Assume we have a two dimensional, incompressible fluid and a moving body (translating and rotating) inside the fluid. The exterior boundary in at infinity (open flow). Then the Navier-Stokes equations are written as
\begin{equation} \begin{array}{l} \left\{ \begin{array}{l} \displaystyle \frac{D\mathbf{u}}{Dt}=\displaystyle-\frac{\nabla p}{\rho_0} + \nu \nabla^2\mathbf{u} , \quad in \quad D\times(0,T), \label{momentum-eq}\\ \displaystyle \nabla\cdot \mathbf{u}=\displaystyle 0, \quad in \quad D\times(0,T), \end{array} \right. \\[0.5cm]\displaystyle \mathbf{u}(\mathbf{x},0)=\mathbf{u}_0(\mathbf{x}), \qquad\mbox{in } D, \\ \displaystyle \mathbf{u}(\mathbf{x}_b)=\mathbf{u}_b , \quad on \quad {\partial D_B}\times(0,T), \nonumber \\ \displaystyle \mathbf{u}(\mathbf{x}) \rightarrow\mathbf{u}_{\infty}(t), \quad as \quad |\mathbf{x}|\rightarrow\infty, \\ \displaystyle p \rightarrow p_{\infty}(t), \;\quad as \quad |\mathbf{x}|\rightarrow\infty, \end{array} \end{equation}
where in the momentum equation $\mathbf{u}$ is the velocity, $p$ is the pressure, $\rho_0$ is the density, $\nu$ is the kinematic viscosity of the specific fluid considered and $\mathbf{u}_b$ is the velocity of the body at $\mathbf{x}_b$ with $\mathbf{u}_b$ given. The problem boundary condition is the no-slip on the body boundary with conditions at infinity for the pressure and the velocity: $\mathbf{u}_\infty$ and $ p_{\infty}$ respectively.
I would like to ask is if there are any references on this problem. I have looked around a bit but without succeeding in finding anything on this specific problem. I am interested also in the vorticity formulation. Thank you very much.