The keyhole formed when a laser is used as the power source for a weld between two pieces of metal contains partially ionized metal vapour as well as other material. This flow is linked with the fluid motion and thermal conduction taking place in the liquid and solid regions surrounding it. A simple model is presented which describes the flow parallel to the axis of the keyhole when the welding speed is in the medium speed range. It is shown that shock waves may be present at its ends. The formulation is time dependent. Investigation of blind keyholes shows that there may be no solutions consistent with the assumptions of the model, but that when there are, solutions exist both with and without shocks present at the keyhole entrance. The character of the solutions can vary considerably. As welding speed increases the process becomes successively more dominated by direct absorption at the keyhole walls.