Only pressure forces and heitional force are significant and dominating. This differential fluid element can be cheated as a particle moving along a streamline as shown here. The pressure force acting on the left force of the fluid element alarmmed the streamline is mentioned here FPL is equal two PDA, where B is the pressure of the fluid at the left face of the element and BA is the area of the face of the element. The pressure force acting on the right face of the fluid element along the streamline is written here, where P+DP is the pressure of the fluid at the right phase of the helment. The weight of the fluid element w has shown in the figure here, acts vertically in a downward direction and is equal to row GK multipl by BS where row is the density of the fluid. G is the excellentation due to gravity and the is the length of the differential fluid element. This w can be resolved or decomposed into two components, the weight component, along the streamline is given here. Now let us take the sign convention for the force acting along the streamline towards the right side is positive and towards the left side is negative. Whereas normal force acting in an upper direction is positive and in a downward direction is negative. However, the normal forces will be balanced by bounded wall reaction. Now, let us apply nutants second law in the space coordinate direction along the streamline on a differential fluidary met. This is presented here from the figure shown here, scientific can be written as you see here. substituting scient thera in the equition shown here and simplify Isa equation shown here. By eliminating TK, which is common in left and right in terms of the equation, the moment term equation can be written as shown here. here, replacing you do you by half day multiplied by you square and dividing the left hand, right hand side terms by row gives the following equusion as depicted here. When this equation is integrated along the streamline for the whole domain, the right hand side down becomes constantly. That is the last two terms of the next side are differential for an incompressible fluid, roof is concert. Then the first term becomes the thick set differential. Therefore, thus equation becomes has shown here, is, and it is popularly known as coronly equation. It is valid for steady, incompressible flow, along a streamline in invisite regions of the flow, applying the bonly equation at location one as shown here, one can have the equation you can see.ain, applying the boundary equation at location two, one can have other equation. Therefore, the two equations depicted here can be equated since the left hand sideumps are equal to concent the corresponding equation is shown here. In this equation, you square by two can be recognized as kinetic energy, whereas zZ can be recognized as B by row is the flow energy that is to push the fluidement through the conduit. Now, let us summarize the barley equation. Barley equation can be stated since the sum of flu energy, kinetic energy, and potational energy is constant for a state flow and incompressible fluid when the net friction effects are negligible. equation is always viewed as the principle of conservation of tennessy. all the three terms in the Bonon equation shown here are mechanical forms of energy terms and no mechanical form of energy term converts into thermal energy, but they convert ham, themselves, keeping the sum of these three terms always constant. Let us write a borrowed equation in towns of, head in meters. dividing by G throughout the bannery equation can be written in terms of head in meters as represented here. Now let us summarize what we have learned in this topic. We derive the panel equation to an ideal flu its tradition, explain specific practical cases where boundary can be applied without any penalty of incurrency, explain the terms involved in the bor equation, applied bonly equation for a profile, flowing through a pipe and final we have written the b equation in

Bernoulli's Equation

Bernoulli's Equation

Describes the behavior of fluid under varying conditions of flow and height, stating that total mechanical energy along a streamline is constant for an incompressible, frictionless fluid.

P

Pressure of the fluid in Bernoulli's equation.

ρ (rho)

Density of the fluid in Bernoulli's equation.

v

Velocity of the fluid in Bernoulli's equation.

g

Acceleration due to gravity in Bernoulli's equation.

h

Height above a reference point in Bernoulli's equation.

Bernoulli's Principle

States that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or potential energy.

Application in Aviation

Understanding lift and the design of aircraft wings using Bernoulli's equation.

Venturi Effect

Utilized in carburetors and atomizers to draw fluid into a spray, based on Bernoulli's principle.

Hydraulic Engineering Applications

Helps in the design of channels and determining flow rates, informed by Bernoulli's equation.

Concept of Conservation of Energy

The principle applied in deriving Bernoulli's equation, considering the work done on the fluid element relative to its energy changes.

Total Mechanical Energy

The sum of pressure energy, kinetic energy, and potential energy along a streamline, as expressed in Bernoulli's equation.