To demonstrate Bernoulli's theorem with simple illustrations.
Conservation of energy
Conservation of energy is the fundamental principle in physics.
It states that "energy can be neither created nor destroyed. It can only be moved from one part of a system to another or changed from one form to another.
When there is a loss of energy in one part of the universe, there will be a gain of energy in another part of the universe.
The simple pendulum is an example which demonstrates the conservation of energy. To know more, click here
Bernoulli's theorem
Swiss physicist Daniel Bernoulli applied the principle of conservation of energy to frictionless laminar flow, resulting in a relationship between pressure and speed in the fluid. This relationship is known as Bernoulli's Equation.
Equation of continuity
Equation continuity states the flow rate of a fluid remains constant. In other words, speed increases as the cross-sectional area decreases and vice versa.
Figure 1: Equation of continuity
The tube narrows from region 1 to 2, occupying the same volume of fluid in both regions. To pass the same volume through region 2 in a given time, the speed must be greater in region 2.
Speed must be higher in region 2, to pass the same volume through region 2 in a given time.
In Figure 2, an incompressible fluid is flowing steadily through the pipe. The pipe has varying areas, and the height of the pipe also changes. According to the equation of continuity, velocity must be different as fluid flows through different areas. A force is required to cause a change in velocity or acceleration.
Figure 2: The flow of an ideal fluid in a pipe of varying cross-section. The fluid in a section of length v1∆t moves to the section of length v2∆t in time ∆t.
Consider that the fluid is initially at B and D. In a time interval, Δt fluid moves to C and E, respectively. According to the equation of continuity, the fluid has different speeds at points B and D due to changes in the area at those points. The speed of the fluid at B is V1, and the speed of the fluid at D is v2.
In a time interval, Δt fluid at B moves v1Δt distance to C and fluid at D moves v2Δt distance to E . (v1Δt is negligibly small, area along v1Δt is constant)
P1 and h1 represent the pressure and height of the fluid, respectively, at point O in the pipe. P2 and h2 are the pressure and height of the fluid, respectively, at the point M in the pipe. ρ is the density of the fluid. A1 and A2 are the areas of the pipe at O and M.
Since there is a change in velocities at O and M, a force is required for acceleration.
Work done on left side BC = P1A1(v1 Δt) = P1ΔV
Work done on right side DE = P2A2(v2 Δt) = P2ΔV
(Work done on right side DE = P1ΔV Since the same volume of fluid flows through both regions BC and DE.)
Total work done on the fluid = (P1 - P2)ΔV
Part of this work goes to change the kinetic energy, and part of this work goes to change the potential energy.
Mass of the fluid passing through the pipe in time Δt, Δm = ρA1v1∆t = ρ∆V
Where ρ is the density of the fluid.
Change in gravitational potential energy = Δmg(h2-h1)
Change in kinetic energy = ½ (Δm(v22 − v12) = ½ ρ∆V(v22 − v12 )
According to the work-energy theorem,
(P1 − P2)∆V = ½ ρ∆V(v22 − v12) + ρg∆V(h2 − h1)
Divide each term by ∆V
(P1 − P2) = ½ ρ(v22 − v12) + ρg(h2 − h1)
Rearranging the above terms to obtain
P1 + ½ ρv12 + ρgh1 = P2 + ½ρv22 + ρgh2
This is Bernoulli's equation. In general, we can write it as
P + ½ ρv2 + ρgh = constant
Bernoulli's equation relates the pressure, velocity and height of a fluid.
Bernoulli's principle: As we move along the streamline sum of pressure, kinetic energy per unit volume and potential energy per unit volume remain constant.
From Bernoulli's equation, pressure is inversely proportional to velocity.
P ∝ 1/v
Aeroplane
Figure 3: Aeroplane
In Figure 2, the upper surface of the aeroplane's wing is curved. The air flowing over the top of the wing must travel a long distance. The top of the wing has a larger area. Air flows at a higher speed over the upper surface. According to Bernoulli's theorem, the pressure will be lower at the top of the wing. At the bottom of the wing, the pressure will be higher. This pressure difference causes the plane to experience an upward lift.
Bunsen burner
When the fuel gas starts to flow through the tube, the speed of the fuel gas inside the tube will increase. According to Bernoulli's theorem, this causes a lower pressure inside the tube. When we open the holes, air from outside flows into the burner. When the fuel gas mixes with oxygen, the colour of the flame changes from yellow to blue.
Figure 4: Bunsen burner a) when the hole is fully closed
Figure 4: b) when the hole is fully opened
Students can