What is the correct dimensionless group formed with the variables ρdensity, Rotational speed, diameter and μcoefficient of viscosity?a) ρNd2 / μb) ρNd / μc) Nd / ρμd) Nd2 / ρμCorrect answer is option 'A'. Can you explain this answer?

Question

What is the correct dimensionless group formed with the variables ρ­density, Rotational speed, diameter and μ­coefficient of viscosity?a) ρNd2 / μb) ρNd / μc) Nd / ρμd) Nd2 / ρμCorrect answer is option 'A'. Can you explain this answer?

Answer

To determine the correct dimensionless group formed with the variables density (ρ), rotational speed (N), diameter (d), and coefficient of viscosity (μ), we can use the Buckingham Pi theorem. This theorem states that if we have n variables (in this case, 4) with m fundamental dimensions, we can form n - m dimensionless groups.

In this case, the fundamental dimensions are:
- Mass (M)
- Length (L)
- Time (T)

The variables given have the following dimensions:
- Density (ρ): M/L^3
- Rotational speed (N): 1/T
- Diameter (d): L
- Coefficient of viscosity (μ): M/(LT)

Since we have 4 variables and 3 fundamental dimensions, there will be 4 - 3 = 1 dimensionless group formed.

We can form this dimensionless group by combining the variables in such a way that the resulting expression is dimensionally homogeneous. This means that all the dimensions in the numerator should cancel out with the dimensions in the denominator.

Let's consider the options given:
a) ρNd^2 / μ
b) ρNd / μ
c) Nd / ρμ
d) Nd^2 / ρμ

We can check the dimensions of each option to see which one is dimensionally homogeneous.

a) ρNd^2 / μ: (M/L^3) * (L) * (L^2) / (M/(LT)) = (M * L^3 * L^2) / (L^3 * T * M) = L

b) ρNd / μ: (M/L^3) * (L) * (L) / (M/(LT)) = (M * L^3 * L) / (L^3 * T * M) = L^2/T

c) Nd / ρμ: (L) * (1/T) / ((M/L^3) * (M/(LT))) = (L * T) / (M * L^3 * M/(LT)) = L^2/(M * T)

d) Nd^2 / ρμ: (L) * (L^2) / ((M/L^3) * (M/(LT))) = (L * L^2) / (M * L^3 * M/(LT)) = L^4/(M^2)

We can see that option a) ρNd^2 / μ is the only one that is dimensionally homogeneous, resulting in the dimension L (length). Therefore, the correct dimensionless group is given by option a).

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