= Added mass of a circular cylinder
{title2=$m_a=\rho\pi a^2$}
An infinitely long <circular cylinder> of radius $a$ translating across its axis through otherwise stationary, inviscid incompressible fluid has <added mass> $m_a=\rho\pi a^2$ per unit length. The decaying, circulation-free <velocity potential> is $\phi=-Ua^2\cos\theta/r$. In moving centre coordinates, the fixed-laboratory derivative is $\phi_t=-\dot U a^2\cos\theta/r-U^2a^2\cos(2\theta)/r^2$. The <Unsteady Bernoulli equation> gives $p-p_\infty=\rho a^2\dot U\cos\theta/r+\rho U^2a^2\cos(2\theta)/r^2-\rho U^2a^4/(2r^4)$. Integrating $-pa\cos\theta$ around $r=a$ yields $F_x=-\rho\pi a^2\dot U$; the transverse force vanishes. Taking a derivative at fixed centre coordinates would omit the necessary convective correction.
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