1. Pressure

Definition: Pressure is the force exerted by the fluid per unit area. It is a measure of the fluid's tendency to exert force on the walls of its container or other objects in the flow.

Units: Pascal (Pa) or N/m² in SI units.

Types of Pressure:

• Static Pressure: The pressure exerted by the fluid at rest or as it flows.
• Dynamic Pressure: The pressure associated with the fluid's motion, related to its velocity.
• Total Pressure: The sum of static and dynamic pressures.

2. Velocity

Definition: Velocity is the speed and direction at which the fluid particles are moving.

Units: Meters per second (m/s) in SI units.

Relevance: Higher velocities often correspond to lower pressures in a flowing fluid (Bernoulli's principle). It is a key factor in determining the kinetic energy of the fluid.

3. Area

Definition: Area refers to the cross-sectional area of the pipe, channel, or duct through which the fluid is flowing.

Units: Square meters (m²) in SI units.

Relevance: Affects the flow rate and velocity of the fluid. According to the continuity equation, changes in area cause changes in velocity and pressure.

Key Relationships Between Pressure, Velocity, and Area

A. Continuity Equation

The continuity equation is derived from the principle of mass conservation and applies to incompressible fluids:

A1 v1 = A2 v2

Explanation:

• If the area decreases (A2 < A1), the velocity must increase (v2 > v1).
• Conversely, if the area increases (A2 > A1), the velocity decreases (v2 < v1).

B. Bernoulli's Principle

Bernoulli's principle describes the conservation of energy in a flowing fluid:

P + (1/2) ρ v² + ρ gh = constant

Explanation:

• In regions where velocity increases (e.g., a constriction in a pipe), static pressure decreases.
• In regions where velocity decreases (e.g., a wider section of a pipe), static pressure increases.

Practical Examples

1. Pipe with Varying Diameter: If a pipe narrows, the velocity of the fluid increases to maintain the same flow rate. As velocity increases, pressure decreases (Bernoulli’s principle).
2. Airplane Wings: The curved shape of the wing causes air to move faster over the top than beneath. This results in lower pressure on top, creating lift.
3. Venturi Effect: In a Venturi tube, a narrowing of the tube increases fluid velocity and decreases pressure, demonstrating both the continuity equation and Bernoulli's principle.

Summary of Relationships

• Velocity increases as cross-sectional area decreases (continuity).
• Pressure decreases as velocity increases in a streamlined flow (Bernoulli’s principle).
• Pressure, velocity, and area are interconnected and dictate how fluids behave in systems ranging from household plumbing to aerodynamics.