Question

In the fully developed region of flow in a circular tube, will the velocity profile change in the flow direction? How about the temperature profile?

Short answer

Answer: No, in the fully developed region of flow in a circular tube, both the velocity profile and the temperature profile will not change along the flow direction, as long as the tube diameter, fluid properties, pressure gradients, boundary conditions, and heat transfer properties remain constant.

Step-by-step solution

Understanding the Fully Developed Flow

A fully developed flow in a circular tube is a steady-state flow where the velocity profile does not change along the flow direction. In other words, the velocity profile remains the same at any axial location along the tube. This occurs when the entrance effects have died out, and the flow has reached equilibrium with the forces acting on it, such as viscous and pressure forces.

Analyzing the Velocity Profile

In a fully developed laminar flow, the velocity profile is parabolic in shape, with the maximum velocity at the center of the tube and zero velocity near the walls due to the no-slip boundary condition. Since the flow is fully developed and has reached equilibrium, the velocity profile will not change along the flow direction as long as the conditions remain constant, such as tube diameter, fluid properties, and pressure gradients.

Analyzing the Temperature Profile

The temperature profile in a fully developed flow depends on the heat transfer mechanisms at play and the boundary conditions. Heat conduction and convection both contribute to the temperature distribution in the fluid. If the wall temperature is held constant or varies predictably, and the heat transfer properties (surface heat flux, convective heat transfer coefficient) are constant, then the temperature profile will also remain steady and not change along the flow direction in the fully developed region.

Conclusion

In summary, in the fully developed region of flow in a circular tube, the velocity profile will not change along the flow direction, as the velocity profile has reached equilibrium with the acting forces. Similarly, if the boundary conditions and heat transfer properties remain constant, the temperature profile will also remain constant and not change along the flow direction.