Functional dependencies of the hydrodynamic fields of an axisymmetric stationary flow of viscous fluid

Abstract

Analysis of the equations of motion for axisymmetric stationary viscous fluid flows in terms of the variables of stream function, vortex, and Bernoulli function shows that, for essentially viscous flows, the rank of the Jacobi matrix of the system of these hydrodynamic variables is equal to 2, which implies the existence a functional relationship between them given by a single expression. A basis for finding this relationship is the equation derived as a corollary of the equations of motion and vortex transfer. It is represented as a linear combination of the gradients of three hydrodynamic fields, the coefficients of which are equal to the second order minors of the Jacobi matrix. Using an integrating factor, the linear combination is reduced to a full gradient of some function that remains constant at least locally on the solution of the original system of hydrodynamic equations. This conserved quantity is used to find an expression for the functional relationship between the Bernoulli function, the modified vortex and the stream function. To this end it is necessary to find the coefficients of a linear combination as the functions of previously-unknown hydrodynamic fields, rather than spatial variables. This requires consideration of a closed system of equations constructed in this paper. The form of the desired functional relationship and the hydrodynamic fields themselves are determined by finding its solutions. Examples of solutions of this kind, which can serve as a basis for testing different algorithms, are given.