Bernoulli Dynamics

Technical Reference Guide // Bernoulli's Principle

REF-07-ENERGY

Conservation in Motion

Bernoulli’s Principle is a statement of the Law of Conservation of Energy applied to flowing fluids. In a steady, non-viscous, incompressible fluid, the total mechanical energy remains constant along a streamline. This energy exists as static pressure, kinetic energy, and potential energy.

The Bernoulli Equation

\[ P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \]

Static Pressure

Dynamic Pressure (KE)

Hydrostatic Head (PE)

The Velocity-Pressure Relationship

The most critical aspect of Bernoulli’s Principle is the inverse relationship between speed and pressure. If the elevation remains constant, an increase in the fluid's velocity must be accompanied by a decrease in its static pressure.

This occurs because energy previously stored as static pressure is converted into kinetic energy to move the fluid faster.

High Velocity

Low Pressure

The Venturi Effect

A Venturi tube is a pipe with a constricted section. To maintain the same mass flow rate (Continuity Equation), fluid must speed up as it enters the narrower section. Bernoulli's Principle states that this high-speed fluid will have lower pressure than the slower fluid.

Technical Schematic: FIG 03-A // Flow Constriction

Observation: P₂ < P₁ because v₂ > v₁

Continuity

Flow rate (\(Q = Av\)) is constant. Smaller Area (\(A\)) requires higher Velocity (\(v\)).

\( A_1 v_1 = A_2 v_2 \)

Energy

The loss in static pressure energy powers the gain in kinetic energy.

\( \Delta P = -\frac{1}{2}\rho \Delta v^2 \)

Aerodynamics: Lift Force

Airfoils generate lift by creating speed differences. A wing shape forces air over the top to travel faster than air underneath, resulting in a low-pressure zone above the wing.

Pressure Differential

Top surface: High Velocity → Low Pressure.
Bottom surface: Lower Velocity → Higher Pressure.
Result: Net upward force (LIFT).

F_LIFT = (P_bottom - P_top) × AREA

Technical Summary Reference

Static vs Dynamic Static pressure + Dynamic pressure (\( \frac{1}{2}\rho v^2 \)) is constant.

Elevation Moving fluid "up" costs pressure or velocity (\( \rho gh \)).

Idealization Applies to steady, laminar, non-viscous flow.

Bernoulli Principle Reading Guide Revised V2

Bernoulli Principle Reading Guide Revised V2

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