# What is the friction force at terminal velocity?

As terminal velocity is inversely proportional to viscosity or fluid friction, then my question is: at terminal velocity, the fluid friction is maximum or zero?

The terminal velocity of an object falling through a fluid is only inversely proportional to the fluid's viscosity when the Reynolds number is small (viscous drag the dominant resistive force). At large Reynolds numbers, terminal velocity is independent of viscosity (see here). In either case, however, the drag force felt by the object is a function of the relative velocity between the object and fluid. At the instant of release the object has zero velocity so, if the fluid is also still, the drag force on the object is zero at this instant. If the object falls under gravity (in a constant gravitational field) and does not lose mass then the impelling force, it's weight, is a constant. Under these conditions, the drag force increases from zero to a maximum value, equal in magnitude to the object's weight and opposite in direction, at which point the object is in a dynamic equilibrium and moves at the terminal velocity.

• @Yasir Mukhtar If this answers your question, please consider marking it as accepted. – Dai Apr 27 '15 at 13:44

When you reach the terminal velocity, you can apply Newton's 2nd Law $F=ma$, and since there is no acceleration (the terminal velocity is constant in time) , there must be a balance in the forces, such that their resultant vanishes.

In your case, the force into play are viscosity and the cause of the initial motion (for example a push, or gravity). When you reach the terminal velocity, the fluid friction is calculated simply as follows (all the relations are meant to be evaluated at equilibrium, i.e., after the particle has reached the terminal velocity):

$$F_{term}=-F_{friction}+F_{ext}=0$$

$$F_{friction}=F_{ext}$$

and since the friction force is (in general cases) of the form $-\eta v$, at the terminal velocity:

$$\eta v_{term}=F_{ext}$$

In particular, in the case of gravity as external force:

$$\eta v_{term}=m g$$

At terminal velocity the resistive forces (drag (you need to push the fluid aside) and friction (the actual friction between the falling body's surface and the surrounding fluid)) are equal to the accelerating forces (in most cases gravity).

Neither. 'Fluid Friction' can be considered as a drag force on a body moving through a fluid which is proportional to the square of the velocity. At terminal velocity the drag force balances the force of gravity - a dynamic equilibrium. If however additional downward force is increased - such as adding additional mass, the velocity will increase and so will the frictional (drag) force.

## protected by Qmechanic♦Apr 27 '15 at 13:56

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).