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.
Learning Objective

Adapt Bernoulli's equation for flows that are either unsteady or compressible
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 steadystate, incompressible Bernoulli equation, can be derived by integrating Newton's 2nd law along a streamline.
Terms

Ideal Fluid
An inviscid and incompressible fluid

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

incompressible
Unable to be compressed or condensed.
Full Text
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 . (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,
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:
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,
This simply adds another term to the above version of the Bernoulli equation and results in
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 . 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.
Syphoning
Syphoning fluid between two reservoirs. The flow rate out can be determined by drawing a streamline from point ( A ) to point ( C ).
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 Reference
 Law
 Appears in these related concepts: TwoComponent Forces, Damped Harmonic Motion, and Models, Theories, and Laws
 Mach number
 Appears in these related concepts: Sonic Booms and Speed of Sound
 Pressure
 Appears in these related concepts: SI Units of Pressure, Physics and Engineering: Fluid Pressure and Force, and Surface Tension and Capillary Action
 SI units
 Appears in these related concepts: Time, Length, and Problem Solving
 application
 Appears in these related concepts: Introduction to Elementary operations and Gaussian Elimination, Physics and Other Fields, and XRay Imaging and CT Scans
 conservative force
 Appears in these related concepts: Gravity, Springs, and Fundamental Theorem for Line Integrals
 dynamic
 Appears in these related concepts: Competitive Dynamics, Translational Equilibrium, and General ProblemSolving Tricks
 element
 Appears in these related concepts: Development of the Periodic Table, Elements and Compounds, and The Periodic Table
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 fluid
 Appears in these related concepts: Pumps and the Heart, Drag, and B.11 Chapter 11
 force
 Appears in these related concepts: Force of Muscle Contraction, Force, and First Condition
 gravitational force
 Appears in these related concepts: Newton and His Laws, Weight of the Earth, and B.3 Chapter 3
 inviscid
 Appears in this related concept: Torricelli's Law
 kinetic
 Appears in these related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Postmodernist Sculpture
 kinetic energy
 Appears in these related concepts: Solid Solubility and Temperature, Pressure, and Types of Energy
 medium
 Appears in these related concepts: Waves, The Role of the Artist, and Refraction Through Lenses
 potential
 Appears in these related concepts: What is Potential Energy?, Conservative and Nonconservative Forces, and Linear Expansion
 potential energy
 Appears in these related concepts: Problem Solving With the Conservation of Energy, Escape Speed, and Defining Graviational Potential Energy
 relative
 Appears in these related concepts: Relative Deprivation Approach, Addition of Velocities, and Relative Velocity
 static
 Appears in these related concepts: LongTerm Approach, Time and Motion, and Alternative Views
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
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