# Application of Bernoulli's Equation: Pressure and Speed

## For "ideal" flow along a streamline with no change in height, an increase in velocity results from a decrease in static pressure.

#### Key Points

• The simplest form of Bernoulli's equation (steady and incompressible flow) states that the sum of mechanical energy, potential energy and kinetic energy, along a streamline is constant. Therefore, any increase in one form results in a decrease in the other.

• Bernoulli's equation considers only pressure and gravitational forces acting on the fluid particles. Therefore, if there is no change in height along a streamline, Bernoulli's equation becomes a balance between static pressure and velocity.

• The steady-state, incompressible Bernoulli equation, can be derived by integrating Newton's 2nd law along a streamline.

#### Terms

• An inviscid and incompressible fluid

• Unable to be compressed or condensed.

• A quantity expressing the magnitude of internal friction in a fluid, as measured by the force per unit area resisting uniform flow.

#### Figures

1. ##### Syphoning

Syphoning fluid between two reservoirs. The flow rate out can be determined by drawing a streamline from point ( A ) to point ( C ).

2. ##### Bernoulli's Principle

A brief introduction to Bernoulli's Principle for students studying fluids.

## Application of Bernoulli's Equation

The relationship between pressure and velocity in ideal fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible and inviscid fluid, the total mechanical energy of the fluid is constant Figure 2. (An inviscid fluid is assumed to be an ideal fluid with no viscosity.)

The total mechanical energy of a fluid exists in two forms: potential and kinetic. The kinetic energy of the fluid is stored in static pressure, $p_s$, and dynamic pressure, $\frac{1}{2}\rho V^2$, where \rho is the fluid density in (SI unit: kg/m3) and V is the fluid velocity (SI unit: m/s). The SI unit of static pressure and dynamic pressure is the pascal.

Static pressure is simply the pressure at a given point in the fluid, dynamic pressure is the kinetic energy per unit volume of a fluid particle. Thus, a fluid will not have dynamic pressure unless it is moving. Therefore, if there is no change in potential energy along a streamline, Bernoulli's equation implies that the total energy along that streamline is constant and is a balance between static and dynamic pressure. Mathematically, the previous statement implies:

$p_s + \frac{1}{2}\rho V^2 = constant$

along a streamline. If changes there are significant changes in height or if the fluid density is high, the change in potential energy should not be ignored and can be accounted for with,

$\Delta PE = \rho g \Delta h$.

This simply adds another term to the above version of the Bernoulli equation and results in

$p_s + \frac{1}{2}\rho V^2 + \rho g\Delta h = constant$.

### Deriving Bernoulli's Equation

The Bernoulli equation can be derived by integrating Newton's 2nd law along a streamline with gravitational and pressure forces as the only forces acting on a fluid element. Given that any energy exchanges result from conservative forces, the total energy along a streamline is constant and is simply swapped between potential and kinetic.

### Applying Bernoulli's Equation

Bernoulli's equation can be applied when syphoning fluid between two reservoirs Figure 1. Another useful application of the Bernoulli equation is in the derivation of Torricelli's law for flow out of a sharp edged hole in a reservoir. A streamline can be drawn from the top of the reservoir, where the total energy is known, to the exit point where the static pressure and potential energy are known but the dynamic pressure (flow velocity out) is not.

### Adapting Bernoulli's Equation

The Bernoulli equation can be adapted to flows that are both unsteady and compressible. However, the assumption of inviscid flow remains in both the unsteady and compressible versions of the equation. Compressibility effects depend on the speed of the flow relative to the speed of sound in the fluid. This is determined by the dimensionless quantity known as the Mach number. The Mach number represents the ratio of the speed of an object moving through a medium to the speed of sound in the medium.

#### Key Term Glossary

application
the act of putting something into operation
##### Appears in these related concepts:
Bernoulli Equation
The equation that Torricelli's Law is derived from.
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conservative force
A force with the property that the work done in moving a particle between two points is independent of the path taken.
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dynamic
Changing; active; in motion.
##### Appears in these related concepts:
element
Any one of the simplest chemical substances that cannot be decomposed in a chemical reaction or by any chemical means and made up of atoms all having the same number of protons.
##### Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
fluid
Any substance which can flow with relative ease, tends to assume the shape of its container, and obeys Bernoulli's principle; a liquid, gas or plasma.
##### Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
##### Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
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incompressible
Unable to be compressed or condensed.
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inviscid
A fluid with zero viscosity (internal friction). In reality viscosity is always present. However, it is often very small compared with other forces (e.g. gravity, pressure) and for common fluids (water and air) the fluid can be approximated as having zero viscosity.
##### Appears in these related concepts:
kinetic
Of or relating to motion
##### Appears in these related concepts:
kinetic energy
The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.
##### Appears in these related concepts:
Kinetic Energy
The energy associated with a moving particle or object having a certain mass.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
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Mach number
The ratio of the velocity of a body to that of sound in the surrounding medium.
##### Appears in these related concepts:
medium
The material or empty space through which signals, waves or forces pass.
##### Appears in these related concepts:
particle
A very small piece of matter, a fragment; especially, the smallest possible part of something.
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potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
##### Appears in these related concepts:
potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
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pressure
the amount of force that is applied over a given area divided by the size of that area
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quantity
A fundamental, generic term used when referring to the measurement (count, amount) of a scalar, vector, number of items or to some other way of denominating the value of a collection or group of items.
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relative
Expressed in relation to another item, rather than in complete form.
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SI units
International System of Units (abbreviated SI from French: Le Système international d'unités). It is the modern form of the metric system.
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static
Fixed in place; having no motion.
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velocity
A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.
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Velocity
The rate of change of displacement with respect to change in time.
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viscosity
A quantity expressing the magnitude of internal friction in a fluid, as measured by the force per unit area resisting uniform flow.
##### Appears in these related concepts:
Viscosity
The property of a fluid that resists the force which tends to cause it to flow.