Final answer:
In a steady uniform flow of an incompressible fluid in a pipe with constant cross-sectional area, the maximum velocity remains consistent once the flow is fully developed; the velocity profile does not change with distance from the entrance after the entrance region. option D is correct.
Step-by-step explanation:
For a steady uniform flow of an incompressible liquid that enters a pipe, one may wonder about the characteristics of the entrance region. The entrance region is where the fluid adjusts from the conditions in the large reservoir or container from which it comes into the conditions characteristic of flow in a pipe. According to the principles of fluid dynamics, specifically the continuity equation for incompressible fluids, the product of the cross-sectional area (A) and the fluid's velocity (v) at any point along the pipe must remain constant, assuming no fluid is added or removed.
In this scenario, we're discussing a pipe where the radius remains constant, implying that the cross-sectional area of the pipe does not change. Thus, the velocity profile must develop as the fluid enters the pipe, reaching a fully developed profile at some distance from the entrance. This development leads to the correct option:
- b. the maximum velocity decreases with distance from the entrance.
Initially, at the very entrance of the pipe, there is a flat velocity profile where the fluid particles move at the same velocity as they leave the large reservoir. As the fluid moves further into the pipe, the velocity profile develops due to the no-slip condition at the walls of the pipe, which essentially means that the fluid at the pipe walls has zero velocity relative to the pipe. This development results in a parabolic profile with a maximum velocity at the center of the pipe. However, since the problem scenario does not indicate a narrowing or expansion of the pipe, the maximum velocity will not increase or decrease, but rather, the velocity profile changes shape until it is fully developed.
Therefore, the most accurate statement is:
- d. The velocity profile does not change with distance from the entrance.
Once the flow is fully developed (beyond the entrance region), the velocity profile remains constant with distance along the pipe.