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Home/Classical Mechanics/Bernoulli Flow

Bernoulli Flow

The Bernoulli Flow simulator visualizes the fundamental principles governing the steady, incompressible flow of an ideal fluid through a pipe with a varying cross-section. At its core, it demonstrates the interplay between two key conservation laws: the continuity equation and Bernoulli's principle. The continuity equation, A₁v₁ = A₂v₂, states that for an incompressible fluid, the product of the cross-sectional area (A) and the flow velocity (v) at any point along the pipe is constant. This means fluid must speed up (v increases) as it enters a constriction (A decreases). Bernoulli's principle, derived from the conservation of energy, relates this change in speed to a change in pressure. Its mathematical form is P + (1/2)ρv² + ρgh = constant, where P is pressure, ρ is fluid density, v is speed, g is gravity, and h is height. For horizontal flow (constant h), the principle simplifies to show that an increase in kinetic energy per unit volume (1/2)ρv² must be balanced by a decrease in pressure (P). The simulator makes several simplifying assumptions: the fluid is ideal (inviscid and incompressible), the flow is steady and laminar, and energy losses due to friction or turbulence are neglected. By interacting with the controls to adjust inlet pressure, pipe geometry, or fluid density, students can observe real-time changes in flow speed, pressure readings, and streamline patterns. This direct manipulation reinforces the conceptual link between geometry, velocity, and pressure, helping learners internalize why wings generate lift, how atomizers work, and why a constricted garden hose has a faster, thinner stream.

Who it's for: High school and introductory college physics students studying fluid dynamics, as well as educators seeking a visual tool to demonstrate the continuity equation and Bernoulli's principle.

Key terms

  • Bernoulli's Principle
  • Continuity Equation
  • Fluid Dynamics
  • Incompressible Flow
  • Pressure Gradient
  • Flow Velocity
  • Cross-sectional Area
  • Conservation of Energy

Pipe & fluid

3 m/s
3
1000 kg/m³
110 kPa

Measured values

v₂ (narrow)9.00 m/s
P₂ (narrow)74.00 kPa
P₂ − P₁-36.00 kPa
½ρ(v₁²−v₂²)-36.00 kPa

How it works

Ideal horizontal flow: A₁v₁ = A₂v₂ (continuity). Bernoulli: P + ½ρv² is constant along a streamline at constant height. In a narrow segment v increases, so static pressure P decreases — the Venturi / airplane-wing effect in a nutshell. The canvas shows steady v₁, v₂ and P; animated dots only illustrate the flow direction and that parcels move faster where the pipe is narrow (same volume flux, different speed).

Key equations

A₁v₁ = A₂v₂ · P₁ + ½ρv₁² = P₂ + ½ρv₂²

Frequently asked questions

If the fluid speeds up in the narrow section, doesn't that mean it has more energy? Why does the pressure drop?
Yes, the fluid gains kinetic energy. According to Bernoulli's principle, which is a statement of energy conservation for a flowing fluid, the total energy per unit volume (pressure energy + kinetic energy + potential energy) must remain constant if no work is done. In a horizontal pipe, the potential energy is constant. Therefore, the gain in kinetic energy (1/2)ρv² must be compensated by an equal loss in pressure energy (P). The energy isn't created; it's converted from one form (pressure) to another (motion).
Does this explain how an airplane wing generates lift?
Bernoulli's principle is a key part of a common explanation for lift, but it is not the complete story. The wing's shape (airfoil) causes airflow to speed up more over the top than the bottom. According to Bernoulli, this creates a pressure difference, with lower pressure on top, resulting in an upward force. However, the full explanation also involves Newton's third law and the turning of the airflow (angle of attack). This simulator models the core Bernoulli effect that contributes to that pressure difference.
Why doesn't the simulator show turbulence or the fluid slowing down due to friction?
This simulator uses an idealized model of an 'ideal fluid,' which is inviscid (has no internal friction) and incompressible. These simplifications allow us to isolate and clearly demonstrate the pure relationship between area, velocity, and pressure described by the continuity equation and Bernoulli's principle. In real-world pipes, viscosity causes friction losses, leading to a gradual pressure drop along the pipe even without a constriction, and high speeds can cause turbulent flow, which violates the assumptions of the model.
Can I use Bernoulli's principle for gases like air?
Yes, Bernoulli's principle applies to gases as well as liquids, provided the flow is steady and the gas can be treated as nearly incompressible. For air, this is a good approximation at speeds significantly below the speed of sound (typically Mach < 0.3). Many real-world applications, like venturi meters, airplane wings, and even some weather phenomena, involve the flow of air and can be analyzed using these principles.