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 ½ the diameter and ¼ 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.