Water flows at a speed of 6 `cms^-1` through a tube of radius 1 cm. coefficient of viscosity of water at room temperature is 0.01 poise. Calculate the Reynolds number. Is it a steady flow.

Explanation

To calculate the Reynolds number for the flow of water through a tube, we will follow these steps:

### Step 1: Convert the given values to standard units
- **Speed (V)**: Given as 6 cm/s, we convert it to meters per second.
\[ V = 6 \, \text{cm/s} = 6 \times 10^{-2} \, \text{m/s} \]

- **Radius (r)**: Given as 1 cm, we convert it to meters.
\[ r = 1 \, \text{cm} = 1 \times 10^{-2} \, \text{m} \]

- **Viscosity (\(\mu\))**: Given as 0.01 poise, we convert it to standard SI units (Newton second per meter squared).
\[ \mu = 0.01 \, \text{poise} = 0.01 \times 0.1 \, \text{N s/m}^2 = 0.001 \, \text{N s/m}^2 \]

### Step 2: Calculate the diameter of the tube
The diameter (d) is twice the radius.
\[ d = 2r = 2 \times (1 \times 10^{-2} \, \text{m}) = 2 \times 10^{-2} \, \text{m} \]

### Step 3: Use the formula for Reynolds number
The Reynolds number (Re) is given by the formula:
\[ Re = \frac{\rho V d}{\mu} \]
where:
- \(\rho\) (density of water) is approximately \(1000 \, \text{kg/m}^3\),
- \(V\) is the velocity,
- \(d\) is the diameter,
- \(\mu\) is the viscosity.

### Step 4: Substitute the values into the formula
Substituting the known values:
\[ Re = \frac{1000 \, \text{kg/m}^3 \times (6 \times 10^{-2} \, \text{m/s}) \times (2 \times 10^{-2} \, \text{m})}{0.001 \, \text{N s/m}^2} \]

### Step 5: Calculate the Reynolds number
Calculating the numerator:
\[ 1000 \times 6 \times 10^{-2} \times 2 \times 10^{-2} = 1000 \times 0.06 \times 0.02 = 1.2 \]
Now, substituting this into the Reynolds number formula:
\[ Re = \frac{1.2}{0.001} = 1200 \]

### Step 6: Determine if the flow is steady
The flow is considered steady if the Reynolds number is less than 2000. Since we calculated:
\[ Re = 1200 < 2000 \]
Thus, the flow is steady.

### Final Answer
The Reynolds number is **1200**, and the flow is **steady**.

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