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# Viscosity-Unit and Dimension, Newton's Law of Viscosity, Newtonian and Non-newtonian Fluids

Let’s imagine that you are going to school and trying to eat your breakfast in a hurry. If you are having pancakes for breakfast and decided to have some honey with it to make it tasty. Now you will observe that honey takes some time to come out of the bottle. This might frustrate you because you don't have enough time as you are getting late for school. Now my question to you is, why is it easier to take out water from a bottle than honey? The answer is honey has higher viscosity than water. Let’s learn about viscosity!

• What is viscosity?
• Unit and dimension of viscosity
• Factors affecting viscosity
• Viscosity of common fluids
• Newton’s law of viscosity
• Newtonian and non-newtonian fluids
• Practice problems
• FAQs

## What is viscosity?

When one layer of a fluid slips or tends to slip on the second layer which is in contact, the two layers exert tangential forces on each other. The direction of this tangential force is such that it opposes the relative motion between the two layers. This property of a fluid which opposes the relative motion between its layers is called viscosity. This opposing force is called viscous force. Therefore, we can say that viscosity is the internal friction of fluid in motion.

Simplest example of viscous flow is the motion of fluid between two parallel plates. The bottom layer is stationary and the top layer is moving with velocity . In the figure you can see the flow speed is decreasing uniformly from top to bottom, shown by arrows.

A fluid having high viscosity resists relative motion between the layers because of strong intermolecular forces that give a lot of internal friction. On the contrary, a fluid with low viscosity flows easily because its molecular forces result in very low friction when it is in motion. Gases also exhibit viscosity, but it is harder to notice in ordinary circumstances.

For example, when we compare honey and water, honey is more viscous than water, i.e., it is more resistant to the flow as compared to that of water. Have a look at the image below. We can easily steer a glass of water compared to honey.

## Unit and dimension of coefficient of viscosity

• SI unit of viscosity is or Pascal-second () commonly known as Poiseuille.
• CGS unit of viscosity is Poise ()
• Dimension of viscosity:

## Factors affecting viscosity

Temperature Dependance:

• For Liquid: with the rise in temperature, the viscosity of liquids decreases. This is due to the decrease in cohesion forces between the molecules.
• For Gases: with the rise in temperature, the viscosity of liquids increases. This is due to the increase in the rate of diffusion of gases from one moving layer to another because random motion of atoms increases.

Pressure Dependance:

• For Liquids: Generally the viscosity of liquids increases with rise in pressure, but for water there is a decrease in viscosity with the rise in pressure (for first few hundred atmospheric pressure)
• For Gases: The viscosity of gases is independent of normal pressure. But for low and high pressure, it is directly proportional to the pressure.

## Newton’s law of viscosity

Now let’s find this force of friction between the layers of fluid because of viscosity. This frictional force is called viscous force. We can calculate the value of viscous force using Newton’s law of viscosity.

According to Newton’s Law of viscosity, viscous force depends on the following factors:

• is directly proportional to the area () of layers in contact
• is directly proportional to the velocity gradient between the layers

So,

Here, is the constant of proportionality called the coefficient of viscosity.

• Negative sign represents that the direction of viscous force () is opposite to the direction of relative velocity of the layer.
• Value of depends on the nature of the fluid.
• For given liquid value decreases with increase in temperature as it weakens the intermolecular forces.
• For gasses value increases with increase in temperature.

## Newtonian and Non-newtonian fluids

From Newton’s law of viscosity,

Now we define,

Therefore,

If we plot the graph between ,

If the nature of the graph for the fluids that follow Newton’s law of viscosity is a straight line as shown below, Then such fluids are called Newtonian Fluids. Water and starch solution are some examples of Newtonian fluids.

The fluids that do not follow Newton’s law of viscosity are known as Non-Newtonian Fluids. Examples of non-Newtonian fluids are blood, toothpaste, and soap solution, etc. Graphs for Non-Newtonian fluids such as Pseudoplastic fluid and Dilatant fluid are shown below.

## Practice Problems

Q. Water is flowing over an inclined plane (thickness of the layer is ). The velocity of the top surface is . Find the force acting on the top surface per unit area? (

A.

Let’s take the magnitude of the force acting per unit area,

Q. A plate 0.02 mm distant from a fixed plate moves at and requires a force of to maintain its speed. Determine fluid viscosity between the plates?

A.

We know, ……….considering only magnitude

Q. A sliding fit cylindrical body of mass of 1 Kg drops vertically down at a constant speed of . Find the viscosity oil?

A. As body moves with constant velocity,

……..forces are balanced

……considering the magnitude of viscous force

Q. A plate of area is placed on the upper surface of castor oil, thick. Taking the coefficient of viscosity to be 15.5 poise, calculate the horizontal force necessary to move the plate with a velocity ?

A. ,

## FAQ

Q. What is the relation between the viscosities of honey and milk?
A.

Q. Is the viscosity intensive property or extensive?
A. Viscosity depends on the nature of fluids, not on the amount of fluid. Therefore viscosity is an intensive property.

Q. How does temperature affect the viscosity of fluids?
A. For liquid viscosity decreases with increase in temperature as it weakens the intermolecular forces. For gasses η value increases with increase in temperature.

Q. What is the nature of the graph between shear stress and velocity gradient for Newtonian Fluids?
A. Straight Line

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