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On exact solutions of the unsteady Navier-Stokes equations - the vortex with instantaneous curvilinear axis. (English) Zbl 0609.76021

The unsteady pseudo-plane motions have been investigated in which each point of the parallel planes is subjected to non-torsional oscillations in their own plane and at any given instant the streamlines are concentric circles.
Exact solutions are obtained and the form of the curve \(\Gamma\), the locus of the centers of these concentric circles, is discussed. The existence of three infinite sets of exact solutions, for the flow in the geometry of an orthogonal rheometer in which the above non-torsional oscillations are superposed on the disks, is established. Three cases arise according to whether \(\omega\) is greater than, equal to or less than \(\sigma\), where \(\omega\) is the angular velocity of the basic rotation and \(\sigma\) is the frequency of the superposed oscillations.
For a symmetric solution of the flow these solutions reduce to a single unique solution. The nature of the curve \(\Gamma\) is illustrated graphically by considering an example of the flow between coaxial rotating disks.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
76M99 Basic methods in fluid mechanics
76U05 General theory of rotating fluids
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References:

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