Departure from spherical shape form resistance

If the laminar-flow condition may thus be assumed to apply to the movements of micro-phytoplankton, probably at all times, it is not at all clear that the Stokes equation can apply other than to spherical organisms, coenobia or colonies, less than 200 ¡m in diameter. In fact, for the majority of phytoplankters that are not spherical, the shape distortion has a significant impact on the rate of settling. Distortion from the sphere inevitably results in a greater surface area to an unchanged volume and density and, hence, a greater volume-specific frictional surface, otherwise known as form resistance. The effect of departure from spherical form was clearly demonstrated by the experiments of McNown and Malaika (1950). In most instances (the 'teardrop' shape being a notable exception), sub-spherical metal shapes sank through oil more slowly than the sphere of the same volume. It is difficult to account quantitatively for these results, as mathematical theory is not so well developed that the effects of distortion can be readily calculated. However, McNown and Malaika (1950) also published their findings on the sinking of spheroids, both oblate (flattened in one axis, like a medicinal pill, or what British readers will understand as 'Smartie'-shaped) or prolate (shortened in two axes, towards the shape of classic airships). These have provided plankton scientists with an important foundation on the impacts of form on algal sinking. Some interesting theoretical or experimental investigations have been pursued using cylinders (Hutchinson, 1967; Komar, 1980), chains of spheres (Davey and Walsby, 1985) and, more recently, some ingenious alga-like shapes fashioned in polyvinyl chloride (PVC) or malleable 'Plasticine' (Padisak etal., 2003a). These have helped to amplify an understanding of the importance of shape in the behaviour of phytoplankton.

In the case of spheroids, the reduction in sinking is related to the ratio of the vertical axis (say, a) to the square root of the product of the other two (*Jbc). The fastest-sinking spheroid is one in which a ~ 2b and b = c: the horizontal cross-sectional area is smaller than that of the sphere of the same volume but with most of the volume in the vertical where it offers less drag, the spheroid actually sinks faster than the sphere. As the ratio is increased [a/(^/bc) to >3], drag increases and velocity falls below that of the equivalent sphere. Analogously, making a < b and a/( bc) < 1, drag increases more than the horizontal cross-sectional area and to >3. Spheroids with the most disparate diameters, that is, the narrower or the flatter they are with respect to the sphere of identical volume and density, offer increased form resistance and up to a twofold reduction in sinking rate.

In the case of cylinders, increasing the length but keeping the cross-sectional diameter constant increases sinking velocity, although this approaches a maximum when length exceeds diameter about five times. Cylinders at this critical length have about the same sinking speed as spheres of diameter 3.5 times the cylinder section (dc). As a cylinder having a length of 7(dc) has the same volume as such a sphere, we may deduce that cylinders relatively longer than this will sink more slowly than the equivalent sphere, so long as all other Stokesian conditions are fulfilled.

It has been suggested in several earlier studies that distortions in shape have another role in orienting the cell, that (for instance) the cylindrical form makes it turn normal to the direction of sinking and that this may be advantageous in presenting the maximum photosyn-thetic area to the penetrating light. Walsby and Reynolds (1980) argued strongly for the counter-view. In a truly turbulence-free viscous medium, a shape should proceed to sink at any angle at which it is set. There is an exception to this, of course, which will apply if the mass distribution is significantly non-uniform: the 'teardrop' reorientates so that it sinks 'heavy' end first. In the experiments of Padisak et al. (2003a), some of the models of Tetrastrum were made deliberately unstable by providing spines on one side only: these, too, reoriented themselves on release but then remained in the new position throughout the subsequent descent. Their observations on Staurastrum models that go on reorienting recalls the observations of Duthie (1965) on real algae of this genus, which reorientated persistently during descent to the extent that they rotated and veered away from a vertical path. At the time, the influence of convection on the sinking behaviour of Staurastrum could not be certainly excluded. The results of Padisak et al. (2003a) suggest that the form of the cells engenders the behaviour as it reproduced in a viscous medium.

For the present, the impact on sinking of distortions as complex as those of Fragilaria or Pedi-astrum coenobia requires more prosaic methods of assessment. The most widely followed of these is to calculate a coefficient of form resistance (^r) by determining all the variables in the Stokes equation (2.14) for a sphere of equivalent volume (having the diameter, ds) and comparing the calculated rate of sinking (ws calc) with the observed rate of sinking by direct measurement, ws. Thus,

The Stokes equation should also be modified in respect of phytoplankton (Eq. 2.16) by including a term for form resistance, accepting that the value of may be so close to 1 that the estimate provided by Eq. (2.14) would have been acceptable.

This approach allows systematic variability in the coefficient to be investigated as a feature of phy-toplankton morphology. Some of the interesting findings that impinge on the evolutionary ecology of phytoplankton are considered in the next section but it is important first to mention the practical difficulties that have been encountered in estimating form resistance in live phytoplank-ton and the ways in which they have been solved. As observed elsewhere (Chapter 1) precise estimates of the volume of a plankter (whence ds is calculable) are difficult to determine if the shape is less than geometrically regular. Making a concentrated suspension and determining the volume of liquid it displaces offers an alternative to careful serial measurements of individuals. Walsby and Xypolyta (1977) gave details of a procedure using 14C-labelled dextran to estimate the unoccupied space in a concentrated suspension. The usefulness of the approach nevertheless depends upon a high uniformity among the organisms under consideration - cultured clones are more promising than wild material in this respect.

The densities of phytoplankton used to be difficult to determine precisely, having to rely on good measurements of mass as well as of volume. Now, it has become relatively easy to set up solution gradients of high-density solutes, introduce the test organisms then centrifuge them until they come to rest at the point of isopycny between the organism and solute (Walsby and Reynolds, 1980). Following Conway and Trainor (1972), Ficoll is frequently selected as an appropriate solute. Being physiologically inert and osmot-ically inactive improves its utility.

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