The largest animals are the rorquals, a group of whales which rapidly engulf large aggregations of small-bodied animals along with the water in which they are embedded, with the latter subsequently expulsed via filtration through baleen. Represented by species like the blue, fin, and humpback whales, rorquals can exist in a wide range of body lengths (8–30 m) and masses (4000–190,000 kg). When feeding on krill, kinematic data collected by whale-borne biologging sensors suggest that they first oscillate their flukes several times to accelerate towards their prey, followed by a coasting period with mouth agape as the prey-water mixture is engulfed in a process approximating a perfectly inelastic collision. These kinematic data, used along with momentum conservation and time-averages of a whale's equation of motion, show the largest rorquals as generating significant body forces (10–40 kN) in order to set into forward motion enough engulfed water to at least double overall mass. Interestingly, a scaling analysis of these equations suggests significant reductions in the amount of body force generated per kilogram of body mass at the larger sizes. In other words, and in concert with the allometric growth of the buccal cavity, gigantism would involve smaller fractions of muscle mass to engulf greater volumes of water and prey, thereby imparting a greater efficiency to this unique feeding strategy.

## I. INTRODUCTION

In the mechanics of living systems, movement and dynamics end up tightly coupled to body physiology and morphology, as animals use muscle to perform the work to swim and collect prey.^{1} Such couplings are spectacularly exemplified by the rorqual whales (Mysticeti: Balaenopteridae), a sub-group of large cetaceans measuring 8–30 m in length and 4000–190,000 kg in mass (Fig. 1).^{2–5} Rorquals, which include the blue whale (*Balaenoptera musculus*), fin whale (*Balaenoptera physalus*), and humpback whale *(Megaptera novaeangliae)* (Fig. 1), are edentulous filter-feeders that forage on aggregations of small prey, typically patches of plankton (krill) or schools of small forage fish (anchovies, capelin, and the like).^{2,4,5} To do this, rorquals have evolved morphologies and adopted a unique prey-acquisition strategy—lunge feeding—which enhances bulk prey collection (Fig. 2). As shown here, the manners in which lunge feeding is carried out, i.e., in terms of foraging durations and swim speeds at the moment of prey capture, become a crucial element for not only determining how much force need be applied, but also for constraining how evolution or growth to large body size might have been suppressed or favored.^{4}

Rorquals feed by first approaching a large aggregation of prey at high fluking frequency to build up speed. This is followed by the whales engulfing both prey and the water in which it is embedded, with the latter being subsequently expulsed out of the then-inflated buccal cavity via through-baleen filtration (Fig. 2).^{3,6} The rorquals' success in capturing enough food to meet the energetic demands of their large body not only depends on prey availability,^{7} but also on their capacity to swim fast enough to defeat the escape strategies of the prey^{8} and, as discussed below, to generate the requisite body forces to engulf and set into forward motion extreme amounts of prey-laden water, namely, up to 240,000 kg at the largest size (Fig. 2).^{2,9–11} Such forces are the fluking thrust generated by the tail musculature, and the forward push onto the engulfed mass via tension of the muscle embedded in the Ventral Groove Blubber (VGB) (Fig. 2).^{10–15}

Rorquals have been observed lunge feeding in multiple ways: Individually or in groups, while lunging along the surface or at depth and generally along an uphill track (Fig. 2); or collectively again, but doing so vertically while breaking the surface after enclosing the prey within a “net” of bubbles.^{16} A distinction is made here between lunges in which the engulfment stage is carried out (Figs. 2 and 3), that is, while fluking (“powered engulfment”) or coasting (“coasting engulfment”). Powered engulfment has been observed mostly against schooling fish near the surface where the escape paths are spatially limited, often by the presence of multiple predators^{17,18} or by curtains of bubbles blown around fish to encourage aggregation.^{16,19} On the other hand, drone video and accelerometer sensors deployed on krill-feeding whales suggest rorquals coasting during most of the engulfment stage, i.e., after carrying out one or two low-frequency fluking strokes as the mouth opens.^{2}

Coasting engulfment is a process analogous to the perfectly inelastic collision of (similar) Velcroed balls studied in introductory physics experiments. Neglecting frictional drag, this is an interaction in which the total momentum of the body and engulfed water is conserved, and also one in which a significant portion of the momentum gained by the whale during prey approach is lost to the prey-water mixture being engulfed. Building-up total momentum prior to *and* during engulfment has been proposed before,^{20} but in powered engulfment scenarios in which the momentum gained is used to reduce fluking intensity in the later stages of engulfment. In the context of the engulfing prey while coasting, a high prey-approach momentum build-up is required to generate the requisite decelerative motion that is to last throughout the duration of engulfment. This is a requirement for a high-speed approach, but one which carries a smaller overhead in drag as discussed here.

Coasting engulfment occurs either at depth or at the surface and is carried out by most rorqual species, particularly when feeding on krill. Such a feeding mode is also used by the blue whale—the largest marine vertebrate and an obligate krill feeder—and therefore presents a useful case study of the impact of physics on the kinematics and dynamics of large body size. A recent study of the feeding energy efficiency by baleen whales foraging on plankton aggregations (including krill) has shown high captured prey energy per units of (predator) metabolic energy expended, in comparison to single-prey item foraging by smaller toothed whales.^{7} A different question is to be explored here, namely, the connection between dynamics, as represented by propulsion and engulfment forces (rather than energy), morphology (body mass and inflated buccal cavity volume^{21,22}), and kinematics (prey approach and engulfment duration and speed); and ultimately, whether this connection scales at large body size to favor gigantism regardless of prey abundance. This shall be established by considering a time-averaged version of the equations of motion for a whale and its engulfed mass, as informed by the swim speeds (at mouth opening) and engulfment durations obtained from bio-logging tags deployed on dozens of free ranging whales (Figs. 1, 4, and 5).^{17} Such tags are equipped with accelerometers, gyroscopes, magnetometers, hydrophones, pressure sensors, and cameras, to document behaviors which, for the most part, have remained out of view (Fig. 5).^{23} The kinematic data thus obtained will be described in Sec. III, following the presentation in Sec. II of general time-averaged force equations for krill-feeding lunges. Section III will also present a new non-dimensional description of engulfment durations used in body-size scaling. Section IV presents new results on engulfment forces, here seen as mediating the (approximated) perfect inelastic collision between a whale and its engulfed mass. These results are used in Sec. V in a scaling study of engulfment capacity and force with respect to body size. Derivations of several new equations are found in Appendices A–D which follow the Concluding Remarks in Sec. VI.

## II. MATERIALS AND METHODS—THE FORCES AT PLAY

The schematics of the forces acting on a whale and the (to be) engulfed mass are shown in Fig. 6. Here, body weight (W) and buoyancy (B) turn out to be unimportant as they nearly cancel each other out, at least near the surface where buoyancy is controlled by lung volume expansion and depression via breathing.^{24} At depths below 60–100 m, compression of the thorax makes the body negatively buoyant (B < W), an imbalance likely compensated for by the lift generated by the tilting of the foil-shaped flippers^{25,26} and head^{27} (in similarity with sharks^{28}).

More important is the fluking thrust and body drag generated in both mouth-open and -closed configurations. As “thunniform” swimmers, whales are hydrodynamically simpler than (undulating) fishes, with thrust production limited to the rear end of the body (the flukes) and drag generation to the rest (including the caudal tail).^{29,30} To this picture, and during engulfment, one adds the so-called “engulfment drag” (*F _{D}^{engulf}*) generated in reaction to the forward push onto the engulfed mass by VGB musculature,

^{15}and “shape drag” (

*F*), which is connected to the friction and longitudinal pressure gradient generated by the flows moving externally to the body

_{D}^{shape}^{10,11}—a force herein shown to be comparatively small (Sec. IV).

Assuming one-dimensional kinematics and during prey approach, a whale's equation of motion is given by *M _{body} a_{whale} = Thrust*, and during engulfment by

*M*(powered) or

_{body}a_{whale}= Thrust – F_{D}^{engulf}*M*(coasting). Another equation will account for prey-water mixture motion via

_{body}a_{whale}= – F_{D}^{engulf}*d*(

*M*. In contrast to coasting engulfment, the total momentum of the whale-mixture system during powered engulfment isn't conserved due to the non-zero impulse by fluking thrust. Moreover, and in both engulfment cases, the system's total kinetic energy isn't conserved either, due to the contribution of the fluking tail (powered engulfment), and the energy spent by the VGB forces (musculature

_{water}U_{water})/dt = + F_{D}^{engulf}^{10–12}and/or elastic

^{20}) to control ventral cavity expansion (both scenarios) (Figs. 1 and 2). Such an expansion is a “deformation” of the whale body both orthogonally and along the direction of motion, and accounts for the extra work spent by the whale-mixture contact forces in work-energy treatments of perfectly inelastic collisions.

^{31}

Between drag and thrust, the former is generally the better-known force and more amenable to formulation in an equation. In a previous study,^{11} time-dependent engulfment forces were expressed in parametric form based on assumed rates of the mouth opening and best-fit parameters (Fig. 3). A simpler, yet more general alternative is used here. Writing down a mathematical expression for the total drag is complicated by the fact that the drag itself is a reaction to the “active” forces of the fluking tail and VGB, i.e., as driven by muscle contractions informed by auditory, visual, and other sensory cues.^{32} But time-averaged values of the Newtonian equation of motion can be derived regardless of muscle contraction specifics. In reference to Figs. 3 and 6, and assuming straight-line trajectories, one has the following in the case of prey approach (a closed-mouth state):

and during coasting engulfment (an open-mouth state),^{33}

Such results aren't very useful where the laws of force are known *a priori*—as in most introductory college physics examples. But in cases of active, muscle-driven forces with temporal variations likely to differ from lunge to lunge and even from individual to individual, Eqs. (1) and (2) become relevant, if not fundamental, for being independent of any force-vs-time profiles of same-duration, initial velocity, and final velocity.

Equations (1) and (2) use tag-derived inputs, namely, parameters *U _{open}, U_{close}*, and

*T*corresponding to the forward speeds at mouth opening and at closure times, and to engulfment duration, respectively; and

_{engulf}*U*and

_{start}*T*, to a whale's speed at the beginning of prey approach, and prey approach duration, respectively (Fig. 3). Here, one assumes

_{approach}*U*, as suggested by the data of Fig. 3 (middle frame) and other tag data.

_{start}≈ U_{close}^{17}Equation (2) informed by the data in Fig. 3 suggests a net force of about 40 kN when generated by a 100,000 kg blue whale engulfing prey and water with speed decrements of ≈ 2m/s over ≈ 5s duration. On the other hand, the average thrust can be estimated from Eq. (1) and the data of Table I, i.e., after calculating the average closed mouth body drag from the approximation described in Appendix A. The manners in which Eqs. (1) and (2) increase with body size will rest with the size-scaling of the durations and speeds discussed in Secs. III and V.

. | Humpback whale . | Blue whale . | Blue whale . | Remarks . |
---|---|---|---|---|

M_{body} (kg) (SD) | 8000 (8000) | 67,273 (23,991) | 129,005 (46,081) | Ref. 21 (blue whale); and Ref. 11 (humpback whale). SD for both species from Ref. 11 |

L (m) Bio-logging Tag Number | 8 mn 160727-11 | 22.72 bw160224-8 | 27.40 bw160727-10 | Ref. 17 |

Number of lunges from Tag sampling | 34 | 4 | 17 | … |

L_{VGB} (m) (SD) | 4.31 (0.21) | 12.99 (0.26) | 16.36 (0.32) | Ref. 21. |

L_{jaw} (m) (SD) | 1.62 (0.13) | 4.34 (0.31) | 5.65 (0.41) | Ref. 21 |

W_{head} (m) (SD) | 1.32 (0.16) | 2.61 (0.15) | 3.27 (0.19) | Ref. 21 |

S_{wet} (m^{2}) | 27.5 | 109.9 | 167.9 | Refs. 29 and 36 |

U_{open} (m/s) (averaged) (SD) | 3.56 (0.26) | 2.78 (0.13) | 3.25 (0.33) | Ref. 17 and Fig. 7 |

M_{water} (kg) (SD) | 4,982 (290) | 90,350 (1,648) | 185,595 (3,446) | Eq. (5); SD from Ref. 21. |

M_{water} /M_{body} (SD) | 0.67 (0.32) | 1.33 (0.02) | 1.40 (0.03) | Ref. 17 |

U_{close} (m/s) (SD) | 2.19 (0.55) | 1.19 (0.15) | 1.33 (0.36) | Eq. (4) |

Engulfment time T_{engulf} (s) ^{*} (SD) | 1.18 (0.16) | 5.65 (0.45) | 6.58 (0.76) | Ref. 17 and Fig. 7 |

Prey-approach time (s) ^{*} (SD) | 16.5 (8.9) | 13.5 (3.9) | 19.519.5 (10.0) | Ref. 17 and Fig. 7 |

Purging time (s) ^{*} | 27.5 (3.7) | 61.9 (12.2) | 48.8 (9.6) | Ref. 17 and Fig. 7 |

U_{open}/T_{engulf} (m/s^{2}) (SD) | 3.07 (0.49) | 0.50 (0.04) | 0.52 (0.07) | Ref. 17 |

U_{open}/T_{approach} (m/s^{2}) (SD) | 0.27 (0.11) | 0.22 (0.06) | 0.20 (0.08) | Ref. 17 |

⟨F⟩ (N) _{thrust} + F_{D}^{shape} | 664 (796) | 7,922 (4,673) | 12,702 (9,272) | U ≈_{start}∼ U Eq. (1); Refs. 17 and 21 for kinematics and morphology _{close} |

⟨F⟩ (N) (SD) _{D}^{engulf} | 9,262 (2,271) | 18,973 (3,704) | 37,590 (13,420) | Eq. (6); Refs. 17 and 21 for kinematics and morphology |

⟨F⟩/M_{D}^{engulf}_{body} (m/s^{2}) (SD) | 1.15 (0.74) | 0.28 (0.03) | 0.29 (0.04) | Eq. (6); Refs. 17 and 21 for kinematics and morphology |

⟨F⟩ (N) (SD) _{D}^{engulf}+ F_{D}^{shape} | 9,288 (10,867) | 18,931 (7,951) | 37,642 (18,068) | Eq. (2); Refs. 17 and 21 for kinematics and morphology |

F (N) (SD) _{D}^{engulf}|_{max} | 24,732 (6,183) | 50,568 (10,114) | 100,365 (35,127) | Eq. (7); Refs. 17 and 21 for kinematics and morphology |

. | Humpback whale . | Blue whale . | Blue whale . | Remarks . |
---|---|---|---|---|

M_{body} (kg) (SD) | 8000 (8000) | 67,273 (23,991) | 129,005 (46,081) | Ref. 21 (blue whale); and Ref. 11 (humpback whale). SD for both species from Ref. 11 |

L (m) Bio-logging Tag Number | 8 mn 160727-11 | 22.72 bw160224-8 | 27.40 bw160727-10 | Ref. 17 |

Number of lunges from Tag sampling | 34 | 4 | 17 | … |

L_{VGB} (m) (SD) | 4.31 (0.21) | 12.99 (0.26) | 16.36 (0.32) | Ref. 21. |

L_{jaw} (m) (SD) | 1.62 (0.13) | 4.34 (0.31) | 5.65 (0.41) | Ref. 21 |

W_{head} (m) (SD) | 1.32 (0.16) | 2.61 (0.15) | 3.27 (0.19) | Ref. 21 |

S_{wet} (m^{2}) | 27.5 | 109.9 | 167.9 | Refs. 29 and 36 |

U_{open} (m/s) (averaged) (SD) | 3.56 (0.26) | 2.78 (0.13) | 3.25 (0.33) | Ref. 17 and Fig. 7 |

M_{water} (kg) (SD) | 4,982 (290) | 90,350 (1,648) | 185,595 (3,446) | Eq. (5); SD from Ref. 21. |

M_{water} /M_{body} (SD) | 0.67 (0.32) | 1.33 (0.02) | 1.40 (0.03) | Ref. 17 |

U_{close} (m/s) (SD) | 2.19 (0.55) | 1.19 (0.15) | 1.33 (0.36) | Eq. (4) |

Engulfment time T_{engulf} (s) ^{*} (SD) | 1.18 (0.16) | 5.65 (0.45) | 6.58 (0.76) | Ref. 17 and Fig. 7 |

Prey-approach time (s) ^{*} (SD) | 16.5 (8.9) | 13.5 (3.9) | 19.519.5 (10.0) | Ref. 17 and Fig. 7 |

Purging time (s) ^{*} | 27.5 (3.7) | 61.9 (12.2) | 48.8 (9.6) | Ref. 17 and Fig. 7 |

U_{open}/T_{engulf} (m/s^{2}) (SD) | 3.07 (0.49) | 0.50 (0.04) | 0.52 (0.07) | Ref. 17 |

U_{open}/T_{approach} (m/s^{2}) (SD) | 0.27 (0.11) | 0.22 (0.06) | 0.20 (0.08) | Ref. 17 |

⟨F⟩ (N) _{thrust} + F_{D}^{shape} | 664 (796) | 7,922 (4,673) | 12,702 (9,272) | U ≈_{start}∼ U Eq. (1); Refs. 17 and 21 for kinematics and morphology _{close} |

⟨F⟩ (N) (SD) _{D}^{engulf} | 9,262 (2,271) | 18,973 (3,704) | 37,590 (13,420) | Eq. (6); Refs. 17 and 21 for kinematics and morphology |

⟨F⟩/M_{D}^{engulf}_{body} (m/s^{2}) (SD) | 1.15 (0.74) | 0.28 (0.03) | 0.29 (0.04) | Eq. (6); Refs. 17 and 21 for kinematics and morphology |

⟨F⟩ (N) (SD) _{D}^{engulf}+ F_{D}^{shape} | 9,288 (10,867) | 18,931 (7,951) | 37,642 (18,068) | Eq. (2); Refs. 17 and 21 for kinematics and morphology |

F (N) (SD) _{D}^{engulf}|_{max} | 24,732 (6,183) | 50,568 (10,114) | 100,365 (35,127) | Eq. (7); Refs. 17 and 21 for kinematics and morphology |

## III. RESULTS—SPEEDS AND DURATIONS

### A. Tag-measured speeds and durations

Bio-logging tags have provided a unique look at krill-feeding at depth (Fig. 5),^{17} with examples of forward speeds at the moment of mouth opening (*U _{open}*) and durations of both prey approach and engulfment shown in Figs. 3 and 7. Speeds are determined from exponential relationships of flow noise and accelerometer vibrations,

^{34}from video-based feeding duration (when available), and from accelerometer signals indicating fluking on approach (

*U*) as well as deceleration during engulfment (

_{start}*U*to

_{open}*U*).

_{close}^{17,18}Interestingly, Fig. 7 shows a surprising degree of kinematic regularity on a lunge-to-lunge and dive-to-dives basis. Also noticeable are the systematic variations among similarly sized blue whales which aren't well understood. Such regularity is likely enabled by slow-moving krill (≈0.1–0.2 m/s)

^{35}living in aggregations that considerably exceed a whale's body size, thereby presenting an essentially immobile target

^{18}when approached at speeds exceeding 2.5–5.0 m/s (top frame). At 6 s on average, engulfment is the shorter of the three stages of lunge feeding (Fig. 7), i.e., versus the 15–20 s during prey approach and the 50–100 s during purging and filtration. The reasons for these different durations are not very well understood, but are likely to depend on the temporal changes of the prey patch's shape and mass density, and occur, e.g., when a whale repeatedly lunges through the same patch during the same feeding dive (Fig. 5).

### B. Non-dimensional formulation

The scatter in *U _{open}* and

*T*(Fig. 7), coupled with relatively small sample sizes currently prevents robust empirical determinations of the relationship between those observables. On the other hand, viewing an engulfing whale as an inflating bag or parachute suggests the idea that, for the same (time-averaged) inlet area, a faster whale engulfs/inflates more rapidly than a slower one. This follows from the conservation of the fluid mass accumulating in the buccal cavity for which fill duration (

_{engulf}*T*) scales as the “to-fill” cavity volume over through-inlet flux, i.e.,

*T*∼ Volume/area ×

*U ∼*length

*/U*, and up to a non-dimensional parameter

*K*:

_{engulf}Equation (3) can also be argued from buccal cavity wall acceleration ( Appendix B), as well as from mandible rotation kinematics which yields the predictions shown in Fig. 8.^{33} Overall, the idea of a constant non-dimensional engulfment time $Kengulf$ appears to make sense for both humpback and blue whales in a lunge-averaged sense.

Variations among the *K*-values of different whales (same species) are at about 30% for reasons likely due to individual size variation of body features such as VGB and mandible lengths (*L _{VGB}* and

*L*) and skull width (

_{mandible}*w*), which are absent in the equation above but present in Eq. (B2) ( Appendix B). In the speed data shown in Fig. 7, the value of

_{skull}*U*appears largely insensitive to body size—at least for the blue whale data sample shown (e.g.,

_{open}*U*= 3.36 m/s (SD = 0.33m/s) at 27.4 m body length; and 3.82 m/s (SD = 0.52m/s) at 23.6 m). It would follow that

_{open}*T*scales proportionally with body size, thereby determining a good part of the force scaling obtained from Eqs. (2) and (6) (below).

_{engulf}## IV. RESULTS—FORCES

### A. Coasting engulfment as a perfectly inelastic collision

Omitting the contribution of shape drag permits approximating coasting engulfment as a perfectly inelastic collision. Applying momentum conservation yields *M _{body} U_{open} = (M_{body} + M_{water}) · U_{close}* and solving for

*U*results in

_{close}a useful formula for the scaling study below. Parameter *M _{water}* is calculated for maximal engulfment which occurs when the mandibles are lowered to the largest angles possible (≈78°) (Fig. 2),

^{36}and by approximating the filled buccal cavity as two juxtaposed quarter-ellipsoid sections spanning the skull's width (

*w*), mandible length (

_{skull}*L*), and VGB length (

_{mandible}*L*)

_{VGB}^{9–11,22}

The factor in brackets corresponds to the cavity volume modeled by the ellipsoids,^{22} and parameter Ψ, a species-dependent adjustment factor corresponding to the departure from the pure ellipsoid, as caused by mandible dislocation during engulfment.^{33} Herein Ψ = 1.17 sin (78°) (blue whale) and = 1.03 sin (78°) (humpback).^{9,11} Using the body dimensions listed in Table I (below), ratio *M _{body}*/(

*M*) ends up varying between 0.6 (

_{water}+M_{body}*L*= 8 m; humpback) and 0.4 (27 m; blue), implying forward speeds at mouth closure reduced to 60% and 40% of the initial speed (

_{body}*U*), respectively.

_{open}Interestingly, both left- (LHS) and right-hand (RHS) sides of Eq. (4) can be checked independently given the known scaling of rorqual morphology,^{21} the recent capability of measuring body length directly via overflying drones,^{37} and the kinematic data collected by bio-logging tags. Averaging over all speed profiles (*U(t)* vs *t*) of the lunges carried out by individual blue whales in Cade *et al.* yields *U _{close}/U_{open}* ≈ 0.43;

^{17}and using the likely values of the body morphology applying to similarly-sized whales (Table I), one obtains

*M*≈

_{body}/(M_{body}+M_{water})*0.41, i.e., a near agreement but with errors in the RHS difficult to assess given the uncertainties connected with body mass (see Table I), and also with body length estimates carried out in the absence of drones. (A preliminary analysis of recently-collected tag and drone data suggests LHS/RHS-ratios of 0.92 (SD = 0.10) in blue whale (20 individuals and 660 lunges), and 1.03 (SD = 0.17) in humpback whales (8 and 268) (W. T. Gough, personal communication).)*

### B. Engulfment drag

Viewing engulfment drag as the reaction to the direct-contact action of the VGB against the water-prey mixture (Fig. 6) allows the calculation here of its time-average ⟨*F _{D}^{engulf}*⟩ in similarity to Eq. (3), while using the equation of motion of the engulfed mass for which the force of the VGB is assumed as dominant. Starting from a state of zero-mass and zero-speed, and ending at mouth-closure time

*T*with a mass

_{engulf}*M*and at a final speed equal to that of the whale (

_{water}*U*=

_{water}*U*=

_{whale}(t = T_{engulf})*U*) (Fig. 2), one obtains ( Appendix C)

_{close}Using Eq. (6) along with Eqs. (4) and (5) and the tag and body morphology data of Table I yields averaged forces in the range of 19 kN–36 kN in blue whales. Interestingly, these are similar to those of the time-averaged total drag calculated via Eq. (2), thereby pointing to small contributions of shape drag.^{11} (Estimates from the recently-collected tag data mentioned above put the ratio $\u27e8FDshape\u27e9/\u27e8FDengulf+FDshape\u27e9$ at 0.14 (SD = 0.15) in blue whales and −0.13 (SD = 0.42) in humpback whales, while showing no intra-species trend versus (adult) body lengths (W. T. Gough, personal communication).)

Being of a pulsed type similar to inflating parachute drag,^{38} the peak value of the engulfment force *F _{D}^{engulf}|_{max}* can also be estimated via the momentum-impulse theorem, which connects the whale's body momentum loss to the impulse of the total (drag) force acting on it ($\Delta Pif=\u222bifFtotaltdt$).

^{38}Rescaling the integral in terms of engulfment duration and maximum total force sustained yields $MwhaleUopen\u2212Uclose=FtotalmaxTengulfI\u2009\u223c\u2009FDengulf|maxTengulfI$, or after solving for maximal engulfment drag

Parameter I is the result of integral $\u222bopencloseFtotal(t)dt/FtotalmaxTengulf\u223c\u222bopencloseFDengulf(t)dt/FDengulf|maxTengulf$, with the last step again assuming engulfment drag as the dominant force.^{11} Integral I is a non-dimensional measure of the shape of the force vs time curve where, for example, I = 1 when F is constant throughout; I ≈ $12$, if shaped like a triangle; or I ≈ 3/8 in the case of engulfment ( Appendix D). Actual values of I do reflect the temporal metering of the muscle-based force of the VGB, but it should be remembered that this is an integrated quantity. Typical values of the maximal engulfment drag are shown in Table I, varying between 25 kN and 100 kN, values which are at least twice as large as the time-averaged values. These are also similar to the time-dependent and parametric forces calculated by Potvin *et al.*^{11}

### C. Engulfment drag vs closed mouth drag

Results of calculations of the various forces at play are shown in Table I, based on the equations above, kinematics from bio-logging tag data,^{17} and morphology.^{21} Generally, the results proportional to *M _{body}* are significantly uncertain since obtaining body mass from stranding events and industrialized whaling involves weighing cut-out body parts while trying to limit significant losses of tissue and fluid in the process. On the other hand, those proportional to the engulfed mass

*M*have smaller uncertainties, being based (via Eq. (5)) on the skull width and lengths of the VGB and mandibles, all of which are measured accurately-enough. In all cases, the variance includes not only measurement uncertainties, but also more crucially, natural variations among individuals of same body length (and species).

_{water}^{21}

Basic comparisons of the drag generated while the mouth is open (Eqs. (2), (6), and (7)) versus closed (Eqs. (1) and (A1)–(A3)) yield the following results. In the case of a 27 m-long blue whale swimming closed-mouth at a speed of 3.25 m/s, and within a factor-2 uncertainty on the value of the tail heaving drag factor ($F\u0303$; Appendix A),^{29,30} the corresponding drag turns out at 3740 N, a small value in comparison to the 37,590 N of engulfment drag sustained (±13,420 N; Table I) while decelerating from $Uopen$= 3.25 m/s to $Uclose$= 1.33 m/s (Fig. 3).^{11,17} On the other hand, and with the same mass and speed, maximal engulfment drag ends up at *F _{drag}^{enulf}* = 100,365 N (±35,127 N; Table I), a value roughly 27 times the closed-mouth drag.

## V. DISCUSSION—ALLOMETRIC SCALING AND DYNAMICAL IMPLICATIONS

### A. Engulfment capacity

Rorqual whale feeding energetics and dynamics have become part of a wider study of the relationships between body size and filter-feeding in aquatic organisms.^{7,39–42} Why this is so rests in good measure with how morphology has coupled with dynamics to exploit prey in manners to insure survival, and for the rorquals, how the coupling has favored large body size. This is modeled here via Eqs. (2) and (6), used along with the known scaling of the morphology^{21} duration and speed (Sec. III), and applied to the blue and fin whales—two closely related species of similar shape and size.^{21}

The calculations first depend on the morphology of the buccal cavity, here assessed with the quotient *M _{water}/M_{body}* of the engulfed mass (Eq. (5)) over body mass shown in Fig. 9.

^{9}The ratio generally increases with size, i.e., starting at

*M*≈ 0.6 where

_{water}/M_{body}*L*= 10 m.

_{body}^{21}This trend tracks similarly in the smaller rorqual species, namely, with

*M*≈ 1.1–1.2 in Bryde's whales (

_{water}/M_{body}*Balaenoptera brydei*) (

*L*= 15 m) and ≈ 0.45–0.50 in minke whales (

_{body}*B. acutorostrata*) (

*L*= 8 m).

_{body}^{21}The figure suggests the ratio scaling as

*∼L*

_{body}^{0.92}(fin whales) and

*∼L*

_{body}^{0.35}(blue), with “∼” signifying equality up to a constant factor, rather than remaining insensitive to size as expected from

*isometric*scaling where volumes scale as

*∼L*and ratios of volumes as

_{body}^{3}*∼L*

_{body}^{0}(Table II). Departure from isometry, or

*allometry*, occurs in all rorqual species, a result of the skull becoming disproportionally longer and wider during growth into adulthood, leading to scaling laws of the type

*L ∼ L*(

_{body}^{α}*α > 1*) (Table II).

^{21,22}Clearly, such allometric scaling is an adaptation that insures larger engulfed volumes and greater harvests of prey to support the metabolic needs of ever-increasing body sizes.

^{22}Such gain comes with the added bonus of reduced speeds imparted to the engulfed mass at the largest scale, namely,

*U*≈ 0.40 ± 0.03 (Fig. 9(b) and Table I), in contrast to the smaller rorquals where

_{close}/U_{open}*U*=

_{close}/U_{open}*M*≈ 0.54 (Bryde's) and 0.64 (minke). Also interesting is the capping of the momentum transferred to the captured prey-water mixture (Fig. 9(c)), revealing the added benefit of body gigantism as a limiting effect on lost body momentum (and kinetic energy), and ultimately, on the body forces at play (Sec. V).

_{body}/M_{body}+M_{water}Z . | Humpback β
. | Fin β
. | Blue β
. | Isometric scaling . |
---|---|---|---|---|

M _{body} | 4.17 | 2.74 | 3.54 | 3 |

L _{VGB} | 1.19 | 1.16 | 1.19 | 1 |

L _{jaw} | 1.21 | 1.29 | 1.47 | 1 |

W _{head} | 1.04 | 1.21 | 1.20 | 1 |

M _{water} | 3.44 | 3.66 | 3.86 | 3 |

Z . | Humpback β
. | Fin β
. | Blue β
. | Isometric scaling . |
---|---|---|---|---|

M _{body} | 4.17 | 2.74 | 3.54 | 3 |

L _{VGB} | 1.19 | 1.16 | 1.19 | 1 |

L _{jaw} | 1.21 | 1.29 | 1.47 | 1 |

W _{head} | 1.04 | 1.21 | 1.20 | 1 |

M _{water} | 3.44 | 3.66 | 3.86 | 3 |

### B. Engulfment force exerted by the body

The scaling of the average engulfment force (Eq. (6)) follows from the scaling laws on engulfment time (Eq. (3)) and swim speed at mouth-opening (*U _{open}*). Apart from the data of Cade

*et al.*

^{17}collected from small populations of humpback and blue whales, the scaling law for

*U*isn't well known. Provisionally using the size-independence suggested in Fig. 7 for blue whales (

_{open}*U*∼

_{open}*L*

_{body}^{0}), one arrives at

*T*∼

_{engulf}*L*and ultimately to the forces $\u27e8Fengulf\u27e9$ scaling as $Lbody2.20$ (fin whale) and $\u223cLbody2.69$ (blue) as shown in Fig. 10(a). Similar trends apply to the maximal engulfment force (Eq. (7)) and fluking thrust (see Eq. (1), after neglecting shape drag). Figure 10(a) shows that these results are also insensitive to the scaling of the mouth-open speed, as demonstrated with the use of an alternative scaling law, e.g.,

_{body}*U*, constructed for correlating data collected a decade ago on humpback, fin, and blue whales.

_{open}= 0.15 L_{body}^{9}Clearly, body mass drives the magnitude of the force, i.e., over that of the acceleration scale

*U*.

_{open}/T_{engulf}Altogether different scaling trends arise when considering the body force generated by per kilograms of mass (*⟨F _{D}^{engulf}⟩/M_{body})*. Equations (5) and (6), and the duration and speed scaling laws used in Fig. 8, lead to the results shown in Fig. 10(b) in the case of the two mouth-open speed scaling scenarios discussed above. Where

*U*∼

_{open}*L*

_{body}^{0}, the specific force decreases with size, in agreement (within uncertainties) with the tagged individuals showcased in Table I: namely,

*⟨F*0.3N/kg for both 23 and 27 m blue whales (Fig. 10(b)), and 1.1 N/kg for the 8 m humpback whale which, incidentally, approached prey at speeds similar to the blue whales' (3.8 m/s, vs 2.8 m/s and 3.2 m/s). Here, and according to Eqs. (6) and (C2), the scaling of the specific force ultimately rests on the scaling of the ratio

_{D}^{engulf}⟩/M ≈*U*, as (likely) determined from the whales' behaviors during prey approach and engulfment.

_{open}/T_{engulf}Although detailed studies of VGB muscle action are still in the future,^{12,13} visuals of buccal cavity expansion generally suggest muscle generating tension during elongation (a.k.a., “eccentric pulls”).^{12} At muscle fiber-level, eccentric tension isn't as sensitive to the elongation rate as for fast contractions which generate significantly less force.^{43} With engulfment drag arising as a reaction to the (longitudinal) pushing action of the VGB onto the engulfed mass, one is left with the latter metering force in proportion to total muscle mass rather than through elongation rate. This, in turns, would lead to the hypothesis in which a smaller specific force arises from the use of smaller proportions of muscle, at least to effect foraging. In other words, the coasting engulfment mechanism described here would make prey collection more efficient in terms of the required body forces and at the largest scale, that is, as long as both larger and smaller individuals of a species approach prey at similar speeds.

## VI. CONCLUDING REMARKS

This paper has shown how coasting engulfment, together with the allometry of the buccal cavity, approach speeds and engulfment duration, combine in ways that favor large body size among the largest rorquals. Whether the scaling of the specific force among the smaller rorquals follows the trend hinted at in Fig. 10 isn't known. Recently collected kinematic data on minke and Bryde's whales (9–14 m) have yet to be fully analyzed and published. Such data are eagerly awaited for further testing of the coasting engulfment paradigm, as well as for uncovering the scaling laws of important ratios such as *U _{open}*/

*T*and

_{engulf}*U*which, in the end, drive the scaling of the specific engulfment force.

_{close}/U_{open}Smaller rorquals such as the minke, Bryde's, and humpback whales have also been observed fluking while engulfing schools of forage fish, but at significantly slower speeds (<2 m/s) (Fig. 3).^{17,18} Clearly, fluking in a high-drag (mouth-open) configuration requires more effort as suggested by the net forces involved when accelerating at speed increments similar in absolute value to those of coasting blue whales (Fig. 3)

Thus with average accelerations generated at *U _{close}* = 3.5 m/s,

*U*= 1.5 m/s, and

_{open}*T*= 6 s, Eq. (8) yields average thrust forces amounting to about twice the drag (and VGB active push), thereby tripling the net body-supplied force (i.e., of the VGB and fluking tail muscle). On the other hand, and as further explored in a sequel paper, the required body forces and energies generated at low speed (<2 m/s) turn out similar to those of coasting engulfment carried out at those same speeds. Whether powered engulfment also yields a dynamical efficiency at large body size remains a question to be further investigated.

_{engulf}Other dynamical aspects favoring large rorquals can be revealed by looking at Life's energy angle.^{7,18} It is known already that mass-specific metabolic expenditures during resting decrease at large body size, i.e., roughly as ∼*1/L _{body}*

^{0.25}for land vertebrates

^{44}and ∼

*1/L*

_{body}^{0.32}in cetaceans.

^{45}However, and during foraging, prey abundance and energy contents become key factors driving the energetic efficiency of feeding, and not surprisingly, the gigantism displayed by both large toothed and edentulous cetaceans.

^{7,46}But, as will be discussed in future work, the energetic efficiency of coasting engulfment is also driven by dynamics, enhanced even, for example when the prey-approach speeds are near the minimum required for coasting over an entire mouth open-closure cycle.

## ACKNOWLEDGMENTS

J.A.G. and D.E.C. were supported in part by grants from the National Science Foundation (NSF) (IOS-1656676), the Office of Naval Research (N000141612477), and a Terman Fellowship from Stanford University; and J.P. by a grant from the NSF (IOS-1656656). The authors thank F. E. Fish, J. H. Kennedy, N. E. Pyenson, and P. Segre for stimulating discussions. They are also grateful to D. J. Albert and A. Boersma for the use of their artwork, and to W. T. Gough for communicating preliminary results from his bio-logging tag data analysis.

### APPENDIX A: CLOSED-MOUTH DRAG (DURING PREY-APPROACH)

Drag for highly streamlined objects is estimated from an expression originally devised for airships,^{47} and later applied to cetaceans^{48,49} after insertion of a correction factor ($F\u0303(U)$) accounting for the heaving of the tail and head during active swimming^{49}

Parameters *S*_{wet}, *U,* and *ρ*_{w} correspond to the wetted body surface area, the whale's swimming speed, and sea water density (1025 kg/m^{3}), respectively. In cetaceans, one uses *S _{wet} = 0.08 m^{2}/kg^{0.65} (M_{body}^{0.65})*.

^{49,50}

*C*is the drag coefficient arising from the viscous friction between the body and its boundary layer (first bracket), and the pressure gradient caused by near-wake turbulence (second bracket).

_{D}^{47,51}In the latter

*L*and

_{body}*w*correspond to the body length and maximum width, respectively. The ratio

_{max}*U(t)L*is the Reynolds number with

_{body}/ν*ν*as the water kinematic viscosity (1.19 × 10

^{−6}m

^{2}/s). The first bracket in Eq. (A2) applies because the whales' large size, swim speed, ratio

*L*/

_{body}*w*, and Reynolds number (12–27 m, 1–3 m/s, 5–10, and >10

_{max}^{6}, respectively) all contribute to the suppression of pressure gradients and attendant flow separation over the body surface.

^{51,52}Finally, and with respect to coefficient $F\u0303$, comparison with direct calculation of the thrust by idealized rigid lunate tails suggests $F\u0303$ ∼ 1–3 at

*Re*∼10

^{7}, hence the value $F\u0303$ ∼ 2.0 used herein.

^{49}Note that Eq. (A2) omits a surface wave drag correction,

^{49}as most of the lunges described here are performed away from the surface (Fig. 5).

To the drag of Eq. (A1), usually valid at constant-speeds, one adds the contribution of the so-called “acceleration reaction” (or “added mass”) whenever a whale is accelerating (*a*)^{52,53}

*V _{body}* is the whale's body volume, and the last step in Eq. (A3) is based on approximating a whale's body density with that of seawater. The added mass coefficient

*k*is calculated from inviscid hydrodynamics.

_{added}^{53,54}As currently unknown in cetaceans, it is approximated by coefficients associated with prolate ellipsoids of revolution, namely,

*k*= 0.059, 0.045, 0.036, and 0.029 for

_{added}*w*∼ 5, 6, 7, and 8 typical of rorqual body aspect ratios.

_{max}/L_{body}^{52,53}Along with the VGB-generated engulfment drag, the acceleration reaction is an inertial source of drag which appears even in the absence of viscosity. It arises from the need to increase, over time, the kinetic energy of the fluid near an accelerating body,

^{54}as imparted by the forward or rearward shift of the pressure gradient along its surface (in comparison to the steady state pressure profile). The first bracket in Eq. (A2) remains valid in accelerated motions, again due to high body fineness and Reynolds numbers, i.e., conditions which (again) limit the importance of surface pressure gradients from head to tail, and most importantly, suppress boundary layer separation and (large) vortex shedding.

^{55}

Approximating *U*(*t*) as the sum *U _{start}* + at (with a = (

*U*−

_{open}*U*)/

_{start}*T*), time-averaging the three equations above during prey approach yields the following result, which can be combined with Eq. (1) to get an estimate of the fluking thrust:

_{approach}### APPENDIX B: INFLATION DURATION SCALING

An expression for the scaling of inflation duration of a bag-like structure, versus inlet fluid speed, can be derived by looking at a cavity as filling with an incompressible fluid of mass density *ρ* entering with speed *U* (time-averaged) through a fixed diameter inlet, to expand and impart its walls with an acceleration *a _{expand}* over a distance scale

*d*and duration

*T*(Fig. 11). Area

_{inflate}*A*characterizes the accelerating sections of the bag walls and, using constant-acceleration kinematics along with zero-initial wall speed, leads to:

*d ∼*$12$

*a*(

_{expand}*T*)

_{inflate}^{2}. In cases analogous to common kitchen garbage bags in which wall elasticity is absent, acceleration during expansion is driven by the internal dynamical pressure ($12$

*ρU*) and leads to

^{2}*a*($12$

_{expand}= A*ρU*)

^{2}*/m*, with

_{wall}*m*corresponding to the mass of the accelerating wall sections. This is a result that describes wall motions controlled solely by internal pressure, rather than by pressure combined with wall elasticity. On the other hand, the hydrodynamic modeling showcased in Fig. 3 is based on VGB muscular contraction forces which are also tuned to dynamic pressure, via a dimensionless “total-force” coefficient

_{wall}*k*.

^{11}In this case, the wall acceleration would become:

*a*=

_{expand}*A*(

*k*$12$

*ρU*)

^{2}*/m*. Merging the latter with constant acceleration kinematics and solving for inflation time will result in

_{wall}with the last step added to emphasize the similarity with Eq. (3). Note that the details of a cavity's architecture and size would enter in the square root (including *k*). With krill feeding whales, Eq. (B1) would apply to the duration of mouth-open-to-maximum-gape, which is basically half of engulfment time,^{17,33} leading to *K _{inflate} =*$12$

*K*. Note that incorporating further morphological details would result in the wall being composed of a number of folds called “furrows” (

_{engulf}*N*) adjacent to thick slats of hardened skin linked by pleated soft tissue.

_{furrow}^{2,14}With a slat width

*w*and mass density close to that of sea water (ρ), and with the ratio

_{slat}*m*/

_{wall}*A*ρ and

*d*approximated as

*w*and

_{slat}*w*/2

_{head}*N*, respectively, coefficient

_{furrow}*K*will read as

_{inflate}This result hints at *K _{inflate}* and

*K*scaling as

_{engulf}*∼ L*

^{0}in an isometric world where both

*w*and

_{furrow}*L*are proportional to

_{mandible}*L*; or as up to

*K*∼

_{expand}*L*in the allometric world of the whales (Table II), leading to the individual-to-individual variations shown in Fig. 8.

^{1.2}### APPENDIX C: TIME-AVERAGED ENGULFMENT DRAG

Viewing lunge-feeding as a colliding two-body system permit non-trivial estimations of the time-averaged value of the engulfment drag as follows. Starting with the general definition $\u27e8F\u27e9=1/T\u222b0TF(t)dt$ in which engulfment drag equal to the rate of momentum change by the engulfed mass over *T = T _{engulf}*, one has $Fengulf(t)=ddtMwater(t)Vwater(t)$. Assuming

*M*and

_{water}(0) = 0*U*yields

_{water}(0) = 0Taking the whale and engulfed mass to be a perfectly inelastic colliding two-body system in which *U _{water} = U_{whale} = U_{close}* (at

*T*) is given in Eq. (4), the above ends up as

_{engulf}### APPENDIX D: SHAPE OF ENGULFMENT DRAG VERSUS TIME

Equation (7) involves evaluating parameter *I*, a non-dimensional measure of the shape of engulfment drag versus time

With engulfment drag being a derived construct rather than a datum obtained from tags, integral *I* is evaluated from the results of more detailed hydrodynamic modeling.^{11} An approximate (linear) rendition of it is shown in Fig. 12, in the form of two juxtaposed triangles for which the area-under-the-curve is readily obtained as ∼*F _{D}^{engulfed}ǀ_{max} T* (

*1/4 + Y/4*) with

*Y*∼ $12$ (Fig. 5 of Ref. 11), leading to

*I*∼ 3/8.