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Recalling the no‐slip rule for a solid body in a flow, it follows logically that a swimming individual will be creating a shear as it moves its “no‐slip” form through the water. The shear will be resisted by the viscosity of the fluid just as in our example with the flat plate, and the magnitude of that resistance will depend on how much surface is exposed to the flow. The resistance is called the friction drag, or skin friction drag, and since friction is something we have all experienced, it is a pretty easy one to comprehend. Friction drag, being in the province of viscosity, is most important at low Reynolds numbers.

Pressure drag may be less easy to appreciate, but it is very important. If you envision a solid object in a flow (Figure 1.3), e.g. a cylinder with its axis normal to the flow, it is an easy matter to see that the pressure on the object will be highest on its upstream face since it is oriented directly into the flow and takes the brunt of the moving fluid. The shape of the pipe results in an acceleration of the fluid as it passes over and under the pipe. Because of Bernoulli’s principle, the increased velocity of the fluid results in a decreased pressure on the downstream side of the cylinder. The differences in pressure from the upstream to the downstream side are rather like pushing from the front and sucking from the rear, and those differences create pressure drag. The expression for dynamic pressure (q) is:

(1.6)

which you may recognize as the “fluid equivalent” of the expression for momentum. In fact, drag is the removal of momentum from a flowing fluid.

All swimming species experience drag as they swim. At low Reynolds numbers, usually in smaller species swimming at slower speeds, friction drag is the more important force. At higher Re, pressure drag is more important. Because drag is difficult to measure and more difficult to predict from theory, total drag is most often used to describe the drag force, without discriminating between friction and pressure drag.

Total drag depends on three elements. The first is the dynamic pressure, as expressed earlier, which itself is a function of the velocity of the fluid and its density. The second is the size of the object in the flow since pressure is a force per unit area. The third is the shape of the object in the flow.

Figure 1.3 Flow‐induced differential pressure. (a) Drag. (b) Lift.

Size can take a few different forms. The most common is the frontal, or the maximum cross‐sectional area of the object in the flow. It can also be the total area of the object in the flow or the wetted surface area. When giving a value for drag, the way size has been expressed is quite important.

The way drag force behaves with different shapes in a flow regime cannot be predicted from first principles except for the simplest shapes, such as a sphere. Instead, drag force must be measured empirically to create a fudge factor known as the drag coefficient or C_d. The drag coefficient in turn depends on the character of the flow regime itself, which is represented by the Reynolds number. So drag is represented by the equations

(1.7)

and

(1.8)

The difficulty of predicting how drag behaves for different shapes in a flow is a real problem for those interested in estimating the cost of overcoming drag to swimming species. You cannot just look up a C_d for your target species unless you happen to be very lucky. Usually you have to use the next best thing, which is a value determined for the closest shape you can find (see e.g. Batchelor 1967).