HYDRAULICS. ENGINEERING HYDROLOGY. HYDRAULIC ENGINEERING

Azimuthal vorticity and stream function in the creeping flow in a pipe

Pages 150-159

The article is devoted to the analytical study of the structure of steady non-uniform creeping flow in a cylindrical channel. There are many papers on the hydrodynamics of such flows, mainly related to the production of polymers. Previously we showed that the structure of steady non-uniform creeping flow in a cylindrical tube is determined by the Laplace equation relative to the azimuthal vorticity. The solution of Laplace's equation regarding the azimuthal vorticity is dedicated to the first half of the article. Fourier expansion allows us to write the azimuthal vortex in the form of two functions, the first of which depends only on the radial coordinate, and the second depends only on the axial coordinate. Fourier expansion can come to the Sturm - Liouville problem with a system of two differential equations, one of which is homogeneous Bessel equation. The radial-axial distribution of the azimuthal vorticity in the creeping flow is obtained on the basis of a rapidly convergent series of Fourier - Bessel. In the next article the radial-axial distribution of the stream function will be discussed. The solution is constructed from the Poisson equation based on the solution for the azimuthal vortex distribution. Fourier expansion can come to the Sturm - Liouville problem with a system of two differential equations, one of which is inhomogeneous Bessel equation. The inhomogeneous Bessel equation is solved through the Wronskian. The distribution of the stream function is obtained in the form of rapidly converging series of Fourier - Bessel.

DOI: 10.22227/1997-0935.2014.4.150-159

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