Final answer:
Bernoulli's equation can be used along a streamline in a pipe with a constant diameter and around a narrowing in the pipe but not across a pump or within a turbulent flow region, as it requires a steady, non-viscous, and incompressible flow without any additional energy input.
Step-by-step explanation:
The question is regarding the use of Bernoulli's equation along a streamline in different flow scenarios. Bernoulli's equation is a principle in fluid dynamics that relates the pressure, velocity, and height of a fluid segment to each other. It assumes that the fluid is incompressible and that there is no energy loss due to friction or other dissipative forces.
Here are the appropriate scenarios for applying Bernoulli's equation:
- A streamline along a pipe with constant diameter is valid because the flow is likely to remain steady and non-viscous if it has a constant diameter, thus satisfying the conditions for Bernoulli's equation.
- A streamline around a narrowing in the pipe is also valid. As Bernoulli's equation can predict the behavior of the fluid when the cross-sectional area changes, resulting in speed and pressure changes.
- A streamline across a pump is not valid because a pump adds energy into the system, which violates the assumptions of Bernoulli's equation.
- A streamline within a turbulent flow region is not suitable for applying Bernoulli's equation due to the chaotic nature of turbulent flow, which introduces energy dissipation and does not satisfy the equation's conditions.