Why a small sphere released from rest to fall through liquid in a deep tank eventually moves with constant speed ?

As the sphere moves in the liquid, it experiences a viscous force upwards that tries to resist its downward motion caused by gravity(it's own weight). This viscous force gradually increases with depth, as the speed rises(because the weight causes a positive acceleration in it's direction). At a point, the viscous force upwards becomes exactly equal to the weight downwards. Then the sphere reaches equilibrium. It doesn't have any external force acting on it, so it can't have any acceleration. So, it's velocity becomes constant, and this velocity is known as the terminal velocity.