If I understand your question about pressure, you want to take a look at the Bernoulli Principle. It states that for steady state laminar incompressible flow:

density * v2 / 2 + density * g * height + pressure = constant

As regards velocity changes with respect to cross section, it is pretty straightforward to use mass conservation to show that

v1A1=v2A2 for incompressible flows.

Cheers!

You're correct about the relationship between the speed and pressure of a liquid in a flowing system, but it's important to clarify the context:

1. When a liquid encounters a constriction:

• Speed increases: As the liquid flows through a narrow section in a pipe, the area available for it to flow through decreases. To maintain the same flow rate (mass of liquid flowing per unit time), the liquid's velocity must increase to compensate for the smaller cross-sectional area. Think of it like squeezing a garden hose; the water comes out faster because there's less space for it to flow.
• Pressure decreases: This phenomenon is explained by Bernoulli's principle. It states that for an ideal, incompressible fluid in steady flow, the total mechanical energy per unit volume remains constant along a streamline. This means the sum of the pressure energy, potential energy, and kinetic energy per unit volume is constant. When the liquid speeds up due to the constriction, the kinetic energy term increases. To compensate and maintain the constant total energy, the pressure energy term (pressure) must decrease.

2. When the cross-sectional area of the pipe increases:

• Velocity decreases: This is the opposite of the first case. As the area available for flow increases, the liquid doesn't need to move as fast to maintain the same flow rate. Imagine widening a river; the water flows slower because there's more space for it to spread out.
• Pressure typically remains constant or changes slightly: In most cases, if the pipe is horizontal and there are no other factors at play, the pressure change will be negligible. This is because the pressure change due to velocity change is usually small compared to the overall pressure in the system. However, in specific situations like open channels or fluids with high viscosity, there might be a slight pressure decrease due to energy dissipation.

Remember, these principles apply to ideal, incompressible fluids in steady flow conditions. Real-world fluids and systems might have additional factors like viscosity, friction, and turbulence that can affect the exact behavior.

Dr Murtadha Shukur thank you for your contribution to the discussion