## Bernoulli’s Equation

**The Bernoulli’s equation** can be considered to be a statement of the **conservation of energy principle** appropriate for flowing fluids. It is one of the most important/useful equations in **fluid mechanics**. It puts into a relation **pressure and velocity** in an **inviscid incompressible flow**. **Bernoulli’s equation** has some restrictions in its applicability, they summarized in following points:

- steady flow system,
- density is constant (which also means the fluid is incompressible),
- no work is done on or by the fluid,
- no heat is transferred to or from the fluid,
- no change occurs in the internal energy,
- the equation relates the states at two points along a single streamline (not conditions on two different streamlines)

Under these conditions, the general energy equation is simplified to:

This equation is the most famous equation in **fluid dynamics**. **The Bernoulli’s equation** describes the qualitative behavior flowing fluid that is usually labeled with the term **Bernoulli’s effect**. This effect causes the **lowering of fluid pressure** in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy. The dimensions of terms in the equation are kinetic energy per unit volume.

## Lift Force – Bernoulli’s Principle

In general, **the lift** is an upward-acting force on an aircraft wing or **airfoil**. There are several ways to explain **how an airfoil generates lift**. Some theories are more complicated or more mathematically rigorous than others. Some theories have been shown to be incorrect. There are theories based on the **Bernoulli’s principle** and there are theories based on directly on the **Newton’s third law**.

The explanation based on the **Newton’s third law** states that the lift is caused by a **flow deflection** of the airstream behind the airfoil. The airfoil generates lift by exerting a downward force on the air as it flows past. According to Newton’s third law, the air must **exert an upward force on the airfoil**. This is very simple explanation.

**Bernoulli’s principle** combined with the **continuity equation** can be also used to determine the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. In this explanation the **shape** of an airfoil is crucial. The shape of an airfoil causes air to **flow faster on top** than on bottom. According to **Bernoulli’s principle**, faster moving air exerts **less pressure**, and therefore the air must exert an **upward force** on the airfoil (as a result of a pressure difference).

**Bernoulli’s principle**requires airfoil to be of an

**asymmetrical shape**. Its surface area must be

**greater on the top**than on the bottom. As the air flows over the airfoil, it is displaced more by the top surface than the bottom. According to the continuity principle, this displacement must lead to an

**increase in flow velocity**(resulting in a decrease in pressure). The flow velocity is increased some by the bottom airfoil surface, but considerably less than the flow on the top surface. The lift force of an airfoil, characterized by the

**lift coefficient**, can be changed during the flight by changes in shape of an airfoil. The lift coefficient can thus be even doubled with relatively simple devices (

**flaps and slats**) if used on the full span of the wing.

The use of **Bernoulli’s principle** may not be correct. The Bernoulli’s principle assumes **incompressibility** of the air, but in reality the air is easily compressible. But there are more limitations of explanations based on Bernoulli’s principle. There are two main popular explanations of lift:

- Explanation based on downward deflection of the flow –
**Newton’s third law** - Explanation based on changes in flow speed and pressure –
**Continuity principle and Bernoulli’s principle**

Both explanations correctly identifies some aspects of the lift forces but leaves other important aspects of the phenomenon unexplained. A more comprehensive explanation involves both changes in flow speed and downward deflection and requires looking at the flow in more detail.

See more: Doug McLean, *Understanding Aerodynamics: Arguing from the Real Physics.* John Wiley & Sons Ltd. 2013. ISBN: 978-1119967514