What is `SI` unit of viscosity coefficient `(eta)` ?

Options

A. pascal
B. `N s m^(-2)`
C. `km^(-2)s`
D. ` N m^(-2)`

Answer

To determine the SI unit of the viscosity coefficient (η), we can derive it from the fundamental quantities involved in its definition.

### Step-by-Step Solution:

1. **Understand the Definition of Viscosity Coefficient (η)**:
Viscosity is a measure of a fluid's resistance to flow. The viscosity coefficient (η) relates the shear stress to the shear rate in a fluid.

2. **Identify the Formula**:
The viscosity coefficient can be expressed using the formula:
\[ \eta = \frac{\text{Force} \times \text{Distance}}{\text{Area} \times \text{Velocity}} \]

3. **Identify the Units of Each Quantity**:
- **Force (F)**: The SI unit of force is Newton (N).
- **Distance (d)**: The SI unit of distance is meter (m).
- **Area (A)**: The SI unit of area is square meter (m²).
- **Velocity (v)**: The SI unit of velocity is meter per second (m/s).

4. **Substitute the Units into the Formula**:
Now, substituting the units into the formula for viscosity coefficient:
\[ \eta = \frac{N \cdot m}{m^2 \cdot \frac{m}{s}} \]

5. **Simplify the Units**:
- The unit of force (N) can be expressed in terms of base units: \(1 \, N = 1 \, kg \cdot m/s^2\).
- Substituting this into the equation gives:
\[ \eta = \frac{(kg \cdot m/s^2) \cdot m}{m^2 \cdot \frac{m}{s}} = \frac{kg \cdot m^2/s^2}{m^2 \cdot m/s} \]
- Now, simplifying further:
\[ \eta = \frac{kg \cdot m^2/s^2}{m^3/s} = \frac{kg \cdot m^2 \cdot s}{m^3 \cdot s^2} \]
- Canceling out the common terms:
\[ \eta = \frac{kg}{m \cdot s} \]

6. **Final Unit**:
The final unit of viscosity coefficient (η) is:
\[ \eta = \text{Newton second per meter square} \, (N \cdot s/m^2) \]

### Conclusion:
The SI unit of viscosity coefficient (η) is **Newton second per meter square (N·s/m²)**.