The experimental realization of thermocapillary flow without return flow is reported. This type of flow (linear flow) was proposed and analyzed theoretically by Smith and Davis (J. Fluid Mech., 132:119–144, 1983). We suppressed the return flow by providing channels and side channels with lower flow resistance compared to that of the return flow. Cooling the layer with linear flow from above results in the Marangoni instability of longitudinal rolls as the most dangerous mode. Strong linear flow stabilizes the system against longitudinal rolls. We report preliminary results on the threshold and on the wavelength of the longitudinal rolls.