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Axisymmetric flows on the torus geometry

Published online by Cambridge University Press:  24 August 2020

Sergiu Busuioc Open the ORCID record for Sergiu Busuioc [Opens in a new window] ,
H. Kusumaatmaja Open the ORCID record for H. Kusumaatmaja [Opens in a new window]  and
Victor E. Ambruş Open the ORCID record for Victor E. Ambruş [Opens in a new window]
Sergiu Busuioc
Affiliation:
School of Engineering, University of Edinburgh, EdinburghEH9 3FB, UK
H. Kusumaatmaja*
Affiliation:
Department of Physics, Durham University, DurhamDH1 3LE, UK
Victor E. Ambruş*
Affiliation:
Department of Physics, West University of Timişoara, Timişoara300223, Romania Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA23529, USA
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

We present a series of analytically solvable axisymmetric flows on the torus geometry. For the single-component flows, we describe the propagation of sound waves for perfect fluids, as well as the viscous damping of shear and longitudinal waves for isothermal and thermal fluids. Unlike the case of planar geometry, the non-uniform curvature on a torus necessitates a distinct spectrum of eigenfrequencies and their corresponding basis functions. This has several interesting consequences, including breaking the degeneracy between even and odd modes, a lack of periodicity even in the flows of perfect fluids and the loss of Galilean invariance for flows with velocity components in the poloidal direction. For the multi-component flows, we study the equilibrium configurations and relaxation dynamics of axisymmetric fluid stripes, described using the Cahn–Hilliard equation. We find a second-order phase transition in the equilibrium location of the stripe as a function of its area ${\rm \Delta} A$. This phase transition leads to a complex dependence of the Laplace pressure on ${\rm \Delta} A$. We also derive the underdamped oscillatory dynamics as the stripes approach equilibrium. Furthermore, relaxing the assumption of axial symmetry, we derive the conditions under which the stripes become unstable. In all cases, the analytical results are confirmed numerically using a finite-difference Navier–Stokes solver.

JFM classification

Mathematical Foundations: General fluid mechanics Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , A9
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Busuioc et al. supplementary movie 1

Development of the instability under azimuthal perturbations of a fluid stripe centred on eq c = 0:86 on the torus with a = 0:4. [See Fig. 15(a) for further details]

Download Busuioc et al. supplementary movie 1(Video)
Video 3 MB

Busuioc et al. supplementary movie 2

Development of the instability under azimuthal perturbations of a fluid stripe centred on hetaeq c = 0:65 on the torus with a = 0:4. [See Fig. 15(b) for further details]

Download Busuioc et al. supplementary movie 2(Video)
Video 1.5 MB
Supplementary material: PDF

Busuioc et al. supplementary material

Supplementary data

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