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A force-free, torque-free microswimmer in a stationary Newtonian fluid, with negligible fluid and body inertia and steady Stokes flow at each instant, gains no net displacement from a reciprocal stroke: one whose sequence of shapes is exactly retraced in reverse. The environment and boundary conditions must also be unchanged during that retracing. Opening a two-shell swimmer and then closing it follows the same sequence backwards; doing the closure faster does not help, because Linearity of Stokes flow changes the instantaneous velocity by the same factor as the stroke rate.

A single real shape coordinate that travels out and back along an interval gives this cancellation, as computed in the force-free two-sphere stroke. Two independently controlled shape coordinates can instead execute a loop that is not retraced, avoiding this obstruction. Continuous rotation also need not retrace an interval: the helical microswimmer with a spherical head provides another route to propulsion. The scallop theorem is a restriction on a stroke under stated conditions, not a claim that small organisms cannot swim. Purcell presented the opening-and-closing scallop and a swimmer with two hinges in Life at low Reynolds number (1977), figures 6–7.

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.

Let the ambient velocity be $\mathbf{Ex}$, with $\mathbf{E}$ symmetric and trace free, and centre a rigid sphere of radius $a$ at the origin. It is force-free and torque-free, with zero translation and rotation. Its disturbance isThe total velocity vanishes at $r=a$, and the leading disturbance is a stresslet. Harmonic dipole and quadrupole potentials in the Papkovich–Neuber representation give the complete field.

Consider two widely separated rigid spheres of equal radius $a$, with separation $R\mathbf{n}$, in Stokes flow. Apply a force $6\pi \mu a{\mathbf{U}}_0$ to the first sphere and let the second be force-free and torque-free. The second sphere samples the rate-of-strain tensor of the first sphere's Stokeslet, $\mathbf{E}=3a({\mathbf{U}}_0\cdot \mathbf{n})(\mathbf{I}-3\mathbf{nn})/(4R^2)$. Its leading disturbance is the stresslet of a sphere in a uniform straining Stokes flow. Evaluating that disturbance back at the first sphere givesThe hydrodynamic mobility matrix decreases at fixed force, consistently with the increase of resistance caused by an additional rigid inclusion.

For a forced sphere and a distant equal force-free, torque-free sphere, the leading incident Stokeslet strain is proportional to $(\mathbf{U}\cdot \mathbf{n})(I-3\mathbf{nn})$. Its reflected stresslet is axisymmetric about the line of centers and has zero vorticity on that line. The nominal fifth-order self-spin coefficient therefore vanishes. Finite-size incident-strain corrections and higher force-free scattering multipoles first supply a generic seventh-order angular velocity. Purely longitudinal forcing has zero spin exactly by axial symmetry. The leading translation correction is fourth order for a generic orientation, but sixth order for purely transverse forcing. These cancellations matter when using the method of reflections for Stokes flow to infer orders from far-field decay alone.

If the second rotor has zero drag, it cannot supply reaction torque. The single remaining helix must be both force-free and torque-free; a positive-definite resistance matrix forces its translation and rotation to vanish. The relative motor motion is absorbed by the zero-resistance rotor.

An axial force balance on fluid between two remote cross-sections of a cylindrical tube giveswhen every immersed body between the sections is force-free.