Bernoulli Equation Applicability
Imagine you're holding a garden hose with your thumb partially covering the opening. Water shoots out much faster than it flows inside the hose. You've just seen Bernoulli's principle in action — the pressure in the narrow opening drops, and speed increases. But here's the catch: if you kink the hose and water stops flowing entirely, Bernoulli's equation no longer describes what's happening. Why? Because the flow is no longer steady.
That's the heart of applicability: Bernoulli's equation is not a universal law of fluids. It's a powerful but conditional tool.
The Intuition First
Bernoulli's equation is really a statement of energy conservation for a moving fluid. Think of a streamline — an imaginary line that follows the path of a fluid particle. As that particle moves along, three forms of energy can change:
- Kinetic energy (from speed)
- Gravitational potential energy (from height)
- Pressure energy (from the fluid being squeezed)
The equation says: along a streamline, the sum of these three remains constant — provided nothing adds or removes energy from the fluid (no friction, no pumps, no turbines).
The "pressure energy" term is not a separate form of energy in thermodynamics. It's the work done by pressure forces to move the fluid. But for problem-solving, treat it as an energy term.
The Precise Statement
For an ideal fluid (incompressible, inviscid) flowing steadily along a streamline, Bernoulli's equation is:
P+21ρv2+ρgh=constant
Where:
- P = static pressure (the actual pressure in the fluid)
- ρ = fluid density (constant for incompressible flow)
- v = flow speed
- g = acceleration due to gravity
- h = height above a reference level
P1+21ρv12+ρgh1=P2+21ρv22+ρgh2
This form is used when comparing two points (1 and 2) on the same streamline.
The Four Conditions for Applicability
You cannot use Bernoulli's equation unless all four conditions are met:
1. Steady Flow
The velocity at any fixed point does not change with time. If you turn a tap on and off, or if a pump pulses, the flow is unsteady — Bernoulli fails.
2. Incompressible Flow
The density ρ is constant. This is excellent for liquids (water, oil) and for gases at low speeds (below about 30% of the speed of sound). For high-speed gas flow (e.g., air around an aircraft wing at Mach 0.8), compressibility matters, and Bernoulli's equation is inaccurate.
3. Inviscid Flow (No Friction)
The fluid has zero viscosity — no internal friction. Real fluids have viscosity, but in many practical situations (flow far from walls, short distances), viscous effects are negligible. Never use Bernoulli's equation for flow through long pipes, where friction dominates.
A common mistake: applying Bernoulli's equation to flow through a pipe with significant friction. The pressure drop due to friction is not accounted for. Use the Darcy-Weisbach equation or the Hagen-Poiseuille equation instead.
4. Along a Streamline
The equation holds along a single streamline. If you pick two points on different streamlines, the constant may differ. However, if the flow is irrotational (no vorticity), the constant is the same everywhere — a special case often used in aerodynamics.
When Can You Relax the Conditions?
- Irrotational flow: If the flow is irrotational (curl of velocity = 0), Bernoulli's constant is the same for all streamlines. This is true for many idealised flows (e.g., flow around a sphere at low speeds, flow through a venturi meter).
- Gradual changes: If friction is small and the flow is nearly steady, Bernoulli gives a good approximation. Engineers often use it with a "loss coefficient" to account for minor friction.
A Quick Decision Tree
Ask these questions in order:
- Is the flow steady? (No time-dependent changes at a point)
- Is the fluid incompressible? (Liquid? Gas at low speed?)
- Is viscosity negligible? (Short distances, no long pipes, no boundary layers) …