Kinetic-energy balance for an incompressible Newtonian fluid

For a fixed domain $\mathcal{D}$, body-force density $f$, and stress tensor $\sigma$, the kinetic energy satisfiesThe last term is the nonnegative rate at which viscosity converts mechanical energy into heat.

For a clean bubble of size $a$ and speed $U$ at large Reynolds number, the outer potential flow violates the stress-free condition by a strain of order $U/a$. A boundary layer of thickness $\delta \sim a{\mathrm{Re}}^{-1/2}$ needs a velocity correction only of order $U\delta /a$, so its dissipation is smaller than the outer-flow dissipation by order $\delta /a$. The leading drag power is therefore obtained from

For the zero-circulation flow around a circle, $e:e=8U^2{a}^4/r^6$. Integration over the exterior plane gives dissipation $8\pi \mu U^2$ per unit axial length and hence drag $D=8\pi \mu U$ per unit length.