The paper considers a one-dimensional Brusselator model with a uniform flow of the mixture of reaction components. An absolute as well as a convective instability can arise for both the Hopf and the Turing modes. The corresponding linear stability analysis is presented and supported by the results of computer simulations of the nonlinear equations. Finally, the condition for spatially undamped tails (the Cherenkov condition) is obtained. This represents a new mechanism for pattern formation in chemical reaction-diffusion systems.