Hans Lord, Ragnar Lagergren, Jan-Erik Svensson, and Nils Lundqvist. 2006. Sexual dimorphism in Bosmina: the role of morphology, drag, and swimming. Ecology 87:788–795.

Appendix B. A description of how drag was calculated for Bosmina models.

Hydrodynamic theory for drop tests of plastic models

For a Bosmina of natural size swimming in water at 20ºC, body drag as a function of speed was calculated using the results of the plastic model experiment.

At terminal velocity the sum of the three forces acting on a falling object equals zero:

(B.1)

where F_g is gravitational force, F_b is buoyancy and D is drag. The equations for F_g and F_b are given by Newton’s law of acceleration, F = ma,

	(B.2)
	(B.3)

where V_mod is volume of the model, _mod is its density, _me is the density of the medium and g is the gravitational acceleration. The drag force is given by

(B.4)

(Vogel 1994) where U is velocity, S_v is an approximation of the area of the object calculated as the two thirds power of its volume (V^2/3), and C_dv is the drag coefficient based on S_v. C_dv is a dimensionless form of drag and a function of Reynolds number (Re), thus equality of Re for a given shape and orientation in flow implies equality of the drag coefficient. If these four equations are combined C_dv can be calculated for the falling plastic models. The Reynolds number was calculated by

(B.5)

where l is the greatest length of the object in the direction of flow and is the dynamic viscosity of the medium (Pa·s ).

Scaling

We chose a short-featured parthenogenetic female B. longispina (model 1a), as our reference morphology when calculating relative drag for the other models. Relative drag was calculated for a body length (l_nat) of 0.5 mm, the observed mean of individuals in the field, and for one constant volume comparison. In the latter calculation, the body volume of model 1a at natural size (0.5 mm) was used as reference. To make constant volume comparisons, we needed to correct for model length. Since the volume of an object of a given shape is proportional to the cube of its length, V = ql³, where q is a constant of proportion, and the plastic model and its corresponding Bosmina of natural size have identical shape, they also have the same q. After combining and solving for l_nat, we estimated

(B.6)

where V_mod.1a is the volume and l_mod.1a is body length of the reference model and l_nat.1a is the body length of the reference animal at natural size. Using Eq. B.6 we could calculate which length at natural size (l_nat) the other animals should have to get the same volume as 1a.

Drag calculations for Bosmina at natural size

For each model we obtained a mean value of Re and C_dv for each glycerine dilution. We used the obtained Re and Eq. 5 to calculate the speed (U_nat) that gave the same Re, as that of a Bosmina at natural size moving in water. Then the drag of the Bosmina moving at U_nat could be calculated by inserting C_dv and U_nat together with the other known parameters in Eq. B.4.

LITERATURE CITED

Vogel, S. 1994. Life in moving fluids, Second Edition. Princeton University Press, Princeton, New Jersey, USA