1. What is the Bernoulli Differential Equation?

The Bernoulli differential equation represents the conservation of energy for a flowing fluid. The differential form (dp + ρ v dv + ρ g dz = 0) shows how pressure, velocity, and elevation changes relate in an inviscid, incompressible flow along a streamline.

2. How Does the Calculator Work?

The calculator verifies the Bernoulli differential equation:

Where:

• \( dp \) — Pressure change (Pa)
• \( \rho \) — Fluid density (kg/m³)
• \( v \) — Flow velocity (m/s)
• \( dv \) — Velocity change (m/s)
• \( g \) — Gravitational acceleration (9.81 m/s²)
• \( dz \) — Elevation change (m)

Explanation: The equation balances pressure energy, kinetic energy, and potential energy changes in fluid flow.

3. Importance of Bernoulli's Equation

Details: The Bernoulli equation is fundamental in fluid mechanics, used in designing piping systems, aircraft wings, and understanding blood flow in arteries.

4. Using the Calculator

Tips: Enter all values in consistent SI units. The result should be close to zero for energy-conserving flows. Significant deviations may indicate measurement errors or violations of Bernoulli's assumptions.

5. Frequently Asked Questions (FAQ)

Q1: What are the assumptions behind Bernoulli's equation?
A: Steady, incompressible, inviscid flow along a streamline with no work or heat transfer.

Q2: When can't I use Bernoulli's equation?
A: For viscous flows, compressible flows, unsteady flows, or across streamlines.

Q3: What does a non-zero result mean?
A: It may indicate energy losses (friction), measurement errors, or violation of Bernoulli's assumptions.

Q4: How is this different from the integral form?
A: The differential form applies to infinitesimal changes along a streamline, while the integral form relates two distinct points.

Q5: What's the practical use of this calculation?
A: It helps verify experimental measurements and theoretical models of fluid flow systems.