We study experimentally the influence of a horizontal flow upon a unidimensional Rayleigh-Bénard roll chain. We show that it produces a roll diameter gradient proportional to the flow rate. Very small flow rates are sufficient to induce big changes in the roll diameters. This effect provides a way of locally obtaining roll diameters which are very different from the critical value, and even of observing a localized instability related to the Eckhaus instability. We can well describe those results by including an advection term in the phase diffusion equation.