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Conically similar viscous flows. Part 1. Basic conservation principles and characterization of axial causes in swirl-free flow

Published online by Cambridge University Press:  20 April 2006

A. F. Pillow  and
R. Paull
A. F. Pillow
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, 4067, Queensland
R. Paull
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, 4067, Queensland
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Abstract

This paper has three main objectives. First, it aims to show that basic general conservation principles for viscous flow can be formulated in terms of diffusion and convection. Secondly, it aims to show that three scalar conservation principles suffice to provide a method for characterizing swirling axisymmetric flows in terms of axial and boundary production of the conserved quantities. Thirdly, it aims to exemplify these two objectives by giving a complete specification of the axial causes for swirl-free conically similar flow in otherwise free space.

This series of papers, overall, is concerned with the analysis and characterization of swirling conically similar flows in terms of the singularities that generate the conserved quantities. In conically similar flows there is no natural lengthscale, and the sole parameters governing the flow are provided by the strengths of the singularities that cause the flow. These are required to have the same dimensions as a power of the kinematic viscosity v. The axisymmetric flow generated by uniform production of swirl angular momentum per unit mass along a half-axis at a constant rate provides a simple example.

In conically similar flow the three conservation principles for axisymmetric flow provide a sixth-order non-autonomous system of two ordinary differential equations governing the flow. Here, in Part 1, these equations are derived for the general case of swirling flow, and are shown to reduce to a fourth-order system when swirl is absent. The two scalar conservation principles describing swirl-free flow are used to classify the basic axial causes for this system.

Part 2 analyses these basic exact one-parameter swirl-free families of solutions, and Part 3 extends the analysis to the remaining one-parameter family of swirling flows associated with uniform swirl angular-momentum production on a half-axis. Each of the families is characterized by a single independent cause, and two of them provide new non-trivial solutions of the Navier–Stokes equations. The effects of nonlinear coupling of these basic one-parameter causes and of conically similar distributions over conical boundaries will be examined in later papers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 155 , June 1985 , pp. 327 - 341
Copyright
© 1985 Cambridge University Press

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