To analyze the situation of an incompressible fluid flowing through a horizontal pipe, we can apply the principles of fluid dynamics, particularly Bernoulli's equation. This equation relates the pressure, velocity, and height of a fluid flowing along a streamline. For a horizontal flow, changes in height can be ignored, simplifying our focus to the pressure and velocity components.

Applying Bernoulli's Principle

Bernoulli's equation states that for an incompressible fluid with negligible viscosity, the total mechanical energy along a streamline is constant. Mathematically, it can be expressed as:

p + 0.5ρv² = constant

Where:

• p = fluid pressure
• ρ = fluid density
• v = fluid velocity

Analyzing Two Points in the Pipe

Let’s denote the two points in the pipe as follows:

• At point 1: Pressure is p1, velocity is v1.
• At point 2: Pressure is p2, velocity is 2v1.

According to Bernoulli's equation, we can set up the following relationship between the two points:

p1 + 0.5ρv1² = p2 + 0.5ρ(2v1)²

Now, let’s simplify the equation:

We can rewrite the second term on the right side:

0.5ρ(2v1)² = 0.5ρ(4v1²) = 2ρv1²

Substituting this back into the equation gives us:

p1 + 0.5ρv1² = p2 + 2ρv1²

Rearranging the Equation

Now, we can rearrange the equation to isolate the pressures:

p1 - p2 = 2ρv1² - 0.5ρv1²

This simplifies to:

p1 - p2 = 1.5ρv1²

Conclusion About Pressures

From this final equation, we can draw an important conclusion: since 1.5ρv1² is a positive quantity (as density and velocity are both positive), it follows that:

p1 > p2

In simpler terms, the pressure at point 1 is greater than the pressure at point 2. This result aligns with our expectations: as the fluid's speed increases (from v1 to 2v1), the pressure must decrease in order to conserve energy within the system. This phenomenon is quite common in fluid dynamics and can be observed in various applications, such as in the design of airplane wings or the flow of fluids in piping systems.