The hydrodynamics of contact of a marine larva, Bugula neritina, with a cylinder

Contact of a marine invertebrate larva with an underwater surface necessarily precedes its attachment to the surface. However, not all larvae that contact a substrate attach to it. Therefore, the probability of contact represents the upper bound of the probability of settlement. The probability of contact is a quantitative characteristic of settlement, which is of great interest in marine biology, particularly if attachment follows the first contact event (Abelson and Denny, 1997; Mullineaux and Butman, 1991; Mullineaux and Garland, 1993).

It is common to distinguish between the settlement of larvae on substrates of infinite extent and settlement on bodies of finite size. Because of the wide variety of larval forms and collector types, it is also common to observe settlement of specific larvae (e.g. bryozoan Bugula neritina) on relatively simple geometric forms, such as plates (Mullineaux and Butman, 1991; Mullineaux and Garland, 1993; Perkol-Finkel et al., 2008), cylinders (Harvey and Bourget, 1997; Rittschof et al., 2007) or inner sides of tubes (Crisp, 1955; Qian et al., 1999; Qian et al., 2000). Here, we studied the contact of B. neritina larvae with a long vertical cylindrical collector.

Most natural larval collectors are covered by microbial films (e.g. Dexter, 1979) or biofilms (e.g. Maki et al., 1989). The effect of microbial films and biofilms on bryozoan larval settlement has been observed both in the laboratory and under natural conditions (Brancato and Woollacott, 1982; Woollacott, 1984; Woollacott et al., 1989; Maki et al., 1989; Callow and Fletcher, 1995; Bryan et al., 1997). Generally, bryozoans are relatively indiscriminate settlers that may also settle on clean surfaces (Ryland, 1976; Dahms et al., 2004; Qian et al., 1999; Qian et al., 2000). We therefore used in our experiments a clean cylindrical collector that does not induce specific cues.

Chemical or physical cues play a central role in the behavioural biotic approach to larval settlement. In this approach a larva is attracted to a collector by cues and deliberately moves toward the collector. In an alternative mechanistic approach to larval settlement, a larva moves in the sea current as a drifter and collides with the collector as a passive particle. The issue of passive versus active contact has been intensively discussed in previous studies (Abelson et al., 1994; Butman, 1987; Butman et al., 1988; Harvey and Bourget, 1997; Harvey et al., 1995; Mullineaux and Butman, 1991; Mullineaux and Garland, 1993; Palmer et al., 2004).

However, the rich variety of models of larval contact with collectors cannot be described solely in terms of the antonyms ‘active–passive’. Consider, for instance, a realistic scenario of a swimming larva that is not aware of a collector. A swimming larva is active by definition, but in the absence of biotic factors influencing its contact with a collector, the larva moves as a mechanical object, i.e. as a microswimmer (e.g. Kirbøe, 2008). Nonetheless, the hydrodynamics and dynamics of such a microswimmer can be rather complex and difficult to describe in detail. Therefore, mathematical modelling of the motion of a larva as a mechanical object is inevitably associated with considerable simplifications, which should, however, retain the most relevant problem parameters, such as the Reynolds number of the cylinder (Re_c) and the Stokes number (St) of the particle–cylinder hydrodynamic system (Fuchs, 1964; Friedlander, 1977).

The Reynolds number Re_c=ρ_fU_∞D_c/μ represents the ratio of the inertial and viscous forces acting on a cylinder. It depends on the fluid density ρ_f, its viscosity μ, the flow velocity far from the cylinder U_∞ and the diameter of the cylinder D_c. The Stokes number represents the ratio of inertial and viscous forces acting on a particle that moves in the velocity field of the cylinder. The Stokes number depends on the parameters that determine Re_c and additionally on the parameters that determine a particle's inertia, its characteristic size d_p and its mean density ρ_p. A measure of the ratio of inertial and viscous forces acting on a particle is its stopping distance, l_p=ρ_p>d²_pU_∞/18μ, the distance at which a particle that starts its motion in a stagnant fluid with speed U_∞ will be stopped by the drag force exerted on the particle by the fluid (Fuchs, 1964). The dimensionless Stokes number is the ratio of the stopping distance of a particle to the characteristic size of the collector, St=l_p/D_c (Fuchs, 1964). Whereas the shape of a collector and its Reynolds number determine the collector's streamlines, the Stokes number characterises the degree of deviation of an inertial particle from the streamlines. The lower the Stokes number, the more a particle is ‘embedded’ in the fluid. Low-inertia particles (St<~0.1) follow streamlines rather closely and can be considered as inertialess particles, which follow streamlines exactly (Fuchs, 1964; Friedlander, 1977).

In contrast with passive particles, even an inertialess self-propelled microswimmer does not follow streamlines exactly. Zilman and colleagues (Zilman et al., 2008) theoretically studied the motion of a three-dimensional spherical microswimmer moving in a linear shear flow, in a channel flow and in a Poiseuille tube flow and calculated the probability of contact of the microswimmer with the walls bounding the flows. Crowdy and Samson (Crowdy and Samson, 2011) and Zöttl and Stark (Zöttl and Stark, 2012) studied trajectories of a two-dimensional and three-dimensional microswimmer moving in linear and Poiseuille shear flows, and took into account not only the re-orientation effect reported by Zilman and colleagues (Zilman et al., 2008) but also the direct hydrodynamic interaction of the microswimmer with a plane substrate.

In this work, we considered a previously unstudied problem, the motion of a low-inertia microswimmer (St<<1) in the velocity field of a large cylinder (Re_c =10²−10⁵). The aim of our study was to clarify how self-propulsion may influence the probability of contact of a microswimmer with a cylinder that does not induce biotic cues.

We observed the motion of B. neritina larvae in the velocity field of a cylinder and formulated a mathematical model of motion of a larva–microswimmer near the cylinder. We parameterised this mathematical model using experimental data, and calculated the probability of contact of larvae with a cylinder for a wide range of realistic problem parameters.

We collected sexually mature colonies of B. neritina (Linnaeus 1758) from floating docks at the Tel-Aviv Marina in spring 2011, 2012. Larvae of B. neritina were maintained in laboratory conditions as described elsewhere (Qian et al., 1999; Wendt, 2000). The shape of B. neritina larvae is close to a prolate spheroid, with a length-to-maximal-width ratio of approximately 1.1. Such a spheroid can be approximated by a sphere of volume equal to the volume of the larva of interest. The diameter of the equivalent sphere approximating B. neritina larva varies as d_p=200–350 μm (Kosman and Pernet, 2009; Wendt, 2000). The sinking velocity of an immobilised B. neritina larva is approximately V_t=1 mm s⁻¹ (Koeugh and Black, 1996). Correspondingly, the ratio of the mean larva density ρ_p to the water density ρ_f is ρ_p/ρ_f=1.02–1.04.

The motion of B. neritina larvae was observed in a transparent experimental flow tank (Fig. 1). Larvae were gently pipetted into the tank, and their trajectories were recorded, both from above and from the side of the tank, using an Optronis (Kehl, Germany) video system with two synchronised digital video cameras (500 frames s⁻¹ and 1280×1024 pixel sensors) equipped with Nikkor (Nikon, Tokyo, Japan) 60 mm/f2.8 or 100 mm/f2.8 macro lenses. The trajectories were digitised using the Matlab Image Processing Toolbox and an open source software package (http://physics.georgetown.edu/matlab).

The flow velocity in the experimental flow tank was measured using the particle image velocimetry system from TSI (Shoreview, MN, USA), which comprises a dual Nd:YAG laser Solo120XT (532 nm, 120 mJ pulse⁻¹, New Wave Research Inc., Fremont, CA, USA), a 4096×2048 pixel CCD camera with dynamic range 12 bits and a Nikkor 60 mm/f2.8 macro lens. Images were analysed using standard fast Fourier transform (FFT)-based cross-correlation algorithms and open-source software (http://www.openpiv.net) for verification purposes.

Typical trajectories of B. neritina larvae in still and moving water are shown in Fig. 2. In still water, a B. neritina larva moves for distances of the order of a few centimetres along a helix-like trajectory with an approximately straight axis but may also randomly change its direction of motion (Fig. 2A). The helical portions of a larva's trajectory can be approximated by a regular helix with a straight axis ox that points in the direction of the vector of the larva's swimming velocity V_S. A larva moves along a helical trajectory with linear velocity V_h and rotates with angular velocity γ.

In a Cartesian coordinate system, oxyz, the coordinates x_h, y_h, z_h of the centroid of a larva moving along a helical path vary with time t as x_h=V_St, y_h=0.5d_hsin(γt) and z_h=0.5d_hcos(γt), where d_h is the diameter of the helix. The projections of the total velocity of a point of a helix V_h(t) on the axes of the coordinate system oxyz can be found as the time derivatives of the coordinates of the point of a helix, dx_h/dt, dy_h/dt and dz_h/dt, thereby yielding the relationship V_S=√(V_h²−d_h² γ²/4).

The diameter of the helix d_h and its temporary period T can be estimated experimentally, as illustrated in Fig. 2B. When a larva moves approximately horizontally, its swimming velocity V_S can be calculated directly. When the trajectory of a larva does not belong to the plane of a lens, Wendt (Wendt, 2000) suggested estimating V_h by filming the motion of the larva in a shallow depth of the field of the lens such that only a small portion of the trajectory is in focus. By calculating the velocity of a larva along this portion, one can estimate V_h.

According to our measurements, V_S≈3–6 mm s⁻¹, which is consistent with Wendt (Wendt, 2000). Once V_S, d_h and γ=2π /T are known, the projections of the velocity V_h on the axes of the coordinate system oxyz can be found as time derivatives of the coordinates of the point of the helix dx_h/dt, dy_h/dt and dz_h/dt.

When a larva moves in a unidirectional flow for which the velocity is much greater than the larva's swimming velocity, the helical trajectory of the larva stretches, straightens and becomes rather close to rectilinear streamlines (Fig. 2C). Seemingly, in such a case, a larva moves as a passive particle. However, the contact problem relates to larval motion in non-uniform velocity fields of a collector, where streamlines are curvilinear and the fluid velocity may be of the same order of magnitude as the swimming velocity of a larva. In the next sections, we compare the motion of a larva and of a passive particle in the velocity field of a cylinder.

We studied trajectories of larvae, not the process of their attachment, because as stated in the Introduction, attachment depends on the physiochemical properties of the surface of a collector. Fig. 3Aillustrates that the fluid velocity field in front of the cylinder is laminar and does not vary significantly between the horizontal planes h=15 mm and h=27 mm, where h is measured from the bottom of the channel. Spherical polystyrene beads of d_p=430 μm diameter and ρ_p=1.05 g cm⁻³ density were pipetted into the flow and allowed to circulate in the tank before settling on the bottom.

Fig. 3B shows typical trajectories of beads. A typical trajectory is characterised by smooth variation of its slope and smooth variation of its curvature; the latter changes the sign at a single inflection point of the trajectory. Using these criteria alone, one can infer that, on many occasions, larvae also move along typical trajectories (Fig. 3C). This similarity does not mean that a larva moves as a passive particle but only that the trajectory of a larva and that of a passive particle resemble one another as long as their slopes and curvatures vary in a similar manner.

However, in addition to the typical trajectories of larvae, we also observed a significant number of trajectories that we signify as atypical (n=23 in 560 tests) (Fig. 4). The atypical trajectories are characterised by an abrupt change in their slope at the points at which the distance between a larva and a cylinder's surface is minimal. No passive particle moves along such a trajectory. Thus, we suggest that atypical trajectories result from larval self-propulsion. Although the number of atypical trajectories is relatively small, they constitute the most salient qualitative manifestation of the influence of self-propulsion on larval trajectories in a non-uniform flow. Therefore, the atypical trajectories represent a considerable interest for our study. One of the aims of this work is to formulate a mathematical model of larval motion that is able to describe not only typical but also atypical trajectories.

A long vertical cylinder of diameter D_c is placed in an unbounded two-dimensional rectilinear current. The vector of current velocity U_∞ is normal to the cylinder's axis and lies in the horizontal plane. The Reynolds number of the cylinder varies between 10² and 10⁵, which implies that the flow at the front part of the cylinder is laminar (Schlichting, 1979). For the further analysis we use the following assumptions. (1) There is no hydrodynamic interaction between individual larvae. (2) A larva is small compared with a collector and with the characteristic linear scale of the spatial flow variations that are induced by the collector in a uniform current. (3) A small larva does not change the velocity field of the cylinder. (4) The sinking velocity of a larva is small compared with the fluid velocity and can be disregarded in the problem of larval contact with the vertical surface of a cylinder. (5) A larva's relative velocity with respect to shear flow is equal to the larva's relative velocity with respect to the stagnant fluid. (6) The velocity field of the cylinder is two-dimensional; the vector of the fluid velocity U lies in the horizontal plane (Fig. 5). (7) The vector of a larva's swimming velocity (V_S) is perpendicular to the axis of the cylinder and lies in the plane of flow (Fig. 5); the direction of V_S does not vary with respect to the rotating larva's body either in stagnant or in moving water. (8) In addition to an intrinsic self-induced rotation, a larva rotates as a small rigid sphere because of the shear-induced viscous torque.

Three primary mechanisms determine the collision of a passive particle: Brownian diffusion, inertial impaction and direct interception (Fuchs, 1964; Kirbøe, 2008). If the diameter of the particle d_p>>1 μm (which is always true for B. neritina larvae), Brownian diffusion does not influence the contact phenomenon under consideration (e.g. Kirbøe, 2008). The inertial impact is determined by the Stokes number of the problem. For the problem parameters adopted here, the Stokes number is much less than the threshold value 1/8, below which inertial impact of a spherical passive particle with a cylinder does not occur (Fuchs, 1964). Correspondingly, in our work, we consider only the mechanism of direct interception. Within the framework of this mechanism a larva follows the streamlines of a collector exactly and collides with the collector because of the larva's finite size.

For the subsequent analysis, we adopt a mathematic model of larval helical motion suggested by Brokaw and generalised by Crenshaw (Crenshaw, 1989), in which the vectors of a larva's swimming velocity V_S and of its angular velocity γ are collinear and are directed along the same axis ox (Fig. 5). Because the vector V_S lies in the horizontal plane, the vector γ is parallel to the horizontal plane. The vector of the angular velocity of the shear-induced rotation, ω=2⁻¹rotU (Lamb, 1945), is perpendicular to vector of fluid velocity U. For a two-dimensional horizontal flow, ω is perpendicular to the horizontal plane and, thus, is perpendicular to γ. The orthogonality of γ and ω implies that shear-induced rotation of a larva does not change its intrinsic rotation about the axis ox.

In the earth-fixed frame of references, the direction of the larva's swimming velocity vector re-orients because of the larva's shear-induced rotation (Zilman et al., 2008). The re-orientation effect of a larva's motion in the shear flow of a cylinder is illustrated in Fig. 6. In the velocity field of a cylinder a larva moves along a curvilinear trajectory that cannot be described as a helix with a straight axis, i.e. as a regular helix. In this respect, the shear flow influences the helical pattern of motion.

To calculate the fluid velocity field near the front part of a cylinder, we use the boundary layer (BL) theory and von Kármán–Pohlhausen method, which is explained in detail elsewhere (Schlichting, 1979). In Fig. 5, we provide a brief description of this method.

The degree of deviation of a microswimmer from the trajectory of a corresponding passive particle depends on the swimmer's velocity and on its initial conditions. Systematic numerical simulations show that depending on initial conditions, swimmers may move along typical or atypical trajectories. To calculate the trajectory of a microswimmer and compare it with an experimental trajectory of a larva, we must know the initial conditions of the larva's motion. Whereas the coordinates (X₀, Y₀) can be measured with high accuracy, measurement of the course angle ϕ₀ is difficult. Therefore, we compare the computed trajectories of a microswimmer with the experimental trajectories of a larva for the same measurable coordinates (X₀, Y₀) but for the track angle ϕ₀ estimated iteratively as a problem parameter (Eykhoff, 1974).

Similarities between the calculated trajectories of a microswimmer and the observed trajectories of larvae (Fig. 7) suggest that the main features of larval motion in the velocity field of a cylinder are faithfully captured by the mathematical model presented here. Although for each atypical trajectory the match between theoretical and experimental data was obtained for a particular initial angle ϕ₀ and a particular coordinate Y₀, the general character of atypical trajectories is determined by the local fluid mechanics in the closest vicinity of a collector, i.e. in its BL.

The trajectory of a larva defines a contact event. Thus, using the same mathematical model we can calculate the trajectory of a larva and the probability of its contact with a cylinder.

In the theory of aerosols (Fuchs, 1964), one of the methods of evaluating the probability of contact (collision) of passive particles with a collector (E₀), is based on the analysis of their trajectories. The trajectory analysis is applied here to calculate the probability of contact of a microswimmer with a collector (E_S). The mathematical details of the trajectory analysis are provided in Fig. 8. Satisfactory agreement between the theoretical and available experimental data of contact probability for passive particles, illustrated in Fig. 9, suggests that the mathematical model we used to calculate the contact probability of passive particles can also be used to calculate the contact probability of microswimmers.

Now, we return to the central question of our work: how does a larva's self-propulsion influence the probability of its contact with a collector if the larva is not aware of the collector? We characterise this influence as the ratio η=E_S/E₀, which is plotted in Fig. 10 for a wide range of realistic problem parameters adopted here.

Here, we observed trajectories of larvae B. neritina and of passive particles that mimic larvae in the velocity field of a vertical cylinder (Figs 3, 4). We revealed a considerable number of larval trajectories that differed markedly from the trajectories of passive particles (Fig. 4). We attributed such trajectories to larval self-propulsions. To explain our experimental observations, we formulated a mathematical model of a larva's motion in the two-dimensional laminar velocity field of a long cylinder (10²<Re_c<10⁵). The validity of our mathematical model was confirmed by satisfactory qualitative agreement between the experimental trajectories of larvae and the simulated trajectories of a microswimmer (Fig. 7) and by satisfactory quantitative agreement between simulated and measured probabilities of contact of passive particles with a cylinder (Fig. 9).

Using trajectory analysis and Monte Carlo simulations, we calculated the probability of contact of a microswimmer with the front part of a cylinder. Mathematical modelling revealed a considerable increase in the probability of contact of the microswimmer with a cylinder compared with the probability of contact with the same cylinder of the same microswimmer but with zero swimming velocity, η=E_S/E₀ (Fig. 10). Regarding orders of magnitude, this increase can be estimated as follows: for V_S/U_∞≈0.01, η≈1; for V_S/U_∞≈0.1, η≈10; and for V_S/U_∞≈1, η≈10². For instance, because of self-propulsion, larvae of B. neritina with swimming velocity ~5 mm s⁻¹ may increase their probability of contact with a cylinder 10-fold in a sea current of ~5 cm s⁻¹ and 100-fold in a sea current ~2.5 cm s⁻¹. Although sea currents of 2.5–5 cm s⁻¹ are rare, our theoretical prediction is consistent with the observations of Qian et al. (Qian et al., 1999; Qian et al., 2000): in tubes with laminar flow, larvae of B. neritina preferred to settle in low-speed currents U≈2.5 cm s⁻¹; whereas for U>~8 cm s⁻¹, the probability of settlement drastically decreased. It should also be noted that some biofouling marine larvae swim much faster than larvae of B. neritina (Table 1). For those larvae, the ratio V_S/U_∞>0.1, which provides a ~10- to 100-fold increase in the probability of contact, may correspond to frequent currents of the order of tens of cm s⁻¹ (Table 1). In contrast, within the framework of a mechanistic approach and according to the results of our mathematical modelling, larvae with V_S<~2 mm s⁻¹ that move in sea current U_∞>5 cm s⁻¹ make contact with a collector as passive particles.

We formulated the problem of larval contact with a collector for a spherical microswimmer moving with low Reynolds numbers. However, such a small sphere and a small spheroid of moderate slenderness ~1.5–2.0 (such as the larvae listed in Table 1) move in a linear shear flow along similar trajectories (Zöttl and Stark, 2012). Given that a BL without separation can be approximated by a linear shear flow for qualitative estimates (Schlichting, 1979), it is not unlikely that a spheroidal swimmer may move in the two-dimensional BL approximately as a spherical swimmer.

We formulated the problem of contact of a microswimmer with a cylinder for laminar flows Re_c>>1. Experimental data regarding settlement (not contact specifically) of marine larvae on a cylinder in a natural turbulent environment were reported by Rittschof and colleagues (Rittschof et al., 2007). We did not find experimental or theoretical work in which the probability of contact of swimmers with a cylinder in turbulent flows was measured or calculated for St<<1 and Re_c>>1. For such flow parameters the available and rather limited experimental data pertain only to contact of passive particles with a cylinder. Asset and colleagues (Asset et al., 1970) and Stuempfle (Stuempfle, 1973) reported that for Stokes and Reynolds numbers such as those studied here, incoming upstream turbulence with an intensity of less than 7–8% practically does not affect the probability of contact of passive particles with a cylinder. In strong turbulence, the swimming speed of a larva may be small compared with the turbulent fluctuations of the fluid velocity. In such cases, a larva's self-propulsion may have little effect on its trajectory except in the vicinity of the collector, where the fluid velocity and its turbulent fluctuations are low (Schlichting, 1979).

The hydrodynamic model of contact of a microswimmer with a cylinder proposed here may be relevant for self-propelled larvae and aquatic larval collectors, such as kelp stems, sea grasses, small artificial reefs, pillars, columns and other engineering structures, that are located in a relatively slow sea current of low-to-moderate turbulent intensity (Abelson et al., 1994). Mathematical modelling of the motion of a larva in the velocity field of a collector located in a fully turbulent environment is beyond the scope of our present work.

In conclusion, the results of our investigation, which are presented in Fig. 10, suggest that for the problem parameters presented here, self-propulsion may greatly increase a larva's odds of making contact with the collector even if the larva does not detect the collector remotely.

The authors are grateful to R. Strathman and M. Hadfield for constructive discussions. L. Kagan, J. Pechenic and L. Shemer read the manuscript and made many valuable comments. G. Gulitsky is acknowledged for the assistance in the design of the experimental flow tank and N. Paz for manuscript preparation.

 
• BL

  boundary layer

 
• D_c

  diameter of a cylinder

 
• d_h

  diameter of a helix

 
• d_p

  equivalent diameter of a larva or particle

 
• E₀

  probability of contact of a particle with a collector

 
• E_s

  probability of contact of a microswimmer with a collector

 
• l_n

  distance between the two limiting (grazing) trajectories

 
• l_p

  the stopping distance a particle (l_p=ρ_pd²_pU_∞/18μ)

 
• n

  number of particles that contact the cylinder

 
• N

  total number of particles used in Monte Carlo simulations

 
• n_S

  number of microswimmers contacting the cylinder

 
• N_S

  total number of microswimmers used in Monte Carlo simulations

 
• oxyz

  helix-fixed Cartesian frame of reference

 
• OXYZ

  earth-fixed Cartesian frame of reference

 
• r

  radius vector of the centre of the particle

 
• R

  radius of a cylinder

 
• Re_c

  the Reynolds number of a cylinder (Re_c=ρ_fU_∞D_c/μ)

 
• St

  the Stokes number (St=l_p/D_c)

 
• t

  time

 
• T

  time period of a helix

 
• U

  flow velocity

 
• U_∞

  flow velocity far from the collector

 
• V

  velocity of motion of a larva or particle

 
• V_h

  velocity of helical motion

 
• V_S

  swimming velocity of a larva

 
• V_t

  sinking velocity of a larva

 
• X₀, Y₀

  initial coordinates of a larva or particle

 
• γ

  intrinsic angular velocity of a larva's helical motion

 
• η

  normalised probability of contact of a larva, E_S/E₀

 
• μ

  water viscosity

 
• ρ_f

  water density

 
• ρ_p

  mean density of a larva or particle

 
• ϕ

  course (track) angle of a larva

 
• ϕ₀

  initial course angle of a larva

 
• ω

  shear-induced angular velocity of a larva or particle