Bernoulli’s Principle Explained for Engineers
In the article Fluid Dynamics Basics for Engineers we introduced the concepts of streamlines, flow tubes, and the continuity equation, showing how velocity is inversely proportional to the cross-sectional area of a pipe.
This leads naturally to the next fundamental concept: the relationship between pressure, velocity, and elevation described by Bernoulli’s principle
Let’s now apply this principle to a typical scenario.
Considering an ideal incompressible fluid in steady motion along a conduit.
As introduced in Fluid Dynamics Basics for Engineers, steady flow is defined as a condition where velocity, pressure, and density at any given point in the system remain constant over time.
Now take two sections of the conduit, S1 and S2 , located at heights h1 and h2 above a horizontal reference plane.
At each section the fluid is characterized by pressure (p1,p2) and velocity (v1,v2).
According to Bernoulli’s principle, the following relation applies:
p1 + 1/2 ρ v12 + ρ g h1 = p2 + 1/2 ρ v22 + ρ g h2
Where:
- p₁, p₂ = pressure at sections S₁ and S₂ [Pa]
- v₁, v₂ = velocity at sections S₁ and S₂ [m/s]
- h₁, h₂ = height of sections S₁ and S₂ above the reference plane [m]
- ρ = fluid density [kg/m³]
- g = gravitational acceleration (≈ 9.81 m/s²)
This relation illustrates how mechanical energy is conserved along a streamline. As the cross-sectional area decreases from S1 to S2, the fluid accelerates (v₂ > v₁), leading to a drop in pressure.
Hydrodynamic Paradox
In the following figure, two manometric measurements are taken along a horizontal pipe where the diameter decreases from section S1 to section S2.
Although the pipe narrows from S1 to S2—causing the fluid velocity to increase—the manometric column measured at S2 is lower than the one at S1. This indicates that the static pressure decreases as the flow accelerates.
This observation may seem contradictory—why would pressure drop where velocity increases?
According to Bernoulli’s principle, the increase in kinetic energy associated with the higher velocity must be balanced by a reduction in pressure energy, since the elevation remains constant. As a result, when the cross-sectional area decreases and the velocity increases, the static pressure necessarily drops.
This phenomenon is historically known as the hydrodynamic paradox. It highlights a fundamental concept in fluid dynamics: pressure cannot be interpreted in isolation, but must always be analyzed together with velocity and elevation within an energy balance framework.
The hydrodynamic paradox illustrates why intuition alone is insufficient when analyzing fluid flow. Instead, engineers must rely on conservation principles to correctly interpret pressure measurements and predict system behavior in real piping networks.
Limits of Bernoulli’s Equation
Although Bernoulli’s principle is fundamental in fluid dynamics, its equation is derived under simplifying assumptions: incompressible, inviscid, and steady flow along a streamline.
Bernoulli’s equation cannot be applied when the flow is unsteady, highly viscous, compressible at significant Mach numbers, or when strong energy interactions occur (such as pumps, turbines, or heat exchange with the surroundings). In these cases, additional terms must be introduced in the energy balance, or different models are required to describe the system correctly.
Example of Bernoulli’s Equation Applied to Pipe Flow
Problem
An ideal liquid flows through a horizontal pipe at an initial velocity of 40 cm/s in a section where the cross-sectional area is 3 cm².
Determine the velocity of the fluid when the cross-sectional area narrows to 2 cm².
Given:
v1 = 40 cm/s = 0.40 m/s
A1 = 3 cm² = 3 × 10⁻⁴ m²
A2 = 2 cm² = 2 × 10⁻⁴ m²
Continuity equation:
A1 × v1 = A2 × v2
Solving for v2:
v2 = (A1 / A2) × v1
v2 = (3 × 10⁻⁴ / 2 × 10⁻⁴) × 0.40
v2 = 1.5 × 0.40 = 0.60 m/s
While Bernoulli’s principle describes how mechanical energy is redistributed in ideal flows, real fluids are always affected by viscosity. When viscous dissipation is taken into account, pressure no longer remains constant along a pipe — even at constant velocity and geometry. This behaviour is described by the law of Poiseuille.
Applications of Bernoulli’s Principle
Bernoulli’s principle is widely used in engineering and physics. Classic applications include:
- Venturi meter – to measure the flow rate in a pipe using differential pressure.
- Orifice plate – a simple and compact device for measuring flow based on pressure drop.
- Pitot tube – used in aerodynamics and HVAC systems to measure fluid velocity.
- Aircraft wings (airfoils) – to explain lift generation due to pressure differences.
- Open tank outflow (Torricelli’s law) – to calculate the velocity of a fluid exiting a tank.
- Ejectors and jet pumps – devices that use fluid acceleration to entrain and transport another fluid.
- Carburetors – to mix fuel and air using pressure drop created by airflow.
- Flettner rotors / Magnus effect – rotational devices using pressure differences for propulsion.
- Industrial piping – for estimating pressure drops and designing efficient flow systems.
Conclusion
Bernoulli’s principle represents one of the most important conceptual milestones in the development of fluid mechanics and engineering thinking.
Not because it provides a formula to be applied mechanically, but because it introduces a structured way of interpreting fluid behaviour in energetic terms.
Thanks to Bernoulli, pressure, velocity, and elevation are no longer independent quantities, but different expressions of the same mechanical energy carried by the fluid. This perspective allows engineers to understand why a fluid accelerates in a restriction, why pressure drops in high-velocity regions, and how energy is continuously redistributed along a flow path.
Even when the ideal assumptions behind Bernoulli’s equation are not fully satisfied, the underlying reasoning remains valid. It is the conceptual framework that enables engineers to read pipes, equipment, and entire plants as energy systems, anticipating where losses occur, where external energy is required, and where a process may become inefficient or unstable.
For this reason, Bernoulli’s principle is not just a topic in fluid dynamics, but a foundation of engineering reasoning applied to real systems. Understanding it means acquiring a common language to analyse fluid flow, bridge theory and plant reality, and design solutions that respect process physics before calculations are even performed.
Ing. Ivet Miranda
Bernoulli’s Principle Quiz
In which situation can Bernoulli’s equation be correctly applied without additional correction terms?
Other Articles You May Find Useful
- First Law of Thermodynamics and Joule’s Experiment
- Second Law of Thermodynamics: PM 2nd Kind
- Fluid Dynamics Basics for Engineers
- Heat Transfer Basics for Engineers
- Unit Operations: A Practical Introduction for Engineers
- What Does a Chemical Engineer Do in Industrial Environments?
- Chemical Engineering Core Disciplines – A Practical Overview
FAQ
What is Bernoulli’s equation?
Bernoulli’s equation describes the conservation of energy in fluid flow. It shows how pressure, velocity, and elevation are related in a moving fluid.
What is Bernoulli effect?
The Bernoulli effect refers to the drop in fluid pressure that occurs when the flow speed increases — a key observation from Bernoulli’s principle.
Bernoulli’s law: what does it state?
Bernoulli’s law states that the total mechanical energy of the fluid remains constant along a streamline: pressure energy + kinetic energy + potential energy.
How does Bernoulli’s principle work?
Bernoulli’s principle works by balancing pressure, kinetic, and potential energy in a fluid. When one increases, at least one of the others must decrease.
Which uses Bernoulli’s principle?
Bernoulli’s principle is used in aircraft wing design, flow meters (like Venturi meters), carburetors, spray bottles, and even chimney ventilation.