The rate of flow (V) of a liquid flowing through a pipe of radius r and pressure gradient (P//I) is given by Poiseuille's equation` V = (pi)/(8)(Pr^4)/(etaI)` Chack the dimensional correctness of this relation.

Explanation

To check the dimensional correctness of the given equation \( V = \frac{\pi}{8} \frac{Pr^4}{\eta L} \), we need to ensure that the dimensions on both sides of the equation match.

### Step-by-Step Solution:

1. **Identify the dimensions of each quantity:**

- **Pressure (P):** Pressure is defined as force per unit area.
\[ [P] = \frac{[F]}{[A]} = \frac{[M^1 L^1 T^{-2}]}{[L^2]} = [M^1 L^{-1} T^{-2}] \]

- **Radius (r):** Radius is a length.
\[ [r] = [L^1] \]

- **Viscosity (\(\eta\)):** Given as \( [\eta] = [M^1 L^{-1} T^{-1}] \).

- **Length (L):** Length is simply \( [L] = [L^1] \).

- **Volume flow rate (V):** Volume per unit time.
\[ [V] = \frac{[L^3]}{[T^1]} = [L^3 T^{-1}] \]

2. **Write the given equation with dimensions:**
\[ V = \frac{\pi}{8} \frac{Pr^4}{\eta L} \]
Since \(\frac{\pi}{8}\) is a dimensionless constant, we can ignore it for dimensional analysis.

3. **Substitute the dimensions of each term into the equation:**
\[ [V] = \frac{[P] [r]^4}{[\eta] [L]} \]

4. **Substitute the dimensions of each quantity:**
\[ [LHS] = [V] = [L^3 T^{-1}] \]
\[ [RHS] = \frac{[P] [r]^4}{[\eta] [L]} = \frac{[M^1 L^{-1} T^{-2}] [L^4]}{[M^1 L^{-1} T^{-1}] [L^1]} \]

5. **Simplify the dimensions on the RHS:**
\[ [RHS] = \frac{[M^1 L^{-1} T^{-2}] [L^4]}{[M^1 L^{-1} T^{-1}] [L^1]} = \frac{[M^1 L^3 T^{-2}]}{[M^1 L^0 T^{-1}]} \]
\[ [RHS] = [M^1 L^3 T^{-2}] \cdot [M^{-1} L^0 T^1] = [M^0 L^3 T^{-1}] \]

6. **Compare the dimensions of LHS and RHS:**
\[ [LHS] = [L^3 T^{-1}] \]
\[ [RHS] = [M^0 L^3 T^{-1}] \]

Since the dimensions on both sides of the equation are equal, the given equation is dimensionally correct.