Bernoulli's Principle (Ideal Flow)

Fluid dynamics simulator for steady-state flow. Assumes an incompressible, inviscid fluid (no friction or roughness losses).

P₁

Pressure 1

Absolute pressure at the pipe inlet

01000

v₁

Velocity 1

Fluid velocity at the inlet

0.01100

h₁

Height 1

Height of the geometric center of section 1

0100

d₁

Diameter 1

Inside diameter of the pipe at point 1

0.1100

Results

Velocity 2

—

m/s

Δv: — m/s

Pressure 2

—

kPa

ΔP: — kPa

Flow Rate

—

m³/s

Fluid Dynamics (Bernoulli's Equation)

Fluid

Select the fluid type (determines density)

Fundamentals & Explanation

Continuity Equation

A_1 \cdot v_1 = A_2 \cdot v_2

The volumetric flow rate Q remains constant along the streamline.

Bernoulli's Equation

P_1 + \frac{1}{2}\rho\, v_1^2 + \rho\, g\, h_1 = P_2 + \frac{1}{2}\rho\, v_2^2 + \rho\, g\, h_2

Venturi Effect

A \downarrow \;\Rightarrow\; v \uparrow \;\Rightarrow\; P_{\text{st}} \downarrow

Qualitative relationship in a cross-section constriction.

1. Theoretical Framework

Bernoulli's Equation is an expression of the Principle of Conservation of Energy applied to fluids in motion. Along a streamline, the total energy per unit volume splits into three contributions:

Pressure energy (flow work): the work exerted by the fluid's pressure on an adjacent volume element.
Kinetic energy (dynamic): proportional to the square of the fluid velocity, reflecting its capacity to do work through motion.
Gravitational potential energy (elevation): associated with the fluid's height above a reference datum.

The Venturi Effect is a direct consequence: when a fluid passes through a constriction, the Continuity Equation demands an increase in velocity to preserve the flow rate. By Bernoulli's equation, this velocity increase translates into a decrease in static pressure, establishing an inversely proportional relationship between the two quantities.

2. Boundary Conditions (Scope & Limitations)

The mathematical model used by this simulator assumes the following idealized conditions, which users should consider when interpreting results:

Ideal, incompressible fluid: the density (ρ) remains constant. For gases at high velocities (Mach > 0.3), compressibility effects become significant and this model loses accuracy.
Steady-state flow: flow properties do not change over time. Transient phenomena such as water hammer are excluded.
Inviscid flow: no internal friction losses or wall friction (head losses) are considered.
No external mechanical work: the action of pumps, compressors, or turbines on the fluid is not modeled.

3. Our Computational & Visual Approach

Edge-case handling: the simulator includes predictive alerts. If the calculated pressure drops below the liquid's vapor pressure, it warns about the risk of cavitation. Additionally, the system mathematically blocks scenarios that would result in a negative absolute pressure (impossible vacuum).

Visual abstraction vs. scale: the dynamic pipe graphic is a schematic, topological representation. The height markers (Δh) and diameters are not drawn to a strict 1:1 visual scale; they are normalized to ensure that conceptual behavior —tangent lines, velocity vectors, and the pressure gradient— remains readable on any device, prioritizing education over hyperrealism.

Çengel, Y. A., & Cimbala, J. M. Fluid Mechanics: Fundamentals and Applications.Munson, B. R., Young, D. F., & Okiishi, T. H. Fundamentals of Fluid Mechanics.CRC Handbook of Chemistry and Physics (fluid properties and vapor pressures).Bernoulli's Principle — Wikipedia