what does bernoullis principle say abt pressure as the velocity of a fluid increases????
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Answers
Answer:
This inverse relationship between the pressure and speed at a point in a fluid is called Bernoulli's principle. Bernoulli's principle: At points along a horizontal streamline, higher pressure regions have lower fluid speed and lower pressure regions have higher fluid speed.
Explanation:
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.
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:
p + 12 ρ v2 + ρgh =constant
Where,
p is the pressure exerted by the fluid
v is the velocity of the fluid
ρ is the density of the fluid
h is the height of the container
Bernoulli’s equation gives great insight into the balance between pressure, velocity and elevation.
Related Articles:
Fluid Dynamics
Continuity Equation
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 is given in the figure below.
Bernoulli's Equation Derivation
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 conservation of gravitational force and change in kinetic energy. The change in kinetic energy of the fluid is given as:
dK=12m2v22−12m1v21=12ρdV(v22−v21)
The change in potential energy is given as:
dU = mgy2 – mgy1 = ρdVg(y2 – y1)
Therefore, the energy equation is given as:
dW = dK + dU
(p1 – p2)dV = 12ρdV(v22−v21) + ρdVg(y2 – y1)
(p1 – p2) = 12ρ(v22−v21) + ρg(y2 – y1)
Rearranging the above equation, we get
p1+12ρv21+ρgy1=p2+12ρv22+ρgy2
This is Bernoulli’s equation.
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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.
Principle of Continuity
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.
Principle of Continuity
Using Bernoulli’s equation at point 1 and point 2, p+12ρv21+ρgh=p0+12ρv22v22=v21+2p−p0ρ+2gh
Generally, A2 is much smaller than A1; in this case, v12 is very much smaller than v22 and can be neglected. We then find, v22=2p−p0ρ+2gh
Assuming A2<<A1,
We get, v2=2gh−−−√
Hence, the velocity of efflux is 2gh−−−√
Bernoulli’s Principle Use
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:
V=a1a22hga21−a22−−−−√
Bernoulli’s Principle Venturimeter
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 on 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 based on the 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 is the result of a change in fluid’s kinetic energy and gravitational potential energy. Bernoulli’s equation can be modified depending on the form of energy that is involved. Other forms of energy include the dissipation of thermal energy due to fluid viscosity.