Based on a one-dimensional fluid model, the collision-dominated ion dynamics in capacitively coupled rf discharges are studied theoretically as well as numerically in the low rf frequency regime omega less than or equal to V-B/l (here omega is the rf frequency, V-B is the Bohm velocity and l is the plasma length). It is found that the ion current entering the sheath varies with time as we have shown in the collisionless sheath [N. Xiang and F. L. Waelbroeck, J. Appl. Phys. 93, 5034 (2003)]. The mechanisms for the ion current modulation in the collision-dominated and collisionless sheaths. however. are different. It is also shown that the sheath dynamics and ion energy distribution are affected significantly by this time-varying ion current. (C) 2005 American Vacuum Society.