I'm trying to model air flow in an area $\Omega$. I'm doing this with the Navier-Stokes equation(s) - I'm primarily interested in recovering a velocity field, so I'm gonna focus on the momentum equation
$\rho \left({\frac {\partial {\mathbf {v}}}{\partial t}}+\left({\mathbf {v}}\cdot \nabla \right){\mathbf {v}}\right)=-\nabla p+\mu \Delta {\mathbf {v}}+{\mathbf {f}}.$
The pressure equation is solved accordingly, both via finite elements.
What I want to model is basically a circular flow in an area shaped like this:
The dark-blue represents a fan, which blows the air with a constant force into the sketched direction. Am I right with modelling this via
$\mathbf{f}(\mathbf {x}) = c \mathbf {e}_{x_2} * \chi_{\mathbf{x} \in A }( \mathbf{x})$
where $c > 0$, $A$ is the dark-blue area, $\mathbf {e}_{x_2}$ is the unit vector in y-direction and $\chi$ is the characteristic function? This basically creates a preasure in $A$, I was just wondering if there was another, maybe more realistic way to model this.
Ideas are appreciated!