Using molecular simulations, we study the properties of a polymer brush in contact with an explicit solvent under Couette and Poiseuille flow. The solvent is comprised of chemically identical chains. We present evidence that individual, unentangled chains in the dense brush exhibit cyclic, tumbling motion and non-Gaussian fluctuations of the molecular orientations similar to the behaviour of isolated tethered chains in shear flow. The collective molecular motion gives rise to an inversion of the hydrodynamic flow direction in the vicinity of the brush-coated surface. Utilising Couette and Poiseuille flow, we investigate to what extent the effect of a brush-coated surface can be described by a Navier slip condition.