  By means of classical density functional theory and its dynamical extension,
we consider a colloidal fluid with spherically-symmetric competing
interactions, which are well known to exhibit a rich bulk phase behavior. This
includes complex three-dimensional periodically ordered cluster phases such as
lamellae, two-dimensional hexagonally packed cylinders, gyroid structures or
spherical micelles. While the bulk phase behavior has been studied extensively
in earlier work, in this paper we focus on such structures confined between
planar repulsive walls under shear flow. For sufficiently high shear rates, we
observe that microphase separation can become fully suppressed. For lower shear
rates, however, we find that e.g. the gyroid structure undergoes a kinetic
phase transition to a hexagonally packed cylindrical phase, which is found
experimentally and theoretically in amphiphilic block copolymer systems. As
such, besides the known similarities between the latter and colloidal systems
regarding the equilibrium phase behavior, our work reveals further intriguing
non-equilibrium relations between copolymer melts and colloidal fluids with
competing interactions.
