Static pressure decreases as velocity increases primarily to maintain a constant total energy within a flowing fluid, according to a fundamental principle in fluid dynamics. This relationship is described in the provided reference from BYJU'S: "If pressure increases, the velocity decreases to keep the algebraic sum of potential energy, kinetic energy, and pressure constant. Similarly, if velocity increases, the pressure decreases to keep the sum of potential energy, kinetic energy, and pressure constant."
Understanding the Inverse Relationship
This inverse relationship is a key concept in fluid mechanics, particularly when considering the flow of incompressible fluids along a streamline. Think of the fluid's energy as being distributed between different forms:
- Potential Energy: Related to the fluid's height or position (often constant in many common scenarios like horizontal flow).
- Kinetic Energy: Related to the fluid's motion or velocity.
- Pressure Energy: Represented by the static pressure of the fluid.
The principle highlights that, in a steady flow where potential energy changes are minimal, if the kinetic energy (related to velocity) increases, the pressure energy (related to static pressure) must decrease to keep the total energy sum constant.
Bernoulli's Principle
The concept described in the reference is a direct application of Bernoulli's Principle. This principle states that for an inviscid (frictionless) and incompressible fluid in steady flow, the total mechanical energy along a streamline is constant. This total energy is the sum of:
- Static pressure (P)
- Dynamic pressure (½ρv², where ρ is fluid density and v is velocity)
- Hydrostatic pressure (ρgh, where g is gravity and h is height)
Mathematically, Bernoulli's equation is often written as:
P + ½ρv² + ρgh = Constant
In many common scenarios, such as flow through a horizontal pipe or over an airplane wing, the height (h) is constant or changes negligibly. In such cases, the ρgh term is effectively constant, simplifying the relationship to:
P + ½ρv² = Constant
This simplified form clearly shows the inverse relationship between static pressure (P) and dynamic pressure (½ρv²), which is directly proportional to the square of velocity (v²). If velocity increases, ½ρv² increases, and therefore P must decrease to keep the sum constant.
How it Works in Practice
Consider fluid flowing through a pipe that narrows. As the pipe narrows, the fluid must speed up (velocity increases) to maintain continuous flow (mass conservation). According to the principle mentioned in the reference and Bernoulli's equation:
- In the wider section: Velocity is lower, so static pressure is higher.
- In the narrower section: Velocity is higher, so static pressure is lower.
| Characteristic | Wider Pipe Section | Narrower Pipe Section |
|---|---|---|
| Velocity (v) | Lower | Higher |
| Kinetic Energy | Lower | Higher |
| Static Pressure (P) | Higher | Lower |
| Total Energy | Constant | Constant |
This pressure difference between the wider and narrower sections is what drives the acceleration of the fluid into the constriction.
Examples and Applications
The inverse relationship between static pressure and velocity is fundamental to many real-world phenomena and engineering applications:
- Airplane Wings: Air flowing over the curved top surface of a wing travels faster than the air flowing beneath it. This results in lower static pressure on top of the wing compared to the bottom, creating lift.
- Carburetors: Air speeds up as it passes through a narrow venturi section, causing a pressure drop that draws fuel into the airflow.
- Pitot Tubes: These devices measure fluid velocity by comparing the static pressure in the flow to the total pressure (static + dynamic) at a stagnation point where velocity is zero.
- Sprayers/Atomizers: Air blown rapidly across the top of a tube dipped in liquid lowers the pressure above the liquid surface, allowing atmospheric pressure to push the liquid up the tube and into the fast-moving air stream, where it is broken into droplets.
In summary, the decrease in static pressure as velocity increases is a consequence of the conservation of energy in a fluid flow, ensuring that the sum of potential energy, kinetic energy, and pressure remains constant, as stated by the principle derived from Bernoulli's equation.
