Why do elephants have big ear flaps?

@article{Narasimhan2008WhyDE,
  title={Why do elephants have big ear flaps?},
  author={A. Narasimhan},
  journal={Resonance},
  year={2008},
  volume={13},
  pages={638-647}
}
In this essay we shall try to answer the question in the title with some geometry and heat transfer thrown in for effect. For doing so, as the joke goes, allow me to assume the elephant as a sphere. The reason will be apparent soon. 
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References

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Heat loss in Dumbo: a theoretical approach.
TLDR
A flat plate model was used to calculate heat loss from the pinnae of the animated elephant Dumbo, suggesting that he may need the large ears to help lose the excess heat produced while flying. Expand
Why do elephants flap their ears
TLDR
The blood flow in the ear of the African elephant Loxodonta africana was measured using the dye dilution technique at the same time as the arterio-venous temperature difference to suggest that this organ is of major importance in thermoregulation under warm environmental conditions. Expand
Heat exchange by the pinna of the African elephant (Loxodonta africana).
  • P. K. Phillips, J. E. Heath
  • Environmental Science, Medicine
  • Comparative biochemistry and physiology. Comparative physiology
  • 1992
TLDR
Thermography indicates that the temperature distribution pattern across the pinna changes with ambient temperature and that areas of specialized motor control exist. Expand
Ambient temperature as a determinant of landscape use in the savanna elephant, Loxodonta africana
Elephants occur in landscapes where temperatures can reach 50 °C. Due to their large size they may face physiological problems of dissipating heat during such high temperatures. In spite of this, noExpand
Convection Heat Transfer
As we consider simultaneous fluid flow and heat transfer in porous media, the role of the macroscopic (Darcean) and microscopic (pore-level) velocity fields on the temperature field needs to beExpand
Dependency of surface temperature regulation on body size in terrestrial mammals
Abstract 1. 1. Infrared scans were taken of 29 mammal species ranging from 20 g white-footed mouse to 4000 kg African elephant and percent of surface area available for heat exchange determined. 2.Expand