Final answer:
The greatest fluid velocity in areas where streamlines are closest together is due to the conservation of mass, dictated by the continuity equation, which requires the fluid to move faster through smaller cross-sectional areas. This is especially true for incompressible fluids. Streamlines in diagrams illustrate these changes in velocity, affected by factors like viscosity and pipe boundaries.
Step-by-step explanation:
The fluid velocity is greatest where streamlines are closest together due to the principle of conservation of mass, often applied as the continuity equation in fluid dynamics. According to this equation, when a non-compressible fluid flows through different cross-sectional areas, the mass flow rate must remain constant.
If the cross-sectional area through which the fluid is flowing decreases, the fluid must travel faster to maintain the same mass flow rate.
This principle can be observed in real-life scenarios, such as when a large pipe narrows into a smaller pipe. The streamlines get closer together, indicating that the fluid's velocity has increased. This effect is particularly noticeable in incompressible fluids where density does not change significantly with pressure.
In the presence of viscosity, which is the fluid's resistance to flow due to internal friction, the speed profile across a pipe can vary, with the greatest speed often at the midpoint. This is due to the drag at the boundaries of the pipe where the fluid experiences resistance.
The concept of streamlines helps visualize these fluid dynamics phenomena, where the density of streamlines can indicate the relative velocities within the flow field.