Problem 59 - Master Resource Book in JEE Main Physics

Question

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes \((a)\), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion For the flow to be streamline, value of critical velocity should be as low as possible. Reason Once the actual velocity of flow of a liquid becomes greater than the critical velocity, the flow becomes turbulent.

Step-by-Step Solution

Answer

(b) Both Assertion and Reason are true, but the Reason is not the correct explanation of the Assertion.

Step 1: Analyze the Assertion

The assertion states that for flow to be streamline (or laminar), the value of critical velocity should be as low as possible. This suggests that a lower critical velocity ensures that the fluid remains in a stable, streamline flow.

Step 2: Evaluate the Reason

The reason given is that once the actual velocity of a fluid exceeds the critical velocity, the flow transitions from streamline to turbulent. This is a true statement because critical velocity is indeed the threshold beyond which flow becomes chaotic.

Step 3: Assess the Relationship

Both the assertion and the reason are true statements individually. However, the reason does not explain why the critical velocity should be as low as possible for streamline flow. The reason describes what happens after the critical velocity is exceeded but does not relate to why it needs to be lower for streamline flow.

Step 4: Determine the Correct Answer

Based on the analysis, both the assertion and the reason are true, but the reason is not the correct explanation of the assertion. Therefore, the correct option is (b).

Key Concepts

Streamline Flow

Streamline flow, also known as laminar flow, is a pattern of fluid motion where every particle of the fluid follows a smooth path. These paths, known as streamlines, never intersect and maintain the same distance relative to one another. This creates an orderly and predictable movement of the fluid. In a streamline flow, fluid particles move in parallel layers, with no disruption between them. This type of flow is common when the fluid moves at lower velocities, where viscous forces are more effective than inertial forces.

Understanding streamline flow is crucial in many engineering applications, as it allows for efficient design of systems like pipes, airfoils, and various fluid transport mechanisms. It's essential because it minimizes energy loss due to friction and allows for smoother flow through conduits. For example, water flowing through a straw in a slow, steady manner is an example of streamline flow.

In mathematics, streamline flow can be described using equations that assume no mixing or whirlpools within the fluid. In this context, the Reynolds number, a dimensionless value used to predict flow patterns in different fluid flow situations, is usually low, indicating a stable flow.

Critical Velocity

Critical velocity is the speed below which fluid flow remains streamlined. It is the threshold velocity that distinguishes between streamline (laminar) and turbulent flow. If the actual velocity of the fluid exceeds this critical threshold, the flow starts to become chaotic and transitions into a turbulent state.

This concept is mathematically defined as the product of the fluid’s kinematic viscosity and a dimensionless number called the Reynolds number, divided by a characteristic length (such as diameter for a pipe). The formula for critical velocity is:\[ V_c = \frac{Re \cdot u}{L} \]

Where:

• \( V_c \) is the critical velocity
• \( Re \) is the Reynolds number
• \( u \) is the kinematic viscosity of the fluid
• \( L \) is the characteristic length

For engineers, determining the critical velocity is crucial when designing systems to ensure the conservation of energy and fluid integrity in industrial processes. In simple terms, maintaining flow below critical velocity means keeping operations stable and efficient without the risk of erratic behaviors that could result from turbulence.

Turbulent Flow

Turbulent flow is characterized by irregular movement of particles in the fluid, which results in mixing and swirling patterns. Unlike streamline flow, turbulent flow is chaotic and involves variations in both speed and direction of fluid particles within the flow.

This occurs when the fluid velocity exceeds the critical velocity. At this point, inertial forces dominate over viscous forces, creating eddies and vortices that disrupt the orderly nature of flow. Sources of turbulence can include obstacles in the path of a flowing fluid, sudden changes in direction or velocity, or high flow speed. All these disturbances can cause the fluid to mix intensively, increasing the energy dissipation through friction and drag.

While turbulent flow increases resistance and energy loss, it can also have beneficial applications. In industries like chemical mixing, having high turbulence can ensure that substances are evenly mixed without the need for additional processes. However, this chaotic flow can also be undesirable in pipelines, where it might cause wear and tear over time.