Bernoulli’s principle formulated by Daniel Bernoulli states that as the speed of a moving fluid increases (liquid or gas), the pressure within the fluid decreases. Although Bernoulli deduced the law, it was Leonhard Euler who derived Bernoulli’s equation in its usual form in the year 1752.
What is Bernoulli’s Principle?
Bernoulli’s principle states that
The total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant.
Bernoulli’s principle can be derived from the principle of conservation of energy.
Bernoulli’s Principle Formula
Bernoulli’s equation formula is a relation between pressure, kinetic energy, and gravitational potential energy of a fluid in a container.
The formula for Bernoulli’s principle is given as follows:
Where p is the pressure exerted by the fluid, v is the velocity of the fluid, ρ is the density of the fluid and h is the height of the container.
Bernoulli’s equation gives great insight into the balance between pressure, velocity and elevation.
Bernoulli’s Equation Derivation
Consider a pipe with varying diameter and height through which an incompressible fluid is flowing. The relationship between the areas of cross-sections A, the flow speed v, height from the ground y, and pressure p at two different points 1 and 2 are given in the figure below.
Assumptions:
- The density of the incompressible fluid remains constant at both points.
- The energy of the fluid is conserved as there are no viscous forces in the fluid.
Therefore, the work done on the fluid is given as:
dW = F1dx1 – F2dx2
dW = p1A1dx1 – p2A2dx2
dW = p1dv – p2dv = (p1 – p2)dv
We know that the work done on the fluid was due to the conservation of change in gravitational potential energy and change in kinetic energy. The change in kinetic energy of the fluid is given as:
The change in potential energy is given as:
dU = m2gy2 – m1gy1 = ρdvg(y2 – y1)
Therefore, the energy equation is given as:
dW = dK + dU
Rearranging the above equation, we get
This is Bernoulli’s equation.
Principle of Continuity
According to the principle of continuity
If the fluid is in streamline flow and is in-compressible then we can say that mass of fluid passing through different cross sections are equal.
From the above situation, we can say the mass of liquid inside the container remains the same.
The rate of mass entering = Rate of mass leaving
The rate of mass entering = ρA1V1Δt—– (1)
The rate of mass entering = ρA2V2Δt—– (2)
Using the above equations,
ρA1V1=ρA2V2
This equation is known as the Principle of Continuity.
Suppose we need to calculate the speed of efflux for the following setup.
Using Bernoulli’s equation at point 1 and point 2,
Generally, A2 is much smaller than A1; in this case, v12 is very much smaller than v22 and can be neglected. We then find,
Assuming A2<<A1,
We get,
Hence, the velocity of efflux is
Applications of Bernoulli’s Principle and Equation
Bernoulli’s principle is used for studying the unsteady potential flow which is used in the theory of ocean surface waves and acoustics. It is also used for approximation of parameters like pressure and speed of the fluid.
The other applications of Bernoulli’s principle are:
- Venturi meter: It is a device that is based on Bernoulli’s theorem and is used for measuring the rate of flow of liquid through the pipes. Using Bernoulli’s theorem, Venturi meter formula is given as:
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\(\begin{array}{l}V=A_{1}A_{2}\sqrt{\frac{2hg}{A_{1}^{2}-A_{2}^{2}}}\end{array} \)
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- Working of an aeroplane: The shape of the wings is such that the air passes at a higher speed over the upper surface than the lower surface. The difference in airspeed is calculated using Bernoulli’s principle to create a pressure difference.
- When we are standing at a railway station and a train comes we tend to fall towards the train. This can be explained using Bernoulli’s principle as the train goes past, the velocity of air between the train and us increases. Hence, from the equation, we can say that the pressure decreases. So the pressure from behind pushes us towards the train. This is base d on Bernoulli’s effect.
Relation between Conservation of Energy and Bernoulli’s Equation
Conservation of energy is applied to the fluid flow to produce Bernoulli’s equation. The net work done results from a change in a fluid’s kinetic energy and gravitational potential energy. Bernoulli’s equation can be modified depending on the form of energy involved. Other forms of energy include the dissipation of thermal energy due to fluid viscosity.
Bernoulli’s Equation at Constant Depth
When the fluid moves at a constant depth that is when h1 = h2, then Bernoulli’s equation is given as:
\(\begin{array}{l}P_{1}+\frac{1}{2}\rho v_{1}^{2}=P_{2}+\frac{1}{2}\rho v_{2}^{2}\end{array} \)
Bernoulli’s Equation for Static Fluids
When the fluid is static, then v1 = v2 = 0, then Bernoulli’s equation is given as:
| When v1 = v2 = 0 | P1 + ρgh1 = P2 + ρgh2 |
| When h2 = 0 | P2 = P1 + ρgh1 |
Bernoulli’s Principle Example
Q1. Calculate the pressure in the hose whose absolute pressure is 1.01 x 105 N.m-2 if the speed of the water in the hose increases from 1.96 m.s-1 to 25.5 m.s-1. Assume that the flow is frictionless and density 103 kg.m-3
Ans: Given,
Pressure at point 2, p2 = 1.01 × 105 N.m-2
Density of the fluid, ρ = 103 kg.m-3
Velocity of the fluid at point 1, v1 = 1.96 m.s-1
Velocity of the fluid at point 2, v2 = 25.5 m.s-1
From Bernoulli’s principle for p1,
Substituting the values in the above equation, we get
p1 = 4.24 × 105 N.m-2
Curve of a Baseball
Have you ever seen a baseball pitcher deliver a curveball? When the baseball takes a curved trajectory as it passes the plate, in most cases, the batter would not be able to judge the path of the ball and miss the ball. One of the crucial aspects of curveball can be explained using a formula typically used to describe fluid flow. Bernoull’s equation can be used to explain the basic aspect of the curve of a baseball (curveball). Usually, Bernoulli’sBernoulli’s equation points to pressure, height, and velocity (“air” is the fluid). At any particular point in the fluid (air), the K (constant) will be equal to the total sum of the other three values (height, pressure, and velocity).
Airfoil and Bernoulli’s Principle
The air over the top of a typical airfoil encounter compressed flow lines and boosted air speed compared to the wing. This introduces a reduction in pressure on the top (as per the Bernoulli equation) and produces a lift force. Aerodynamicists use the Benaoulli concept to explain the pressure calculations made in wind tunnels. They show that when pressure calculations are done at multiple places around the airfoil and added together, it is similar to the observed lift.
