The `SI` unit of viscosity coefficient is

Options

A. `Nms^(-1)`
B. `Nm^(2) s^(-1)`
C. `Nm^(-2) s`
D. None of these

Explanation

To determine the SI unit of viscosity coefficient, we can analyze the definition of viscosity and its relation to force, area, and time.

### Step-by-Step Solution:

1. **Understanding Viscosity**:
- Viscosity is a measure of a fluid's resistance to flow. It quantifies how thick or sticky a fluid is.

2. **Defining Viscosity Coefficient**:
- The viscosity coefficient (often denoted as η) is defined as the ratio of shear stress to shear rate.

3. **Shear Stress**:
- Shear stress (τ) is defined as the force (F) applied per unit area (A). The formula is:
\[ \tau = \frac{F}{A} \]
- The SI unit of force (F) is Newton (N), and the SI unit of area (A) is square meters (m²). Therefore, the unit of shear stress is:
\[ \text{Unit of shear stress} = \frac{\text{N}}{\text{m}^2} = \text{N/m}^2 \]

4. **Shear Rate**:
- Shear rate (γ) is defined as the rate of change of velocity (v) with respect to distance (y). The formula is:
\[ \gamma = \frac{dv}{dy} \]
- The unit of velocity (v) is meters per second (m/s), and the unit of distance (y) is meters (m). Thus, the unit of shear rate is:
\[ \text{Unit of shear rate} = \frac{\text{m/s}}{\text{m}} = \text{s}^{-1} \]

5. **Combining Units**:
- The viscosity coefficient (η) can be expressed as:
\[ \eta = \frac{\tau}{\gamma} = \frac{\text{N/m}^2}{\text{s}^{-1}} \]
- This simplifies to:
\[ \eta = \text{N} \cdot \text{s/m}^2 \]

6. **Final Unit Conversion**:
- Since 1 Newton (N) can be expressed as \( \text{kg} \cdot \text{m/s}^2 \), we can rewrite the unit of viscosity coefficient as:
\[ \eta = \frac{\text{kg} \cdot \text{m/s}^2 \cdot \text{s}}{\text{m}^2} = \frac{\text{kg}}{\text{m} \cdot \text{s}} \]

7. **Conclusion**:
- Therefore, the SI unit of viscosity coefficient is:
\[ \text{kg/(m} \cdot \text{s)} \]