Find the critical velocity for air flowing through a tube of 2cm diammeter .For air `rho=1.3xx10^(-3)gcm^(-3) and eta=181xx10^(-6)` poise

Answer

To find the critical velocity for air flowing through a tube of 2 cm diameter, we will use the formula for Reynolds number and the given parameters. The critical velocity is the velocity at which the flow transitions from laminar to turbulent, typically at a Reynolds number of 2000.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Diameter of the tube, \( d = 2 \, \text{cm} = 0.02 \, \text{m} \)
- Density of air, \( \rho = 1.3 \times 10^{-3} \, \text{g/cm}^3 = 1.3 \times 10^{-3} \times 10^3 \, \text{kg/m}^3 = 1.3 \, \text{kg/m}^3 \)
- Viscosity of air, \( \eta = 181 \times 10^{-6} \, \text{poise} = 181 \times 10^{-6} \times 0.1 \, \text{Pa.s} = 1.81 \times 10^{-5} \, \text{Pa.s} \)

2. **Write the Formula for Reynolds Number**:
The Reynolds number \( Re \) is given by the formula:
\[ Re = \frac{\rho v d}{\eta} \]
where:
- \( v \) is the critical velocity (which we need to find),
- \( d \) is the diameter of the tube,
- \( \rho \) is the density of the fluid,
- \( \eta \) is the dynamic viscosity.

3. **Set the Reynolds Number for Critical Flow**:
For critical flow, we set \( Re = 2000 \):
\[ 2000 = \frac{\rho v d}{\eta} \]

4. **Rearrange the Equation to Solve for Critical Velocity \( v \)**:
Rearranging gives:
\[ v = \frac{Re \cdot \eta}{\rho \cdot d} \]

5. **Substitute the Known Values**:
Substitute \( Re = 2000 \), \( \eta = 1.81 \times 10^{-5} \, \text{Pa.s} \), \( \rho = 1.3 \, \text{kg/m}^3 \), and \( d = 0.02 \, \text{m} \):
\[ v = \frac{2000 \cdot (1.81 \times 10^{-5})}{1.3 \cdot 0.02} \]

6. **Calculate the Critical Velocity**:
First, calculate the denominator:
\[ 1.3 \cdot 0.02 = 0.026 \, \text{kg/m}^2 \]
Now calculate \( v \):
\[ v = \frac{2000 \cdot 1.81 \times 10^{-5}}{0.026} \]
\[ v = \frac{3.62 \times 10^{-2}}{0.026} \approx 1.39 \, \text{m/s} \]

7. **Convert to cm/s**:
To convert \( v \) from m/s to cm/s:
\[ v \approx 1.39 \, \text{m/s} \times 100 \approx 139 \, \text{cm/s} \]

### Final Answer:
The critical velocity for air flowing through a tube of 2 cm diameter is approximately **140 cm/s**.

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