## 7. Objects in fluids – the effects of size

Observing micro-organisms moving through fluids, it is striking that as soon as they stop swimming they come to an abrupt halt. Contrast this with a supertanker that can take many kilometres to come to a stop. Both the microbes and the ship are moving through water that has very similar properties, so why do they behave differently?

Size is clearly important. On the one hand it determines the momentum of the body – the tendency to keep moving – and this is related directly to volume (size cubed). However, friction between the surface of the object and the water is a property of surface area (size squared). For different-sized bodies of similar density moving through the same fluid, the ratio between surface area and volume determines the ratio between friction (drag) and momentum.

Figure 4. The rod-shaped bacteria are very similar in shape to the submarine, and both exist in similar fluid media. However, size determines how the two particles interact dynamically with their environment.

In the following table, we have calculated the SA:V ratio for a cylinder with hemispherical ends – the shape of both the bacteria and the submarine illustrated in Figure 4. You can see that a 1 µm bacteria has an SA:V ratio that is a hundred million times that for the 100 m submarine. Note that the huge values of SA:V for small objects is because the calculations are being performed with the same base unit (metre) throughout – however the same size-related differences would have been apparent if everything had been calculated in millimetres or micrometres.

 Overall length Overall length (m) Width (m) Surface area (m2) Volume (m3) SA:V (m-1) 1 µm 1 × 10-6 5 × 10-8 3.46 × 10-13 8.25 × 10-21 4.17 × 107 10 µm 1 × 10-5 5 × 10-7 3.46 × 10-11 8.25 × 10-18 4.17 × 106 100 µm 1 × 10-4 5 × 10-6 3.46 × 10-9 8.25 × 10-15 4.17 × 105 1 mm 1 × 10-3 5 × 10-5 3.46 × 10-7 8.25 × 10-12 4.17 × 104 10 mm 1 × 10-2 5 × 10-4 3.46 × 10-5 8.25 × 10-9 4.17 × 103 100 mm 1 × 10-1 5 × 10-3 3.46 × 10-3 8.25 × 10-6 4.17 × 102 1 m 1 × 100 5 × 10-2 3.46 × 10-1 8.25 × 10-3 4.17 × 101 10 m 1 × 101 5 × 10-1 3.46 × 101 8.25 × 100 4.17 × 100 100 m 1 × 102 5 × 100 3.46 × 103 8.25 × 103 4.17 × 10-1
Note: If you are not familiar with the 'scientific' notation (eg 3.46 x 10-11) used in this table, refer to the helpsheet 'Powers and logarithms'