According to the equation $v²=GM/r$ to decrease the orbital radius of a satellite currently in orbit, you'd want to increase its speed. However intuitively this doesn't make sense to me, surely if you decreased the speed (e.g. by ejecting a mass in the direction of movement) the resultant force would cause the satellite's path to curve inwards? Please someone explain this to me. I'd like to know the practical way of decreasing the orbit and the intuition behind why it works.
Suppose a satellite is in a circular orbit at some altitude $h$ and is supposed to change its orbit to a circular orbit of $h-\Delta h$, where $\Delta h$ is positive. The orbital velocity will necessarily increase.
This seems paradoxical, but to accomplish this, this spacecraft has to reduce its orbital velocity, and then reduce it again half of an orbit later. The first reduction in orbital velocity results in an elliptical orbit with apogee at the original circular altitude and perigee at some lower altitude, dictated by how much the satellite reduced its orbital velocity.
The satellite will reach perigee half of an orbit after the initial burn. At this point, its orbital velocity will be greater than that of a satellite orbiting circularly at that altitude. This means the satellite in question has to once again fire its thrusters to reduce its orbital velocity, only to find that its orbital velocity has increased compared to the original orbital velocity.