= Solution
For <inviscid flow> that is also <incompressible flow> and <irrotational flow> with <velocity potential> $\mathbf u=\nabla\phi$, the <Euler equations> integrate to the <Unsteady Bernoulli equation>
$$
\boxed{\left.\frac{\partial\phi}{\partial t}\right|_{\text{fixed laboratory point}}+\frac12|\nabla\phi|^2+\frac p\rho+gz=F(t).}
$$
The same $F(t)$ applies throughout a connected irrotational fluid region; an additive time-dependent potential can absorb it. With gravity absent, fluid at rest at infinity and $\phi\to0$, it is $p_\infty(t)/\rho$.
Take the cylinder centre at laboratory position $(X(t),0)$ with $\dot X=U(t)$, and define $r,\theta$ relative to this moving centre. The circulation-free <potential flow> satisfies <Laplace's equation>, decays at infinity, and has surface condition $\phi_r(a,\theta)=U\cos\theta$. The $\cos\theta$ harmonic solution is consequently
$$
\boxed{\phi=-\frac{Ua^2}{r}\cos\theta.}
$$
The velocities are $u_r=Ua^2\cos\theta/r^2$ and $u_\theta=Ua^2\sin\theta/r^2$, hence $|\mathbf u|^2=U^2a^4/r^4$.
The time derivative must be taken at fixed laboratory coordinates, not fixed $r,\theta$. At a fixed fluid point $\dot r=-U\cos\theta$, $\dot\theta=U\sin\theta/r$, so
$$
\left.\phi_t\right|_{\rm lab}=-\frac{\dot Ua^2}{r}\cos\theta-\frac{U^2a^2}{r^2}\cos2\theta.
$$
Substituting into the <Unsteady Bernoulli equation> gives the entire pressure field
$$
\boxed{p(r,\theta,t)=p_\infty(t)+\frac{\rho a^2\dot U}{r}\cos\theta+\frac{\rho U^2a^2}{r^2}\cos2\theta-\frac{\rho U^2a^4}{2r^4}.}
$$
At the cylinder surface this is $p_\infty+\rho a\dot U\cos\theta+\rho U^2(\cos2\theta-1/2)$. Pressure acts against the outward radial normal of the cylinder, so the force per unit length is
$$
F_x=-a\int_0^{2\pi}p(a,\theta,t)\cos\theta\,d\theta=-\rho\pi a^2\dot U,\qquad
F_y=-a\int_0^{2\pi}p(a,\theta,t)\sin\theta\,d\theta=0.
$$
All steady terms cancel by trigonometric orthogonality. Therefore
$$
\boxed{\mathbf F=-\rho\pi a^2\dot U\,\mathbf e_x.}
$$
This identifies the <added mass of a circular cylinder> as the mass of displaced fluid per unit length. The solution assumes the ambient fluid is initially at rest and has no separately prescribed circulation; these select the usual pure translating-cylinder flow.
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