Added mass

Added mass is the effective inertia contributed by fluid accelerated together with an immersed body.

Fluid added mass increases inertia but need not alter the same restoring force. A submerged spring's constant buoyancy shifts equilibrium, whereas a pendulum's buoyancy reduces its restoring torque. Thus an unbounded-fluid sphere has spring frequency factor $\sqrt{\rho /(\rho +{\rho }_f/2)}$ and downward-pendulum factor $\sqrt{(\rho -{\rho }_f)/(\rho +{\rho }_f/2)}$. The latter stable configuration requires $\rho >{\rho }_f$.

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.

A sphere of volume $V$ translating through unbounded inviscid incompressible fluid has fluid kinetic energyIts added mass is therefore one half of the displaced fluid mass.

A sphere of density ${\rho }_s$ falling in inviscid fluid of density $\rho$ has effective inertia $({\rho }_s+\rho /2)V$ and net weight $({\rho }_s-\rho )Vg$. Hence