# How to formulate a 3D version of the Navier-Stokes equations in 2D for numerical solution?

I'm considering a topology optimization problem where the design domain is a rotating annulus (in $\mathbb{R}^2$). One of the adjoint equations of the Navier-Stokes system whose solution is to be used in the sensitivity analysis has a weak form with righthand side: $$\int_{\Gamma_{\text{out}}} \rho(r\times u) \cdot \omega)n\cdot v + \int_{\Gamma_{\text{out}}} \rho(r\times \omega \times r)\cdot \omega)n \cdot v + \int_{\Gamma_{\text{out}}} (\rho(\omega\times r)\bigotimes u)n\cdot v,$$ where $v$ is a test function, $r$ is "perpendicular to the axis of rotation" (I've assumed that this is a position vector, see below), $\omega$ is the angular velocity (which is constant, say $|\omega| = 500$), $\Gamma_{\text{out}}$ is the outflow boundary (the outer boundary of the annulus), $n$ is the outward normal to $\Gamma_{\text{out}}$ (i.e. $n(x) = \frac{x}{||x||}$), $u$ is the velocity computed by solving the Navier-Stokes equations, and $\rho$ is the mass density (scalar). I'm wondering how I can formulate this problem for a 2D problem. What I'm doing now is writing $$r(x) = \begin{bmatrix} x_1 & x_2 & 0\end{bmatrix}^\top,\\\omega(x) = \frac{500}{||x||}\begin{bmatrix} -x_2 & x_1 & 0\end{bmatrix}^\top,$$ and so on, treating every 2D vector as a 3D vector in the $xy$-plane, doing the computations by hand and then coding them explicitly (since of course the PDE software I'm using doesn't know how to interpret a cross product of vectors in $\mathbb{R}^2$ either). I'm wondering if this approach is correct or there's some other way of formulating this problem that I'm missing. I'm also not sure that I'm interpreting $r$ or $\omega$ properly, so if there's anything I'm missing there I'd be grateful for some feedback. I'm not an engineer or physicist by training, so I'm not aware of the conventions for these things!