Fluid Dynamics

Continuity equation, Bernoulli's theorem, and energy transport along streamlines.

General idea of the topic

Fluid dynamics studies fluids in motion. In the ideal model, the fluid is incompressible, the flow steady and there is no viscosity, and the behavior is governed by two laws: conservation of mass, which gives the continuity equation, and conservation of energy, which gives Bernoulli's.

The topic develops both, explores their technical applications, such as the Venturi effect, the Pitot tube and lift, and presents the equations of motion, with Navier-Stokes as the general framework. For more general flows, the Reynolds transport theorem is added, which turns the conservation laws into balances on a control volume, along with vorticity and circulation, which measure the local spin of the fluid.

What you will truly understand

By the end you will be able to relate velocity and cross-section in a pipe, apply Bernoulli between two points on a streamline and explain how speeds are measured and lift is generated.

You will also know when these equations stop being valid.

Recommended learning route

1. Continuity Equation: conservation of mass flow rate and the relation between cross-section and velocity.
2. Bernoulli's Equation: energy balance along a streamline.
3. Applications of Bernoulli's Equation: Venturi, Pitot and lift as uses of Bernoulli.
4. Equations of Motion: Euler and Navier-Stokes equations governing flow in general.
5. Reynolds Transport Theorem: how to pass from balances on a fixed mass to balances on a control volume.
6. Vorticity and Circulation: local rotation of the fluid, circulation around a curve and its relation to lift.

Physical criterion you must acquire

The essential criterion is to check the assumptions before applying Bernoulli: steady, incompressible, inviscid flow along a streamline. If there are pumps, turbines or friction losses, energy terms must be added.

Connection with real problems

• pipes, pumps and turbines.
• airplanes and wings.
• carburetors and speed probes.
• blood circulation.

Final goal of the topic

The final goal is to apply continuity and Bernoulli to simple real flows, justify their assumptions and compute pressures and velocities.