Dimensions of Coefficient of Viscosity: Dimensional Formula Explained

Correct Answer: (A) ML⁻¹T⁻¹

Understanding the Coefficient of Viscosity

The coefficient of viscosity is a physical quantity that measures the internal resistance offered by a fluid to the relative motion of its adjacent layers. When one layer of a fluid moves over another, the fluid opposes this relative motion because of internal friction. This resistance to flow is known as viscosity.

A highly viscous liquid offers greater resistance to flow, while a less viscous liquid flows more easily. For example, honey flows much more slowly than water because honey has a greater coefficient of viscosity. The coefficient of viscosity is generally represented by the Greek letter η (eta).

To determine the fundamental dimensions of the coefficient of viscosity, we use Newton’s law of viscosity and express every physical quantity involved in terms of mass M, length L and time T.

Newton’s Law of Viscosity

According to Newton’s law of viscosity, the tangential force acting between two adjacent layers of a fluid is directly proportional to the area of contact and the velocity gradient between the layers.

Mathematically:

F = ηA(dv/dx)

where F is the viscous force, η is the coefficient of viscosity, A is the area of contact between the fluid layers and dv/dx is the velocity gradient.

The velocity gradient describes how rapidly the velocity changes with perpendicular distance between adjacent fluid layers. By rearranging Newton’s law of viscosity, the coefficient of viscosity can be written as:

η = F/[A(dv/dx)]

We can now determine the dimensions of each quantity on the right-hand side and obtain the dimensional formula of η.

Step-by-Step Derivation of the Dimensional Formula

Dimensions of Force

Force is the product of mass and acceleration:

Force = Mass × Acceleration

The dimensions of mass are M, while the dimensions of acceleration are LT⁻². Therefore:

[F] = M × LT⁻²

Hence:

[F] = MLT⁻²

Dimensions of Area

Area is the product of two length dimensions. Therefore:

[A] = L²

Dimensions of Velocity Gradient

The velocity gradient is represented by:

dv/dx

The dimensions of velocity are:

[v] = LT⁻¹

The dimensions of distance are:

[x] = L

Therefore, the dimensions of the velocity gradient are:

[dv/dx] = LT⁻¹/L

After cancelling the length dimension:

[dv/dx] = T⁻¹

Thus, the velocity gradient has the dimensions of inverse time.

Calculating the Dimensions of the Coefficient of Viscosity

We have:

η = F/[A(dv/dx)]

Substituting the dimensions of force, area and velocity gradient:

[η] = [MLT⁻²]/[L² × T⁻¹]

Combining the terms in the denominator:

[η] = [MLT⁻²]/[L²T⁻¹]

Subtracting the powers of corresponding fundamental dimensions gives:

[η] = ML¹⁻²T^{−2−(−1)}

Therefore:

[η] = ML⁻¹T⁻¹

Hence, the dimensional formula of the coefficient of viscosity is ML⁻¹T⁻¹, making option (A) the correct answer.

Alternative Derivation Using Shear Stress

Newton’s law of viscosity can also be expressed in terms of shear stress:

Shear stress = η × Velocity gradient

Therefore:

η = Shear stress/Velocity gradient

Shear stress is force per unit area. Therefore, its dimensions are:

[Shear stress] = [Force]/[Area]

= MLT⁻²/L²

Hence:

[Shear stress] = ML⁻¹T⁻²

The dimensions of velocity gradient are T⁻¹. Therefore:

[η] = [ML⁻¹T⁻²]/[T⁻¹]

Thus:

[η] = ML⁻¹T⁻¹

This alternative approach gives exactly the same dimensional formula.

SI Unit of the Coefficient of Viscosity

The dimensional formula of the coefficient of viscosity is:

[η] = ML⁻¹T⁻¹

Replacing the fundamental dimensions with their corresponding SI units gives:

kg m⁻¹ s⁻¹

Therefore, the SI unit of the coefficient of viscosity is:

kg m⁻¹ s⁻¹

This unit can also be written as:

Pa s

or:

N s m⁻²

All these expressions represent the same SI unit of dynamic viscosity.

Why the Coefficient of Viscosity Contains a Negative Power of Length

The negative power of length in the dimensional formula arises because the viscous force depends on the area of contact between fluid layers and the velocity gradient. The area contributes L² in the denominator when the formula is rearranged for viscosity, while force contributes only one power of length in the numerator.

As a result, the net power of length becomes:

L/L² = L⁻¹

Similarly, the time dependence comes from the dimensions of force and velocity gradient. After simplification, the net time dimension becomes T⁻¹.

Detailed Analysis of Each Option

Option (A): ML⁻¹T⁻¹

Option (A) is correct. According to Newton’s law of viscosity:

F = ηA(dv/dx)

Rearranging this relation and substituting the dimensions of force, area and velocity gradient gives:

[η] = ML⁻¹T⁻¹

This is the standard dimensional formula of the coefficient of dynamic viscosity.

Option (B): ML⁻¹T⁻²

Option (B) is incorrect. The dimensions ML⁻¹T⁻² correspond to pressure or stress. Pressure is force divided by area, so its dimensional formula is obtained as MLT⁻²/L², which gives ML⁻¹T⁻².

The coefficient of viscosity is not simply shear stress. It is the ratio of shear stress to velocity gradient. Dividing by the velocity gradient introduces an additional time factor and changes T⁻² to T⁻¹.

Option (C): ML⁻²T⁻²

Option (C) is incorrect because it contains the wrong powers of both length and time. The coefficient of viscosity has one inverse power of length, not two, and one inverse power of time, not two.

Applying Newton’s law of viscosity directly gives ML⁻¹T⁻¹, so ML⁻²T⁻² cannot represent the dimensions of the coefficient of viscosity.

Option (D): ML⁻²T⁻¹

Option (D) is incorrect because the exponent of length is −2 instead of −1. When the dimensions of force are divided by the product of area and velocity gradient, one power of length remains in the denominator.

Therefore, the correct length dependence is L⁻¹, not L⁻².

Difference Between Dynamic and Kinematic Viscosity

The coefficient of viscosity discussed in this question refers to dynamic viscosity, represented by η. Its dimensional formula is:

[η] = ML⁻¹T⁻¹

Kinematic viscosity is a different physical quantity. It is defined as the ratio of dynamic viscosity to fluid density:

ν = η/ρ

Since the dimensions of density are ML⁻³, the dimensions of kinematic viscosity are:

[ν] = [ML⁻¹T⁻¹]/[ML⁻³]

Therefore:

[ν] = L²T⁻¹

Thus, dynamic viscosity and kinematic viscosity are different quantities and have different dimensional formulas.

Physical Meaning of a High Coefficient of Viscosity

A fluid with a high coefficient of viscosity offers strong resistance to the relative motion of its layers. Such a fluid flows slowly because greater force is required to produce a given velocity gradient.

A fluid with a low coefficient of viscosity offers less internal resistance and flows more easily. Therefore, the coefficient of viscosity provides a quantitative measure of the internal friction present in a fluid.

This property is important in fluid mechanics, lubrication, blood flow, industrial pipelines and many other applications involving the motion of liquids and gases.

Final Answer

According to Newton’s law of viscosity:

F = ηA(dv/dx)

Therefore:

η = F/[A(dv/dx)]

Using:

[F] = MLT⁻²

[A] = L²

[dv/dx] = T⁻¹

we obtain:

[η] = [MLT⁻²]/[L²T⁻¹]

Therefore:

[η] = ML⁻¹T⁻¹

Hence, the correct answer is (A) ML⁻¹T⁻¹.