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Hydrodynamics of water flow in front of and through the gills of skipjack tuna
Camp. Liiochem. Physiol. Printed in Great Britam
Vol. 83A.
0300-9629/86 $3.00 + 0.00 ‘Q 1986 Pergamon Press Ltd
No. 2. pp. 255-259, 1986
HYDRODYNAMICS AND THROUGH
OF WATER FLOW IN FRONT OF THE GILLS OF SKIPJACK TUNA
E. DON STEVENSand E. N. LIGHTFOOT* Department of Zoology. University of Guelph, Guelph, Ontario. Canada NIG 2Wl. Telephone:
(519) 824-4120
and *Department Madison,
of Chemical Engineering, Wisconsin, USA
(Rewired
23 Mq
University
of Wisconsin,
1985)
The structure of the gills of skipjack tuna (Kafsuwonus pelamis) is reviewed and the pattern of water flow in front of, and through, the gills is described. The tuna system is attractive for analysis of
Abstract-l.
this type because water flow is continuous rather than pulsatile and the gill system is more rigid than that of other fish. In addition, oxygen uptake rates are the extreme case (i.e. highest values) for water-breathers, 2. Flow into the slits between the gill bars is rectilinear and nearly uniform, and therefore irrigation at the gill sieve should be nearly uniform. 3. Reynolds numbers are so low that turbulent flow is unlikely and entrance effects are negligible.
INTRODUCTION In this paper we bring together measurements of the structure and function of tuna gills to analyse (in a preliminary fashion) the pattern of water flow through them. Hills and Hughes (1970) concluded resistance to oxygen diffusion in the water in the gill pore is 5-lo-fold greater than through gill tissue. Theoretical analysis of the same problem also shows that most of the resistance to oxygen diffusion resides in the gill pore (Piiper, 1980; Piiper and Schied, 1984). If this is the case then an understanding of the hydrodynamics of water flow seems essential in order to describe the important features that limit oxygen movement in water. Hughes (1966, 1984) applied the Poiseuille equation for rectangular tubes (ignoring boundary layer effects):
fundamental assumption of the Hughes type model of flow. As Hughes (1966) stated: “It is assumed that the pressure gradient across them (the pore) remains constant and equal to the mean of the differential pressure measured simultaneously “. The fact that there is a lag between peak velocity and peak differential pressure suggests that inertial effects are important (they account for the lag). It also means that models based on such an assumption may be in error. An alternative explanation suggested by Lauder (1984) is that the phase difference is due to the movement of the gill bars. It is unlikely that this explanation applies to tuna because they irrigate their gills by holding the mouth open while continuously swimming and “ramming” water through the gill sieve. METHODS AND
where P, and PZ are pressures (dynes/cm2) on either side of the sieve, p is bulk viscosity (poise), d is pore width, I pore length, b pore height (cm), and Q (ml/pore/set) is flow. Total flow for the gill is calculated by multiplying Q by the number of pores. Total flow calculated in this way is about one order of magnitude too high and some reasons for this are detailed by Steen (1971). Holeton and Jones (1975) measured water velocity during the respiratory cycle using a blood flow probe on three large (3-3.5 kg) lightly anesthetized carp (C?iprinus carpio). They reported that: (I) velocity normal (i.e. perpendicular) to the probe was negligible; (2) peak velocity in front of the gill rakers was about 16 cm/set; (3) peak velocity more than doubled when ventilation was increased during hypoxia, and (4) velocity in front of the sieve was pulsatile. They also made simultaneous measurements of pressures in the buccal and opercular cavities and showed that peak velocity in front of the sieve lags behind peak differential pressure. We stress the significance of this observation because it raises some doubt regarding a
MATERIALS
Morphometric analysis was based on drawings that were made from sections of the head of a skipjack tuna (fork length 40 cm, weight 1.2 kg). The tish was freshly caught in Hawaiian waters and then perfused with formalin. It was decalcified in Jenkin’s solution (6 weeks, 3 changes/week), and embedded in paraffin. Sections of the whole head were made with a sliding microtome (1&20pm), and stained with H and E. The drawings were made from composite sections.
RESULTS
In the tuna gill, a large portion of each lamellae is fused to the lamella above or below it so that pore height is constant (filaments cannot be forced apart by high pressures or velocities). In addition, there are continuous supporting elements in front of and behind each filament. In Table 1 we present a summary of morphometric and flow data for a skipjack tuna gill. Some of the morphometric data are calculated from measurements by Muir and Hughes (1969) and so the analysis is based on a fish the same size as used by them (1667 g body weight). The skipjack tuna has the highest weight-specific gill surface exchange area
256
E. DON STEVENS and E. N. LIGHTFOOT Table I. Dimensions,
water velocity
and Reynolds
numbers tuna
for flow in front of and in the gill sieve of skipjack Actwe
A Mass Oxygen uptake (cm’ O:!Sec) B Ventdation volume (cm’ HzO/sec) C Non, in from of gills in slirs between rhr mchrs Total areaof slit5 at front of gill (cm’) n Average filament length with pores (cm) E I.‘2 width between arches (1.325 cm/l6 slits) F Average filament length pores + arch (cm) G Face velocity between arch (cm:sec) (C)O) H I R, = F x H ~ 0.01 Pore dmmsionc (cm) d = wdth J K h = height I = length L Total number of pores per fish M n; Total pore area (cm’) J x K x M Water path length at exchange surface (cm) 0 Flow wrll within rhe siew pores Average velocity (cm;sec) (C - N) R, = 0.5 x J x P - 0.01 ; Average tune water is in pore (set) (0 -P) R s Entrance length (cm) 7 Entrance length (“,, of pore length) u Estimated head pressure (dynes/cm’)
and the highest oxygen uptake of any fish, so these calculations represent the extreme case for fish. Oxygen uptake values are taken directly from Gooding ez ul. (1981). For the maximum value we use 1.5 mg O,/g/hr (they reported “the first five determinations of oxygen-consumption rate were 1.5mg OJg/hr or more”). Minimum rates are from their data extrapolated to zero swim speed. V, was calculated assuming that 7O:O of the oxygen is removed from the water as it passes over the gills (Stevens,
Standard
1667 0.486 143
0.0837 24.63 14.28 I .9 0.083 2.7
10.0 83
I .I? 14 0.0021 0.0127 0.146 7183683 I92 0.12
0.745 0.08 0.16 1.5 x 10-s 0.01 2370
0.128 0.01 0.94 2.5 x IO 0.002 408
’
1972) so that ti8 = PO, + 0.0034. The differential hydrostatic pressure across the gills during routine oxygen uptake is about 2cm water (Stevens, 1972). Water flow from mouth to slits between gill hurs Water velocity is greatest at the mouth and slows as cross-sectional area increases in the buccal cavity. The area of the mouth (1 cm*) relative to that in front of the gill slits is illustrated in Fig. IA in which the elements are drawn to scale. The arrows indicate the
A
Fig. 1. Schematic illustrating the water path in tuna gills. Figure IA is a front-on (i.e. looking into the buccal cavity from the mouth) view of the gill apparatus. Gill bars numbered with Roman numerals; water must enter the slits between the bars. Elements are drawn in relative proportion, although in lift there is a small variation in total length of the gill bars. The circle indicates the relative area at the entrance region (the mouth). In reality the apparatus is bent so that the top and bottom portions are closer to the reader. Figure I B is a drawing made from histological slides; top view. right side only. The path of water flow entering the slits between the gill bars is indicated by the arrows, Figure IC is a schematic of part of IB with the angles between the slits exaggerated to show rectilinear flow in the slit between adjacent hemibranchs and a typical streamline. The dashed lines delimit the boundaries of one unit cell of flow.
in tuna gills
Fig. 2. Drawing to show the relative size of five lamellae and the four pores between them. All elements drawn to scale. Arrows indicate path of water flow. The inset is exactly the same view as Fig. lB, but at higher magnification; the dashed line demarcates the structural non-exchange area in front of the lamellae. In the threedimensional drawing only the lamellae on top of the filament are drawn; a mirror image exists below. The lamellae shown are fused to those from the filament above in the front flat section.
path of water flow from the mouth (located about 10 cm above the page) into the slits between the gill bars. In the fish, the top and bottom of the gill bars are bent anterior at an angle of about 70”, about the level where the circle is drawn. The cross-sectional area in front of the slits is 14.28 times that at the mouth so that velocity is reduced more than one order of magnitude. The basic unit of flow in front of the sieve is illustrated schematically in Fig. le. There are 16 such flow units in a fish, one for each hemibranch. For each unit v is the velocity, and x is the distance in the direction of flow measured along the axis of the flow from the entrance between gill arches. Total area in front of slits was calculated from average slit length (10.78 cm, Muir and Hughes, 1969) and sum of the slit widths (1.325 cm) measured from histological sections. Average velocity within the sieve was calculated from pore area and water flow. One-half the entrance width in front of the sieve was used as the characteristic length to estimate Reynolds numbers. Engineers invariably use one-half slit width as the characteristic length for flow at low velocity passing through small slits (for examples and details, see Lightfoot, 1974). The kinematic viscosity of water (I x lo- * cm*/sec) is such that the Reynolds numbers for flow in front of the gill filaments tend to be relatively small (Table 1). Water ,flow within the sieve The morphometric data for sieve flow were calculated from the data of Muir and Hughes (1969). Pore width is the reciprocal of the number of lamellae per cm (323) minus the thickness of one lamella (10 pm). Pore height was calculated from the number of filaments on the second arch (390) and its length (11.5 cm). Hughes and Grimstone (1965) report
257
filament thickness as twice pore width (2 x 0.0021 = 0.004). Thus, pore height is (11.5/390 - 0.004)/2 = 0.0127 cm. Pore length is interpolated from data from 11 skipjack tuna (weight range 965-7010 g) in Muir and Brown (1971); where lamella length (cm) = -0.183 + 0.444 In weight. Muir and Brown suggested that the average water path length (0.12 cm) was slightly shorter than pore length (0.146 cm) (see arrows. Fig. 2). They suggested that the slight curve of the lamellae would direct the water flow obliquely across the surface. It is possible that water may flow less obliquely at low flow rates (R, = 0.01) and that path length may shorten as water velocity increases (R, = 0.08). The entrance effects are calculated and it is shown that they are negligibly small relative to total pore length. As flow enters the sieve its velocity is nearly uniform across the pore. Gradually a gradient develops with velocity greatest at the centre of the pore and flow is said to be “fully developed”. The entrance length is the length to the point at which the axial velocity is 99yd of the final velocity (Vogel, 1981). At the low Reynolds numbers for flow within the gill slits, the entrance length is approximately equal to the pore width (Lightfoot, 1974); still less than 2% of the pore length. Some of the numbers do not agree with those previously reported for tuna (Langille et al., 1983) because the present analysis incorporates new and better estimates of the variables. Ignoring details the following generalizations emerge: (1) Water velocity decreases more than an order of magnitude from the mouth to the entrance in front of the gill at the slits between the arches, and decreases another order of magnitude upon entering the gill sieve so that velocity in the sieve is about 0.005 of what it is upon entering the mouth. Face velocity varies from about 1 to 10 cm/set at the slits between gill arches. (2) During maximum oxygen uptake water velocity in the pore increases about six-fold over the minimum value. (3) On average a molecule of water is in the sieve about 1 set at low ventilation rates, and about 0.16 set at the highest ventilation rates. (4) Reynolds numbers are so low that turbulent flow is extremely unlikely. (5) Entrance effects are negligibly small. DISCUSSION
Reynolds numbers express the ratio of inertial forces to viscous forces. The low Reynolds numbers for gill water flow mean that inertia tends to be small compared to drag, so that when propulsion ceases then motion ceases. Negligible inertia means that no dynamic pressure is developed by a change in speed or direction so that water goes around curves and into the pores of the sieve with indifference (by this we mean it is solely influenced by viscous forces), and that there is no significant complex secondary or turbulent flow, i.e. flow is non-inertial and laminar. Laminar flow means that flow occurs in lamina or layers and that mixing between the layers is exceedingly difficult so that streamlines are predictable in detail. Generally, flow in tubes is laminar if
E.
258
DON STEVENSand
E. N.
LIGHTFOOT
of the entrance less than 2%). 3000
region within the gill sieve itself (i.e.
Microscopic ,jlow kvithin the gill sieve P‘ressure
= 22.7 x Flow
Flow of this type is fully developed creeping flow. If the flow profile is parabolic (the most reasonable assumption) then axial velocity in the center of the pore equals 1.5 times the average velocity. Fully developed flow means the form of the velocity profile is rapidly established and does not change as water flows through the pores. Thus, the hydrodynamic pressure gradients are uniform along the secondary lamellae; the pressure gradient across the pore near the front of the lamella is similar to that near the back of the lamella. Consequently, a study of gas exchange is somewhat simplified because it is not necessary to account for changes in the velocity profile as water flows past the secondary lamellae. Flow is described as two-dimensional slit flow.
GE Y r & 2000 0, r 7 t t 1000
20
40
60
60
100
120
140
Water Flow (cm3 H20/s)
v
Fig. 3. Relation between total volume of water flowing across the gills and total pressure head driving the flow. Data is from five skipjack tuna and details of methods are
given by Stevens (1972). Line is least squares regression through the origin; triangles are the extreme values estimated for total flow in this species.
R, < 1000. However, for the flow around bluff objects inertial effects always exist, begin to become appreciable at R, = 1.0, and dominate at R, > 1000. Water flow through tuna gills is complex and varied, with inertial conditions at the intake, and very nearly pure creeping flow in the sieve. One must therefore analyse each flow section separately and with care. The first step is to describe the flow pattern at the slits in front of the gill filaments then flow within the pores can be described. Wuter JOM’ at slits betkeen
gill bars
Assume that the fluid is directed through the buccal cavity and through the gill sieve by a pressure gradient, called suction by convention in fluid dynamics. Properties such as density and viscosity are assumed to be invariant. For the present analysis we theorize that flow in the wedge between the filaments is very nearly uniform rectilinear flow with negligible pressure gradients and therefore that perfusion at the sieve should be nearly uniform. A first approximation to macroscopic flow in the wedge between the gill arches in front of the sieve is given by: V = 6x V, for upstream V = 6x’V,, for downstream
flow,
(2)
flow.
This satisfies the equation of motion and yields uniform pressure on each side of the gill; it is probably a reasonable approximation. However, it does not satisfy the “no-slip” condition at the wall. It treats the screen-like entrance to the sieve as a plane surface, and neglects the small tangential stress necessary to deflect the fluid from flow in the negative-x direction to flow perpendicular to the sieve surface to enter the pores. However, departures from the predicted flow will be limited to a region with the thickness of the order of rl, and errors will not be significantly larger than those resulting from neglect
= (0.5d)‘AP
(3) 3 PI where d is pore width, I is pore length, AP is pressure drop across the sieve, p is dynamic viscosity. Using data in Table 1 then ll,g
1/ = (0.00105)’ x AP ocp 3 x 0.01 x 0.12
(4)
In order to estimate Vucxwe need an estimate of AP. Pressure drop, AP, and total flow across the gills of skipjack tuna have been measured by Stevens (1972) (Fig. 3). The least squares equation (forced through the origin because AP is zero at zero flow) is AP = 22.7 tig
(5)
where AP is in dynes/cm2, r’, is in cm’ H,O/sec. If we extrapolate equation 5 to the minimum and maximum flow rates from Table I (24.63 and 143 cm’ H,O/sec) then our estimates of AP are 559 and 3246 dynes/cm2. However, the pressure drop measured by Stevens included total pressure from inlet to outlet and it has been estimated that sieve losses are 0.73 of total losses (Brown and Muir, 1970). Thus, the estimates for BP for sieve flow reduce to 408 dynes/cm2 at minimum flow and 2370 dynes/cm’ at maximum flow. Substituting these values into equation 3 yields Vorlp= 0.73 cmjsec for maximum flow and Vnlp = 0.12 cm/s for minimum flow. These values are within lo:/, of the values in Table I. The closeness of the calculated and actual values suggests the approach of equation 3 is reasonable and that Brown and Muir’s estimate of relative sieve losses are reasonable. in summary, water velocity is reduced in order of magnitude as water enters the slits between the arch; thus Reynolds numbers are low and flow approaching the sieve can be treated as though it were approaching a two-dimensional screen. Flow is rectilinear on either side of the screen so that perfusion at the sieve should be nearly uniform. Flow within the sieve is fully developed creeping flow with negligible entrance effects and no significant turbulence. Inasmuch as velocities and Reynolds numbers are greater in tuna than other fish (Langille et a/., 1983) then it is also probably true that water flow within the gill
Water
flow in tuna gills
sieve of other fish is likely to be fully developed creeping flow with negligible entrance effects and no significant turbulence. Acknowledgements-We thank Dr Rich Brill National Marine Fisheries Service for perfusing and ing the fixed tuna specimen. Mr M. Baker-Pearce greatly in the preparation of the histological slides. thank Mary Anne Finkbeiner for typing.
of the providassisted We also
REFERENCES Brown C. E. and Muir B. S. (1970) Analysis of ram ventilation of fish gills with application to skipjack tuna. J. Fish. Res. Bd. Can. 27, 163771652. Gooding R. M., Neil1 W. H. and Dizon A. E. (1981) Respiration rates and low oxygen tolerance limits in skipjack tuna. Fish Bull. 79, 31-48. Hills B. A. and Hughes G. M. (1970) A dimensional analysis of oxygen transfer in the fish gill. Respir. Physiol. 9, 126-140. Holeton G. F. and Jones D. R. (1975) Water flow dynamics in the respiratory tract of the carp (Cyprinus carpio L.). J. exp. Biol. 63, 537-549. Hughes G. M. (1966) The dimensions of fish gills in relation to their function. J. exp. Biol. 45, 1777195. Hughes G. M. (1984) General anatomy of the gills. In Fish Physiology (Edited by Hoar W. S. and Randall D. J.), Vol. IO, pp. I-72. Academic Press, New York.
259
Hughes G. M. and Grimstone A. V. (1965) The fine structure of the secondary lamellae in the gills of Gadus pollachius. Q. J. Micr. Sri. 106, 343-353. Lauder G. V. (1983) Pressure and water flow patterns in the respiratory tract of the bass. J. exp. Biol. 113, 151-164. Langille B. L., Stevens E. Don and Anantaraman A. (1983) Cardiovascular and respiratory flow dynamics. In Fish Biomechanics (Edited by Webb P. W. and Weihs D.). pp. 92-139. Praeger, New York. Lightfoot E. N. (1974) Trunsport Phenomena and Living Sysfems. Wiley, New York. Muir B. S. and Brown C. E. (I 97 I) Effects of blood pathway on the blood-uressure drop in fish gills, with special reference to tunas. J. Fish. -Rex. Bd. C;?n. 28, 947-555. Muir B. S. and Hughes G. H. (1969) Gill dimensions for three species of tinny. J. exp. Bioi 51, 271-285. Piiper J. (1980) A model for evaluating diffusion limitation in gas exchange organs of vertebrates. In A Companion fo Animal Physiology (Edited by Taylor C. R., Johansen K. and Bolis L.), pp. 49-64. Cambridge University Press, New York. Piiper J. and Scheid P. (1984) Model analysis of gas transfer in fish gills. In Fish Physiology (Edited by Hoar W. S. and Randall D. J.), pp. 230-262. Academic Press, New York. Steen J. B. (I97 I) Comparative Physiology of Respiratory Mechanisms. Academic Press, New York. Stevens E. D. (I 972) Some aspects of gas exchange in tuna. J. exp. Biol. 56, 809-823. Vogel S. (198 I) L[fe in Moving Fluids. Willard Grant Press, Boston.
Vol. 83A.
0300-9629/86 $3.00 + 0.00 ‘Q 1986 Pergamon Press Ltd
No. 2. pp. 255-259, 1986
HYDRODYNAMICS AND THROUGH
OF WATER FLOW IN FRONT OF THE GILLS OF SKIPJACK TUNA
E. DON STEVENSand E. N. LIGHTFOOT* Department of Zoology. University of Guelph, Guelph, Ontario. Canada NIG 2Wl. Telephone:
(519) 824-4120
and *Department Madison,
of Chemical Engineering, Wisconsin, USA
(Rewired
23 Mq
University
of Wisconsin,
1985)
The structure of the gills of skipjack tuna (Kafsuwonus pelamis) is reviewed and the pattern of water flow in front of, and through, the gills is described. The tuna system is attractive for analysis of
Abstract-l.
this type because water flow is continuous rather than pulsatile and the gill system is more rigid than that of other fish. In addition, oxygen uptake rates are the extreme case (i.e. highest values) for water-breathers, 2. Flow into the slits between the gill bars is rectilinear and nearly uniform, and therefore irrigation at the gill sieve should be nearly uniform. 3. Reynolds numbers are so low that turbulent flow is unlikely and entrance effects are negligible.
INTRODUCTION In this paper we bring together measurements of the structure and function of tuna gills to analyse (in a preliminary fashion) the pattern of water flow through them. Hills and Hughes (1970) concluded resistance to oxygen diffusion in the water in the gill pore is 5-lo-fold greater than through gill tissue. Theoretical analysis of the same problem also shows that most of the resistance to oxygen diffusion resides in the gill pore (Piiper, 1980; Piiper and Schied, 1984). If this is the case then an understanding of the hydrodynamics of water flow seems essential in order to describe the important features that limit oxygen movement in water. Hughes (1966, 1984) applied the Poiseuille equation for rectangular tubes (ignoring boundary layer effects):
fundamental assumption of the Hughes type model of flow. As Hughes (1966) stated: “It is assumed that the pressure gradient across them (the pore) remains constant and equal to the mean of the differential pressure measured simultaneously “. The fact that there is a lag between peak velocity and peak differential pressure suggests that inertial effects are important (they account for the lag). It also means that models based on such an assumption may be in error. An alternative explanation suggested by Lauder (1984) is that the phase difference is due to the movement of the gill bars. It is unlikely that this explanation applies to tuna because they irrigate their gills by holding the mouth open while continuously swimming and “ramming” water through the gill sieve. METHODS AND
where P, and PZ are pressures (dynes/cm2) on either side of the sieve, p is bulk viscosity (poise), d is pore width, I pore length, b pore height (cm), and Q (ml/pore/set) is flow. Total flow for the gill is calculated by multiplying Q by the number of pores. Total flow calculated in this way is about one order of magnitude too high and some reasons for this are detailed by Steen (1971). Holeton and Jones (1975) measured water velocity during the respiratory cycle using a blood flow probe on three large (3-3.5 kg) lightly anesthetized carp (C?iprinus carpio). They reported that: (I) velocity normal (i.e. perpendicular) to the probe was negligible; (2) peak velocity in front of the gill rakers was about 16 cm/set; (3) peak velocity more than doubled when ventilation was increased during hypoxia, and (4) velocity in front of the sieve was pulsatile. They also made simultaneous measurements of pressures in the buccal and opercular cavities and showed that peak velocity in front of the sieve lags behind peak differential pressure. We stress the significance of this observation because it raises some doubt regarding a
MATERIALS
Morphometric analysis was based on drawings that were made from sections of the head of a skipjack tuna (fork length 40 cm, weight 1.2 kg). The tish was freshly caught in Hawaiian waters and then perfused with formalin. It was decalcified in Jenkin’s solution (6 weeks, 3 changes/week), and embedded in paraffin. Sections of the whole head were made with a sliding microtome (1&20pm), and stained with H and E. The drawings were made from composite sections.
RESULTS
In the tuna gill, a large portion of each lamellae is fused to the lamella above or below it so that pore height is constant (filaments cannot be forced apart by high pressures or velocities). In addition, there are continuous supporting elements in front of and behind each filament. In Table 1 we present a summary of morphometric and flow data for a skipjack tuna gill. Some of the morphometric data are calculated from measurements by Muir and Hughes (1969) and so the analysis is based on a fish the same size as used by them (1667 g body weight). The skipjack tuna has the highest weight-specific gill surface exchange area
256
E. DON STEVENS and E. N. LIGHTFOOT Table I. Dimensions,
water velocity
and Reynolds
numbers tuna
for flow in front of and in the gill sieve of skipjack Actwe
A Mass Oxygen uptake (cm’ O:!Sec) B Ventdation volume (cm’ HzO/sec) C Non, in from of gills in slirs between rhr mchrs Total areaof slit5 at front of gill (cm’) n Average filament length with pores (cm) E I.‘2 width between arches (1.325 cm/l6 slits) F Average filament length pores + arch (cm) G Face velocity between arch (cm:sec) (C)O) H I R, = F x H ~ 0.01 Pore dmmsionc (cm) d = wdth J K h = height I = length L Total number of pores per fish M n; Total pore area (cm’) J x K x M Water path length at exchange surface (cm) 0 Flow wrll within rhe siew pores Average velocity (cm;sec) (C - N) R, = 0.5 x J x P - 0.01 ; Average tune water is in pore (set) (0 -P) R s Entrance length (cm) 7 Entrance length (“,, of pore length) u Estimated head pressure (dynes/cm’)
and the highest oxygen uptake of any fish, so these calculations represent the extreme case for fish. Oxygen uptake values are taken directly from Gooding ez ul. (1981). For the maximum value we use 1.5 mg O,/g/hr (they reported “the first five determinations of oxygen-consumption rate were 1.5mg OJg/hr or more”). Minimum rates are from their data extrapolated to zero swim speed. V, was calculated assuming that 7O:O of the oxygen is removed from the water as it passes over the gills (Stevens,
Standard
1667 0.486 143
0.0837 24.63 14.28 I .9 0.083 2.7
10.0 83
I .I? 14 0.0021 0.0127 0.146 7183683 I92 0.12
0.745 0.08 0.16 1.5 x 10-s 0.01 2370
0.128 0.01 0.94 2.5 x IO 0.002 408
’
1972) so that ti8 = PO, + 0.0034. The differential hydrostatic pressure across the gills during routine oxygen uptake is about 2cm water (Stevens, 1972). Water flow from mouth to slits between gill hurs Water velocity is greatest at the mouth and slows as cross-sectional area increases in the buccal cavity. The area of the mouth (1 cm*) relative to that in front of the gill slits is illustrated in Fig. IA in which the elements are drawn to scale. The arrows indicate the
A
Fig. 1. Schematic illustrating the water path in tuna gills. Figure IA is a front-on (i.e. looking into the buccal cavity from the mouth) view of the gill apparatus. Gill bars numbered with Roman numerals; water must enter the slits between the bars. Elements are drawn in relative proportion, although in lift there is a small variation in total length of the gill bars. The circle indicates the relative area at the entrance region (the mouth). In reality the apparatus is bent so that the top and bottom portions are closer to the reader. Figure I B is a drawing made from histological slides; top view. right side only. The path of water flow entering the slits between the gill bars is indicated by the arrows, Figure IC is a schematic of part of IB with the angles between the slits exaggerated to show rectilinear flow in the slit between adjacent hemibranchs and a typical streamline. The dashed lines delimit the boundaries of one unit cell of flow.
in tuna gills
Fig. 2. Drawing to show the relative size of five lamellae and the four pores between them. All elements drawn to scale. Arrows indicate path of water flow. The inset is exactly the same view as Fig. lB, but at higher magnification; the dashed line demarcates the structural non-exchange area in front of the lamellae. In the threedimensional drawing only the lamellae on top of the filament are drawn; a mirror image exists below. The lamellae shown are fused to those from the filament above in the front flat section.
path of water flow from the mouth (located about 10 cm above the page) into the slits between the gill bars. In the fish, the top and bottom of the gill bars are bent anterior at an angle of about 70”, about the level where the circle is drawn. The cross-sectional area in front of the slits is 14.28 times that at the mouth so that velocity is reduced more than one order of magnitude. The basic unit of flow in front of the sieve is illustrated schematically in Fig. le. There are 16 such flow units in a fish, one for each hemibranch. For each unit v is the velocity, and x is the distance in the direction of flow measured along the axis of the flow from the entrance between gill arches. Total area in front of slits was calculated from average slit length (10.78 cm, Muir and Hughes, 1969) and sum of the slit widths (1.325 cm) measured from histological sections. Average velocity within the sieve was calculated from pore area and water flow. One-half the entrance width in front of the sieve was used as the characteristic length to estimate Reynolds numbers. Engineers invariably use one-half slit width as the characteristic length for flow at low velocity passing through small slits (for examples and details, see Lightfoot, 1974). The kinematic viscosity of water (I x lo- * cm*/sec) is such that the Reynolds numbers for flow in front of the gill filaments tend to be relatively small (Table 1). Water ,flow within the sieve The morphometric data for sieve flow were calculated from the data of Muir and Hughes (1969). Pore width is the reciprocal of the number of lamellae per cm (323) minus the thickness of one lamella (10 pm). Pore height was calculated from the number of filaments on the second arch (390) and its length (11.5 cm). Hughes and Grimstone (1965) report
257
filament thickness as twice pore width (2 x 0.0021 = 0.004). Thus, pore height is (11.5/390 - 0.004)/2 = 0.0127 cm. Pore length is interpolated from data from 11 skipjack tuna (weight range 965-7010 g) in Muir and Brown (1971); where lamella length (cm) = -0.183 + 0.444 In weight. Muir and Brown suggested that the average water path length (0.12 cm) was slightly shorter than pore length (0.146 cm) (see arrows. Fig. 2). They suggested that the slight curve of the lamellae would direct the water flow obliquely across the surface. It is possible that water may flow less obliquely at low flow rates (R, = 0.01) and that path length may shorten as water velocity increases (R, = 0.08). The entrance effects are calculated and it is shown that they are negligibly small relative to total pore length. As flow enters the sieve its velocity is nearly uniform across the pore. Gradually a gradient develops with velocity greatest at the centre of the pore and flow is said to be “fully developed”. The entrance length is the length to the point at which the axial velocity is 99yd of the final velocity (Vogel, 1981). At the low Reynolds numbers for flow within the gill slits, the entrance length is approximately equal to the pore width (Lightfoot, 1974); still less than 2% of the pore length. Some of the numbers do not agree with those previously reported for tuna (Langille et al., 1983) because the present analysis incorporates new and better estimates of the variables. Ignoring details the following generalizations emerge: (1) Water velocity decreases more than an order of magnitude from the mouth to the entrance in front of the gill at the slits between the arches, and decreases another order of magnitude upon entering the gill sieve so that velocity in the sieve is about 0.005 of what it is upon entering the mouth. Face velocity varies from about 1 to 10 cm/set at the slits between gill arches. (2) During maximum oxygen uptake water velocity in the pore increases about six-fold over the minimum value. (3) On average a molecule of water is in the sieve about 1 set at low ventilation rates, and about 0.16 set at the highest ventilation rates. (4) Reynolds numbers are so low that turbulent flow is extremely unlikely. (5) Entrance effects are negligibly small. DISCUSSION
Reynolds numbers express the ratio of inertial forces to viscous forces. The low Reynolds numbers for gill water flow mean that inertia tends to be small compared to drag, so that when propulsion ceases then motion ceases. Negligible inertia means that no dynamic pressure is developed by a change in speed or direction so that water goes around curves and into the pores of the sieve with indifference (by this we mean it is solely influenced by viscous forces), and that there is no significant complex secondary or turbulent flow, i.e. flow is non-inertial and laminar. Laminar flow means that flow occurs in lamina or layers and that mixing between the layers is exceedingly difficult so that streamlines are predictable in detail. Generally, flow in tubes is laminar if
E.
258
DON STEVENSand
E. N.
LIGHTFOOT
of the entrance less than 2%). 3000
region within the gill sieve itself (i.e.
Microscopic ,jlow kvithin the gill sieve P‘ressure
= 22.7 x Flow
Flow of this type is fully developed creeping flow. If the flow profile is parabolic (the most reasonable assumption) then axial velocity in the center of the pore equals 1.5 times the average velocity. Fully developed flow means the form of the velocity profile is rapidly established and does not change as water flows through the pores. Thus, the hydrodynamic pressure gradients are uniform along the secondary lamellae; the pressure gradient across the pore near the front of the lamella is similar to that near the back of the lamella. Consequently, a study of gas exchange is somewhat simplified because it is not necessary to account for changes in the velocity profile as water flows past the secondary lamellae. Flow is described as two-dimensional slit flow.
GE Y r & 2000 0, r 7 t t 1000
20
40
60
60
100
120
140
Water Flow (cm3 H20/s)
v
Fig. 3. Relation between total volume of water flowing across the gills and total pressure head driving the flow. Data is from five skipjack tuna and details of methods are
given by Stevens (1972). Line is least squares regression through the origin; triangles are the extreme values estimated for total flow in this species.
R, < 1000. However, for the flow around bluff objects inertial effects always exist, begin to become appreciable at R, = 1.0, and dominate at R, > 1000. Water flow through tuna gills is complex and varied, with inertial conditions at the intake, and very nearly pure creeping flow in the sieve. One must therefore analyse each flow section separately and with care. The first step is to describe the flow pattern at the slits in front of the gill filaments then flow within the pores can be described. Wuter JOM’ at slits betkeen
gill bars
Assume that the fluid is directed through the buccal cavity and through the gill sieve by a pressure gradient, called suction by convention in fluid dynamics. Properties such as density and viscosity are assumed to be invariant. For the present analysis we theorize that flow in the wedge between the filaments is very nearly uniform rectilinear flow with negligible pressure gradients and therefore that perfusion at the sieve should be nearly uniform. A first approximation to macroscopic flow in the wedge between the gill arches in front of the sieve is given by: V = 6x V, for upstream V = 6x’V,, for downstream
flow,
(2)
flow.
This satisfies the equation of motion and yields uniform pressure on each side of the gill; it is probably a reasonable approximation. However, it does not satisfy the “no-slip” condition at the wall. It treats the screen-like entrance to the sieve as a plane surface, and neglects the small tangential stress necessary to deflect the fluid from flow in the negative-x direction to flow perpendicular to the sieve surface to enter the pores. However, departures from the predicted flow will be limited to a region with the thickness of the order of rl, and errors will not be significantly larger than those resulting from neglect
= (0.5d)‘AP
(3) 3 PI where d is pore width, I is pore length, AP is pressure drop across the sieve, p is dynamic viscosity. Using data in Table 1 then ll,g
1/ = (0.00105)’ x AP ocp 3 x 0.01 x 0.12
(4)
In order to estimate Vucxwe need an estimate of AP. Pressure drop, AP, and total flow across the gills of skipjack tuna have been measured by Stevens (1972) (Fig. 3). The least squares equation (forced through the origin because AP is zero at zero flow) is AP = 22.7 tig
(5)
where AP is in dynes/cm2, r’, is in cm’ H,O/sec. If we extrapolate equation 5 to the minimum and maximum flow rates from Table I (24.63 and 143 cm’ H,O/sec) then our estimates of AP are 559 and 3246 dynes/cm2. However, the pressure drop measured by Stevens included total pressure from inlet to outlet and it has been estimated that sieve losses are 0.73 of total losses (Brown and Muir, 1970). Thus, the estimates for BP for sieve flow reduce to 408 dynes/cm2 at minimum flow and 2370 dynes/cm’ at maximum flow. Substituting these values into equation 3 yields Vorlp= 0.73 cmjsec for maximum flow and Vnlp = 0.12 cm/s for minimum flow. These values are within lo:/, of the values in Table I. The closeness of the calculated and actual values suggests the approach of equation 3 is reasonable and that Brown and Muir’s estimate of relative sieve losses are reasonable. in summary, water velocity is reduced in order of magnitude as water enters the slits between the arch; thus Reynolds numbers are low and flow approaching the sieve can be treated as though it were approaching a two-dimensional screen. Flow is rectilinear on either side of the screen so that perfusion at the sieve should be nearly uniform. Flow within the sieve is fully developed creeping flow with negligible entrance effects and no significant turbulence. Inasmuch as velocities and Reynolds numbers are greater in tuna than other fish (Langille et a/., 1983) then it is also probably true that water flow within the gill
Water
flow in tuna gills
sieve of other fish is likely to be fully developed creeping flow with negligible entrance effects and no significant turbulence. Acknowledgements-We thank Dr Rich Brill National Marine Fisheries Service for perfusing and ing the fixed tuna specimen. Mr M. Baker-Pearce greatly in the preparation of the histological slides. thank Mary Anne Finkbeiner for typing.
of the providassisted We also
REFERENCES Brown C. E. and Muir B. S. (1970) Analysis of ram ventilation of fish gills with application to skipjack tuna. J. Fish. Res. Bd. Can. 27, 163771652. Gooding R. M., Neil1 W. H. and Dizon A. E. (1981) Respiration rates and low oxygen tolerance limits in skipjack tuna. Fish Bull. 79, 31-48. Hills B. A. and Hughes G. M. (1970) A dimensional analysis of oxygen transfer in the fish gill. Respir. Physiol. 9, 126-140. Holeton G. F. and Jones D. R. (1975) Water flow dynamics in the respiratory tract of the carp (Cyprinus carpio L.). J. exp. Biol. 63, 537-549. Hughes G. M. (1966) The dimensions of fish gills in relation to their function. J. exp. Biol. 45, 1777195. Hughes G. M. (1984) General anatomy of the gills. In Fish Physiology (Edited by Hoar W. S. and Randall D. J.), Vol. IO, pp. I-72. Academic Press, New York.
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Hughes G. M. and Grimstone A. V. (1965) The fine structure of the secondary lamellae in the gills of Gadus pollachius. Q. J. Micr. Sri. 106, 343-353. Lauder G. V. (1983) Pressure and water flow patterns in the respiratory tract of the bass. J. exp. Biol. 113, 151-164. Langille B. L., Stevens E. Don and Anantaraman A. (1983) Cardiovascular and respiratory flow dynamics. In Fish Biomechanics (Edited by Webb P. W. and Weihs D.). pp. 92-139. Praeger, New York. Lightfoot E. N. (1974) Trunsport Phenomena and Living Sysfems. Wiley, New York. Muir B. S. and Brown C. E. (I 97 I) Effects of blood pathway on the blood-uressure drop in fish gills, with special reference to tunas. J. Fish. -Rex. Bd. C;?n. 28, 947-555. Muir B. S. and Hughes G. H. (1969) Gill dimensions for three species of tinny. J. exp. Bioi 51, 271-285. Piiper J. (1980) A model for evaluating diffusion limitation in gas exchange organs of vertebrates. In A Companion fo Animal Physiology (Edited by Taylor C. R., Johansen K. and Bolis L.), pp. 49-64. Cambridge University Press, New York. Piiper J. and Scheid P. (1984) Model analysis of gas transfer in fish gills. In Fish Physiology (Edited by Hoar W. S. and Randall D. J.), pp. 230-262. Academic Press, New York. Steen J. B. (I97 I) Comparative Physiology of Respiratory Mechanisms. Academic Press, New York. Stevens E. D. (I 972) Some aspects of gas exchange in tuna. J. exp. Biol. 56, 809-823. Vogel S. (198 I) L[fe in Moving Fluids. Willard Grant Press, Boston.