Reynolds number for a liquid of density `rho`, viscosity `eta` and flowing through a pipe of diameter D, is given by

Options

A. `(rho vD)/(eta)`
B. `(rho eta D)/(v)`
C. `(rho v eta)/(D)`
D. `(eta v D)/(rho)`

Answer

To find the Reynolds number (Re) for a liquid with density (ρ), viscosity (η), and flowing through a pipe of diameter (D), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Reynolds Number:**
The Reynolds number is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is defined as the ratio of inertial forces to viscous forces.

2. **Identify the Relevant Forces:**
- **Inertial Force (F_inertial):** This can be expressed as the product of density (ρ), velocity (V), and the cross-sectional area (A) of the pipe. Mathematically, it is given by:
\[ F_{\text{inertial}} = \rho V^2 A \]
- **Viscous Force (F_viscous):** This can be expressed as the product of the area (A), velocity (V), viscosity (η), and is divided by the diameter (D) of the pipe. Mathematically, it is given by:
\[ F_{\text{viscous}} = A \cdot V \cdot \frac{\eta}{D} \]

3. **Set Up the Ratio for Reynolds Number:**
The Reynolds number (Re) is defined as:
\[ Re = \frac{F_{\text{inertial}}}{F_{\text{viscous}}} \]

4. **Substitute the Expressions for Forces:**
Substitute the expressions for inertial and viscous forces into the Reynolds number formula:
\[ Re = \frac{\rho V^2 A}{A \cdot V \cdot \frac{\eta}{D}} \]

5. **Simplify the Expression:**
- Cancel the area (A) from the numerator and denominator.
- Cancel one velocity (V) from the numerator and denominator.
This simplifies to:
\[ Re = \frac{\rho V D}{\eta} \]

6. **Final Expression for Reynolds Number:**
The final expression for the Reynolds number is:
\[ Re = \frac{\rho V D}{\eta} \]

### Conclusion:
Thus, the Reynolds number for a liquid of density ρ, viscosity η, and flowing through a pipe of diameter D is given by:
\[ Re = \frac{\rho V D}{\eta} \]