### 6.3 Bernoulli's Theorem

Quiz

##### Daniel Bernoulli

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, (A fluid which has no viscosity; it therefore can support no shearing stress, and flows without energy dissipation. Also known as ideal fluid; non-viscous fluid; perfect fluid) an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

Bernoulli's principle is named after the Dutch-Swiss mathematician Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.

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.

Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume is the same everywhere.

Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

A flow of air into a Venturi meter.

The kinetic energy increases at the expense of the fluid pressure, as shown by the difference in height of the two columns of water.

In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid and a small viscosity often has a large effect on the flow.

Bernoulli's principle can be used to calculate the lift force on an aerofoil if the behaviour of the fluid flow in the vicinity of the foil is known.

Condensation develops in the lower pressure areas created by the aircraft.

For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli's principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lift force. Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli's equation.

Daniel Bernoulli established his theorem over a century before the first man-made wings were used for the purpose of flight. However the principle does not explain why the air flows faster past the top of the wing and slower past the underside.

##### Bernoulli's Equation

Bernoulli's principle states Total Pressure is made up of two mathematical components termed Static Pressure and Dynamic Pressure.

Total Pressure = Static Pressure + Dynamic Pressure

The equation shown is the incompressible or slow speed form of Bernoulli’s equation. If energy is transferred within a body of air, which causes dynamic pressure to increase, then there must be a decrease in static pressure to maintain a constant total pressure.

For example if you face into a strong wind, you can feel the force of dynamic pressure acting against you. Dynamic Pressure is the pressure a moving air mass would possess if it were brought to a stop, relative to air density and an object.

##### Venturi Tube

Visualise what happens to pressure as air flows through a venturi tube.

The air accelerates over the curved shape of the venturi.

A pressure differential force is generated by the local variations of static and dynamic pressures on the curved surface of the venturi. As a result, dynamic pressure increases with acceleration, causing static pressure to decrease perpendicular to the curved shape of the tube.