Suspension feeders rely on filter structures of a variety of forms to capture food particles. Much effort has been devoted to examining the operation of such filters, but mechanistic evaluations have generally represented filter elements with artificially stiff cylinders. We extended this previous work to investigate how bending affects the function of flexible cylindrical filter elements. Scaled models of filters were constructed from materials with elastic moduli comparable to material stiffnesses of invertebrate appendages (1–177 GPa). These models were mounted on a sled to mimic the protrusion of filters away from an animal's body or from the substratum, and were towed through a vat of syrup to generate relative fluid motion at low Reynolds numbers (Re <10–3, based on cylinder diameter and tow speed). Flow between filter elements was quantified at multiple positions along their lengths, and a hydrodynamic index of filter performance (`leakiness') was calculated. Leakiness generally increased with cylinder Re and distance from the filter base. At higher flexibilities, however, streamwise bending and lateral narrowing of the filter reduced projected area and slowed flow between elements. This effect decreased leakiness and reversed the otherwise monotonic trend for increased leakiness at higher cylinder Re. Additional experiments showed that filters composed of stouter elements were less susceptible to bending but experienced lower leakiness because of their reduced ability to transcend boundary layers formed over surfaces to which they attached. These findings indicate that filter bending can strongly alter the performance of particle capture apparatus in suspension feeders.
Suspension feeders provide an important link in both benthic and pelagic marine food webs by capturing particulate organic matter from the water column and by serving as food themselves for many consumers (e.g. Gili and Coma, 1998). Recognizing their ecological significance, numerous researchers have examined flow effects on particle capture by these animals (e.g. Koehl and Strickler, 1981; Leonard et al., 1988; Hunter, 1989; Patterson, 1991; Lacoursière and Craig,1993; Larsson and Jonsson,2006) (reviewed by Wildish and Kristmanson, 1997). The traditional point of departure for understanding the mechanics of particle capture has been hydrosol filtration theory, commonly applied to idealized collector arrays composed of perfectly rigid cylinders (Rubenstein and Koehl,1977; LaBarbera,1984; Shimeta and Jumars,1991). A key index emerging from this conceptualization is`leakiness', a measure of filter efficacy quantified as the proportion of water passing through an externally held feeding array, relative to the flow impinging on it (Cheer and Koehl,1987). Based on many studies (e.g. Hansen and Tiselius, 1992; Loudon et al., 1994; Koehl, 1995; Mead and Koehl, 2000), much is now known about how leakiness varies as a function of geometry and collector Reynolds number (Re=Udρ/μ; a measure of the relative importance of inertial and viscous forces in a flow past a collector, where U is an appropriate velocity scale, d is the collector diameter, ρ is the density and μ is the dynamic viscosity).
Despite these advances, notable gaps remain in understanding even the most basic features of suspension feeding. One important omission has been a systematic appreciation for the consequences of deformation. The structural flexibility of appendages of real animals enables them to deflect in flow,creating deviations from the idealized case. For example, filter bending can reduce imposed forces and expand the range of flow conditions under which feeding is possible (Patterson,1984; Harvell and LaBarbera,1985; Sponaugle and LaBarbera,1991). Deflection can also have negative effects because the area of a filter array facing flow is diminished by bending.
To begin to isolate the consequences of structural flexibility for filter function, we examined leakiness through simple but deformable physical models composed of multiple cylindrical elements. These artificial filters extended away from a solid surface, mimicking the situation faced by animals that hold filter arrays away from the substratum or from their own support structures. Because we wanted to distinguish effects of filter bending from other factors,we focused on relatively low cylinder Re (∼10–5to ∼10–3, with U set equal to the free-stream velocity outside local boundary layers adjacent to animal bodies or the seafloor). Reynolds numbers of suspension feeders extend across a much broader range (<10–6 to >10)(Vogel, 1994), but the higher end of this range is characterized by leakiness values that depend strongly on hydrodynamic processes that interact with other factors(Koehl, 1995; Loudon and Koehl, 2000). Our results, derived for the more easily interpreted range of lower Reconditions, indicate clearly that the bending of feeding appendages can strongly influence filter leakiness and the capacity for particle capture. Effects of collector deflection at low Re may apply to a variety of suspension-feeding animals that experience flows as slow as 0.1 mm s–1 and possess small filtering structures (∼0.1–10μm diameter), including flagellated protozoans, cladocerans, bryozoans and serpulid polychaetes, among others (e.g. Turner et al., 1988; Scardino et al., 2008).
MATERIALS AND METHODS
|Study .||Elastic modulus (GPa) .||Tank size (m3) .||Re .||Number of cylinders .||Gap to diameter ratio .||Length to diameter ratio .|
|Hansen and Tiselius (1992||∼160*||0.005, 0.012||0.005–40||2–12||1–80||25–108|
|Loudon et al. (1994)||160*||0.13||10–4–1||2||2.5–30||∞ (entire depth)|
|Mead and Koehl (2000)||>180*||0.13||0.13–1.68||3 in each of 4 rows†||1†||26–32|
|Study .||Elastic modulus (GPa) .||Tank size (m3) .||Re .||Number of cylinders .||Gap to diameter ratio .||Length to diameter ratio .|
|Hansen and Tiselius (1992||∼160*||0.005, 0.012||0.005–40||2–12||1–80||25–108|
|Loudon et al. (1994)||160*||0.13||10–4–1||2||2.5–30||∞ (entire depth)|
|Mead and Koehl (2000)||>180*||0.13||0.13–1.68||3 in each of 4 rows†||1†||26–32|
Estimated from known engineering properties of cylinder materials used by the author
Parallel rows of cylinders were spaced at distances of either 7.5 or 16 cylinder diameters
In more complex geometries beyond solitary cylinders, including multiple cylinders in arrays, additional dimensionless parameters become important. They include the ratio of gap width between cylinders to cylinder diameter,the ratio of the width of a multiple-cylinder array to cylinder diameter, and(in the case of laboratory experiments) ratios of tank or other apparatus dimensions to cylinder diameter. In situations where cylinders protrude from rough surfaces into turbulent regions of benthic boundary layers, two further parameters arise (the ratio of roughness height to cylinder diameter, and another Reynolds number based on U*, the so-called`friction velocity') (Eckman and Nowell,1984).
We focused on a subset of the full parameter space spanned by the governing variables identified above. For each of three values of collector Rebetween approximately 10–5 and 10–3, we tested cylinder aspect ratios, L/d, of 50 and 100. Given the dimensional analysis described above, the use of Reynolds numbers based on free-stream velocities, rather than `face velocities' incident on the filter array, simplified the approach. However, we acknowledge that this convention[also used by other researchers (e.g. Cheer and Koehl, 1987)] interferes in some circumstances with easy interpretation. Values of E have not been measured for intact suspension-feeding appendages, yet natural polymer composites that make up these structures have elastic moduli on the order of 0.1–100 GPa(Wegst and Ashby, 2004). In aquatic environments characterized by relative velocities of 10–4–1 m s–1, it follows that actual values of B range from approximately 105 to 1016 for most suspension feeders, but we restricted experiments to the upper end of this range to avoid a disproportionate emphasis on consequences of extreme bending. We held the parameter x/dconstant in all experiments. We considered only one ratio of gap width to cylinder diameter, one ratio of array width to cylinder diameter, and single ratios of apparatus dimensions to cylinder diameter. These restrictions obviously limit our ability to generalize across the full breadth of conditions occurring in nature, so we focused on phenomena that are likely to be general across a wide choice of cylinder arrays, emphasizing qualitative results that ensued from the quantitative measures we undertook using specific filter sizes and geometries.
Low Reynolds number tank
The rectangular tank (Fig. 1) we used for experiments was filled with 416 l of high-viscosity corn syrup (ADM LSI Specialty Products, Liquidose 434, San Leandro, CA, USA). The tank was 1.25 m long, 0.57 m wide and 0.61 m high (larger than chambers used in previous studies) which reduced wall effects as much as was feasible. Although it was not possible to entirely eliminate such effects(Vogel, 1994), filter structures inherently operate in close proximity to other solid surfaces (e.g. bodies, neighbors or substratum) and our intention was to compare filter performance in the context of biologically reasonable geometries. Dynamic viscosity of the syrup was measured with a digital viscometer (Brookfield Engineering Labs, Model HBDV-E, Middleboro, MA, USA) and density was calculated by weighing a known volume. The syrup was thoroughly mixed 10 days prior to experiments so that large entrained air bubbles, which would have interfered with model visualization, had time to float to the surface and be removed. A nearly uniform suspension of small bubbles [mean (± s.d.)diameter=123 (±33) μm; N=100] remained trapped in the syrup, rising only exceptionally slowly (<2 mm day–1) and thereby serving as tracer particles for flow visualization (see Filter leakiness, below).
Model filter arrays
Scale models of organism filter arrays were constructed using four parallel and vertically oriented, cylindrical rods spaced evenly in a row perpendicular to the direction of motion. Cylinders were inserted into predrilled holes in the central region of a transparent acrylic sled (230 mm long, 160 mm wide, 5 mm thick) that rested on the fluid surface. The longest axis of the sled was aligned with the direction of motion, and the leading and trailing edges of the sled were curved upward to facilitate travel across the fluid surface. Filter arrays were centered on the sled at a distance of 56 mm (∼35 cylinder diameters) from the leading edge, and all cylinders had a diameter(d) of 1.59 mm. The cylinders extended either 159 or 79.5 mm into the flow (referred to as `slender' and `stout' arrays, respectively, where L is cylinder length). The gap between adjacent cylinder edges was 7.95 mm (5d), yielding a total array width of 30.2 mm (19d)for all filters.
Each array contained cylinders composed of one of the following materials:stainless steel (type 316), brass (alloy 260), Garolite-G10 (glass-cloth laminate plastic), Garolite-XX (paper-based laminate plastic), or nylon (6/6). Materials were selected to represent the natural range of structural flexibility found in materials composing biological fibers, hairs and setae of real animals (Wegst and Ashby,2004). The elastic modulus (E) of each model material was measured using three-point bending tests conducted on an Instron materials testing device (Model 3345, Norwood, MA, USA).
Relative fluid motion through and around the filter arrays was generated by towing the model filters through the syrup using a micro-stepping motor(Oriental Motor Company, Model RK564AA, Torrance, CA, USA) driven by motion-control software (National Instruments, NI-Motion v.1.2). Filters were towed at each of three velocities (U=0.3, 3.0, or 30 mm s–1). After filter elements reached a steady state, the filters were photographed through the walls of the tank using a 35 mm camera(Nikon D70s, 60 mm lens). The spatial coordinates of each filter array (in both side and end views) were recorded under conditions of no motion and at all tow velocities, in order to determine the extent of streamwise bending of the array, lateral compression or narrowing of the array, and the resultant projected filter area as viewed from directly upstream.
Fluid motion between individual filter elements was measured by quantifying displacements of suspended tracer particles relative to the filter over time,a technique known as particle tracking velocimetry. Overhead images were collected by mounting the 35 mm camera directly above the sled and focusing it on a plane located at a known distance below the base of the filter array. Trials were run in the dark and fluid motion at the target height was illuminated with a light sheet (5–7 mm thick) produced by shining a synchronized flash (Nikon Speedlight SB-800) through a 1-mm slit at the upstream end of the tank. As before, the sled was towed at the chosen test velocity until filter elements reached steady-state deflection. Four successive images then were collected at regular intervals during each trial to allow sufficient particle displacement for velocity determination. In experiments with slender filter elements (L/d=100), images were collected at 10, 30, 50, 70, 90, 110, 130 and 150 mm from the filter base. In experiments with stout filter elements (L/d=50),images were collected at 10, 30, 50 and 70 mm from the filter base. The order of the three tow velocities was determined randomly for each light sheet position. Replicate trials (three per condition) were collected on different days and this same protocol was applied to filter arrays composed of each of the five materials examined in the study, resulting in 540 total experimental trials.
Images were calibrated to convert from pixels to actual dimensions for each combination of filter material, aspect ratio and measurement height (ImageJ 1.37). Coordinates of cylinder locations were recorded and particles for tracking were selected in frame one from those located within in the narrow sector between adjacent cylinders (Fig. 2). Flow past highly bent cylinders was sometimes deflected vertically, carrying particles out of the imaging plane. To avoid positional ambiguity associated with such vertical movements, only particles that were illuminated through all four frames were tracked and used in estimating velocity. Total horizontal (streamwise) displacement was calculated for each particle and divided by the time interval between the first and fourth frames to obtain particle velocities.
Two distinct leakiness quantities were calculated for each model filter. Estimates of `planar' leakiness were obtained at each of multiple discrete distances from the filter base by dividing (at each distance) the spatially integrated velocity profile across the width of the array by the product of array width (without deformation) and free-stream velocity. This proportionality equaled the amount of fluid that actually passed through the filter in a plane at a given distance from the base, normalized by the amount of fluid that would have passed through the same space (i.e. the region spanned by the undeformed filter) if the filter were absent. Replicate estimates of planar leakiness were averaged for each filter material and within each combination of velocity and vertical position. Taking each set of planar leakiness values (one value for each of a series of distances from the base of the filter array), the `whole-model' leakiness was then computed for the entire array in order to account for the proximal-to-distal variation in planar leakiness along the length of the filter. Note that the whole-model leakiness was estimated by averaging the values of planar leakiness across all measurement distances out to L, including those beyond the reduced height of the bent array. This convention effectively normalized the amount of fluid passing through the bent array by the amount of fluid that would, in the absence of the filter, pass through the region spanned by it in its undeflected configuration (as opposed to the amount of fluid that would, in the absence of the filter, pass through the smaller region spanned by it in its bent configuration). Such values of whole-model leakiness thereby captured effects of both decreases in fluid passage through the array and array deformation, when providing an index of overall filter performance.
Fluid and material properties
Syrup density remained nearly constant throughout the study (mean ±s.d.; ρ=1456±27 kg m–3; N=9), but unavoidable changes in room temperature caused modest variation in dynamic viscosity [μ=118±19 Ns m–2 (N=13) for trials with arrays of slender filter elements (L/d=100);μ=161±5 Ns m–2 (N=10) for trials with arrays of stout filter elements (L/d=50)]. These conditions resulted in average Re values of 5.1×10–6,5.1×10–5 and 5.1×10–4, although for simplicity's sake these experimental conditions are referenced using order-of-magnitude designations (i.e. Re of 10–5,10–4 and 10–3). The elastic moduli of the cylinder materials (mean ± s.d.; N=4 of each material) spanned a more than 100-fold range in stiffness (stainless steel, 176.50±1.97 GPa; brass, 127.61±0.14 GPa; Garolite-G10, 21.03±1.25 GPa;Garolite-XX, 8.00±0.07 GPa; nylon, 1.37±0.05 GPa) and yielded average B values of 1.0×109 to 1.4×1015 across conditions.
Although Eqns 1, 2, 3 dictate that flow patterns in our experiments vary formally with Re, L/d and B(recall that we held x/d constant), the parameter Bincludes components associated with both organism material properties(E) and free-stream (=tow) velocity (U). These two variables can be confusing to manipulate mentally at the same time, which led us to depict filtration consequences of flexibility as a function of absolute stiffness (E) instead of normalized stiffness (although values of B are available by straightforward substitution). The resultant data demonstrated that model filter arrays of a given modulus exhibited greater bending as tow velocities increased (Fig. 3). Over the range of flow conditions tested, the tip angle of deflection [=arctan(δ/L)] of the cylinders averaged across all four filter elements in an array varied from 0.1 to 2.9 deg. for the most rigid material (E∼177 GPa), and from 3.7 to 70.4 deg. for the most flexible material (E∼1 GPa). This streamwise bending decreased vertical filter height, as viewed from upstream, by as much as 0.5%,1.1%, 7.1%, 21.8% or 68.2%, for filters ranging from least to most flexible. The oblique orientation of deflected filter elements also changed the cross-sectional shape of elements penetrating a given horizontal plane (e.g. Fig. 2E), increasing the amount of element surface bordering the planar flow. The two outer filter elements bent farther than the inner elements, probably because the outer elements experienced faster relative velocities and thus higher drag (e.g. Fig. 2E). The latter differential bending was detectable in many of the filter materials and flow conditions but was maximal in the flexible arrays exposed to higher-velocity flows.
Array width also decreased as a function of Re, distance from the base of the filter (Fig. 4),and as values of elastic modulus declined. This cross-stream narrowing of an array was expressed through changes in gap width between individual elements and a corresponding increase in the proportion of projected area occluded by the elements themselves. In one case, the outer two filter elements bent inward enough to make contact with the central two elements, thereby minimizing flow through the outer gaps of the array in the region of contact(Fig. 4C; E∼8 GPa,distance from the filter base=0.7 L).
The two effects outlined above, streamwise bending and cross-stream narrowing, together caused a substantial decrease in total projected filter area (Fig. 5). Relative to stationary arrays, filter deformation reduced projected area by as much as 1.0, 4.0, 20.7, 46.0 or 72.1%, for models ranging from least to most flexible. Filter area was similar among arrays composed of all five materials when flow velocities and Re were lowest. In the intermediate Recondition, only the most flexible array (E∼1 GPa) deformed appreciably, and in the highest Re condition the observed reductions in filter area were inversely correlated with material stiffness. For the most flexible array (E ∼1 GPa) in the highest Re condition,streamwise bending dominated the reduction in projected area because extreme bending restricted the vertical filter span to only three measurement distances from the base (e.g. Fig. 3C; Fig. 4C).
Planar leakiness averaged across the width (three gaps) of each array varied systematically with distance from the array base, Re, and elastic modulus (Fig. 6). Maximum planar leakiness associated with any of the test conditions was only 0.17 because of pronounced velocity retardation in the boundary layer over the sled and substantial flow diversion around the filter arrays, as would be expected for low Re conditions. All filters exhibited comparable leakiness for planes immediately beneath the sled where the local angle of cylinder deflection was negligible. The general increase in leakiness with distance from the array base was attributed to the velocity gradient established over the sled, which led to faster relative flow speeds in planes farther from the sled.
For the most rigid filter arrays (E∼128, 177 GPa), filter leakiness was dependent on current speed, with the greatest leakiness occurring in the highest Re flow(Fig. 6A,B). This positive trend in planar leakiness with increasing Re was reversed for filters composed of the other three materials (E∼1, 8, 21). In these more flexible arrays, leakiness was reduced in fast flows. This reversal was most apparent in the highest Re condition where strong streamwise bending prevented filtration far from the filter base in the two most flexible arrays(E∼1, 8), producing leakiness values that were effectively zero for the most distal measurement planes(Fig. 6D,E).
Further examination of planar leakiness revealed a subtler effect of cylinder bending that was independent of changes in filter area. For any given filter array and distance from the filter base, one might expect that variations in planar leakiness would correspond to variations in array width. This expectation held for model filters that experienced relatively little deflection, but when streamwise bending was more pronounced, observed reductions in planar leakiness were modified. We evaluated this trend by calculating a `renormalized' value of planar leakiness that controlled for the influence of changing array width. For every combination of filter material,tow velocity and distance from the base, values of renormalized planar leakiness were obtained by dividing the spatially integrated velocity profile across the width of the array by the product of narrowed array width (i.e. accounting for deformation) and free-stream velocity. The resultant estimates of renormalized planar leakiness were then plotted as a function of distance from the array base for all filter materials and Re conditions(Fig. 7). If changes in array width alone had accounted for the previously observed reductions in planar leakiness through more flexible filters(Fig. 6), then renormalized planar leakiness values should have been identical for all filters operating at a given Re, regardless of their material stiffness. This default outcome, however, did not occur. Instead, additional reductions in water flow between filter elements arose in association with the oblique orientation of deflected elements, probably as a result of the combination of vertical diversion of flow around a bent filter array (resulting in lower horizontal velocities impinging on the filter) and increased resistance to flow through the filter due to more element surface bordering each gap (cf. elliptical cross sections in Fig. 2E).
Effects of element flexibility on renormalized planar leakiness also appeared to intensify with increasing Re and filter deflection(Fig. 7B,C). However, careful inspection revealed an experimental artifact associated with extreme levels of element bending due to our method of flow visualization. Beyond a certain distance from the filter base, the substantial reductions in renormalized planar leakiness associated with filter deformation became relatively less pronounced. This shift can be explained by the observation that bent cylinders within an array did not stabilize at precisely the same height. Particularly at large distances from the base of a flexible array, bending of the outer elements exceeded that of the inner ones (e.g. Fig. 2E) and resulted in vertical separation of elements that caused the absolute distance between adjacent elements to become greater than the observed (horizontal) separation when looking straight down on the filter array. This vertical separation of filter elements changed the three-dimensional flow patterns such that measured fluid motion through the `observed gap' became greater than expected.
Effects of aspect ratio
Additional tests using arrays of shorter and proportionally thicker(stouter) filter elements (L/d=50) of the same materials showed a similar but less pronounced pattern than the slender arrays(Fig. 8). Maximum planar leakiness (through the most rigid filters) was 0.13, representing a 14%reduction from the maximum leakiness through the slender arrays. Planar leakiness near the base was similar among model filters (∼0.03) but the increase in leakiness with distance from the base was suppressed by filter deformation, although this effect was weaker than that observed for the slender arrays. Relative to average maximum leakiness through the rigid filters, leakiness at the most distal measurement plane decreased by 16% and 11% for stout arrays in which E∼8 and 21 GPa, respectively,representing a milder consequence of bending than was observed in slender arrays (e.g. Fig. 6C,D). In the highest Re, the most flexible stout array (E∼1 GPa)deformed enough to prevent filtration far from the filter base, consequently reducing leakiness relative to the lower Re conditions(Fig. 8E), but again this reduction in leakiness was far less than that in slender arrays where more than 50% of the filter span was effectively eliminated by bending(Fig. 6E).
Whole-model leakiness, computed by integrating planar leakiness across the full height of the filter array (Fig. 9), illustrated the overall effect of structural flexibility for filter function. One critical trend was the different behavior of rigid versus flexible filters, especially for the slender arrays. In general, whole-model leakiness of rigid filters (E∼128, 177)increased with Re. In our study, average whole-model leakiness through the most rigid arrays of slender elements (L/d=100)increased by 24% between Re of 10–4 and 10–3, compared to an increase of only 5% between Reof 10–5 and 10–4(Fig. 9A). In marked contrast,the three more flexible arrays exhibited the opposite trend in whole-model leakiness. For instance, the most flexible slender array (E∼1)exhibited a dramatic, 41% decline in leakiness between Re of 10–5 and 10–4 and an additional decrease of 60% between Re of 10–4 and 10–3. The effect was similar, although somewhat muted, in the second most flexible(E∼8) and intermediately flexible (E∼21) arrays. These filter arrays experienced decreases in leakiness of 55% and 10%,respectively, between the intermediate and high Re conditions. The most pronounced reductions in leakiness resulted primarily from effective shortening of the filters, which led to zero filtration at fixed distances far from the filter base in the two most flexible arrays(Fig. 6D,E).
A similar, but less dramatic pattern also arose in filters composed of stouter elements (L/d=50; Fig. 9B). Whole-model leakiness of the most flexible stout array showed a 28% decrease across the range of Re conditions examined. Importantly, even though flexibility-induced reductions in leakiness with increased Re were smaller in stout arrays as compared to slender arrays, stout arrays still often processed less fluid (i.e. had lower overall leakiness). This trend reflects the commonly(but not universally) poorer filtering abilities of stout arrays, most apparent at larger Re in the rigid-element data. For instance,average whole-model leakiness of the two most rigid filters(E∼128, 177) in the highest Re condition was 0.08 for stout arrays versus almost 0.12 for slender arrays. Given that cylinder diameters, gap widths, array width, and fraction of projected area occluded by filter elements were equivalent among the most rigid filter arrays, observed reductions in leakiness through the stout arrays can be fully attributed to their lower position in the boundary layer formed over the tow sled.
Results indicate that flexible filter arrays process less water than more rigid arrays (Fig. 9). This decrease in leakiness derives from changes to projected filter area (Figs 3, 4, 5) and reduced flow through deflected arrays (Fig. 7). The reduction in filter area is due to both streamwise bending and cross-stream narrowing, but the influence of bending dominates under conditions of extreme deflection. Structural deformation also reduces the degree to which the filter extends through the boundary layer and it leads to slower velocities impinging on the filter.
Patterns of whole-model (integrated) leakiness also demonstrate that filter flexibility can alter the relationship between filter leakiness and Re, particularly in arrays composed of slender elements. Prior studies of leakiness through rigid filter arrays have found that leakiness increases gradually over Re from 10–5 to 10–3, in contrast to filters operating at slightly higher Re where leakiness is very sensitive to Re(Cheer and Koehl, 1987; Koehl, 1995). However, data presented here suggest that deformation of feeding appendages may incur a cost of reduced leakiness across a range of low Re, potentially altering expectations of when critical transitions in filter function may occur. Our results also suggest the need to re-evaluate the notion that slender (as opposed to stout) elements in filter arrays always ensure greater leakiness,given that this trend may disappear in the presence of substantial deformation(compare case for E∼1 GPa at the highest Re condition in Fig. 9A,B). Such neglected aspects of filter function may require some adjustment to existing perceptions regarding linkages between morphology and ecology in suspension feeders.
Although strong trends are apparent in the results above, some care should also be applied in their interpretation. Findings are relevant to low-Re conditions representative of suspension feeders on the small end of the size spectrum, and quantitative values underlying the overall qualitative patterns may apply rigorously only to specific geometries. Examples of organisms that operate at Re ∼10–5whose filter structures resemble those studied here include nanoplanktonic flagellates (e.g. Actinomonas mirabilis; L/d∼60; g/d ∼5; x/d ∼20)(Fenchel, 1982). Suspension feeders that function at Re ∼10–4 include cladocerans (e.g. Penilia avirostris, with L/d∼45; g/d ∼6; x/d ∼15)(Turner et al., 1988). There are many organisms that feed at Re ∼10–3 and use filters analogous in shape to those of this study (e.g. Euphausia superba, with L/d ∼ 60; g/d ∼4; x/d ∼10)(McClatchie and Boyd,1983).
Complete effects of reduced filter area also may not be evident in our analysis because flow was measured though a finite number of horizontal planes. We were therefore forced to assume that leakiness through interior portions of arrays where flow was not quantified (i.e. regions between measurement heights) could be reasonably approximated by interpolation. However, contributions to leakiness from the extreme upper, and sometimes lower, portions of each filter were not represented. Material imperfections in individual filter elements also may have contributed to variability in the leakiness calculations, as they occasionally caused slight non-uniformities in bending. These inconsistencies likely resulted from asymmetries introduced during manufacture of the model elements, although analogous effects could also play a role in the bending trajectories of actual animal feeding appendages. We further note that although observations of invertebrate suspension feeders reveal that bending of their particle capture apparatus is common, quantitative evaluations of the moduli of these structures are generally unavailable. For our purposes, we assumed that the stiffnesses of suspension-feeding structures are similar to those of the `building-block'materials from which they are composed, and for which there are data. For example, skeletal components such as chitin and keratin have elastic moduli of 1–100 GPa, and structures reinforced with calcium carbonate have elastic moduli less than 100 GPa (Wegst and Ashby,2004). Assuming stiffnesses across this range, the materials we employed in our experiments corresponded to biologically relevant ranges of nondimensional stiffness (B). However, it is also possible that some suspension feeders produce particle capture apparatuses that diverge from values of B used in our experiments.
Filters in the natural environment
Leakiness patterns presented here reveal new insights into the function of filters operating in steady flows at very low Re (e.g. in pelagic environments or in tidal currents where changes occur over timescales of hours). Our findings may be less applicable to flows associated with rapidly evolving waves or turbulence, such as those found in intertidal or shallow subtidal habitats (Gaylord,1999; Gaylord,2000; Gaylord,2008). In these latter environments, filter deformation and its implications for particle capture could be either suppressed or intensified depending on the combined effect of fluid reversals and vortex attachment or shedding. During periods of accelerating or decelerating flow, for example,the bending trajectories of even moderately flexible filters might lag behind flow reversals and thus damp fluctuations in relative velocities. Some microhabitats where benthic suspension feeders settle will also experience velocity bursts as the effects of turbulent eddies penetrate all the way to the seafloor (Cantwell, 1981; Dade, 1993). Individuals living in topographical microhabitats where velocity transients are common could experience disproportionate consequences of filter deflection. We also note that the bending of biological filters may influence the function of array-like structures in organisms beyond marine suspension feeders. Low-Re filters are employed for respiration, propulsion and olfaction in a wide variety of freshwater and terrestrial invertebrates (see e.g. Loudon and Koehl, 2000; Sunada et al., 2002; Barta and Weihs, 2006). Deformation-induced effects on leakiness probably play paramount roles in these organisms as well.
Our results concern leakiness, and not particle encounter or capture (e.g. Palmer et al., 2004), and therefore do not address the full suite of factors driving food acquisition by mechanisms of hydrosol filtration. However, fluid transmission through a filter array is one of the fundamental gatekeepers for particle capture. Here,we show that structural flexibility can compromise leakiness across a range of low Re conditions. Detrimental effects of filter deformation on leakiness generally increase with Re and are offset in some cases for stouter filter elements. Such consequences of material properties and element shape for filter leakiness are likely to have important implications for suspension feeders of a variety of morphologies. Future studies should examine effects of flexibility across a greater spectrum of filter geometries and at higher Re, and should examine other steps in the feeding process(encounter and retention). Such work will facilitate the development of more comprehensive predictions of how food acquisition varies with collector morphology, flow conditions, and particle size and concentration.
LIST OF SYMBOLS
We are grateful to P. Jumars and an anonymous reviewer for feedback on previous versions of the paper. M. Petty assisted with design and construction of the low Reynolds number tank and tow sled, and enthusiastically helped produce the thinnest possible light sheet. This research was funded by NSF grants OCE-021447 and OCE-0523870, and the University of California Marine Council Coastal Environmental Quality Initiative grant 04-T-CEQI-08-0048. Contribution 2432 Bodega Marine Laboratory, University of California at Davis.