Hydrodynamic torque ($\mathbf{T}$)

The hydrodynamic torque on a body about a reference point is $T={\int }_{S}r\times (\sigma \cdot n)dS$, with $r$ measured from that point and $n$ pointing from the body into the fluid. The stress includes pressure and viscous contributions, and $\sigma \cdot n$ is the fluid traction on the body. Changing the reference point by $d$ changes the torque by $-d\times F$, where $F={\int }_S\sigma \cdot ndS$ is the fluid force. A torque-free body has zero resultant torque, including any external torque.