Results

Suction feeding may be simply described mathematically. The pressure-driven flow of a fluid of density ρ and viscosity μ along a tube of radius a, with mean speed u, is described by Newton’s second law: [1]where g is the gravitational acceleration and ΔP the pressure difference applied at the height h of the nectar. For active suction, ΔP is mainly generated by cibarial muscles (1, 2), while for capillary suction, ΔP ∼ σ/a results from curvature pressure, where σ is the surface tension (3). A cornerstone of biomechanics is that the force that a creature of characteristic size l can generate (10) F ∼ l²; thus, one expects the suction pressure generated by muscles, ΔP ∼ F/l² ∼ l⁰, to be independent of scale and to be of comparable magnitude for all creatures [e.g., ΔP ∼ 10 kPa for both mosquitoes (4) and humans (11)]. One can thus assess the tube scale a ∼ σ/ΔP ∼ 10 μm below which curvature pressure dominates the applied suction pressure ΔP. For most suction feeders, the radius a of the proboscis is of order 100 μm (12, 13), so the curvature pressure is less than the pressure applied in active suction. Nevertheless, capillary suction is employed by certain creatures (Fig. 1) for which active suction is precluded by virtue of geometrical and physiological constraints such as the open, passive tongue of the hummingbird (13). We further note that most suction feeders have tubes of characteristic length L ∼ 1 cm (12, 13); consequently, ρgL/ΔP < 0.1, and the effect of gravity on the flows is negligible. Finally, the ratio of inertial to viscous terms scales as ρa²f/μ < 0.1, where f ∼ 10 Hz is the typical suction frequency (14, 15), indicating negligible inertial effects. Neglecting the gravitational and inertial terms in Eq. 1 yields 8μhu = a²ΔP.

In active suction, the nectar motion is described by Poiseuille flow, for which the volumetric flow rate is given by Q = πa²u = πa⁴ΔP/8 μL. By measuring the dependence of flow rate on sugar concentration, Pivnick (2) inferred that butterflies apply constant suction power in drinking, regardless of nectar concentration. The work per unit time required to overcome the viscous friction on the wall or power output of the pump is given by . Expressing ΔP in terms of Q then yields the dependence of volume flux on viscosity: . In capillary suction, ΔP = 2σ cos θ/a, where θ denotes the contact angle, and the height of the nectar is time-dependent: h = h(t) and u = h^′(t) (Fig. 1). The solution of the force balance, 4μhh^′ = aσ cos θ, with initial condition h(0) = 0 is given by h(t) = (aσt cos θ/2μ)^1/2. Capillary suction consists of repeated cycles of tongue insertion and retraction. The whole time for a cycle is thus the sum of the time to absorb the nectar, T, and the time to unload it, T₀. The average volumetric flow rate per cycle, , is given by , where T^1/2/(T + T₀) depends weakly on viscosity (16), and so . Thus, for all suction mechanisms, we anticipate Q ∝ μ^-1/2.

To test these proposed scalings against experimental data, we introduce a general relation between Q and μ: Q = Xμⁿ, where X is a geometry-dependent prefactor that we expect to be different for each species. If we plot Q as a function of μ on a log scale, n and X represent the slope and the offset on the y axis, respectively. For each species, we calculate an average value based on the measured dependence of flow rate on viscosity. In Fig. 2, red and blue points, respectively, indicate the dependence of on μ for active and capillary suction. The convincing collapse of the data, plus the fact that, for each species, the slopes are close to -1/2, together support the proposed scalings.

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Fig. 2.

The dependence of scaled volumetric flow rate on nectar viscosity μ (2, 5, 8, 9, 12, 13, 16, 24, 26, 28, 29, 31, 32). The red points represent data for active suction, the blue points for capillary suction, and the green points for viscous dipping. The slopes of the expected lines for suction and viscous dipping are -1/2 and -1/6, respectively. Inset: Optimal concentrations for suction feeding (33%) and viscous dipping (52%), calculated from the dependence of relative energy intake rate on nectar viscosity, are denoted by vertical bands. Characteristic error bars are shown.

The energy intake rate is given by the product of the energy content per unit mass of sugar c, the sucrose concentration s, and the volumetric flow rate . For the sake of simplicity, density is treated as constant because its variation with sugar concentration is much less than that of viscosity. Considering the known dependence of nectar viscosity μ(s) on s (2), the dependence of on s can be computed as shown in the inset of Fig. 2 and reveals an optimal concentration of 33% as inferred by Pivnick and McNeil for butterflies (2) and Kingsolver and Daniel for hummingbirds (3). These predicted optimal concentrations are consistent with the results from the experimental studies reported in Fig. 1.

We proceed by presenting a model for feeding in which the nectar intake relies on viscous entrainment by the outer surface of the tongue (Fig. 3 and Movie S2). Viscous dipping is generally characterized by an extendible tongue being immersed into nectar, coated, then extracted as shown in Fig. 3C, where a honeybee (Apis) drinks nectar from a reservoir. One expects the volume entrained to be proportional to the area of the immersed tongue surface and the thickness e of the nectar layer. As in capillary suction, the feeding by viscous dipping consists of repeated cycles. If T and T₀ represent, respectively, the time needed for tongue retraction and the interval between each cycle, then the volumetric flow rate is given by , where u represents the average tongue retraction speed.

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Fig. 3.

Bees uptake nectar via viscous dipping. (A) A bumblebee (Bombus) drinking. While normally protruding straight from the galeae, the tongue here bends to lap up nectar from the substrate, a paper towel soaked in a sucrose solution. (B) Scanning Electron Microscope (SEM) image of the bumblebee’s tongue. (C) A schematic illustration of the experiment that allows us to visualize the viscous dipping of a honeybee (Apis) with a long-distance microscope and a high-speed camera operating at 250 frames per second. Here, the bee’s tongue is dipped into a 40% sucrose solution, then withdrawn (see Movie S2).

Encouraged by its success in the modeling of suction feeding, we introduce the assumption that the work rate applied in viscous dipping remains constant with respect to nectar concentration. The movement of the tongue in the fluid requires the power P_v ∼ μLu² to overcome the viscous drag, where L is the tongue length (Fig. 3). The power required for tongue acceleration P_t ∼ mu^′u ∼ ρa²u³, where m ∼ ρa²L is the tongue mass. The ratio P_t/P_v ∼ ρua²/μL ≪ 1, so the effect of P_t is negligible. Assuming constant applied power P_v thus suggests that u ∝ μ^-1/2. One does not expect T/(2T + T₀) to depend strongly on viscosity because if T is shorter in less viscous nectar due to a faster retraction, the unloading time T₀ would also be shorter, so that T ∝ T₀. Thus, the average volumetric flow rate may be expressed as .

The thickness of the fluid layer entrained by a cylinder of radius a depends explicitly on three dimensionless groups: the Bond number Bo = ρga²/σ (the ratio of hydrostatic to capillary pressures), the Weber number We = ρu²a/σ (the ratio of inertial to curvature pressures), and the Capillary number Ca = μu/σ (the ratio of viscous stresses to curvature pressures). For bees, We ∼ 10^-3 ≪ 1, Bo ∼ 10^-3 ≪ 1, and Ca < 0.1 for s < 65%, so the thickness of the liquid layer on a tongue is prescribed by the Landau–Levich–Derjaguin theory (17) that predicts e ∼ Ca^2/3a. We thus anticipate that . In Fig. 2, this proposed scaling is validated by the data for all creatures that employ viscous dipping. The energy intake rate, , thus scales as . In the inset of Fig. 2, the energy intake rate is plotted as a function of the sucrose concentration and peaks at a concentration of 52%, which is consistent with the data presented in Fig. 1. Our analysis thus provides rationale for the different optimal concentrations reported for creatures using suction and viscous dipping. For example, we can now rationalize the observation that orchid bees that employ active suction have optimal concentrations of 35%, while honeybees and bumblebees that use viscous dipping, 50–60% (18).