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First published online January 3, 2006
Journal of Experimental Biology 209, 238-248 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.01974

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Unifying constructal theory for scale effects in running, swimming and flying

Adrian Bejan^1,* and James H. Marden²

¹ Duke University, Department of Mechanical Engineering and Materials Science, Durham, NC 27708-0300, USA
² Pennsylvania State University, Department of Biology, University Park, PA 16802, USA

View larger version (14K): [in a new window]   Fig. 1. (A) The periodic trajectory of a running animal, (B) model of foot contact with soft ground, and (C) history of vertical movement during one cycle, where F=0 refers to the time after t1 when the foot no longer exerts a force on the ground.

View larger version (26K): [in a new window]   Fig. 2. Comparison of theoretical predictions with the speeds, stroke frequencies, and force outputs of a wide variety of animals (Iriarte-Diaz, 2002; Drucker and Jensen, 1996; Pennycuik, 1975; Heglund et al., 1974; Marden and Allen, 2002; Taylor et al., 2003; Rohr and Fish, 2004; Bartholomew and Casey, 1978; Marsh, 1988; Wakeling and Ellington, 1997; Marden et al., 1997; Tennekes, 1997; Kiceniuk and Jones, 1977; May, 1995; Arnott et al., 1998; Peake and Farrell, 2004; Muller and Leeuwen, 2004; Bartholomew et al., 1985; Blickhan and Full, 1987; Full and Tu, 1991). The theoretical predictions are based on scale analysis, which neglects factors of order 1 and therefore should be accurate in an order of magnitude sense.

View larger version (14K): [in a new window]   Fig. 3. (A) The periodic trajectory of a flying animal, and (B) the cyclical progress of a swimming animal.

View larger version (65K): [in a new window]   Fig. 4. The blue line (A) shows schematically and in exaggerated form the slight elevation of the water surface over a large area (which requires total work=MgLb per body length travelled) that is centered over the anterior of a swimming body. (B,C) Reproductions of summary diagrams from two studies (Nauen and Lauder, 2002; Muller et al., 1997) of hydrodynamics of fish swimming. (B) The downward angle (–3°) of the central jet through the wake vortices formed by the tail of a mackerel, and the downward force (shown as a torque by the green arrow in this diagram) on the head. Both of these features are consistent with water lifted at the anterior end that falls at the posterior end. (C) A two-dimensional representation of water pressure and motion around a swimming mullet, with high pressure (P) at the anterior end and low pressure (S) on alternating sides of the vortices formed along the posterior. (D) Photos of a trout swimming slowly forward into a gentle current while held near the surface by a fine fishing line (not visible). Elevation of the water surface occurs over an arc at the anterior of the fish, with apparent patterns of flow and pressure quite similar to those depicted in C. Vertical displacement of the water surface during swimming at greater depth is spread over a larger area and harder to detect, but is fundamentally the same.