Abstract
The three-dimensional (3-D) analytic-function theory whose foundation was developed in Part I is further developed and applied to representing ideal flow solutions in fluid dynamics. A 3-D complex velocity and 3-D complex potential are defined in terms of analytic functions of a 3-d variable. Components of the complex potential relate to a 3-D flow velocity potential and 3-D stream functions. It was shown in Part I that the parts of every analytic function of one 3-D variable provide the components of a solenoidal and irrotational ( S and I) harmonic vector. In this paper, S and I harmonic vectors are shown also to be provided by the vector terms of a certain trigonometric-series expansion of every elementary 3-D analytic function. (“Elementary” here refers to an analytic function extended from the set of ordinary complex analytic functions.) The “primary part” (defined as the first real vector term in such an expansion) of every elementary analytic function of one 3-D variable produces an axisymmetric-flow solution. Specific simple examples are shown, including the functions whose respective primary parts give the velocity potential and Stokes stream function of elementary flows including a uniform stream, stagnation flow, source flow, flow due to a doublet (dipole) and flow over a sphere.
Published Version
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