Steady Incompressible Laminar Flow between Two Infinite Parallel Stationary Plates

In a steady incompressible laminar flow between two infinite parallel stationary plates, the flow is characterized by the absence of turbulence and a constant velocity profile across the cross-section of the flow. This type of flow is commonly encountered in various engineering applications such as lubrication, flow in pipes, and boundary layer flows.

Shear Stress Variation

The shear stress is the force per unit area acting tangentially to the flow direction. In this flow configuration, the velocity of the fluid varies across the channel, resulting in a velocity gradient. This velocity gradient gives rise to a shear stress distribution within the fluid.

Linear Variation of Shear Stress

The correct answer to the given question is option 'B': linear with zero value at the center. This means that the shear stress variation across the channel is linear, and the maximum shear stress occurs at the walls of the channel, while the shear stress is zero at the center.

Explanations

1. Boundary Conditions: At the walls of the channel (i.e., at the plates), the fluid velocity is zero due to the no-slip condition. This implies that the fluid particles at the walls are stationary, and therefore there is no relative motion between the fluid and the walls. As a result, the shear stress is zero at the walls.

2. Velocity Profile: Due to the no-slip condition, the fluid velocity is maximum at the center of the channel and decreases linearly towards the walls. This velocity profile leads to a linear velocity gradient across the channel.

3. Shear Stress Calculation: The shear stress can be calculated using the equation: shear stress = dynamic viscosity × velocity gradient. In this flow configuration, the velocity gradient is linear, and therefore the shear stress variation is also linear.

4. Zero Shear Stress at the Center: At the center of the channel, the velocity gradient is zero, as the velocity is constant. Consequently, the shear stress is also zero at the center.

Conclusion

In a steady incompressible laminar flow between two infinite parallel stationary plates, the shear stress variation is linear with zero value at the center. This is due to the no-slip condition at the walls, which results in a linear velocity gradient across the channel. Understanding the shear stress variation is essential for analyzing fluid flow and designing various engineering systems.