The factors influencing the optical path stability in a dye laser flow cell are studied numerically and experimentally. A specially designed curved metallic dye flow cell providing a gain medium of 25 mm x 0.5 mm x 0.2 mm along with a compact resonator mechanical assembly is used in the study. The same configuration with gain medium of 15 mm x 0.5 mm x 0.2 mm is successfully used for single mode dye laser. The effects of flow induced vibrations on dye flow cell are studied with and without mechanically coupling it with the resonator structure for flow speeds varying from 1.33 m/s to 6.67 m/s at laser pump position. The effect of the mechanical instability, velocity fluctuation and temperature fluctuations in flowing dye solution on dye laser performance are studied at different flow speeds. These results are compared with the dye laser output parameters and found to be in good agreement. This study is useful in designing a high stability narrowband dye laser.