On Gray's paradox and efficiency measures for swimming

In 1936 Gray reported that the ``drag'' power of dolphins was substantially larger than the estimates of muscle power. We revisit this ``paradox'' in the context of undulatory swimming. We consider larval zebrafish as a model system. We question the basic premise of comparing drag power to muscle power. There are two reasons. First, we recognize that it may not be possible decompose the net force on an undulatory swimmer into drag and thrust. If it becomes possible, as we show in our case, the drag power, which represents the work done on the fluid due to motion in the swimming direction, is exactly balanced by the thrust power, which represents the work done by the fluid. Thus, the total power in the swimming direction, computed in this way, is zero. Second, we show that most of the muscle energy is dissipated in causing the lateral motion of the body - not in overcoming the ``drag'' in the swimming direction. This will be shown based on a power balance equation. Thus, we argue that efficiency measures, that relate the drag power to muscle power, or the Froude efficiency, are not recommended. Instead non-dimensional cost-of-transport could be a useful measure to compare efficiencies of organisms at different scales.