Final answer:
The pressure difference in a pipe carrying a viscous fluid is highest where the fluid velocity is highest, which occurs when the pipe's cross-sectional area decreases; based on principles of fluid continuity and Bernoulli's equation.
Step-by-step explanation:
The pressure difference across a segment of the pipe through which a viscous fluid flows is greatest at the point where the fluid velocity is highest. This happens because, according to the principle of continuity, the mass of fluid entering a pipe section must be equal to the mass leaving it, implying that fluid velocity increases as the cross-sectional area decreases. Furthermore, Bernoulli's equation indicates that an increase in fluid velocity will result in a decrease in pressure, leading to a greater pressure difference. It's essential to think about the cross-sectional area and flow rate because when a fluid goes from a larger to a smaller pipe, the streamlines become closer, the fluid velocity increases due to the smaller cross-sectional area, and the pressure correspondingly changes. Fluid dynamics concepts such as the relationship between pressure difference, velocity, and cross-sectional area are central to understanding the flow behavior in such situations.
For example, if a fluid is flowing through a pipe that narrows down, the streamlines will converge and the fluid will move faster in the narrow section. This increase in velocity indicates a decrease in pressure, and therefore, the pressure difference across that narrow section would be the greatest compared to sections where the pipe has a uniform or larger diameter.