A force-free swimmer with a reciprocal shape stroke cannot achieve net locomotion in an inertia-free Newtonian fluid. A single real shape coordinate that retraces an interval gives a reciprocal cycle. The conclusion concerns free swimming; externally imposed translations can instead force and pump the fluid.
For spheres linked by a length , axial mobility and imply for their midpoint. The displacement over a periodic stroke is . Unequal radii permit instantaneous midpoint motion but not net translation. This is an explicit one-shape-coordinate example of the scallop theorem.
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The Scallop Theorem is a concept from the field of mathematical biology, specifically in the study of the dynamics of movement in organisms. It addresses the limitations of locomotion in certain types of organisms, particularly those that are at or near the microscopic scale, like small aquatic animals or microorganisms. The theorem states that certain types of organisms cannot swim effectively by using only passive movements in their appendages, such as flagella or cilia.