Viscosity: Concept and Mathematical Expression

Basic concept of viscosity (η)

Anything which flows is termed a fluid. Gases and liquids both flow and hence are fluids. Any fluid, by nature, exists in the form of layers. During the fluid motion, the relative drag between the layers of the fluid results in resistance in the flow of the fluid. The resistance experienced by the fluids in motion is termed viscosity. The drag or internal friction between the layers of fluids in motion is the clear indicator of the measure of viscosity, as the large intermolecular forces of attraction resist the flow. Thus, fluid thickness, i.e. the number of layers in the fluid contributes to the viscosity significantly.

Liquids with a low magnitude of viscosity flow easily as compared to liquids with a high magnitude of viscosity. The clear reason for the smoother flow is the low magnitude of intermolecular forces of attraction, which results in lesser drag and hence, lesser friction. The concept of viscosity is also present in gases but is not as prominent as in the case of liquids in ordinary environments.

Mathematical expression to calculate viscosity (η)

Considering the laminar flow of the liquids, shearing stress plays an important role in the calculation of the viscosity of the fluids. Viscosity is the measure of the ratio of the shearing stress and velocity gradient of the fluid in motion. The motion of any object when dropped in the fluid describes the effect of viscosity and hence helps in calculating the magnitude of the viscosity of the fluid. In the ideal case, the object dropped is a sphere. Mathematically viscosity is calculated as:

η=2ga29v

Here, denotes the difference in densities of fluid and the object dropped in the fluid. The ideal object to get the formula of viscosity is a sphere. ‘a’ denotes the radius of the sphere, ‘g’ denotes the acceleration due to gravity and ‘v’ denotes the velocity of the sphere.

• The unit of the viscosity is newton-second per square metre (NSm^-2) or pascal-second (Pas), whose SI unit of viscosity is Poiseuille (Pl)
• Dimensional formula of viscosity is [ML^-1T^-1]

There is another way to measure the viscosity of the fluid, without dropping an object in the fluid. A scientific instrument used to measure the viscosity of any fluid is a viscometer. A U-shaped tube apparatus is also known as the Ostwald viscometer.

Types of Viscosity

As viscosity is defined as the measure of the friction in the flow of the fluid, it is categorized in two ways:

• Dynamic or absolute viscosity
• Kinematic viscosity

Dynamic or absolute viscosity comes into play when the external force is applied to the fluid. It is the measure of the external force applied to the fluid.

Kinematic viscosity on the other hand is the measure of the resistance experienced by the fluid under the weight of the gravity.

Effect of temperature on the viscosity of fluids

As we know that liquids and gases are both fluids and exhibit the concept of viscosity, the effect of temperature on the viscosity of both gases and liquids is significantly different. With the increase in temperature, the viscosity of liquids decreases considerably, whereas the viscosity of gases increases. An increase in temperature in the case of liquids results in the breaking of intermolecular forces of attraction and hence reduces the drag. In the case of gases, an increase in temperature increases the randomness of the gas, particles create hindrance in each other’s flow and hence increase the drag.

The viscosity of fluids is independent of the quantity of matter contained in the body, hence it is an intensive property.

Effect of pressure on the viscosity of fluids

With the increase in pressure on both liquids and gases, the intermolecular forces of attraction become even stronger. This results in an increase in the viscosity of both liquids and gases.

Types of Fluids

We now understand the effect of temperature and pressure on the viscosity of the fluids. When the viscosity of any fluid does not change with pressure, the fluid is termed a Newtonian fluid. The viscosity of non-Newtonian fluids, however, can be changed by changing the external stress and temperature. Pressure has no impact on the viscosity of water, hence, water is a Newtonian fluid, whereas, external pressure has an impact on the motion of fluid-like toothpaste, hence toothpaste is a clear example of a non-Newtonian fluid.