We report experimental and numerical analysis of expontentially shaped long Josephson junctions with lateral current injection. Quasi-linear flux flow branches are observed in the current-voltage characteristic of the junctions in the absence of magnetic field. A strongly asymmetric response to an applied magnetic field is also exhibited by the junctions. Experimental data are found in agreement with numerical predictions and demonstrate the existence of a geometry-induced potential experienced by the flux quanta in nonuniform width junctions. ; Comment: 16 pg, 8 figures, Submitted in PRB March 4