Added mass of a circular cylinder ($m_a=\rho \pi a^2$)

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 $\varphi =-U{a}^2\cos \theta /r$. In moving centre coordinates, the fixed-laboratory derivative is ${\varphi }_t=-\dot{U}{a}^2\cos \theta /r-U^2{a}^2\cos (2\theta )/r^2$. The Unsteady Bernoulli equation gives $p-p_{\infty }=\rho a^2\dot{U}\cos \theta /r+\rho U^2{a}^2\cos (2\theta )/r^2-\rho U^2{a}^4/(2r^4)$. 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.