Unit of Coefficient of Viscosity: SI Unit and Mathematical Formula

The unit of coefficient of viscosity is a very important topic in physics. Viscosity may not be seen with the eyes, but it is involved in almost all the things which we see around us. It plays a vital role, whether it is the flow of oil in the engines or it is the blood flow in our body. However, the viscosity is not related to how thick a substance is. The unit of Coefficient of Viscosity shows how much effort is required to make a layer of the substance move over another layer. This means the unit of coefficient of viscosity measures the amount of resistance a liquid offers to its flow.

Table of Contents

• What is the Coefficient of Viscosity?
• SI Unit of Coefficient of Viscosity

To get a clearer idea of how liquids resist motion, let’s first understand what the coefficient of viscosity really means.

What is the Coefficient of Viscosity?

Have you ever noticed how honey flows so slowly compared to water? That’s because of viscosity; it’s basically how much a liquid resists moving. When a liquid flows, its layers don’t all move at the same speed. The top layers might slide quickly, while the bottom ones lag behind. The resistance between these layers is what we call the coefficient of viscosity.

In simple words, the coefficient of viscosity tells us how much “push” is needed to make one layer of a liquid slide over another at a steady speed. The thicker the liquid, the more push it needs. That’s why honey has a higher viscosity than water.

We can put this idea into a formula:

η=FA×rv

Here:

• η is the coefficient of viscosity
• F is the tangential force applied
• A is the area of the layer
• v is the velocity difference between the layers
• r is the distance between the layers

So basically, viscosity depends on how much force you apply, how fast one layer moves compared to the other, and how far apart the layers are.

Now, let’s see how this connects to basic physical quantities like mass, length, and time. Using the dimensions of each term in the formula:

• Force (F) = M¹L¹T⁻²
• Area (A) = L²
• Distance (r) = L¹
• Velocity (v) = L¹T⁻¹

We get:

η\eta = \frac{M¹L¹T⁻²}{L²} \times \frac{L¹}{L¹T⁻¹} = M¹L⁻¹T⁻¹ 

So the dimensional formula of viscosity is [M¹ L⁻¹ T⁻¹]. In simple terms, this tells us that viscosity depends on mass, length, and time and links it with other physical properties of fluids.

Have you ever noticed how some liquids flow easily while others move slowly and thickly? 

For instance, water pours smoothly, but honey or oil takes its time. That difference comes from a property called viscosity, and the measure of it is known as the coefficient of viscosity.

Simply put, the coefficient of viscosity of a liquid tells us how much a liquid resists flowing. It is defined as the viscous force acting per unit area between two layers of a liquid, where the velocity gradient is normal to the direction of flow.

Mathematically, it can be written as:

η=FA⋅dvdx

Where,

• F = viscous force
• A = area of contact
• dv/dx = velocity gradient

The SI unit of the coefficient of viscosity is kg·m⁻¹·s⁻¹ or N·s/m².

Now, you might wonder: how do we actually measure this?

That’s where Poiseuille’s law comes in. It explains how a liquid flows through a thin capillary tube and helps calculate its viscosity using this formula:

η=πPr4t8LV

Here,

• P = pressure difference between the two ends of the tube
• L = length of the capillary
• r = inner radius of the capillary
• V = volume of liquid flowing through
• t = time taken for flow
• η = coefficient of viscosity

Every liquid behaves differently. Thicker liquids like oil or glycerine resist motion more, while lighter liquids like water flow easily. That’s why the coefficient of viscosity isn’t the same for all liquids.

For example, at 20°C, water has a coefficient of viscosity of about 1.002 × 10⁻³ N·s/m² (Pa·s).

But what happens if the temperature increases? As the liquid gets warmer, its molecules move faster, the internal resistance drops, and the liquid flows more freely. That’s why warm water pours faster than cold water.

Now that we know how viscosity works, it’s important to see how we measure it. This brings us to the SI unit of the coefficient of viscosity and its equivalents.

SI Unit of the Coefficient of Viscosity

The SI unit of the coefficient of viscosity is Newton-second per square meter (N·s/m²), which is also written as Pascal-second (Pa·s).

This unit tells us how much force is needed to move one layer of fluid over another with a given velocity difference.

We can also write:

1 Pa\cdotps=1 N\cdotps/m2=1 kg\cdotpm−1s−11 \, \text{Pa·s} = 1 \, \text{N·s/m}^2 = 1 \, \text{kg·m}^{-1}\text{s}^{-1}1Pa\cdotps=1N\cdotps/m2=1kg\cdotpm−1s−1

In the CGS system, the unit of viscosity is poise (P), where

\text{poise} = 0.1 \, \text{Pa·s}1poise=0.1Pa\cdotpsse 

Take a glance at these units of the coefficient of viscosity:

System	Unit	Equivalent Expression
SI Unit	Pascal-second (Pa·s)	kg·m−1s−1
CGS Unit	Poise (P)	g·cm−1s−1
Relation	1 P = 0.1 Pa·s	-

In this article, we learnt that the unit of coefficient of viscosity is a fundamental concept that helps us understand how liquids resist flow. By knowing its dimensional formula and practical applications, we can better describe and predict the behaviour of fluids in real life, from engine oils and paints to blood flow in our bodies. Understanding viscosity not only strengthens our grasp of physics but also helps in choosing the right materials for everyday applications.