Fluid Mechanics

Mechanical Engineering department

Created by:Manthan Kanani

Verification of Bernoulli's Theorem

And Continuity Equation

Manthan kanani 140050119028Ishant kalra 140050119027

BITs edu campus

Bernoulli's EquationIn fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738.

Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of Bernoulli's equa-tion; there are different forms of Bernoulli's equation for different types of flow. The simple form of Bernoulli's equation is valid for incompressible flows

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 energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy, potential energy and internal energy remains constant. Thus an increase in the speed of the fluid – implying an increase in both its dynamic pressure and kinetic energy – occurs with a simultaneous decrease in (the sum of) its static pressure, potential energy and internal 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 (the sum of pres-sure and gravitational potential ρ g h) 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 be-cause 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.

Bernoulli's EquationIn continuum mechanics there are two laws to attention.

Conservation

Energy

Mass

Momentum

InequalitiesEntropy

In fluid mechanics we should be keep mind on this given mem-ber of liquid, gas.FLUID - Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure.LIQUID - Surface tension Capillary action

GAS -Atmosphere Boyle's law Charles's law Gay-Lussac's law Combined gas law.

Proof of Bernoulli’s theorem

where F denotes a force and an x a dis-placement. The second term picked up its negative sign because the force and dis-placement are in opposite directions.

We examine a fluid section of mass m traveling to the right as shown in the schematic above. The net work done in moving the fluid is

.......Eq.(1)

Proof of Bernoulli’s theoremPressure is the force exerted over the cross-sectional area, or P = F/A. Rewriting this as F = PA and substituting into Eq.(1) we find that

The displaced fluid volume V is the cross-sectional area A times the thick-ness x. This volume remains constant for an incompressible fluid, so

Eq.(3)

Eq.(2)

Using Eq.(3) in Eq.(2) we have

Eq.(4)

Proof of Bernoulli’s theoremSince work has been done, there has been a change in the mechanical energy of the fluid segment. This energy change is found with the help of the next dia-gram.

The energy change between the initial and final positions is given by

.......Eq.(5)

Here, the the kinetic energy K = mv²/2 where m is the fluid mass and v is the speed of the fluid. The potential energy U = mgh where g is the acceleration of gravity, and h is average fluid height.

The work-energy theorem says that the net work done is equal to the change in the system energy. This can be written as

Proof of Bernoulli’s theorem

Eq.(6)

Substitution of Eq.(4) and Eq.(5) into Eq.(6) yields

Eq.(7)

Dividing Eq.(7) by the fluid volume, V gives us

Eq.(8)

whereEq.(9)

Proof of Bernoulli’s theoremis the fluid mass density. To complete our derivation, we reorganize Eq.(8).

Eq.(10)

Finally, note that Eq.(10) is true for any two positions. Therefore,

Eq.(11)

Equation (11) is commonly referred to as Bernoulli's equation. Keep in mind that this expression was restricted to incompressible fluids and smooth fluid flows.

A continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and particularly powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quant-ity. Since mass,energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of phys-ical phenomena may be described using continuity equations.

continuity theorem

Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of en-ergy is fixed. This statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge at any point can only change by the amount of electric current flowing into or out of that point.

Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always con-served, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a con-tinuity equation for the number of people alive; it has a "source term" to ac-count for people being born, and a "sink term" to account for people dying.

continuity theorem

Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point.

Proof of continuity theorem

Proof of continuity theorem-Streamlines represent a trajectory and do not meet or cross-Velocity vector is tangent to streamline

-Volume of fluid follows a tube of flow bounded by streamlines

-Streamline density is proportional to velocityFlow obeys continuity equation

Volume flow rate ( /s) Q = A·v ( x m / s ) is constant along flow tube.

A1v1 = A2v2

Thank you• KaNaNi MaNtHaN 140050119028• IsHaNt KaLrA 140050119027