When a spherical body falls through a viscous fluid, it experiences a viscous force. The magnitude of viscous force increases with the increase in velocity of the falling body under the action of its weight. As a result, the viscous force soon balances the driving force (weight of the body) and the body starts moving with a constant velocity known as its terminal velocity.
When a solid sphere falls through a highly viscous liquid, it experiences three forces.
One is the gravitational force (i.e. weight of the body) which acts downwards, W = mg = V?g
Where, V is volume of the object, and ? is the density of material of the object.
Another is buoyant force, which always act in upward direction, and equal to the weight of the volume displaced by the object.
Thus, FB = V?g, where, ? is the density of the liquid.
It also suffers an upward drag force due to viscosity of the liquid which increases while going down which is given by,
F = 6??rv, where, ? is coefficient of viscosity, r radius of sphere, and v velocity at a given instant.
As it depends on the value of v, it increases as the sphere falls down.
At one stage the drag force becomes so strong that the all forces achieve an equilibrium, and the body falls at constant speed, this velocity is called terminal velocity.
At equilibrium, V?g = V?g + 6??rvt
or 6??rvt = Vg (? -?)
This velocity vt is called terminal velocity.