Reynold number `N_R` a dimensionless quantity determines the condition of laminar flow of a viscous liquied through a pipe. `N_R` is a function of density `rho` of liquid, average speed `upsilon` and coeff. Of viscosity `eta.` Given that `N_R prop D,` diameter of pipe. Show by the method of dimensions that `N_R prop(rhoupsilonD)/(eta)`

Explanation

To show that the Reynolds number \( N_R \) is proportional to \( \frac{\rho \upsilon D}{\eta} \) using dimensional analysis, we will follow these steps:

### Step 1: Understand the Variables
We know that the Reynolds number \( N_R \) is a dimensionless quantity that depends on:
- \( \rho \): Density of the liquid (units: \( \text{kg/m}^3 \))
- \( \upsilon \): Average speed of the liquid (units: \( \text{m/s} \))
- \( D \): Diameter of the pipe (units: \( \text{m} \))
- \( \eta \): Coefficient of viscosity (units: \( \text{Pa} \cdot \text{s} = \text{kg/(m} \cdot \text{s}) \))
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