Lagrangian Solutions of the Equations of Motion in Fluid Mechanics by Integrating Certain Types of Nonlinear Ordinary Differential Equations

Abstract

The classical problems of the forced motion of a fluid particle induced by the motion of solid bodies immersed in an incompressible and nonviscous fluid of unbounded extent are solved by integrating nonlinear equations of the type: (d2r/dt2) + P(r,t)(dr/dt)3 + Q(r,t)(dr/dt)2 + R(t)dr/dt = 0, and (d2r/dt2) + M(r)(dr/dt)2 + N(r,t)dr/dt + S(r,t) = 0. The solutions of the Lagrangian equations of motion instead of the usual Euler's equations furnish better physical insight. The second integrals are evaluated in terms of both elliptic integrals and series for the problem of cylinder, and hyperelliptic integrals and series for the sphere. A systematic method of integrating nonlinear equations of this type is discussed. The problem of a half-body is also discussed.