Question

Water flows from a tap of diameter 1.5 cm with velocity \( 7.5 \times 10^{-5} \, \text{m}^{3}\text{s}^{-1} \). The coefficient of viscosity of water is \( 10^{-3} \) Pas. The flow is:

• A. Turbulent with Reynolds number less than 6000
• B. Steady flow with Reynolds number less than 2000
• C. Turbulent with Reynolds number greater than 6000
• D. Steady flow with Reynolds number more than 6000

Hint:

For flow problems involving Reynolds number, use the formula \( Re = \frac{\rho V D}{\eta} \) and classify the flow as turbulent if \( Re>6000 \) or laminar if \( Re<2000 \).

Answer 1

The Correct Option is C

To determine whether the flow is steady or turbulent, we need to calculate the Reynolds number (\( Re \)). The Reynolds number is a dimensionless quantity used to predict flow patterns in fluid dynamics. It is given by: \[ Re = \frac{\rho v d}{\eta}, \] where:
\( \rho \) is the density of the fluid (for water, \( \rho = 1000 \, \text{kg/m}^3 \)),
\( v \) is the velocity of the fluid,
\( d \) is the diameter of the pipe or tap,
\( \eta \) is the coefficient of viscosity of the fluid.

Step 1: Calculate the velocity \( v \)
The volumetric flow rate \( Q \) is given by: \[ Q = v \cdot A, \] where \( A \) is the cross-sectional area of the tap. The area \( A \) is: \[ A = \pi \left(\frac{d}{2}\right)^2. \] Given:
Diameter \( d = 1.5 \, \text{cm} = 0.015 \, \text{m} \),
Volumetric flow rate \( Q = 7.5 \times 10^{-5} \, \text{m}^3/\text{s} \). Substitute \( d \) into the area formula: \[ A = \pi \left(\frac{0.015}{2}\right)^2 = \pi \left(0.0075\right)^2 = 1.767 \times 10^{-4} \, \text{m}^2. \] Now, solve for \( v \): \[ v = \frac{Q}{A} = \frac{7.5 \times 10^{-5}}{1.767 \times 10^{-4}} \approx 0.424 \, \text{m/s}. \] 

Step 2: Calculate the Reynolds number Substitute the values into the Reynolds number formula: \[ Re = \frac{\rho v d}{\eta}. \] Given:
\( \rho = 1000 \, \text{kg/m}^3 \),
\( v = 0.424 \, \text{m/s} \),
\( d = 0.015 \, \text{m} \),
\( \eta = 10^{-3} \, \text{Pas} \). Substitute these values: \[ Re = \frac{(1000)(0.424)(0.015)}{10^{-3}}. \] Simplify: \[ Re = \frac{6.36}{10^{-3}} = 6360. \] 

Step 3: Interpret the Reynolds number
If \( Re<2000 \), the flow is steady (laminar).
If \( 2000<Re<6000 \), the flow is transitional.
If \( Re>6000 \), the flow is turbulent.
Here, \( Re = 6360 \), which is greater than 6000. Therefore, the flow is turbulent. Final Answer:
The flow is:
Turbulent with Reynolds number greater than 6000