Estimation of sinking velocities using free-falling dynamically scaled models: foraminifera as a test case

The velocity of settling particles is an important determinant of distribution in extinct and extant species with passive dispersal mechanisms, such as plants, corals, and phytoplankton. Here we adapt dynamic scaling, borrowed from engineering, to determine settling velocities. Dynamic scaling leverages physical models with relevant dimensionless numbers matched to achieve similar dynamics to the original object. Previous studies have used flumes, wind tunnels, or towed models to examine fluid flows around objects with known velocities. Our novel application uses free-falling models to determine the unknown sinking velocities of planktonic foraminifera – organisms important to our understanding of the Earth’s current and historic climate. Using enlarged 3D printed models of microscopic foraminifera tests, sunk in viscous mineral oil to match their Reynolds numbers and drag coefficients, we predict sinking velocities of real tests in seawater. This method can be applied to study other settling particles such as plankton, spores, or seeds. Summary Statement We developed a novel method to determine the sinking velocities of biologically important microscale particles using 3D printed scale models.


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The transport of organisms and biologically derived particles through fluid environments strongly 24 influences their spatiotemporal distributions and ecology. In up to a third of terrestrial plants (Willson 25 et al., 1990), reproduction is achieved through passive movement of propagules (e.g., seeds) on the 26 wind. In aquatic environments, propagules of many sessile groups from corals (Jones et al., 2015) to 27 bivalves (Booth, 1983) are dispersed by ambient currents, eventually settling out of the water column 28 to their final locations. Furthermore, most dead aquatic organisms (from diatoms to whales) sink, 29 transporting nutrients to deeper water and contributing to long term storage of carbon (De La Rocha 30 & Passow, 2007). In the case of microfossils, sinking dynamics of the original organisms even 31 influences our reconstructions of the Earth's paleoclimate (Van Sebille et al., 2015). Crucially, the 32 horizontal distances over which all these biological entities are transported, and therefore their 33 shape -for which settling speed is the key unknown parameter -presents a unique challenge to 44 experimental design that we overcome in this work. 45 Engineering problems such as aircraft and submarine design often are approached using scaled-down 46 models in wind tunnels or flumes to examine fluid flows around the model and the resulting fluid 47 dynamic forces it is subjected to. To ensure that the behaviour of the model system is an accurate 48 representation of real life, similarity of relevant physical phenomena must be maintained between 49 the two. If certain dimensionless numbers (i.e., ratios of physical quantities such that all dimensional 50 units cancel) that describe the system are equal between the life-size original and the scaled-down 51 model, "similitude" is achieved and all parameters of interest (e.g., velocities and forces) will be 52 proportional between prototype and model (Zohuri, 2015). Intuitively, the model and real object must 53 be geometrically similar (i.e., have the same shape), so that the dimensionless ratio of any length 54 between model and original, ℎ ℎ ⁄ , is constant -this is the scale factor ( ) of the 55 model. Less obvious is the additional requirement of dynamic similarity, signifying that the ratios of 56 all relevant forces are constant. For completely immersed objects moving steadily through the 57 surrounding fluid, dynamic similarity is achieved by matching the Reynolds number ( ). 58 is a measure of the ratio of inertial to viscous forces in the flow (Batchelor, 2000; within a biological 59 context Vogel, 1994), and is typically defined as: 60 where is density of the fluid (kg m -3 ); is a characteristic length (m) of the object; is the 63 object's velocity (m s -1 ); and is the dynamic viscosity ( −2 , or ) of the fluid. In cases where 64 is large compared to , e.g. fish, birds, and whales ( ≈ 3×10 6 -3×10 9 , Vogel, 1994

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Similitude and Settling Theory 116 We assume that the size (i.e., -defined as the maximum length parallel to the settling direction, 117 -defined as the projected frontal area, and -the particle volume not including any fluid-filled 118 cavities), 3D shape ( , here treated as a categorical variable due to our consideration of arbitrarily 119 complex morphologies), and density ( ) of the original sinking particle are known, while the 120 sinking speed ( ) is unknown. The properties of the fluid surrounding the original particle (i.e. , 121 ) are also known, and our goal is to design experiments in which we sink a scaled-up model particle 122 in a working fluid of known and in order to determine the model particle's sedimentation 123 speed and, via similitude, of the original particle. While previous work (Berger et al., 1972 fluid (e.g., seawater of , ). While the fluid dynamics of flow around a particle of particular 132 shape can be considered theoretically over a range of , only the dynamics at and will 133 represent the operating point corresponding to the life size particle settling speed . 134 When a particle is sinking steadily at its terminal velocity, the sum of the external forces acting on the 135 particle is zero (Eqn 2); that is, the upward drag force ( , Eqn 3) and buoyant force ( , 136 Eqn 4) must balance the weight of the particle ( where we use the fact that for a model, = and = 2 . While one would also expect = 182 3 for 3D printed models, limitations of our 3D printer led to variation in that we overcame using 183 a more general empirical relationship between S and based on mass measurements -see 3D Printer 184 Limitations. Eqn 7 represents a constraining relationship between and for the sinking particle, 185 which we use to collect ( , ) experimental data points at several S. Once sufficient data are 186 collected, we can construct a new, empirical relationship (e.g., a cubic spline fit) between and 187 for a particular particle shape, which we term ( ). Finally, we can solve for the operating , 188 , and by finding the intersection point between the ℱ ( ) constraint curve specific to life-size 189 particles sinking in seawater (i.e., Eqn 7 with = 1 and , , ) and our empirical ( ) 190 spline curve valid for a particular particle shape moving steadily through any fluid. 191

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To construct an empirical ( ) curve for a particular test morphology, we started with 3D scans of 193 individual specimens (SI, Table 1). These data allowed us to easily fabricate scaled-up (scale factor ) 194 physical models of each specimen using a desktop 3D printer (SI 2). The models were then released in 195 a tank of mineral oil ( = 830 kg m -3 , = 0.022 Pa S) and imaged by two digital cameras oriented at 90 196 degrees to each other to record 3D trajectories (SI 3, SI Fig. 7). There were two primary difficulties 197 with these experiments: the first associated with the resolution of our 3D printer, and the second 3D Printer Limitations 200 Whilst in principle, the volume of a printed model should simply scale according to = 3 , due to 201 inherent limitations of the 3D printer as well as difficulty in removing excess resin from small models, 202 we found that this expectation was usually not satisfied, and weighing the models showed that 203 Fig. 3). Therefore, we estimated of each model by weighing on an Entris 204 224-1S mass balance (±0.001 g) and assuming was 1121.43 ± 13.73 kg m -3 , based on the 205 average mass of five 1 cm 3 cubes of printed resin. Furthermore, whenever a predicted value for at 206 a given scale factor was needed, i.e. in Eqn 7 (see Remaining iterations), we based this on cubic 207 spline interpolation of our ( ) data for existing models when sufficient data were available, with 208 extrapolation based on cubic scaling of ( ) if required (see SI Wall effects 218 At low , the effects of artificial walls in an experimental (or computational) system can be 219 nonintuitively large and lead to substantial errors if not accounted for (Vogel, 1994). Acting as an 220 additional source of drag, the walls several tens of particle diameters away can slow down a sinking 221 particle and increase its apparent drag coefficient. We designed our experiments to minimise wall 222 effects by using an 0.8 m diameter tank (SI Fig. 6)  Here, = ⁄ where is the diameter of the sinking particle and is the tank diameter; we take 231 = . While Eqn 9 is not exact, it substantially reduces the error otherwise incurred if one were to 232 neglect wall effects entirely. We applied this correction by taking any experimentally determined 233 to equal , and using ∞ estimated according to Eqn 9 for subsequent calculations as detailed 234 below. 235 Iterative approach  An empirical cubic spline curve ( ) can now be fitted (D'Errico, 2009) to these three initial (Re, 261 ) data points, constrained to be monotonically decreasing and concave up within the limits of the data 262 to match expectations for drag on objects at low to moderate . Three optimally spaced spline knots 263 were used since this yielded excellent fits to the data as the number of data points increased. These 264 details of the spline as well as its order (i.e., cubic vs linear) are somewhat arbitrary but we ensured 265 that our results were sufficiently converged as to be insensitive to them (see Remaining iterations). 2) the variation in calculated between the fitting of a linear spline and cubic spline was no 289 greater than 5%, and 290 3) the variation between the predicted and the closest experimentally measured was 291 less than 15%. 292 Through this method we calculated the sinking velocities of 30 species of planktonic foraminifera 293 (Table 1)

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Our basic methodology was first validated by 3D printing a series of spherical models (10 -20 mm in 299 diameter) for which the theoretical ( ) relationship is already well-known. In order to achieve low 300 density (and thus low sinking velocity and low ), these spheres were hollow and filled with oil via To quantify errors in our approach even more directly, we then considered hypothetical hollow 307 spherical particles with the same material density as foram tests and a range of sizes ( = 308 750 − 1150 μm, similar to the species we studied) settling in seawater. This size range corresponds 309 to = 12 − 27, the area where our ( ) curve is most divergent from ( ). We compared 310 predictions of the operating based on our empirical ( ) curve versus the theoretical ( ) 311 curve for spheres as outlined above, substituting Eqn 10 for ( ) in the latter case. Maximum 312 relative error in predicted was 11.5% at Re = 16 (corresponding to a sphere 860 µm in diameter) 313 while the minimum difference was 6.5% at Re = 27 (corresponding to a sphere of 1150 µm in diameter, 314 SI Fig. 5). This level of error is much smaller than the variation in we predicted across the 30 foram 315 species we investigated (Table 1). 316 There was little variation in the number of iterations required to reach convergence (mean 4, range 317 3-6, see Table 1), despite the morphological complexity of some species (e.g. Globigerinoides 318 fistulosus). We suspect the higher end of this range was due to these species having forms that were 319 particularly challenging to clean residual resin from, or the incomplete removal of air bubbles once 320 submerged in oil. In Fig. 4 we present an example of convergence of our method to the operating , 321

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Here we present a novel method of determining settling speed by leveraging dynamically scaled 339 models falling under gravity rather than being towed at a controlled speed. Applying our method to 340 foraminifera-inspired spherical particles, we predict settling speeds within 11.5% of theoretical 341 expectations (SI Fig. 5). Our predicted speeds of real foraminifera also fall within existing data for 14 342 species (SI Fig. 1, Fok well with known sinking speeds for other particles of comparable size and density (e.g. faecal pellets, 344 (Table 3, Iversen & Ploug, 2010), various phytoplankton (Fig. 1, Smayda, 1971)). 345 Key questions in biological oceanography revolve around the sedimentation of both living and dead 346 planktonic organisms. Sedimentation of microscale plankton has been measured both in situ (e.g. 347 Waniek, Koeve and Prien, 2000) and in the laboratory (e.g. Smayda, 1971; Miklasz and Denny, 2010). 348 By settling dense suspensions of microorganisms, these studies provided a population sinking rate 349 (Bienfang, 1981) which could be two to three times lower than the settling velocity of an isolated  Table 1) to enable sufficiently large models to be produced (SI Table 2). The method can also be 379 applied to terrestrial systems such as settling spores ( ≈ 50 e.g. Noblin, Yang and Dumais, 2009) and 380 dispersing seeds ( ≈ 10 3 Azuma and Yasuda, 1989), again by using 3D printed models based on (often 381 existing) µCT data. 382 Whilst our method pertains to settling in a quiescent fluid, it would be relatively simple to conduct 383 similar experiments using a flume to calculate threshold resuspension velocities (i.e. the horizontal 384 flow speed required to lift a particle off the substrate), important in the study of wind erosion and 385 particle transport and deposition (Bloesch, 1995;Bagnold, 1971   Van Sebille, E., Scussolini, P., Durgadoo, J. V., Peeters, F.J.C.C., Biastoch, A., Weijer, W., Turney, C., (√ 2 + 2 + ,1 2 + √ 2 + 2 + ,2 2 ) 671 Each model was sunk five times and a mean was calculated from these replicates. Replicates beyond 672 a threshold of ±5% of the median sinking velocity were discarded from this average. Each model was 673 dropped one additional time and photographed using a Canon 1200D DSLR camera (Tokyo, Japan) 674 mounted on a tripod close to the tank, to obtain high resolution (18 megapixels) images which were 675 used to determine model orientation (and thus and ) during settling (SI Fig. 4). 676 677 SI Fig. 6. Schematic of the tank and model retrieval system. All measurements in mm.