Spatially-periodic steady solutions to the three-dimensional Navier-Stokes equation with the ABC-force

For an appropriately scaled ABC-forcing the ABC-flow is a steady space-periodic solution to the three-dimensional Navier-Stokes equation for an arbitrary Reynolds number R. We are investigating both numerically and analytically different branches of steady solutions in the case A = B = C when the equation has a group of symmetries isomorphic to the rotation group of the cube. Two families of steady flows, each comprised of three mutually symmetric branches, were detected numerically, One was shown to persist for 7.9 less than or equal to R less than or equal to 2000 and another for 149 less than or equal to R less than or equal to 1000, Branches of the first family intersect twice with the ABC-flow in a bifurcation generic to a system with symmetry group D-3. Other possible bifurcations of the ABC-flow and the possibility of existence of branches of steady flows with other symmetries are considered. (C) 1999 Elsevier Science B.V. All rights reserved.