Estimation of sinking velocity using free-falling dynamically scaled models: Foraminifera as a test case

The transport of organisms and biologically derived particles through fluid environments strongly influences their spatiotemporal distribution and ecology. In up to a third of terrestrial plants (Willson et al., 1990), reproduction is achieved through passive movement of propagules (e.g. seeds) on the wind. In aquatic environments, propagules of many sessile groups from corals (Jones et al., 2015) to bivalves (Booth, 1983) are dispersed by ambient currents, eventually settling out of the water column to their final locations. Furthermore, most dead aquatic organisms (from diatoms to whales) sink, transporting nutrients to deeper water and contributing to long-term storage of carbon (De La Rocha and Passow, 2007). In the case of microfossils, the sinking dynamics of the original organisms even influences our reconstructions of the Earth's paleoclimate (Van Sebille et al., 2015). Crucially, the horizontal distances over which all these biological entities are transported, and therefore their distribution, are affected by their settling velocity (Ali et al., 2011).

Measuring the individual settling velocity of small particles directly is challenging, especially when they are too small to be imaged easily without magnification (e.g. Walsby and Holland, 2006). Here, we applied dynamic scaling, an approach commonly used in engineering, to circumvent this difficulty and accurately quantify the kinematics of sub-millimetre-scale free-falling particles using enlarged physical models. We used scaled-up physical models in a high-viscosity fluid, enabling easy measurement of settling speed, orientation and other parameters using inexpensive standard high-definition web cameras. While dynamically scaled models have previously been employed to study a number of problems in biological fluid mechanics (e.g. Vogel et al., 1973; Vogel, 1987, 1994; Koehl, 2003), the study of freely falling particles of complex shape – for which settling speed is the key unknown parameter – presents a unique challenge to experimental design that we overcome in this work.

Engineering problems such as aircraft and submarine design often are approached using scaled-down models in wind tunnels or flumes to examine fluid flows around the model and the resulting fluid dynamic forces it is subjected to. To ensure that the behaviour of the model system is an accurate representation of real life, similarity of relevant physical phenomena must be maintained between the two. If certain dimensionless numbers (i.e. ratios of physical quantities such that all dimensional units cancel) that describe the system are equal between the life-size original and the scaled-down model, ‘similitude’ is achieved and all parameters of interest (e.g. velocities and forces) will be proportional between prototype and model (Zohuri, 2015). Intuitively, the model and real object must be geometrically similar (i.e. have the same shape), so that the dimensionless ratio of any length between model and original, Length_model/Length_real, is constant – this is the scale factor (S) of the model. Less obvious is the additional requirement of dynamic similarity, signifying that the ratios of all relevant forces are constant. For completely immersed objects sinking steadily at terminal velocity (achieved quickly for most small particles, see Materials and Methods, ‘Time to terminal velocity’), dynamic similarity is achieved by matching the Reynolds number (Re).

Dynamically scaled physical models exhibiting the same Re as the original systems have been used in a number of biological studies. Vogel and La Barbera (1978) outline the principles of dynamic scaling: to obtain the same Re when enlarging small organisms, the fluid flow must be slower and/or the fluid more viscous, and when making smaller models of large organisms, the fluid flow must be faster and/or the fluid less viscous. For instance, Vogel (1987) used air in place of water flowing at lower speeds when investigating the refilling of the squid mantle during swimming by scaling up a model 1.5 times relative to the animal's actual size. More recently, Stadler et al. (2016) investigated sand inhalation in skinks with 3D-printed enlarged models, using helium instead of air (thereby increasing viscosity) as the experimental fluid. Koehl and colleagues have studied crustacean antennule flicking (lobsters: Reidenbach et al., 2008; mantis shrimp: Stacey et al., 2002; and crabs: Waldrop et al., 2015) as well as the movements of copepod appendages (Koehl, 1995) with enlarged models, using mineral oil in place of water. Finally, perhaps the largest change in scale was employed by Kim et al. (2003), who modelled the bundling of E. coli flagella at a scale factor of ∼61,000, submerged in silicone oil (10⁵ times more viscous than water), and rotated at 0.002 rpm compared with the 600 rpm observed in real bacteria (Sowa and Berry, 2008).

In all the above studies, basic kinematics such as speed in the original system were relatively easy to measure, and the experiments aimed to reveal the forces involved (e.g. hydrodynamic drag) or details of the fluid flow such as the pattern of streamlines. As the representative speed (U) of the original system was known, designing experiments to achieve similitude was relatively straightforward because the Re was also known a priori – in these cases, the model size, speed and working fluid properties were simply interrelated through Re (Eqn 1). For instance, once a working fluid and the model size were chosen, the required towing speed was obvious. However, in the case of sedimentation of small particles (e.g. spores, seeds, plankton), the sinking speed (U) is the key unknown. With an unknown sinking speed, the operating Re is also unknown, so it is not straightforward to design experiments that achieve similitude with the original system. Here, we present an iterative methodology leveraging 3D printed dynamically scaled models that allows determination of the sinking speed of small objects of arbitrarily complex shape.

We used planktonic organisms of the subphylum Foraminifera (hereafter referred to as the group ‘foraminifera’) as an example of a small (200–1500 µm) biological particle for which the settling velocity is important and typically unknown. Foraminifera are marine amoeboid protists (Gupta, 2002; Schiebel and Hemleben, 2005). By secreting calcium carbonate, foraminifera produce a multi-chambered shell (test) which, in planktonic foraminifera, can grow up to 1500 µm in diameter, and which frequently exhibits a complex shape (Table 1). Once the organism dies or undergoes reproduction, the empty test sinks to the ocean floor, and so oceanic sediment contains substantial numbers of foraminifera tests. Foraminifera account for 23–56% of the oceans’ production of carbonate (CO₃) (Schiebel, 2002), an important factor in climate change models (Passow and Carlson, 2012). Of particular interest for climate predictions is calculating the flux of tests reaching the ocean floor (Schiebel, 2002; Jonkers and Kučera, 2015). While there are more than 30 extant species and over 600 species in the fossil record, settling velocities are known for only 14 species of foraminifera (Fok-Pun and Komar, 1983; Takahashi and Be, 1984; Caromel et al., 2014: 3.41×10⁻⁴ to 6.8×10⁻² m s⁻¹, Re≈18–55).

We assume that the size (i.e. L – defined as the maximum length parallel to the settling direction, A – defined as the projected frontal area, and V – the particle volume not including any fluid-filled cavities), 3D shape (ψ, here treated as a categorical variable because of our consideration of arbitrarily complex morphologies; see Table 1) and density (ρ_particle) of the original sinking particle are known, while the sinking speed (U) is unknown. The properties of the fluid surrounding the original particle (i.e. ρ_fluid, μ) are also known, and our goal is to design experiments in which we sink a scaled-up model particle in a working fluid of known ρ_fluid and μ in order to determine the model particle's sedimentation speed and, via similitude, U of the original particle.

While previous work (Berger and Piper, 1972; Fok-Pun and Komar, 1983; Takahashi and Be, 1984; Caromel et al., 2014) suggests that the Re of sinking foraminifera should be 10⁰−10², the exact value of Re for morphology ψ is assumed to be unknown. Hence, it is not immediately clear what size the model should be (i.e. the scale factor S=L_model/L_real) in order to match this Re in the experiments and ensure similitude. Solving for both Re and S simultaneously requires additional mathematical relationships beyond Eqn 1.

Throughout, we use a superscript O to refer to the original values of dimensioned variables at life size (e.g. L^O, V^O, A^O, ⁠, U^O) and Re^O, for the values of the dimensionless Reynolds number and drag coefficient (defined below) corresponding to real particles sinking in the original fluid (e.g. seawater of ⁠, μ^O). While the fluid dynamics of flow around a particle of particular shape ψ can be considered theoretically over a range of Re, only the dynamics at Re^O and will represent the ‘operating point’ corresponding to the life-size particle settling speed U^O.

To construct an empirical curve for a particular test morphology, we started with 3D scans of individual specimens from 30 different species (Table 1). The majority of the species were selected from the University of Tohoku museum's database, eForam Stock (http://webdb2.museum.tohoku.ac.jp/e-foram/), with a micro-computed tomography (µCT) scan resolution between 2.5 and 3.6 pixels µm⁻¹, and were exported as 3D triangular mesh (STL format) files. Specimens of an additional three species were scanned using synchrotron radiation-based micro-computed tomography (SRµCT). Imaging was performed at the Imaging Beamline P05 (IBL) (Greving et al., 2014; Haibel et al., 2010; Wilde et al., 2016) operated by the Helmholtz-Zentrum-Geesthacht at the storage ring PETRA III (Deutsches Elektronen Synchrotron – DESY, Hamburg, Germany). Specimens were imaged at a photon energy of 14 keV and with a sample to detector distance of 17 mm. For each tomographic scan, 900 projections at equal intervals between 0 and π were recorded. Tomographic reconstruction was done via a classical filtered back projection using the RECLBL library (Huesman et al., 1977). For processing, raw projections were binned 2 times resulting in an effective pixel size of the reconstructed volume of 1.44 µm. These scans were segmented and rendered using SPIERS (Sutton et al., 2012), and again exported in STL format and are available from MorphoSource (https://www.morphosource.org/Detail/ProjectDetail/Show/project_id/1167). Meshes of all foraminifera were manually checked in Meshlab (Callieri et al., 2012) for integrity.

For species where more than one scan was available, the scan that contained the best-preserved specimen was chosen. By only including one specimen per species, this approach neglects phenotypic plasticity which is demonstrated in planktonic foraminifera (e.g. Lohmann, 1983; Morard et al., 2013), but was chosen because of limitations of µCT scan availability and time constraints on the project.

The 3D scans allowed us to easily fabricate scaled-up (scale factor S) physical models of each specimen using a FormLabs Form1+ (Formlabs, Somerville, MA, USA) 3D printer, using FormLabs Clear Resin Version 2 with a layer thickness of 50 µm (see Fig. 2D–I for examples) and x–y resolution of 200 µm. Models were washed and flushed with isopropanol to remove excess resin following Formlabs' guide and allowed to air dry. Support material was removed (Fig. 2I), and the models lightly sanded with 400 grit Wet ‘n’ Dry paper, followed by a final isopropanol wash to remove any remaining residue. Once dry, models were filled with mineral oil in preparation for sinking. Clear resin was chosen to allow each model to be checked for bubbles (which would increase the buoyancy of the model). Any bubbles were removed using a 30-gauge needle and syringe.

Following convention, when defining the area A_particle used in the definition of C_D (Eqn 3), we measured the projected area of the sinking foraminifera. Referring to high-resolution images of the sinking model (Fig. S2D), a digital model of the foraminifera was manually aligned to measure the projected area in a plane perpendicular to the sinking direction (Fig. S2D). We used the same procedure to measure the maximum length parallel to the flow (L) for the calculation of Re (Fig. S2D). These choices facilitated objective comparisons of C_D across morphologically diverse species, to be detailed in a future study.

The models were released in a cylindrical acrylic tank (0.9 m in diameter and 1.2 m in height) of mineral oil (‘Carnation’ white mineral oil, Tennants Distribution Limited, Cheetham, Manchester, UK; ρ=830 kg m⁻³, μ=0.022 Pa S) filled to a depth of 1.18 m (approximately 750 l). The tank was fitted with a custom-designed net and net retrieval system (Fig. S2A) to allow easy retrieval of the models after their descent, allowing each model to be sunk 5 times. Integrated into the net retrieval system was the release mechanism, which was held centrally over the tank, with the grasping parts submerged below the oil level. This ensured that each model was released in a controlled and repeatable fashion.

To minimize reflections, the tank was surrounded by a black fabric tent-like structure. This also served as a dark background to facilitate visualization of the model during descent. The tank was illuminated with a single 800 lumen LED spotlight placed beneath it and, as the Formlab Clear Resin is UV-fluorescent, two 20 W ‘Blacklight’ UV fluorescent tubes were placed above the tank.

The sinking models were recorded using two Logitech C920 HD webcams (Logitech, Lausanne, Switzerland), placed at 90 deg to each other (Fig. S2A) and recording at 960 pixels×720 pixels and ∼30 frames s⁻¹, allowing monitoring of the position and orientation of the particle in 3D as it fell. As these consumer-grade webcams use a variable frame-rate system, a custom-written MATLAB script was used to initiate camera recording, recording both frames and frame time stamps. Videos were recorded for 500 frames (∼17 s). Sinking velocity was calculated over a central 0.8 m depth range, ensuring the model was at terminal velocity (see ‘Time to terminal velocity’, below) whilst also avoiding end effects which could slow the model as it reached the bottom of the tank. Based on observations of suspended dust, there was no discernible convection in the tank during any trials that might potentially affect sinking velocity. The curved walls of the tank introduced distortion, which was removed using the MATLAB toolbox ‘Camera Calibrator’ (McAndrew, 2004). Pixel size was 1.06 pixels mm⁻¹ with a mean reprojection error of 0.5 pixels; therefore, distance measurements (for calculating sinking velocity) were accurate to within 0.5 mm (0.06% of the traversed depth).

We applied this correction by taking any experimentally determined C_D to equal ⁠, and using estimated according to Eqn 10 for subsequent calculations as detailed below. In our experiments, λ ranged from 0.0027 to 0.0173, yielding K between 1.0057 and 1.0377. Wall effects were therefore quite small, with ranging from 0.993 to 0.994.

While morphologically complex particles such as foraminifera tests (Table 1) are not expected to behave like ideal spheres, Eqn 11 should be sufficient to provide initial guesses, after which we iterate to find the solution. We note that if the particle shapes of interest were all most similar to some other well-studied geometry (e.g. cylinders, discs, etc.), using a known C_D(Re) relationship for that shape could provide better initial guesses and faster convergence.

Substituting Eqn 11 into Eqn 7 (with S=1, V=V^O, and ⁠, substituted) and moving all terms to one side, we can numerically solve (MATLAB's fzero function) for our first estimate of the operating Re^O. Substituting this Re back into Eqn 7 or Eqn 11 yields an estimate of the operating ⁠. We aimed to reproduce this Re and C_D in the first experiment, except that we accounted for wall effects by distinguishing between and expected to occur in the tank. Hence, we could again substitute this Re into Eqn 7 but now with ρ_particle corresponding to the resin model and ρ_fluid and μ corresponding to mineral oil, and combine this expression with Eqn 10, assuming our estimated C_D=⁠, and λ=SL^O/D. The resulting expression can be solved numerically for the first scale factor, termed S₁. Two more scale factors (S₂ and S₃), one smaller and one larger than S₁, were chosen to span expected Re values for foraminifera from published literature (e.g. Fok-Pun and Komar, 1983; Takahashi and Be, 1984; Caromel et al., 2014) as well as Re^O for other species which had reached convergence. This procedure was intended to bound the correct S value that reproduces the operating Re^O and of the settling particle. The three models were printed, their actual volumes V measured via weighing, and their settling velocities U experimentally measured as detailed in the preceding sections.

An empirical cubic spline curve can now be fitted (http://www.mathworks.co.uk/matlabcentral/fileexchange/24443-slm-shape-language-modeling) to these three initial (Re, C_D) data points, constrained to be monotonically decreasing and concave-up within the limits of the data to match expectations for drag on objects at low to moderate _­Re. Three optimally spaced spline knots were used as this yielded excellent fits to the data as the number of data points increased. These details of the spline as well as its order (i.e. cubic versus linear) are somewhat arbitrary but we ensured that our results were sufficiently converged as to be insensitive to them (see ‘Remaining iterations’, below).

The operating point (Re^O, ⁠) corresponding to the particle settling in the natural environment can be visually represented as the intersection point of the curve defined by Eqn 7 (with S=1 and ⁠, ⁠) and the empirical relationship based on our experimental data. Algebraically, the operating point is the solution to ⁠. We solved for Re^O numerically using a root finding algorithm (MATLAB's fzero function) on the objective function and then obtained by substituting Re^O into Eqn 7. Finally, U^O was easily determined from the definition of Re^O (Eqn 1 with U^O, LU^O and substituted).

Because our first three empirical data points and fitted spline corresponded to guessed model scale factors S, our initial operating point prediction (Re^O, ⁠) often was not located near any of these initial points or sometimes even within the bounds of these data (in which case linear extrapolation of was used to estimate the operating point). Therefore, to ensure the accuracy of our predicted U^O, we continued iterating with additional experiments.

The model scale factor for the Nth experiment was chosen by combining Eqns 7, 8 and 10 with Re=Re^O and (from the previous iteration), and V=V^predicted, and numerically solving for S. A model close to this new scale was printed and sunk, its settling velocity U recorded and Re and C_D computed, and a more accurate spline constructed by including this new data point. The calculation of (Re^O, ⁠) detailed in the previous section was then repeated, yielding a more accurate operating point. Overall, the aim was to tightly bound the predicted operating point with experimental data to maximize confidence in the fitted spline in this region.

The iterative process (visualized as a flowchart, Fig. 1B, with a specific example of convergence given in Fig. 2B,C) was repeated until: (1) the predicted operating point was not extrapolated beyond our existing data, (2) the variation in calculated U^O between the fitting of a linear spline and cubic spline was no greater than 5%, and (3) the variation between the predicted Re^O and the closest experimentally measured Re was less than 15%.

In many cases, the difference between results based on four versus three data points was very small (Fig. 2B,C), indicating rapid convergence and the possibility of streamlining the method further in the future. Through this method, we calculated the sinking velocity of 30 species of planktonic foraminifera (Table 1).

Our basic methodology was first validated by 3D printing a series of spherical models (10–20 mm in diameter) for which the theoretical C_D(Re) relationship is already well known. In order to achieve low density (and thus low sinking velocity and low Re), these spheres were hollow and filled with oil via two small holes (of diameter 0.8% of the sphere diameter). Our empirically generated curve compares favourably with the theoretical curve (Morrison, 2010) (R²=0.875, Fig. 2A), with the distance between the curves approximately constant above Re≈25. While the error grows larger at lower Re, we expected most foraminifera species to operate at Re≈18–55 based on previous work (Berger and Piper, 1972; Fok-Pun and Komar, 1983; Takahashi and Be, 1984; Caromel et al., 2014).

To quantify errors in our approach even more directly, we then considered hypothetical hollow spherical particles with the same material density as foraminifera tests and a range of sizes (L^O=750–1150 µm, similar to the species we studied) settling in seawater. This size range corresponds to Re=12–27, the area where our curve is most divergent from ⁠. We compared predictions of the operating U^O based on our empirical curve versus the theoretical curve for spheres as outlined above, substituting Eqn 11 for in the latter case. Maximum relative error in predicted U^O was 11.5% at Re=16 (corresponding to a sphere 860 µm in diameter) while the minimum difference was 6.5% at Re=27 (corresponding to a sphere of 1150 µm in diameter, Fig. 2A). This level of error is much smaller than the variation in U^O we predicted across the 30 foraminifera species we investigated (Table 1).

This study was concerned with predicting steady sinking speed, but in our experiments, each model foraminifera took a finite amount of time to accelerate from rest at the point of release to its terminal sinking velocity. As this transient portion of the sinking trajectory could introduce errors into our analysis, it is important to determine whether it affected any of our recorded data.

It is well known that as Re approaches zero in the limit of inertia-less Stokes flow, unsteadiness can only occur as a result of time-varying boundary conditions. Thus, a microorganism that stops actively swimming will almost instantly come to a stop, and a heavy micro-particle released from rest will almost instantly begin sinking at its terminal velocity (Purcell, 1977). As Re increases and inertia becomes increasingly important, the transient period of acceleration becomes longer. Therefore, a reasonable worst case to examine here is the foraminifera model that sank at the highest Re.

We found Globorotalia (Truncorotalia) truncatulinoides to operate at Re=42 (Table 1) but here we conservatively chose the largest scale model used to generate its spline for which S=16 and Re=90. Inserting this model's length L, area A, and measured volume V into Eqn 14, we obtain solutions for the time-varying speed and depth of a sphere approximating this model's geometry (Fig. S2B). The depth corresponding to where speed equals 99.9% of the terminal velocity is approximately 4.6 cm, which is much smaller than the 19 cm between where the models were released and the edge of the cameras' field of view for data collection. Hence, the transient acceleration of each model foraminifera should have had no effect on our data or results. Most of our models should have reached terminal velocity even sooner as they sank at lower Re, e.g. within 2.2 cm for Catapsydrax dissimilis operating at Re=36 (Table 1).

Here, we present a novel method of determining settling speed by leveraging dynamically scaled models falling under gravity rather than being towed at a controlled speed. Applying our method to foraminifera-inspired spherical particles (Fig. 2A), we predict settling speeds within 11.5% of theoretical expectations (Fig. S2E). In Fig. 2B,C we present an example of convergence of our method to the operating Re^O, and U^O of a typical foraminifera species. There was little variation in the number of iterations required to reach convergence (mean 4, range 3–6; see Table 1), despite the morphological complexity of some species (e.g. Globigerinoidesella fistulosa). We suspect the higher end of this range was due to these species having forms that were particularly challenging to clean residual resin from, or to the incomplete removal of air bubbles once submerged in oil.

Our predicted sinking speeds of foraminifera fall within aggregated existing data for 14 species (Fig. S1; Fok-Pun and Komar, 1983; Takahashi and Be, 1984; Caromel et al., 2014) and compare well with known speeds for other particles of comparable size and density (e.g. faecal pellets: table 3 of Iversen and Ploug, 2010; phytoplankton: fig. 1 of Smayda, 1971). However, it should be noted that our predicted speeds are higher than published values for five out of the seven foraminifera species for which direct comparisons are possible (Fig. S1). This could be due to our ability to observe enlarged models of sinking foraminifera more accurately compared with actual specimens, and the lack of control for wall effects in previous work, which would tend to underestimate sinking speeds. There could also be considerable natural variation, which our single specimen per species (excluding Sphaeroidinella dehiscens) does not capture.

Sedimentation of microscale plankton has been measured both in situ (e.g. Waniek et al., 2000) and in the laboratory (e.g. Smayda, 1971; Miklasz and Denny, 2010). By settling dense suspensions of microorganisms, these studies provided a population sinking rate (Bienfang, 1981) which could be 2–3 times lower than the settling velocity of an isolated particle in the typically dilute ocean (Miklasz and Denny, 2010). Other studies have, as in the present study, used enlarged models of microscale plankton to facilitate observations. Padisák et al. (2003) used handmade models of plankton to examine drag, but there was no attempt to accurately match Re. Holland (2010) used mechanical pencil leads as models of sinking diatom chains, keeping Re<1 in an improvement over Padisák et al. (2003). However, neither study calculated sinking velocity for real organisms. Our dynamic scaling approach ensured that we accurately recreate the fluid flows around settling organisms – a requirement for the correct prediction of sinking speed. We also improved on previous methodologies by effectively eliminating wall effects, basing our models on µCT scans, and using inexpensive cameras to observe natural sinking orientation.

By design, our dynamic scaling approach yields an interpolated C_D(Re) curve that describes the flow dynamics (and thus sinking speeds) that would occur if various fluid and/or particle parameters were varied, offering a degree of flexibility not seen in other studies. For example, phytoplankton blooms can increase both the density and viscosity of water due to exudates (Jenkinson et al., 2015), while increasing global temperatures have the opposite effect. The density and viscosity of seawater also naturally vary with latitude. Understanding how these variations affect sinking rates can offer insights into the evolutionary pressures on plankton. Our approach also allows us to isolate the effects of shape on sinking, even across species of widely varying size, density, etc., by comparing C_D of different species all hypothetically sinking at the same Re; a manuscript focused on such biological questions relating to foraminifera is currently in preparation. Differential settling speeds of foraminfera also have implications for nutrient cycling, paleoclimate reconstruction (Kucera, 2007) and the marine calcite budget (Schiebel, 2002).

Our method can easily be modified to study sedimenting particles operating at any Re, providing the system's Re range can be experimentally replicated. Other sinking marine particles include diatoms (Re≈10⁻² to 1; Botte et al., 2013) and radiolaria (Re≈10–200; Takahashi and Honjo, 1983), for which one could use digital models as we have in conjunction with a suitably viscous fluid (high viscosity silicone oil; see Table S1) to enable sufficiently large models to be produced (25 cm; see Table S1). The method can also be applied to terrestrial systems such as settling spores (Re≈50; e.g. Gómez-Noguez et al., 2016; Noblin et al., 2009) and dispersing seeds (Re≈10³; Osuki et al., 2017; Azuma and Yasuda, 1989), again by using 3D printed models based on (often existing) µCT data.

Whilst our method pertains to settling in a quiescent fluid, one could conduct similar experiments using a flume to calculate threshold resuspension velocity (i.e. the horizontal flow speed required to lift a particle off the substrate), important in the study of wind erosion and particle transport and deposition (Bloesch, 1995; Bagnold, 1971). Similarly, studying particles suspended in shear flow could be achieved using a treadmill-like device (e.g. Durham et al., 2009) or a Taylor–Couette apparatus (e.g. Karp-Boss and Jumars, 1998). While additional dimensionless groups beyond Re and C_D would need to be matched to achieve similitude in these systems, we hope that our study provides a starting point for the experimental study of these and other more complex problems.

Our thanks go to Michael Kucera for supplying samples of planktonic foraminifera. We thank Alex Blanke for preparing the samples and taking them to Hamburg. We also thank the rest of the Physical Ecology Lab (School of Life Sciences, University of Lincoln: Òscar Guadayol i Roig, Fouad El Baidouri, Mariona Segura, Jackson Sage and Sei Suzuki-Teller) for their support and help during the project. Our thanks go to Gregory Sutton and David Smith for their comments on a draft of the manuscript and we also thank Mark Denny, Matthew A. Reidenbach and one anonymous reviewer for their comments.