How to handle the formula for computing net thrust of closely coupled engine systems for an engine with a nacelle that has non-circular dimensions? I am working on a paper and I wish to use this equation for calculating the thrust of an engine in a nacelle:
$$ Fn= \displaystyle \oint V_x\rho\vec V\cdot\vec n \ dA + \displaystyle \oint(P-P_{\infty})\vec n_x \ dA - \displaystyle \oint \vec \tau_x \ dA$$

This equation was given for a simplified case: the common circular nacelle. However, I would like to apply it to the flattened bottom nacelle as on the 737.



I have attempted to fit an equation to this by adjusting the parameters of a cardiod in polar coordinates. I'm having some trouble changing the above to polar integrals, however I'm not sure as to how this conversion would affect the rest of the equation. I could apply the rectangular form instead and keep the above integrals, but I don't think there is a away to express y explicitly in terms of x; it must be done implicitly and so that integral wouldn't really be possible.
In review of my earlier question, @Penguin and my math professor's commentaries agree on the point that the equation $$ Fn= \displaystyle \oint V_x\rho\vec V\cdot\vec n \ dA + \displaystyle \oint(P-P_{\infty})\vec n_x \ dA$$ is the meat of the problem and that when you get the piecewise pressure and velocity, the third term $$- \displaystyle \oint \vec \tau_x \ dA$$ was inconsequential, as it (as far as I can mathematically deduce) describes the core cowl and the external nacelle. As it is an independent characteristic, I'm confident I can just adjust the modelling for a different shape. Thanks again to @Pengiun for providing an outside perspective to my dilemma.