Table 2.

Drag at low Reynolds numbers

Our everyday experiences of movement and of the behaviour of fluids are not necessarily applicable in situations where viscosity, not inertia, dominates. We are used to the effects of inertia, where stopping and starting require some time to occur, and where a swimming animal imparts rearward momentum to the surrounding fluid in order to move forwards. In contrast, small, slow organisms exist in a world where inertia can effectively be ignored, and viscosity dominates. This has many implications [12,68,69], but the most relevant here is that drag becomes much more important than inertia, such that when propulsion ceases, so does movement. Under such conditions, the component of drag due to the friction between the fluid and the object's surface greatly exceeds that due to pressure differences between the front and back of the object. In this case, the extra surface area realised by changing from a sphere to what we consider a 'streamlined shape', such as that of a fish, can outweigh the reduction in pressure drag.

To illustrate, figure 2 shows the relative difference in drag between a sphere and a 'streamlined' body (in this case a prolate spheroid), similar to many spermatozoan heads, at low Reynolds number. Drag with respect to volume (drag per unit volume) is likely to be most important in this context, as volume most probably determines the payload (DNA) or energy stores (mitochondria or their analogues) available to the spermatozoan, so we compare spheroids of equal volume. The conclusion is that drag on a prolate spheroid differs by maximum of 4.44% (for a 2:1 length:diameter ratio) from that of a sphere, and that for ratios higher than 4:1, drag on the 'streamlined' shape is higher than that for a sphere of equivalent volume. We suggest that it may be possible to use this relationship as a null model against which to test whether head morphology is under selection for hydrodynamic or non-hydrodynamic aspects of fertilisation success.