A liquid flows between two parallel plates along the x-axis. The difference between the velocity of two layers separated by the distance `dy` is `dv`. If` A` is the area of each plate, then Newton's law of viscosity may be written as

Options

A. `F=-etaA(dv)/(dx)`
B. `F=+etaA(dv)/(dx)`
C. `F=-etaA(dv)/(dy)`
D. `F=+etaA(dv)/(dy)`

Answer

To derive the expression for Newton's law of viscosity in the context of a liquid flowing between two parallel plates, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Consider two parallel plates separated by a distance `dy`.
- The velocity difference between two layers of fluid at these plates is denoted as `dv`.

2. **Applying Newton's Law of Viscosity**:
- According to Newton's law of viscosity, the shear stress (τ) is proportional to the velocity gradient (dv/dy).
- Mathematically, this can be expressed as:
\[ \tau = -\mu \frac{dv}{dy} \]
- Here, `μ` (mu) is the dynamic viscosity of the fluid.

3. **Relating Shear Stress to Force**:
- Shear stress is defined as the force (F) applied per unit area (A):
\[ \tau = \frac{F}{A} \]
- Therefore, we can substitute this into our previous equation:
\[ \frac{F}{A} = -\mu \frac{dv}{dy} \]

4. **Rearranging the Equation**:
- Rearranging the equation to express the force in terms of the viscosity and the velocity gradient gives:
\[ F = -\mu A \frac{dv}{dy} \]

5. **Final Expression**:
- This equation represents Newton's law of viscosity for a fluid flowing between two parallel plates, where the force is proportional to the area of the plates and the velocity gradient.

### Final Result:
\[ F = -\mu A \frac{dv}{dy} \]