Question

Given the variables: velocity (V), density ($\rho$), viscosity ($\mu$), and characteristic length (L), how many dimensionless groups can be formed?

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

In dimensional analysis, always select $3$ repeating variables ($m = 3$) representing geometric, kinematic, and dynamic properties to construct your dimensionless $\pi$ groups.
These repeating variables are typically $L$, $V$, and $\rho$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the number of base parameters or repeating variables required to form the dimensionless groups from the given physical variables: velocity ($V$), density ($\rho$), viscosity ($\mu$), and characteristic length ($L$).

Step 2: Key Formula or Approach:
According to dimensional analysis and the Buckingham $\pi$ theorem, the variables are analyzed using their fundamental dimensions (Mass, Length, Time).
The number of repeating variables ($m$) selected to form the dimensionless groups is determined by the number of fundamental dimensions involved in the system:
\[ m = 3 \]

Step 3: Detailed Explanation:

• Let us analyze the dimensions of the given physical variables:
- Velocity ($V$): $[\text{L T}^{-1}]$
- Density ($\rho$): $[\text{M L}^{-3}]$
- Dynamic viscosity ($\mu$): $[\text{M L}^{-1} \text{T}^{-1}]$
- Characteristic length ($L$): $[\text{L}]$

• The fundamental dimensions required to express these four variables are Mass ($\text{M}$), Length ($\text{L}$), and Time ($\text{T}$).

• Therefore, the number of primary dimensions is $3$.

• In the formulation of dimensionless groups, we select $3$ repeating variables that collectively contain all the fundamental dimensions.

• These three repeating variables must represent:
1. A geometric property (Characteristic length, $L$).
2. A kinematic property (Velocity, $V$).
3. A dynamic property (Density, $\rho$).

• Thus, these $3$ variables form the essential base groups from which any independent dimensionless relationship (such as the Reynolds number) is constructed.

Step 4: Final Answer:
The number of base parameters or repeating variables required to form the dimensionless groups is 3.