What is dimensional homogeneity?

The term dimensional homogeneity means that in an equation, the dimensions of each term on both sides are equal, which means the dimensions of the left-hand side (LHS) of an equation are similar to the dimensions of the right-hand side (RHS). For a dimensionally homogeneous equation, the powers of fundamental dimensions will be identical on both sides of the equation.

Dimensional Analysis

Dimensional analysis is the mathematical technique used in research works for conducting model tests. The dimensional analysis deals with a physical quantity involved in the research. The dimensions of quantities are derived based on their units. There are two types of quantities –

• Fundamental or Primary quantities
• Derived or Secondary quantities

Fundamental or Primary quantities

Quantities independent and invariable of any other quantity are known as fundamental quantities. They have a fixed dimension. The seven fundamental quantities are - length, mass, time, temperature, electric current, luminous intensity, and amount of matter. The dimensions of the seven fundamental quantities are L, M, T, θ, A, I, and N respectively.

Derived or Secondary quantities

Quantities that possess more than one fundamental quantity and are dependent on the fundamental quantities are derived or secondary quantities. For example, the force has the dimensions $ML{T}^{-2}$. Hence, force is a derived quantity.

The dimensions of two different quantities can be similar for example, the dimensions for both acceleration and angular acceleration are $L{T}^{-2}$. The dimensions of both velocity and angular velocity are $L{T}^{-1}$.

Dimensional homogeneity

Dimensional homogeneity exists when the dimensions assigned to variables on both sides of an equation are equal. The dimensions are found based on units of quantities. The dimensional homogeneity is useful in deriving the formulas for various physical quantities. Let us understand this better with a few examples-

Example 1-

The velocity of flow is given as, v = $\sqrt{2.g.h}$ 

where, v is the velocity of flow, g is the gravitational constant, and h is the head of flow.

The unit of velocity is $m{s}^{-1}$. Hence, the dimensions of velocity are $L{T}^{-1}$.

The unit of g is $m{s}^{-2}$. Hence, the dimensions of g are $L{T}^{-2}$.

The unit of h is m. Hence, the dimensions of h is L.

To check the dimensional homogeneity, 

Dimensions of LHS = Dimensions of velocity = $L{T}^{-1}$

Dimensions of RHS = Dimensions of $\sqrt{2.g.h}$ = ${(L{T}^{-2} . L)}^{\frac{1}{2}}$ = $L{T}^{-1}$

Hence, dimensional homogeneity exists in the above equation.

Example 2-

As per the kinematic equation, v = u + a.t

where u is the initial velocity, a is the acceleration having unit meter per second squared, t is the time, and v is the final velocity.

Dimensions of LHS = Dimensions of velocity =$L{T}^{-1}$

Dimensions of RHS = Dimensions of u + a.t = $L{T}^{-1}$ + $\left(L{T}^{-2}\right).\left(T\right)$ = $L{T}^{-1}$

Hence, dimensional homogeneity exists in the above kinematic equation.

Methods of Dimensional Analysis using Dimensional Homogeneity

If there is more than one variable involved in an equation, the relation among the variables can be derived using the following two methods:

• Rayleigh's method
• Buckingham's method

Rayleigh's method

The Rayleigh's method determines the expressions for variables that depend on a maximum of four independent variables. According to this method, if X is a variable dependent on X₁, X₂, and X₃, then X is a function of X₁, X₂, and X₃. Mathematically, the equation for the same can be formed as, X = f(X₁, X₂, X₃).

The above equation is also written as, $X = k.X_1^a . X_2^b . X_3^c$

where, a, b, and c are arbitrary powers and k is an additive term, known as the non-dimensional constant.

Example

It is required to find the equation for drag force on a sphere of diameter D, moving with velocity v, in a fluid of density $\rho$ and dynamic viscosity $\mu$, considering that the equation will be dimensionally homogenous. The step-by-step solution is as follows.

Hence, the drag force F = $k.D^a.v^b.{\rho }^c.{\mu }^d$

The LHS and RHS are dimensionally homogenous. Hence we will equate dimensions of variables on both sides.

The unit of force is Newton, whose dimension is $ML{T}^{-2}$.

The unit of density is $kg/m^3$, whose dimension is $M{L}^{-3}$.

The unit of dynamic viscosity is $N-s / m^2$, whose dimension is $M{L}^{-1}{T}^{-1}$.

Hence, $ML{T}^{-2}$ = $k. L^a . {\left(L{T}^{-1}\right)}^b . {\left(M{L}^{-3}\right)}^c . {\left(M{L}^{-1}{T}^{-1}\right)}^d$

Equating the powers of M, L, and T on the LHS and RHS,

we get, 1 = c+d

1 = a+b-3c-d

and

-2 = -b-d respectively.

Substituting all the values in terms of d, the equation is obtained as,

F = $k. D^{2-d}. v^{2-d}. {\rho }^{1-d}. {\mu }^d$

Hence, F = $k.\rho .D^2.v^2.{\left(\frac{\mu }{\rho .v.d}\right)}^d$

Hence, F = $k.\rho .D^2.v^2. \varphi \left(\frac{\mu }{\rho .v.d}\right)$

where, $\varphi$ denotes the function and k is the additive term, dimensionless constant. The above equation is the required equation.

Buckingham's method

Buckingham's method is used when more than four dependent or independent variables exist in an equation. As per this theorem, if there are 'n' variables in a physical quantity and if these variable contains 'm' fundamental dimensions, then the variables are arranged in the (n-m) dimensionless terms. Let, X₁, X₂, .., X_n are the variables and X₁ is the dependent variable, and X₂, X₃, ..., X_n are the independent variables on which X₁ depends, then, as per this theorem, X₁ = f(X₂, X₃, ..., X_n). The equation can also be stated as f₁(X₁, X₂, X₃,..., X_n). Buckingham's method is also known as Buckingham's pi-method as all the (n-m) terms are substituted in the form of pi-terms.

Dimensionless groups

Dimensionless groups are the set of variables without dimensions or units that have an important role. There are two types of constants- dimensional constants and dimensionless constants. Dimensional constants have dimensions, such as gravitational constant g. The dimensionless constants do not have dimensions, such as Reynolds number. The dimensionless quantities are also known as pi-terms. The dimensionless quantities are mostly associated with fluid mechanics. The common dimensionless quantities used are-

• Reynolds number
• Froude's number
• Euler's number
• Weber's number
• Mach's number

Context and Application

The dimensional homogeneity studied under the following courses-

• Bachelors of Technology (Civil Engineering)
• Masters of Technology (Water Resources Engineering)
• Masters of Technology (Fluid Mechanics)

Practice Problems

1. Which of the following is not a fundamental quantity?

1. Force
2. Length
3. Mass
4. Time

Answer- a

Explanation- Force is not a fundamental quantity.

2. Which of the following is not a dimensionless quantity?

1. Reynolds number
2. Froude's number
3. Gravitational constant
4. Mach's number

Answer- c

Explanation- Gravitational constant is not a dimensionless quantity.

3. As per the principle of dimensional homogeneity, what should be the relation between dimensions of LHS and RHS?

1. Dimensions on LHS = Dimensions on RHS
2. Dimensions on LHS Dimensions on RHS
3. Dimensions on LHS > Dimensions on RHS
4. Dimensions on LHS < Dimensions on RHS 

Answer- a

Explanation- As per the principle of homogeneity Dimensions on LHS = Dimensions on RHS.

4. What is the unit of dynamic viscosity?

a) $N^3-s^2 / m^2$

b) $N / m^2 . s$

c) $N^2-s / m^2$

d) $N-s / m^2$

Answer – d)

Explanation – The unit of dynamic viscosity is $N-s / m^2$.

5. Which of the following is a derived quantity?

1. Velocity
2. Luminous intensity
3. Temperature
4. Amount of matter

Answer- a

Explanation- Velocity is a derived quantity.