## Bernoulli's Equation

**INTRODUCTION**: In Hydrodynamics the Motion of Liquid and the Forces causing Flow are covered. In this post, I will cover some topics of Hydrodynamics. Bernoulli’s principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers.

**Incompressible flow equation**: In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible and these flows are called incompressible flow. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is

Where:

Is the fluid flow speed at a point on a streamline?

Is the acceleration due to gravity?

Is the elevation of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration?

Is the pressure at the chosen point, and

Is the density of the fluid at all points in the fluid?

For conservative force fields, Bernoulli's equation can be generalized as:

Where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's gravity Ψ = gz.

The following two assumptions must be met for this Bernoulli equation to apply:

- the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;
- Friction by viscous forces has to be negligible.

By multiplying with the fluid density , equation (A) can be rewritten as:

Or

Where: Dynamic pressure:

piezometric head or hydraulic head:

Total pressure:

The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head H: