Fig 1.
External and schematic (dorsal cutaway) views showing position of paired racks of 300 serial baleen plates between tongue and lips.
In dorsal cutaway view with oral roof removed (bottom of figure), blue arrows indicate direction of water flow though and around baleen filtering apparatus in life as well as in experimental flow tank trials and computational modeling calculations (hypothetical but predicted from data of current study and previously published experiments [3, 5]). Water can flow anteroposteriorly (AP) within mouth along the tongue (APT channel) or the lip (APL channel). Filtered water exits the mouth via paired posterior openings (PO). Oropharyngeal opening which leads to esophagus lies near oral floor caudal to the tongue root. Dashed red box indicate the approximate location of the shortened mini-rack studied in a previous flow tank study [9]; dotted green box shows the system under consideration in this paper.
Fig 2.
Schematic definitions of oral cavity inlet and outlet surface areas.
Equivalent inlet area versus actual area, in two cases of lip emplacement and mandibular opening: semi-closed (A) and wide open (B). Definition of the flow exit area at the Posterior Opening (C).
Fig 3.
Schematic buccal cavity (right half, dorsal view) in BHC model of whale with four baleen plates and without flows into the oropharynx.
Crosses mark locations of flow speeds (U) and pressures (P) calculated in the BHC model. Arrows represent the cross section-averaged flow speeds (the longer the arrow the faster). On the lingual side, anteroposterior flow decelerates because of the mass loss incurred by intra-baleen flow. On the labial side, flow accelerates from the mass gained from intra-baleen flows. Variables Ain and Apo correspond to surface areas of the gap between the baleen rack and tongue’s side wall and to the posterior opening (Fig 2).
Table 1.
BHC input data—Morphology.
Table 2.
BHC input data—Environmental characterization.
Table 3.
Hydrodynamic parameters of interest (10m whale; data from Tables 1 & 2).
The flow speeds used here correspond to medium rack obstruction (C = 0.4). Flow speeds at lower obstruction (C > 0.4) would be higher.
Table 4.
BHC input: Supplementary definitions and constraints.
Fig 4.
Calculated flow speeds and pressure drops for a 10m-whale swimming at Uwhale = 1.0m/s.
The IB, APT and APL flow speeds correspond to the dash-dot, continuous and dashed lines respectively; and the pressure profiles in the APT and APL canals likewise corresponding to the continuous and dash-dot lines. The inputs used in both frames are: 10m body length, dbaleen = 0.01m, cbaleen = 0.07m, hHT = 1.5m and Din = 0.5m, WPO = 0.435m, a = 0.071 and C = 0.44 (“with prey” [9]). Further input details are listed in Tables 1 and 2.
Table 5.
Flow rate conservation (left) and flow splitting (right) equations in the 4-baleen BHC system.
Indices correspond to the locations shown in Fig 3.
Table 6.
APT and IB flow solutions of the 4-baleen BHC system.
Indices correspond to locations in Fig 3. AIBchannel is the cross section area of the intra-baleen space and equal to the product hHt cbaleen.
Table 7.
Pressure drop equation in the APT (left) and IB (right) canals in the simplified 4-baleen BHC system.
Indices correspond to the locations shown in Fig 3.
Fig 5.
Canal friction coefficient for a 10m-whale swimming at Uwhale = 1.0m/s.
In the top frame the values of kIB, kAPL and kAPT are represented by the dashed, continuous and dot-dashed lines respectively. The values of the last two are shown again in the bottom frames. These are calculated with the same input data as in Fig 4. The figure shows a trend seen at all speeds, morphology and body size.
Fig 6.
IB and APT flow speeds versus flow-splitting coefficient C.
The larger the value of the coefficient, the larger the baleen obstruction by trapped prey. Shown are C = 0.30 (continuous curve), C = 0.44 (dash-dot; “with prey” [9]) and C = 0.53 (dotted; “no prey” [9]). All three cases involve the same mouth inlet flow rate. The rest of the inputs used in both frames where: 10m body length, Uwhale = 1.0m/s, dbaleen = 0.01m, cbaleen = 0.07m, hHT = 1.5m and Din = 0.5m. Further details are listed in Tables 1 and 2.
Fig 7.
BHC flow speeds versus body size.
Shown are values corresponding to body lengths of 10m (continuous line) and 15m –case A (dashed line). Both cases with simulated with Uwhale = 1.0m/s and C = 0.40. At a 10m-length, dbaleen = 0.01m, cbaleen = 0.07m, hHT = 1.5m and Din = 0.5m; and at 15m, dbaleen = 0.015m, cbaleen = 0.105m, hHT = 2.6m and Din = 0.65m.
Fig 8.
BHC flow speeds versus mean tongue wall distance from baleen (Din).
Shown are values for widths of Din = 0.5m (continuous line) and 1.00m (dashed line), in the case of a 10m whale swimming at Uwhale = 1.0m/s. The rest of the inputs in both frames are as follows: dbaleen = 0.01m, cbaleen = 0.07m, hHT = 1.5m and C = 0.44.
Fig 9.
Calculated mouth friction drag versus the square of the swim speed and body size.
Calculated via Eq 3 using the morphological inputs of Table 2 in the case C = 0.44 (“with prey” [9]). (Note—with 10m, dbaleen was set at 0.01m). The body lengths considered were 8m (starburst and dash-dot line), 10m (squares and continuous line), 15m-case B (diamonds and dotted line) and 15m-case A (triangles and dashed line) respectively. The slopes corresponding to the lines are as follows: 71.26 kg/m (8m; R2 = 0.99), 207.95 kg/m (10m; R2 = 0.99), 407.79 kg/m (15m-case B; R2 = 0.99) and 431.98 kg/m (15m-case A; R2 = 0.99). The presence of the straight lines in this plot suggest a swim speed-independent but body size-dependent drag coefficient for the drag generated by the internal flows of the oral apparatus (Eq 6b).
Fig 10.
Mouth friction drag versus the square of the swim speed and mouth inlet parameters (Din).
Calculated for 10m body length via Eq 3. Here dbaleen = 0.01m, cbaleen = 0.07m, hHT = 1.5m, and C = 0.44 (“with prey” [9]). The mouth inlet widths considered were Din = 0.30m (starburst and dash-dot line), Din = 0.50m (squares and continuous line) and Din = 0.72m (diamonds and dotted line) respectively. The slopes corresponding to the lines are as follows: 105.57 kg/m (8m; R2 = 0.99), 207.95 kg/m (10m; R2 = 0.99), 337.04 kg/m (15m-case B; R2 = 0.99) and 431.98 kg/m (15m-case A; R2 = 0.99). As with Fig 9, the presence of the straight lines in this plot suggest a swim speed-independent but Din-dependent drag coefficient for the drag generated by the internal flows of the oral apparatus (see Eq 6b).
Fig 11.
Scaling of the pressure plus kinetic energy difference between both ends of the oral cavity, versus swim speed.
This is the first term in the curly brackets of Eq 3, also called T1 in the text, normalized into kg/m. Data at each body size are represented as follows: 8m (starbursts), 10m (circles), 15m-case A; (squares) and 15m-case B (triangles). Note that each body size may include cases with different values of Din and hHT.
Fig 12.
Coefficient of total (CDtotal) versus swim speed and body size.
Calculated from Eqs 1–3 and 6a and for C = 0.44 (“with prey” [9]). Shown are the following body sizes: 8m (blue triangles); 10m (all black symbols, with Din = 0.50m (crosses), Din = 0.72m (times) and Din = 0.30m (starbursts); and 15m (yellow diamonds for case A and brown diamonds for case B). This numerical data is compared with the drag coefficient calculated from kinematic data collected during feeding (red filled circles labeled “F”) and non-feeding transport (red filled circles labeled “T”) [18]. Note that the foraging swim speed of balaenids has been observed in the range of 0.6 to 2m/s [18, 19–22].
Fig 13.
Viscous energy dissipation rate in each IB canals.
Example of the 10m whale featured in Figs 4 and 5. Data summation yields the total energy dissipated by the baleen system and yields a contribution to mouth drag (Eq 20) identical to the hydrofoil model of Eq 3.
Fig 14.
APT flow speeds near the oropharyngeal wall decrease with larger filtering surface area (Afilter) relative to mouth area (Ain).
Calculated from Eq 16. Note that Afilter is equal to the ratio Nb AIBcanal/Ain and thus proportional to the number of baleen plates (Nb). Full scale plot (top) versus reduced scale plot (bottom). The continuous and dashed curves correspond to AIBcanal/Ain = 0.02 and 0.20 respectively.
Fig 15.
Input values for the coefficients fAPT and xIB necessary for calculating kAPT and kIB.
These inputs are used in Eqs 13 and 15. In both frames the circles correspond to a 10m body size (Din = 0.50m (black), 0.30m (red) and 0.72m (yellow)); squares to 15m-case A and triangles for 15m-case B; and starburst for 8m. In all of these cases values of yrostr and zrostr have been set to yrostr = 0.60 to 0.70 and zrostr = 0.0085.
Fig 16.
BHC boundary layer thickness ratio at the end of each IB canal (top), and along the APT canal (bottom).
The ratio is defined as δ/(0.5*dbaleen), with δ calculated from Eqs 14 and 18. The data shown corresponds to a 10m whale swimming at Uwhale = 1.0m/s. The rest of the input parameters are the same as those of Fig 4.