Question

Someone claims that in fully developed turbulent flow in a tube, the shear stress is a maximum at the tube surface. Do you agree with this claim? Explain.

Short answer

Answer: Yes, in a fully developed turbulent flow, the shear stress is maximum at the tube surface due to the higher velocity gradient near the surface as a result of resistance against the tube wall.

Step-by-step solution

1. Understanding turbulent flow

Fully developed turbulent flow is characterized by chaotic fluctuations of velocity throughout the flow. In a tube, the turbulent flow has a parabolic velocity profile and the shear stress occurs due to the resistance between the layers of fluid.

2. Understanding shear stress

Shear stress is defined as the force per unit area acting parallel to the surface of the tube. In fluid dynamics, shear stress is related to the fluid viscosity and the velocity gradient (or rate of strain) perpendicular to the surface.

3. Analyze shear stress distribution in turbulent flow

In a fully developed turbulent flow, the velocity profile is parabolic, with the maximum velocity at the center of the tube and gradually decreasing towards the tube surface. As the fluid flows through the tube, the fluid layers near the tube surface experience resistance due to drag against the wall, leading to a gradient in the velocity.

4. Derive the relationship between turbulent flow and shear stress

Shear stress in a fluid flow is given by the equation: τ = μ(du/dy), where τ is the shear stress, μ is the fluid viscosity, du is the change in velocity, and dy is the change in the distance perpendicular to the wall. In turbulent flow, the velocity gradient is highest near the tube surface as it decreases rapidly due to resistance.

5. Determine if shear stress reaches maximum at the tube surface

From the equation for shear stress, we can see that the shear stress (τ) is directly proportional to the velocity gradient (du/dy). In fully developed turbulent flow, the velocity gradient is highest near the tube surface due to the presence of resistance against the tube wall. This causes the shear stress to be maximum at the tube surface.

6. Conclusion

Based on the analysis of fully developed turbulent flow and the relationship between shear stress and velocity gradient, we can agree with the claim that shear stress is a maximum at the tube surface. This is due to the higher velocity gradient near the surface as a result of resistance against the tube wall.

Key concepts

Fully Developed Turbulent Flow

When we talk about fully developed turbulent flow, we're looking at a situation where fluid moves through a tube, such as a pipe, in a chaotic and unpredictable manner. Unlike laminar flow, where the motion of the fluid is orderly and smooth, turbulent flow is characterized by random and violent fluctuations. This happens due to the high flow rate or when the fluid moves through a large or rough tube.

Turbulent flow reaches a fully developed state after a certain distance from the tube entrance; at this point, the flow doesn't change much as it moves downstream. One of the interesting properties of fully developed turbulent flow in a tube is that it displays a velocity profile, where the fluid moves fastest at the center and slows down as it approaches the tube wall due to friction. The interplay between the fluid layers moving at different speeds leads to a phenomenon known as shear stress.

Shear Stress in Fluid Dynamics

In the world of fluid dynamics, shear stress is a critical concept. Imagine spreading butter on toast using a knife; the action of the knife against the butter surface is a good analogy for shear stress. In fluids, shear stress arises from the force exerted by fluid layers sliding past one another at different velocities.

This force per unit area is exerted along the plane of the fluid and can deform the fluid. It plays a fundamental role in the behavior of fluids moving within confines, such as pipes or channels. In turbulent flow within a tube, the shear stress is indeed highest at the wall of the tube where the fluid's velocity change is most significant due to the drag interaction with the solid surface.

Velocity Gradient

To understand velocity gradient, let's refer back to our butter analogy. When you spread butter, the knife moves smoothly across the top but not the bottom – creating a 'gradient' or change in the speed at which the butter is moving. Similarly, in fluid flow, a velocity gradient is the change in speed (velocity) of the fluid between two points.

This phenomenon is crucial in fluids because it measures how quickly the fluid speed changes from one layer to the next. In turbulent flow, you'll find the sharpest velocity gradients close to solid boundaries, such as walls of tubes, where the fluid is briefly 'slowed down' or 'held back' by friction before being flung into the center of the stream.

Fluid Viscosity

If you've ever compared pouring honey to pouring water, you've observed fluid viscosity firsthand. Viscosity describes a fluid's resistance to flow or shear. Thicker (more viscous) fluids like honey resist shear stress more than thinner (less viscous) fluids like water.

In a fully developed turbulent flow, viscosity becomes super important because it contributes to the formation of shear stress. The fluid's viscosity essentially determines how much stress is needed to generate a certain rate of strain or velocity gradient in the fluid. So, understanding the viscosity of a fluid tells us a lot about how it will behave in different conditions, including how it will react to shear stress as it flows through a tube.