A pump according to the present invention has a circular diaphragm 4 placed at the bottom of a casing 2. At the bottom of the diaphragm 4, a piezoelectric element 6 is installed in contact with the diaphragm 4. A narrow space between the diaphragm 4 and the top wall of the casing 2 constitutes a pumping chamber 8. An inlet flow path 12 and an outlet flow path 14 are open to the pumping chamber 8, wherein a check valve 10 is installed in the inlet flow path 12. Immediately downstream of the pumping chamber, the outlet flow path 14 has a narrow segment 16. The narrow segment 16 of the outlet flow path has the diameter and the cross sectional area of the outlet flow path 14. The outlet flow path 14 has a return inlet 22, which is connected to a return outlet 23 in the inlet flow path via an active valve 24. The active valve 24 is opened and closed freely by an actuator 26 made of shape-memory alloy.