Thursday 06 Oct 2016: Scalar description of three-dimensional flows of incompressible fluid
Prof Yury Stepanyants - University of Southern Queensland, Australia
An essential progress in investigation of flows of incompressible fluid may be achieved with the help of scalar functions, e.g., the velocity potential or stream-function. Indeed, flow description by means of one scalar function is much simpler than the description based on the three-dimensional vector field. Many interesting and physically important problems were solved by this way. However, the traditional usage of the velocity potential or stream-function is restricted by certain assumptions – in the former case the flow is assumed to be ideal and potential, whereas in the latter case the flow may be viscous, but consisting of two-components only with only one component of vorticity. Such restrictions essentially limit the range of applicability of the traditional approaches.
Here we propose another approach, also based on the introduction of only one scalar function. However, we show that with this scalar function a wide class of non-stationary three-dimensional flows can be described. This class of flows includes both potential and vortex flows. In the latter case, the corresponding vorticity field may consist of two-components, in general. Characteristic features of such flows are described in details. Particular examples of flows are presented in the explicit form. We also derive the Bernoulli integral for this class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical non-stationary motions of a viscous incompressible fluid. We suggest a generalisation of the developed concept which allows one to describe a certain class of 3D flows with 3D vorticity.
Yury A. Stepanyants (1) and Evsey I. Yakubovich (2)
(1) Faculty of Health, Engineering and Sciences, University of Southern Queensland, Toowoomba, Australia
(2) Department of Geophysical Research, Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia