When a particle moves in a Newtonian flow at a low Reynolds number, inertia is irrelevant, and a linear relationship exists between velocities and forces. For incompressible flows, any force distribution $\mathbf{f}\left(\mathbf{r}\right)$ acting in the fluid bulk induces flow and motion only through its solenoidal component. For force distributions that are spatially localized (i.e., vanish sufficiently fast at infinity), we derive the representation of the rigid-body motion as an explicit linear functional of $\mathbf{\nabla }\times \mathbf{f}$, which automatically selects the solenoidal component of $\mathbf{f}$. This alternative representation complements the usual one in terms of $\mathbf{f}$ and has the advantage of having the incompressibility constraint built-in. We illustrate its utility in avoiding certain ambiguities that arise, for example, when implementing approximations for swimmers.

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• Low Reynolds number swimmers
• Stokesian dynamics