Question

A glass tube of uniform cross-section is connected to a tap with a rubber tube. The tap is opened slowly. Initially the flow of water in the tube is streamline. The speed of flow of water to convert it into a turbulent flow is [radius of tube = $1\text{ cm}$, $\eta = 1 \times 10^{-3}\text{ Ns/m}^2$, $R_n = 2500$ and density of water = $10^3\text{ kg/m}^3$]

• A. $0.15\text{ m/s}$
• B. $0.125\text{ m/s}$
• C. $0.3\text{ m/s}$
• D. $0.2\text{ m/s}$

Hint:

Always double-check whether the parameter provided is radius or diameter. A very frequent pitfall is directly inserting $r$ instead of $d = 2r$ into the denominator of the Reynold equation, which incorrectly doubles the calculated velocity.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks for the critical speed ($v_c$) at which the fluid flow inside a uniform glass tube transitions from smooth streamline motion into turbulent flow.
We are given the tube's radius ($r = 1\text{ cm}$), the dynamic viscosity of water ($\eta = 1 \times 10^{-3}\text{ Ns/m}^2$), the transition Reynold number ($R_n = 2500$), and the fluid density ($\rho = 10^3\text{ kg/m}^3$).

Step 2: Key Formula or Approach:
The dimensionless Reynold number $R_n$ governing fluid transition limits is calculated using the formula:
$$R_n = \frac{v_c \rho d}{\eta}$$ Rearranging this relationship directly to isolate the critical velocity boundary yields:
$$v_c = \frac{R_n \eta}{\rho d}$$ where $d = 2r$ is the inner diameter of the circular glass tube.

Step 3: Detailed Explanation:
First, convert the dimensions explicitly into standard SI metric units:
Radius, $r = 1\text{ cm} = 1 \times 10^{-2}\text{ m}$.
Diameter, $d = 2r = 2 \times 10^{-2}\text{ m}$.
Now substitute the numerical values into the rearranged critical velocity equation:
$$v_c = \frac{2500 \times (1 \times 10^{-3})}{10^3 \times (2 \times 10^{-2})}$$ Simplify the numerator:
$$2500 \times 10^{-3} = 2.5$$ Simplify the denominator:
$$10^3 \times 2 \times 10^{-2} = 2 \times 10^1 = 20$$ Divide the simplified terms to find $v_c$:
$$v_c = \frac{2.5}{20} = \frac{25}{200} = 0.125\text{ m/s}$$

Step 4: Final Answer:
The speed of flow required to convert the streamline flow to a turbulent flow is $0.125\text{ m/s}$, matching option (B).