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

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.

Some similarity solutions for three-dimensional boundary layers

“…Guided by our experience with flows linear in a transverse coordinate [11,12], we now investigate whether or not this solution is unique. The technique we use is to add a fictitious forcing term αF(η) to one or more of our equations and to find the solution as the continuation parameter α varies.…”

“…These are readily solved numerically in §4, giving rise to a typical boundary layer profile. Drawing on experience with other flows linear in one coordinate [11,12], we use path-continuation techniques to locate a second solution, with a more elaborate structure. Solving a time-dependent problem in §5 reveals that the second of these solutions is unstable.…”