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‘Unforced’ Navier–Stokes solutions derived from convection in a curved channel

Published online by Cambridge University Press:  11 June 2018

R. H. Vaz Open the ORCID record for R. H. Vaz [Opens in a new window] ,
F. A. T. Boshier  and
A. J. Mestel Open the ORCID record for A. J. Mestel [Opens in a new window]
R. H. Vaz*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
F. A. T. Boshier
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
A. J. Mestel
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

Steady Boussinesq flow in a weakly curved channel driven by a horizontal temperature gradient is considered. Linear variation in the transverse direction is assumed so that the problem reduces to a system of ordinary differential equations. A series expansion in $G$, a parameter proportional to the Grashof number and the square root of the curvature, reveals a real singularity and anticipates hysteresis. Numerical solutions are found using path continuation and the bifurcation diagrams for different parameter values are obtained. Multivalued solutions are observed as $G$ and the Prandtl number vary. Often fields with the imposed structure that satisfy all the governing equations are insensitive to the boundary conditions and can be regarded as perturbations of the homogeneous (or ‘unforced’) problem. Four such unforced solutions are found. In two of these the velocity remains coupled with temperature which, formally, scales as $1/G$ as $G\rightarrow 0$. The other two are purely hydrodynamic. The existence of such solutions is due to the unbounded nature of the domain. It is shown that these occur not only for the Dean equations, but constitute previously unreported solutions of the full Navier–Stokes equations in an annulus of arbitrary curvature. Two additional unforced solutions are found for large curvature.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Convection: Convection Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 848 , 10 August 2018 , pp. 676 - 695
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Copyright
© 2018 Cambridge University Press 

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