The coefficient of viscosity `(eta)` of a fluid moving steadily between two surface is given by the formula `(f) = etaA dV//dx` where f is the frictional force on the flluid, A is the area in the fluid, and `dN//dx` is velocity gradient inside the fluid at that area. The SI unit of viscosity is given as :

Options

A. `kg m^(-1) s^(-1)`
B. `Nm^(-2) s`
C. Nil
D. Newtons

Explanation

To find the SI unit of viscosity (η) from the given formula, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The formula given is:
\[ F = \eta A \frac{dV}{dx} \]
where:
- \( F \) is the frictional force,
- \( \eta \) is the coefficient of viscosity,
- \( A \) is the area,
- \( \frac{dV}{dx} \) is the velocity gradient.

2. **Rearranging the Formula**: We can rearrange the formula to solve for η:
\[ \eta = \frac{F}{A \frac{dV}{dx}} \]

3. **Identify Units**:
- The SI unit of force \( F \) is Newton (N).
- The SI unit of area \( A \) is square meters (m²).
- The velocity \( V \) is in meters per second (m/s), and the distance \( x \) is in meters (m). Thus, the velocity gradient \( \frac{dV}{dx} \) has units of:
\[ \frac{m/s}{m} = s^{-1} \]

4. **Substituting Units into the Equation**:
Now substituting the units into the equation for η:
\[ \text{Unit of } \eta = \frac{\text{Unit of } F}{\text{Unit of } A \cdot \text{Unit of } \frac{dV}{dx}} = \frac{N}{m^2 \cdot s^{-1}} \]

5. **Simplifying the Units**:
We know that \( 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \), so:
\[ \text{Unit of } \eta = \frac{kg \cdot m/s^2}{m^2 \cdot s^{-1}} = \frac{kg \cdot m}{s^2 \cdot m^2} \cdot s = \frac{kg}{m \cdot s} \]

6. **Final Unit of Viscosity**:
Therefore, the SI unit of viscosity (η) is:
\[ \text{Unit of } \eta = \text{Pascal-second} \, (Pa \cdot s) = \frac{N \cdot s}{m^2} = \frac{kg}{m \cdot s} \]

### Conclusion:
The SI unit of viscosity (η) is \( \text{Pa} \cdot \text{s} \) or \( \frac{kg}{m \cdot s} \).