The optimum shapes of bodies moving in a medium taking friction into account

Within the limits of the model of local interaction of a body and a medium, the special features of the design of optimum three-dimensional bodies are investigated taking friction into account. It is assumed that the pressure on the body surface is described by a two-term formula which is quadratic in the velocity and has constant terms representing the strength of the medium. Three models of the friction are used to represent the shear stresses: constant friction, friction proportional to the pressure, and mixed friction. A comparative analysis is carried out of solutions of problems of optimizing the body shape with respect to drag and with respect to penetration depth, obtained in the class of three-dimensional configurations for the different models of the friction. It is shown that, if the base area of the body is given, the optimum shapes for all the friction models are those in which the normal at each point makes a constant optimum angle with the direction of motion. This angle is independent of the base area and is determined by the velocity of motion and the parameters of the model, which depend on the characteristics of the medium. The influence of the parameters of the model on the optimum shapes is demonstrated and, for each model, formulae are derived relating the velocity of motion and the characteristics of the medium with the optimum angle.