

where the first three describe its position and orientation in space that are to be determined, and the rest describe its internal shape configuration that is prescribed a priori.

The problem then is to find such \((\dot{x},\dot{y},\dot{\theta})\) that together with the given set \((\dot{s_{1}}.\dot{s_{2}},\dots)\) would be consistent with the momentum balance at every instance in time during the motion. The translation and the rotation of the body must be such that the immersed boundary code would give zero net force.

But since we do not know \((\dot{x},\dot{y},\dot{\theta})\) a priori, we cannot rely on the immersed boundary code to compute the force. This is why we need to rely on geometric mechanics to find the “mechanical connection”, namely how a trajectory in shape space\((s_{1}(t),s_{2}(t),\dots)\) determines the trajectory in position space \((x(t),y(t),\theta(t))\).