Added mass

Remarks

Example: a sphere of radius $a$ has $\phi_i = -\frac{a^3 x_i}{2\abs{x}^3}$, and the added mass tensor is $m_{ij} = \tfrac{1}{2}\rho \cdot \tfrac{4}{3}\pi a^3 \, \delta_{ij}$, half the mass of the displaced fluid. Check: on $\abs{x} = a$ with $n = x/a$, $\partial \phi_i / \partial n = \frac{a^3 x_i}{\abs{x}^4} = n_i$. Non-example: a body moving through fluid that carries its own vortices is not covered. The flow is then not a potential flow, and the energy does not split as above. For a body of mass $M$ accelerating in a fluid otherwise at rest, the fluid exerts the force $-\sum_j m_{ij} \, dU_j/dt$, so the body moves as if its mass matrix were $M\delta_{ij} + m_{ij}$. The tensor is positive definite, since $\sum m_{ij} c_i c_j = \rho \int_\Omega \abs{\nabla(\sum c_i \phi_i)}^2 \, dV > 0$ for $c \neq 0$. In two dimensions the same definition per unit length gives $\rho \pi a^2$ for a circular cylinder of radius $a$.