A bifurcation study of viscous flow through a rotating curved duct

Abstract

The combined effects of system rotation (Coriolis force) and curvature (centrifugal force) on the bifurcation structure of two-dimensional flows in a toroidal geometry of rectangular cross-section are examined. The problem depends on the Rossby number, Ro = U/bΩ, the Ekman number, Ek = ν/b2Ω, the aspect ratio, γ = b/h and the radius ratio, η = ri/ro; here U is the velocity scale, b is the channel width in the spanwise direction, Ω is the rotational speed, (ri, ro) are the inner and outer radii of the duct, h = ro – ri is the channel gap in the radial direction and v is the kinematic viscosity of the fluid. A pseudospectral method is devised to discretize the two-dimensional Navier–Stokes equation in stream-function form. Continuation schemes are used to track the solution paths with Rossby number as the control parameter. Extended systems are used to determine the precise location of the singular points of the discretized system. The loci of such singular points are tracked with respect to curvature of the duct. Unlike the findings of Miyazaki (1973) on the same problem, curvature is found to have profound effects on the solution structure; flow mutations take place through a tilted cusp at (Ro = 7.122, η = 0.678) and a transcritical bifurcation point at (Ro = 1.357, η = 0.349).