A numerical study of the processes that occur in a small cylindrical vessel with rigid walls, by the dissolution of a surfactant droplet placed on the capillary tip under the free water interface, is presented. Periodically arising Marangoni instability reveals itself in auto-oscillations of the surface tension. The effect of geometrical factors-namely, capillary immersion depth, capillary and droplet radii, and container height and radius-on the auto-oscillation characteristics is analyzed. The numerical and experimental results are compared and discussed.