Printed in Great Britain THE MECHANICS OF LABRIFORM LOCOMOTION II. AN ANALYSIS OF THE RECOVERY STROKE AND THE OVERALL FIN-BEAT CYCLE PROPULSIVE EFFICIENCY IN THE ANGELFISH

It is likely that the ' undulatory propulsors ' 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 7-1070 in a time (tr, the recovery stroke duration time) of o-i s. The angular velocity (Q, 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 (/'), mid-points (r') and masses (m'e) of which are the same as those for /, r and me respectively, used in analysis of the power-stroke (Blake, 1979). The components of velocity perpendicular to the major axis of the fin, v'p, and parallel to the major axis, v'a are given by: v'p = Qr'+Vsmy', (1) Ve = VcO&y'. (2) The values of v'v and v'a are plotted against time in Fig. 2. The spanwise (dFe) force on the fin is given by: dF8 = i /2^ ()C,, (3