The writer measured the torque between co-axial cylinder walls exerted by the water flow when the inner cylinder rotates. Furthermore, the writer carried out experiments on the co-axial cylinders having circumferential grooves on its inner cylinder surface. The results obtained are as follows : 1) When grooves and an axial flow are absent, the torque coefficient C_f increases as the clearance between two cylinders increases, and when the peripheral Reynolds number R_w is large, the clearance has little effect on the value of C_f. 2) The axial flow extends the range of laminar state of R_w, where the value of C_f decreases, but when the axial flow increases beyond that range the value of C_f increases again. 3) When there are grooves on the cylinder, it becomes difficult to keep the flow laminar, so that the torque hardly decreases with an increase of the axial flow.