Solve Bernoulli's equation for velocity, pressure, or height between two flow points. Energy balance visualization and fluid reference table included.
Bernoulli's equation is one of the most important relationships in fluid mechanics, relating pressure, velocity, and height at two points along a streamline in steady, incompressible, inviscid flow. It is the mathematical expression of energy conservation for flowing fluids.
This calculator solves Bernoulli's equation for any of the three unknowns at point 2: velocity, pressure, or height. Enter the conditions at point 1 (pressure, velocity, height) along with the known values at point 2, select what to solve for, and get the result instantly. The tool also computes dynamic pressure, hydrostatic pressure, and total head.
Presets for common scenarios — garden hose nozzles, venturi meters, water towers, airplane wings, and fire hoses — demonstrate how the equation applies across engineering and everyday physics. An energy balance bar visualizes the relative contributions of static pressure, dynamic pressure, and hydrostatic pressure. It keeps the input and output states visible together so the pressure-velocity tradeoff is easier to follow. Check the example with realistic values before reporting.
Bernoulli's equation is essential for pipe sizing, nozzle design, venturi flow meters, and aerodynamics. This calculator eliminates the tedious algebra of rearranging the equation for different unknowns and provides immediate unit conversions.
The energy balance visualization helps students intuitively understand how static pressure, dynamic pressure, and elevation head trade off along a streamline — making it an excellent teaching tool.
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂. Where P = static pressure (Pa), ρ = fluid density (kg/m³), v = velocity (m/s), g = 9.80665 m/s², h = height (m).
Result: v₂ = 18.73 m/s
Water at 275,790 Pa and 2 m/s at 1 m height exits at atmospheric pressure (101,325 Pa) at ground level. Bernoulli gives v₂ = 18.73 m/s — typical of a garden hose nozzle.
Bernoulli's equation is applied in venturi flow meters (measuring flow rate from pressure drop), pitot tubes (measuring aircraft airspeed), carburetors (mixing fuel and air), perfume atomizers, and water distribution systems. The Torricelli theorem for fluid draining from a tank is a direct consequence of Bernoulli's equation with the top surface at atmospheric pressure.
The ideal Bernoulli equation assumes no friction, no heat transfer, and incompressible flow. In reality, engineers use extended versions: the modified Bernoulli equation adds a friction loss term, and for compressible flows (Mach > 0.3), the compressible Bernoulli or isentropic flow equations must be used instead.
Daniel Bernoulli published this principle in his book Hydrodynamica in 1738, though the mathematical formulation commonly used today was developed later by Leonhard Euler. The equation represents one of the earliest applications of energy conservation to fluid mechanics and remains one of the most widely used equations in engineering.
It states that the total mechanical energy per unit volume of a flowing fluid remains constant along a streamline: P + ½ρv² + ρgh = constant. When velocity increases, pressure decreases, and vice versa.
It applies to steady, incompressible, inviscid (frictionless) flow along a streamline. It is a good approximation for low-speed liquid flow with minimal friction losses.
Energy conservation: if fluid speeds up (higher kinetic energy), it must lose pressure energy to keep the total constant. This is the venturi effect.
Air flowing faster over the curved top of a wing has lower pressure than the slower air below. The pressure difference creates an upward net force (lift). Bernoulli's principle partially explains this.
Real pipe flows have friction losses. Engineers add a head-loss term to the equation (Darcy-Weisbach equation) to account for viscous dissipation in long pipes.
Yes, for low-speed air flow (below Mach 0.3). Above Mach 0.3, compressibility effects become significant and the incompressible Bernoulli equation loses accuracy.