ABSTRACT
It is likely that the ‘undulatory propulsora’ employed by many fish are capable of producing a usefully directed thrust force over most of the tail-beat cycle. In contrast, fish which employ the ‘paddling pectoral fin propulsor’ only produce thrust during the power stroke phase of their fin-beat cycle, after which a recovery stroke occurs, when little usefully directed thrust force is produced. However, in order to gain a complete understanding of the mechanics of the paddling propulsor it is necessary to investigate the recovery stroke fully.
It is likely that the ‘undulatory propulsora’ employed by many fish are capable of producing a usefully directed thrust force over most of the tail-beat cycle. In contrast, fish which employ the ‘paddling pectoral fin propulsor’ only produce thrust during the power stroke phase of their fin-beat cycle, after which a recovery stroke occurs, when little usefully directed thrust force is produced. However, in order to gain a complete understanding of the mechanics of the paddling propulsor it is necessary to investigate the recovery stroke fully.
The basic movements performed by the pectoral fins during the recovery stroke have been qualitatively described and schematically illustrated (Blake, 1979). The present analysis is based on kinematic information derived from one representative stroke, taken from the same steady swimming sequence (forward velocity, V = 0·04 ms-1) selected for the power stroke analysis (Blake, 1979).
The leading edge of the right side pectoral fin moved from a positional angle (y′, the angle between the projection of the leading edge of the fin on to the horizontal plane) of about 7–107° in a time (tr, the recovery stroke duration time) of 0·1 s. The angular velocity (Ω, the angular velocity of the fin projected on to the horizontal plane) during the stroke is shown in Fig. 1.
We recognize four blade-elements (e′I-e′4); the values for the lengths (l′), midpoints (r′) and masses 
of which are the same as those for l, r and me respectively, used in analysis of the power-stroke (Blake, 1979).
, and parallel to the major axis, 
are given by: 
and 
are plotted against time in Fig. 2.
The component of velocity perpendicular to the major axis of the fin (•, e′1 ; ◼, e′2; ◯ e′ 3 and ▴, e′4) and the spanwiae velocity component (
, □) plotted against time.
: 
The power required to overcome the drag force in the direction of the body plotted against time.
is given by: 
is calculated as previously described (Blake, 1979). Here, however, the analysis is simplified by only considering the component of velocity that acts in a direction that is perpendicular to the fin’s major axis: 
amounts to about 3·0 × 10-7 J.
The impulse of the drag force acting in the direction of the body of the Angelfish during the recovery stroke is about 1 /20th of that associated with the hydrodynamic thrust force generated during the power stroke. The mean power associated with it is approximately one fifteenth of that required to produce the thrust force of the power stroke phase. The overall efficiency (ηc = 0·16) is about 11% less than the value calculated for the power stroke phase only (η ′ = 0·18; Blake, 1979). Comparable information on other animals employing the paddling propulsor is lacking, so comparisons can not be made at this stage.
Lighthill (1969, 1970) draws a distinction (applicable to animals swimming in the undulatory mode at high Reynolds Numbers) between those animals swimming with a high Froude efficiency (η > 0-5) and those which swim with a low Froude efficiency (η < 0·5). Lighthill’s analytical studies indicate that it is probable that fusiform fish swimming in the carangiform modes operate over most of their range of swimming speeds at levels of Froude efficiency similar to those expected of well designed screw propellers (η > 0·75).
Webb (1971) estimated the propulsive efficiency of trout (Salmo gairdeneri) from respirometric data and compared the values he obtained with those predicted on the basis of Lighthill’s reactive models. Good agreement was found at preferred cruising and high swimming speeds (where inertial effects dominate and the models designed to apply), with η > 0·7.
Using a very effective method of wake visualization, McCutchen (1975) calculated the Froude propulsive efficiency of a Zebra Dardo (Brachydanio rerio, length = 3·15 cm) during steady swimming and in the ‘push and coast’ mode. Values of about 0·8 were obtained during steady swimming over a range of speeds. However, an upper limit of 0·56 was calculated for the ‘push and coast’ mode.
It is likely that the pectoral fin propulsor studied here operates at a Froude efficiency (ηc = 0·15 –0·3) that is lower than values typical of fish swimming in the carangiform modes at their preferred cruising speeds. Webb (1975) estimates the propulsive efficiency of Cymatogaster aggregata (length = 14·3 cm, velocity = 55 cm s-1) to be between 0·6 and 0·65 ; showing that the lift-based mechanism of pectoral fin propulsion operates at a higher level of propulsive efficiency than the drag-based one studied here.
However, the efficiency of the ‘undulatory body and caudal fin’ modes of swimming fall off rapidly as swimming speeds decrease. At low forward speeds the paddling propulsor becomes more efficient than the undulatory mechanism (Blake, 1979) and it is probable that many fusiform fish switch over from the undulatory mode of swimming to a pectoral fin propulsion system when this happens.
ACKNOWLEDGEMENTS
I am grateful to Dr K. E. Machin, Professor Sir James Lighthill and Mr C. P. Ellington for their interest in this work. J would like to thank Mr G. G. Runnails for help with Photography and the N.E.R.C. for financial support.
