Fluid Mechanics
Bernoulli Calculator
Apply Bernoulli's equation P + ½ρv² + ρgh = const to find pressure or velocity at any point in steady fluid flow.
Point 1 (Known)
Point 2 (Solve For)
Result at Point 2
P + ½ρv² + ρgh = const
Step-by-Step Solution
What is Bernoulli's Equation?
Bernoulli's equation is a statement of energy conservation for fluid flow. Along a streamline in steady, incompressible, non-viscous flow, the sum of pressure energy, kinetic energy and potential energy per unit volume remains constant. It explains how aircraft wings generate lift, how carburettors mix fuel, and why a shower curtain blows inward.
P + ½ρv² + ρgh = constant
P = pressure (Pa) | ρ = fluid density (kg/m³) | v = flow velocity (m/s) | h = height (m)
💡 Bernoulli's equation shows that where fluid flows faster, pressure is lower. This is the key to aerodynamic lift: air over the curved upper wing surface moves faster, creating lower pressure, producing an upward force.
Real-World Applications
Aircraft Wings
Wing shape forces air to move faster over the top surface, reducing pressure and creating lift.
Sailing
Sails work like vertical wings. Wind moving faster over the convex side creates lower pressure, pulling the sail forward.
Venturi Meters
A constriction in a pipe speeds up flow and drops pressure. Measuring this drop reveals the flow rate.
Shower Curtain
Fast-moving shower air has lower pressure, causing the higher-pressure outside air to push the curtain inward.
Frequently Asked Questions
What are the assumptions of Bernoulli's equation?
(1) Steady flow — velocity at each point does not change with time. (2) Incompressible fluid — density is constant (valid for liquids; approximate for slow air flow). (3) Non-viscous — no internal friction. (4) Flow along a streamline. Real fluids violate these to some extent.
How does Bernoulli's equation explain lift?
A wing creates a situation where air above the wing moves faster (longer path, same transit time — though this equal-transit-time explanation is simplified). Faster flow → lower pressure (Bernoulli). The pressure difference between bottom and top creates net upward force = lift.
What is the continuity equation?
For incompressible flow, A₁v₁ = A₂v₂ (volume flow rate is conserved). A narrower pipe means faster flow. Combined with Bernoulli, this explains why a garden hose with a thumb over the end shoots water farther.