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A class of generalized analytic functions arising from the Oseen equations in the axially symmetric case has been identified. For this class of functions, the generalized Cauchy integral formula has been obtained, and a series representation for the region exterior to a sphere has been constructed. The velocity field of the axially symmetric Oseen flow has been represented in terms of two generalized analytic functions, and it has been shown that for an exterior Oseen flow problem, those functions are uniquely determined, provided that they both vanish at infinity. Also the pressure and vorticity have been determined as the real and imaginary parts of the two functions representing the velocity field, and the drag exerted on a solid body of revolution in the axially symmetric Oseen flow has been expressed in terms of one of the involved generalized analytic functions. The problem of the axially symmetric Oseen flow past a solid body of revolution has been reduced to an integral equation based on the generalized Cauchy integral formula. The integral equation has been shown to have computational advantage over an integral equation based on the Oseenlets. The developed framework of generalized analytic functions has been illustrated in solving the problem of the Oseen flow past a solid sphere and solid bispheroids. For different Reynolds numbers, a minimum drag spheroid of fixed volume has been found. [ABSTRACT FROM AUTHOR]

An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the optimal shape for the fixed-volume nose of a solid torpedo-shaped body encountering minimum drag along its axis of revolution. In the later problem, the dependence of the nose's optimal shape on the torpedo's length has been investigated. [ABSTRACT FROM AUTHOR]

The framework of generalized analytic functions arising from the related potentials (so-called k-harmonically analytic functions) has been developed in application to three-dimensional (3D) axially symmetric Stokes flow problems. Cauchy's integral formula for the class of k-harmonically analytic functions has been obtained, and series representations for k-harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived. As the central result in the developed framework, a solution form representing the velocity field and pressure for 3D axially symmetric Stokes flows has been constructed in terms of two 0-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, the problem of 3D Stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary-value problem for two 0-harmonically analytic functions, and the resisting force exerted on the body has been expressed in terms of a 0-harmonically analytic function entering the solution form. For regions in which Laplace's equation admits separation of variables, the boundary-value problem can be solved analytically via representations of 0-harmonically analytic functions in corresponding curvilinear coordinates. This approach has been demonstrated for the axially symmetric translation of solid sphere and solid prolate and oblate spheroids. As the second approach, the boundary-value problem has been reduced to an integral equation based on Cauchy's integral formula for k-harmonically analytic functions. As an illustration, the integral equation has been solved for the axially symmetric translation of solid bispheroids and the solid torus of elliptical cross-section for various values of a geometrical parameter. [ABSTRACT FROM AUTHOR]