# Coefficient of Viscosity

## Introduction

We all might have noticed that it takes a long time for the honey to reach the mouth of the bottle when the honey bottle is about to empty. Such liquid behaviour in the flow possesses some internal property which is explained by the term viscosity. It is the property of a liquid by which an internal force comes into play between different layers whenever there is a relative motion between these layers of the liquid. In simple words, viscosity is related to the thickness of the liquid. For example, honey is thicker than water so it will take more time to flow as compared to water as it has less resistance to flow. In technical terms, viscosity is the amount or quantity of opposition or resistance to flow provided by a liquid when it is under an applied force. Here, in the above cases, water has less force of attraction as compared to honey particles. Therefore, the water has more fluidity, less resistance to flow and lower viscosity, whereas honey has less fluidity, more resistance to flow and higher viscosity.

## What is Viscosity?

• It is a quantity of the opposition or resistance of a fluid to move under an applied force.

• The viscosity is the opposite or reciprocal of fluidity.

• When a liquid or gas has a low viscosity it tends to flow more easily as its molecular interaction causes very little friction when moving.

• When a liquid or gas has a high viscosity it flows with difficulty due to its internal friction caused by its sticky, thick, and semi-fluid consistency nature.

• Mathematically viscosity is defined by this formula :

$$\mathrm{\eta =Viscosity=\frac{\tau}{\lambda }=\frac{shear\:stress}{shear\:rate}}$$

where, shear stress means a parallel force acting on the surface of an object, and shear rate is the rate of change in speed when one layer of fluid exceeds the adjacent layer.

## Viscosity Measurement

• It is the quantity of a substance's (fluids) opposition or resistance to a motion under an applied force.

• It is measured in terms of $\mathrm{(\frac{dynes-sec}{cm^{2}})}$

• The fundamental unit of viscosity measurement is poise.

• A material requiring shear stress of one dyne per centimetre to produce a shear rate of one reciprocal second has a viscosity of one poise or 100 centipoises.

• The final result is expressed in centipoise (cP), i.e,

$$\mathrm{1cP=10^{-3}Pa.S=1mPa.s}$$

There are different types of Instruments to Measure Viscosity are -

• Rotational Rheometry.

• Capillary Viscometers.

• Non-Contact Rheology.

• Vibrating Viscometers.

• Viscosity Measurements with Formulation.

• Microfluidic Rheometers.

## What is a Viscous Gradient?

• It is the difference in velocity between the adjacent layers of the fluid. If more force is applied by the upper layer to move forward the more will be the viscous gradient.

• It is represented by $\mathrm{\frac{v}{x}}$, where v is the velocity difference and x will be the difference in distance between the two layers.

• The formula for the viscous gradient is :

$$\mathrm{\tau _{yx}=\mu \cdot \frac{dv_{x}}{dy}=\mu \cdot \gamma _{yx}}$$

Where,

$$\mathrm{\tau _{yx}=shear \:stress;\mu =Viscosity;}$$

$$\mathrm{\frac{dv_{x}}{dy}\:or\: \cdot \gamma _{yx}=velocity\:gradient\:or\:viscous\:gradient\:or\:shear\:rate}$$

• It affects the Rate of Flow of Liquids

• As viscosity increases, flow rates decrease.

• As viscosity decreases, flow rates increase.

## The Coefficient of Viscosity Formula

It is the measure of the degree to which a fluid opposes or resists flow under an applied force. It is denoted by the symbol $\mathrm{\eta}$ . It is the ratio of shearing stress to the strain rate. Generally, the gas viscosity is lower than the liquid viscosity. This is because of its intermolecular force of attraction. The inverse of viscosity is called fluidity. Thus, a fluid with higher viscosity tends to flow slow and a fluid with lower viscosity tends to flow fast.

### The formula for the Coefficient of Viscosity is given by

$$\mathrm{(\eta )=\frac{F\times r}{A\times v}}$$

where, $\mathrm{\eta}$ = coefficient of viscosity, F = tangential force, r = distance between the layers, A = area, v = velocity

### It has a Dimensional Formula

F or Force = $\mathrm{M^{1}L^{1}T^{-2}}$

A or Area = $\mathrm{L^{2}}$

Velocity gradient = $\mathrm{\frac{dv}{dx}=[LT^{-1}]/[L]=[T^{-1}]}$

Therefore,

$$\mathrm{\eta =\frac{[MLT^{-2}]}{[L^{2}]\times[T^{-1}] }=[ML^{-1}T^{-1}]}$$

## SI Unit of Coefficient of Viscosity

It is a measure of the resistance force caused by a liquid to stop the related movements between the layers of the liquids.

It is first derived from resistance force, the formula numerically represented as :

$$\mathrm{F=\eta A\frac{dv}{dz}\cdot \cdot \cdot (a)}$$

Or

$$\mathrm{\eta =\frac{F}{A\cdot \frac{dv}{dz}}}$$

Where,

$$\mathrm{\eta =Coefficient\:of\:viscosity}$$

$$\mathrm{F=Forces\:of\:resistance}$$

$$\mathrm{A=Area\:of\:contact}$$

$$\mathrm{\frac{dv}{dz}=Velocity\:of\:gradient}$$

Therefore, the units for the following are:

$$\mathrm{F=1\:Newton}$$

$$\mathrm{A=metre\:per\:square}$$

$$\mathrm{\frac{dv}{dz}=\frac{metre}{second}\times \frac{1}{metre}=\frac{1}{second}=second^{-1}}$$

Put these values in the above expression (a).

We get,

$$\mathrm{1\:newton=(\eta)(metre^{2})(second^{-1})}$$

Hence,S.I unit is:

$$\mathrm{\eta =\frac{1\:Newton\times second}{metre^{2}}\:or\:\frac{N.s}{m^{2}}\:or\:Pa.s}$$

## Unit of Coefficient of Viscosity

In terms of Units
S.I or International System of Units $\mathrm{\frac{N\times second}{metre^{2}}or\:Pa.s}$
CGS or centimeter-gram-second unit $\mathrm{\frac{dyne-sec}{cm^{2}}or\:1\:poise}$
MKS or meter-kilogram-second unit $\mathrm{\frac{Kgf-sec}{m^{2}}}$

## Viscosity Examples

Some of the examples are

Sl.No Substances Viscosity or $\mathrm{\eta}$ of substances (Pa.s)
1 Air $\mathrm{10^{-5}}$
2 Water $\mathrm{10^{-3}}$
3 Ethyl alcohol $\mathrm{1.2\times 10^{-3}}$
4 Mercury $\mathrm{1.5\times 10^{-3}}$
5 Ethylene glycol $\mathrm{20\times 10^{-3}}$
6 Olive oil 0.1
7 Honey 10
8 Corn syrup 100
9 Bitumen $\mathrm{ 10^{8}}$
10 Molten glass $\mathrm{ 10^{12}}$

## The Viscosity of Water in SI Units

The coefficient of viscosity ($\mathrm{\eta}$) of water can be determined by using Poiseuille’s law.

Poiseuille's liquid flow equation determines the volume of fluid flowing through the capillary tube per unit of time. The equation will be as follows

$$\mathrm{V=\frac{\pi \Delta pr^{4}t}{8\eta L},}$$

Therefore, the coefficient of viscosity of water will become

$$\mathrm{\eta=\frac{\pi \Delta pr^{4}t}{8V L}}$$

Where, V = Volume of the Liquid, r = radius of the vessel, t = time, $\mathrm{\eta}$ = coefficient of viscosity,$\mathrm{\delta p}$ = change of pressure, L = length of the vessel

Therefore

$$\mathrm{\eta=\frac{Newton\times second}{metre^{2}}}$$

The SI unit of viscosity of water is $\mathrm{Ns.m^{-2}or\:Pa.s.}$

## Conclusion

The term viscosity is defined as the resistance of fluids (gas or liquid) to flow. In simple words, the reciprocal of the viscosity is called fluidity. Fluidity is the tendency or ease of the fluid to flow, whereas viscosity is the quantity or measure of the resistance of fluids to flow. It also depends on the state of the fluids, such as thier pressure, rate of deformation and temperature. A liquid with high viscosity is opposing the movement of fluids to flow because of its strong intermolecular forces and gives it a lot of internal friction, resisting the movement of more than one layer over another. In contrast, a liquid with low viscosity flows easily because its molecular interactions cause very little friction when moving. Gases also show viscosity but are difficult to detect in normal conditions. But due to its lower intermolecular force of attraction, it has larger fluidity.

## FAQs

Q1. What happens to the coefficient of viscosity when temperature increases?

Ans. Viscosity depends on temperature, when temperature increases the molecules of fluids are free to move, thereby decreasing the coefficient of viscosity and increasing the flow of fluids. It means the coefficient of viscosity decreases when temperature increases.

Q2. Why do gases have low viscosity?

Ans. The intermolecular force of attraction is least in gases and molecules are free to flow easily. When gaseous molecules are free to flow means it has a low viscosity.

Q3. Does the viscosity of air change with pressure?

Ans. The viscosity increases with an increase in pressure. This is because an increase in pressure will increase the force of attraction between the air particles, therefore, making air difficult to flow.

Q4. Which flows faster: oil or water?

Ans. According to the concept of viscosity, more resistance for fluids to flow means more viscosity or thicker the substance. Hence water has less resistance as compared to oil, therefore, water flows faster.

Q5. Does impurity affect viscosity?

Ans. The impurities like $\mathrm{CO_{2}}$ will affect both density and viscosity and also affects the flow properties of the fluid.

Updated on: 28-Apr-2023

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