Chiral symmetry breaking and entropy production in Dean vortices

Abstract

In toroidal pipes, the secondary flow in cross section is a mirror symmetric pair of counter-rotating axially oriented Dean vortices. This mirror symmetry is broken in helical pipes. We investigate in detail the mirror symmetry breaking in these secondary flows in going from toroidal to helical geometries. We quantify the degree of mirror symmetry breaking in helical flows by calculating both an (i) order-parameter − 1 ≤ χ ≤ 1 that measures the net integrated chirality of vortices in section and (ii) the entropy production due to both viscous shear forces and convection for Dean vortices as the Dean number and pitch of the helix are varied. We prove that the entropy production due to convective processes is always greater than that due to viscous shear, for stationary incompressible flows in the absence of body forces. For the same pipe radius and pipe curvature, fluid density, viscosity, and entrance flows, the vortex entropy production in the stationary state is minimized for helical conduits (for a given Dean number) with respect to that of toroidal pipes (zero pitch). The dissipation in the fluid flow due to Dean vortices decreases in going from a toroidal to a helical geometry, while the chiral order parameter tends to χ = ± 1 for finite values of the pitch as the Dean number is decreased.