Elevator Pitch: Dumbo’s EARodynamics

During their glider tests, the Wright brothers utilised an equation for lift that incorporated Smeaton’s coefficient—a value representing air pressure force—but their gliders underperformed, prompting them to question the accuracy of this coefficient and Lilienthal’s lift data. Through wind tunnel experiments in 1901, they determined that the accepted Smeaton coefficient of 0.005 was too high, finding a more accurate value of 0.0033. They discovered that wing shape significantly affects lift, rendering Lilienthal’s data inapplicable to different geometries. The brothers tested over fifty different models to determine how lift and drag are affected by various design parameters and used this data to invent the world’s first powered aeroplane in 1903. Their aircraft, the Flyer, travelled approximately 37 metres in 12 seconds, opening the way for aviation to flourish[1]. Forty years later, with the creation of Walt Disney’s Dumbo, another milestone in aviation was achieved. Following its historic flight in 1941, the elephant completely overthrew the long-established laws of physics.

In this work, the physics behind Dumbo’s flight are explored in greater detail. First, one must consider the issue of Dumbo’s ear size and its correlation to his ability to fly with his original weight. Secondly, it would be interesting to consider what weight Dumbo would have had to be able to fly with his original ear size.

The author of this article has to acknowledge the fact, that this article goes where someone else has gone before, which means, that it relies heavily on the values already published by Philipps & Heath in their paper about a heat loss model for Dumbo. In their groundbreaking work, they used scale drawings provided by the Walt Disney Company to ascertain Dumbo’s statistics and concluded that Dumbo must have been 0.762 m tall. By using this scale of reference, his ears must have measured 0.686 m by 0.457 m. The then calculated area of his ears would be 0.3135 m² (one side). In addition to that, they plotted the known heights and mass of several elephants to estimate Dumbo’s actual weight. With that, they concluded that Dumbo must have weighed around 75 kg[2]. However, for our calculations, we have to make a few simplifications (or shall I say: we have to „wing“ it). In the following calculations, Dumbo is treated rather as a plane than an elephant to simplify the lift equation:

where v is velocity, A is the wing/ear area, C_L is the lift coefficient, and  is the density of air.

In order to lift an object from the ground, the upward force needs to be greater than its weight. Dumbos weight is converted from kg to N (Newtons) with the following equation, where 9.8 m/s² is the gravitational acceleration.

with inserting Dumbos statistics:

To lift Dumbo from the ground, the lift (L) has to be greater than 735 N.

According to the ISO International Standard Atmosphere, the standard sea level density of air () at 101.325 kPa and 15 °C is 1.2250 kg/m³[3]. It is assumed, that Dumbo is gliding with a speed of around 10 m/s (36 km/h). The lift coefficient (C_L) describes the dependency of object shape on lift. The lift coefficient also contains the effects of air viscosity and compressibility. For simplicity, a C_L of 0.3 (similar to a Boeing 747)[4] is assumed:

This means, that with his original weight, Dumbo would have needed an ear area larger than 40 m² in order to fly, which corresponds to about 20 m² per ear. This is 64 times larger than his originally depicted ear size. However, the lift equation assumes stable gliding flight rather than flapping movement, resulting in a rough estimation of the ear size.

In order to determine the weight that Dumbo would need to be able to fly with his actual ear size, we must revisit the lift equation (1):

This means, that Dumbo must have weighted 1.2 kg in order to fly with his original ear size. Using the same regression as Philipps & Heath (weight = 170 * height³), the elephant’s height must have been about 0.224 m or 22.4 cm. In contrast to the Flyer, which achieved a gliding speed of 8—11 m/s and weighed approximately 115 kg (depending on which Wright brother was piloting), the required wing area was 93 m²—about 2.3 times larger than that of Dumbo. For perspective, a Boeing 747-100 weighs roughly 333 390 kg (about 4 445 times Dumbo’s original weight), cruises at 252 m/s (25 times faster), and has a wing area of 511 m²— approximately 13 times larger.

As stated in the article, three aircraft were referenced. Each of these aircraft can be regarded as an example of innovative aviation engineering. However, it is only Dumbo that has fundamentally altered the laws of physics with its remarkable ear-powered flight. He is the exception that proves the rule.

[1] Padfield GD, Lawrence B. The birth of flight control: An engineering analysis of the Wright brothers’ 1902 glider. The Aeronautical Journal. 2003;107(1078):697-718. doi:10.1017/S0001924000013464

[2] Phillips, P., & Heath, J. (2001). Heat loss in Dumbo: a theoretical approach. Journal of Thermal Biology, 26(2), 117–120. doi:10.1016/s0306-4565(00)00031-0

[3] Advanced Aircraft Design: Conceptual Design, Analysis and Optimisation of Subsonic Civil Airplanes, First Edition. Egbert Torenbeek. ©2013 by Egbert Torenbeek. Published 2013 by John Wiley & Sons, Ltd.}.

[4] Ahmad, M., Hussain, Z. L., Shah, S. I. A. & Shams, T. A. (2021). Estimation of Stability Parameters for Wide-Body Aircraft Using Computational Techniques. Applied Sciences, 11(5), 2087. https://doi.org/10.3390/app11052087