Viscous Force - Difference between Viscosity and Friction for JEE

When fluid moves in the form of layers relative to each other, an internal resistance occurs between the moving layers which restricts the motion of the fluid layer. Viscosity is the property of the fluid that offers resistance to the relative velocity of the fluid. In viscosity, the internal resistance force developed between the layers of the fluid which restricts the relative motion between them is known as ‘viscous force.’  

In this article, attempts have been made to give brief information about viscous force, the dimensional formula of viscous force, the dimension of viscous force, and the difference between viscosity and friction. 

What is Viscous Force?

A viscous force is the internal resistance force that develops between the fluid layers to restrict the relative motion between them. The property of the fluid which offers  resistance between the fluid layers is known as ‘viscosity’. An example of viscous force is glycerin moving slowly, as it is more viscous than water. Viscosity is determined with the help of Newton's law of viscosity.

Newton’s Law of Viscosity

According to Newton's law of viscosity, the shear stress developed on the fluid layer is directly proportional to the velocity gradient. Mathematically, it is written as:

• $\tau \propto \dfrac{d u}{d y}$

• $\tau=\mu \dfrac{d u}{d y}$

In the above relation, the constant of proportionality is known as the ‘coefficient of viscosity.’ According to fluid mechanics, it is the coefficient of kinematic viscosity specifically.

Velocity gradient means the difference in velocity between adjacent layers of the fluid. In simple terms, it is the rate of change of velocity with respect to the change in height of fluid from the base of the fluid to the top layer of the fluid.

• $\tau=$Shear stress developed due to viscosity along with the fluid layers.

• $\mu=$Coefficient of viscosity.

• $\dfrac{d u}{d y}=$Velocity gradient of fluid.

Viscous Force Formula 

Viscous force formula is derived from the above relationship which originated from Newton's law of viscosity. The above relationship is given as:

• $\tau=\mu \dfrac{d u}{d y}$

• It is known that shear stress is the ratio of shear force and shear area. Therefore, 

• $\begin{align} &\dfrac{F}{A}=\mu \dfrac{d u}{d y} \\ &F=A \mu \dfrac{d u}{d y} \end{align}$

In the above relation, the shear force is the viscous force, which is the only internal resistance due to which shear stress develops along with the layers of the fluid. Therefore, the viscous force formula will be given as

• $F=A \mu \dfrac{d u}{d y}$

• F = The viscous force

• A = Area of the fluid layer 

From the above relations,

The viscous force per unit area is known as ‘shear stress’. Mathematically, it is written as

• $\dfrac{F}{A}=\mu \dfrac{d u}{d y}$

• $\begin{align} &\tau=\mu \dfrac{d u}{d y} \\ &\tau=\dfrac{F}{A} \end{align}$

Dimensional Formula of Viscous Force

The dimensional formula for the viscous force is derived using the above relation solved for the viscous force. The relation of viscous force is given as,

• $\begin{align} &\dfrac{F}{A}=\mu \dfrac{d u}{d y} \\ &F=A \mu \dfrac{d u}{d y} \end{align}$

• The dimensional formula for the restive terms are substituted in the above relation and solved for determining the dimensional formula for viscous force.

• $\begin{align} &A=\left(M^{0} L^{2} T^{0}\right) \\ &\mu=\left(M^{1} L^{-1} T^{-1}\right) \\ &d u=\left(M^{0} L^{1} T^{-1}\right) \\ &d y=\left(M^{0} L^{1} T^{0}\right) \end{align}$

Therefore, the dimensional formula for viscous force is obtained as follows:

• $\begin{align} &F=\dfrac{\left(M^{0} L^{2} T^{0}\right)\left(M^{1} L^{-1} T^{-1}\right)\left(M^{0} L^{1} T^{-1}\right)}{\left(M^{0} L^{1} T^{0}\right)} \\ &F=\left(M^{1} L^{1} T^{-2}\right) \end{align}$

Difference Between Viscosity and Friction

Conclusion

This article concludes with some interesting information about viscosity and viscous force. It also describes Newton's law of viscosity, which explains the relationship of shear stress developed in the fluid layer due to the internal resistance of the adjacent fluid layers. It also derived the dimensional formula for viscous force and shares its analogy with friction. This article covers the maximum portion of the viscous force.