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For the outer flow from part (i), the rate-of-strain tensor in polar coordinates has componentsThereforeThe leading dissipation per unit axial length isBalancing this with drag power $DU$ gives the drag on a two-dimensional circular bubble from outer-flow dissipationper unit axial length, directed opposite to the bubble velocity.

The momentum equation isTaking its scalar product with $u$ givesFor the Newtonian fluid stress tensor $\sigma =-pI+2\mu e$, its contraction with the velocity gradient isbecause incompressibility removes the pressure term and the symmetric rate-of-strain tensor is orthogonal to the antisymmetric part of $\nabla u$. Integration over $\mathcal{D}$ and the divergence theorem therefore give the kinetic-energy balance for an incompressible Newtonian fluid:The terms are respectively the rate of change of kinetic energy, outward advective flux of kinetic energy, power supplied by the body force, power supplied by surface traction, and irreversible viscous dissipation. Since $f=-\nabla \psi$, the body-force power may also be written $-\nabla \cdot (\psi u)$ and interpreted as exchange with potential energy.

Differentiating the Stokeslet gives its rate-of-strain tensor and vorticity, displayed above for $r=|\mathbf{x}|>0$. The pressure is $p=\mathbf{F}\cdot \mathbf{x}/(4\pi r^3)$. Integrating the Newtonian fluid stress tensor over a sphere enclosing the origin gives traction resultant $-\mathbf{F}$, fixing the point-force normalization.

For two equal spheres of radius $a$ separated by $\mathbf{R}$, let the first have applied force $\mathbf{F}$ and let the second be force-free and torque-free. With ${\mathbf{U}}_0=\mathbf{F}/(6\pi \mu a)$, the leading reflected stresslet changes the first velocity byThe distant sphere's freely translating and rotating rigid motions remove its monopole force and torque fields; its incident rate-of-strain tensor produces the first self-mobility correction.