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

For an appropriately scaled ABC-forcing the ABC-flow is a steady space-peri odic solution to the three-dimensional Navier-Stokes equation for an arbitr ary Reynolds number R. We are investigating both numerically and analytical ly different branches of steady solutions in the case A = B = C when the eq uation has a group of symmetries isomorphic to the rotation group of the cu be. Two families of steady flows, each comprised of three mutually symmetri c branches, were detected numerically, One was shown to persist for 7.9 les s than or equal to R less than or equal to 2000 and another for 149 less th an or equal to R less than or equal to 1000, Branches of the first family i ntersect 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 pos sibility of existence of branches of steady flows with other symmetries are considered. (C) 1999 Elsevier Science B.V. All rights reserved.