🔴 PHY103. BERNOULLI'S PRINCIPLES

 BERNOULLI'S PRINCIPLE

 Bernoulli principle formulated by Daniel Bernoulli states that as the speed of a moving fluid🌊 increases (liquid or gas), the pressure within the fluid decreases.
  In other words, 
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.
 
 Conservation of energy is applied to the fluid flow to produce Bernoulli’s equation. 
 
Question.
 Does mechanical energy not consist of just kinetic and potential energy?
 
Answer
 You're right but since the energy is associated with fluid thereby releasing energy in form of pressure
 
 According to Bernoullis principle, pressure is the mainly observed factor
 
 Moving on, 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.
 
 As for the 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 + 1/2 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
 
   The equation is derived as thus, considering 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.
 
 A lot of you will meet diagrams like this and questions arising based on it
 
 So based on assumptions,
The density of the incompressible fluid remains constant at both points.
Also,the energy of the fluid is conserved as there are no viscous forces in the fluid
 
 It could be ranging from work done, change in KE and PE, and general energy...
 
 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= 1/2m2v2^2−1/ 2m1v1^2

 = 1/2ρdV(v2^2−v1^2)
 
Question
 This equation consist of Force, pressure, and acceleration
Answer
 Yeah
Force of the fluid, pressure exerted by the fluid, and acceleration as the fluid moves
 
 The change in potential energyis given as:
dU = mgy2 – mgy1 = ρdVg(y2 – y1)
 
 Therefore, the energy equation is given as:
dW = dK + dU

(p1 – p2)dV = 1/2ρdV(v2^2−v1^2)+ ρdVg(y2 – y1)
(p1 – p2) = 1/2ρ(v2^2−v1^2)+ ρg(y2 – y1)
   
 Rearranging the above equation, we get
 *p₁+1/2ρv₁²+ρgy₁=p₂+1/2ρv₂²+ρgy₂*
 
 Moving on 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)
 
 So, the mass of fluid entering A1 per time is equal to that of A2. Yes!
 
 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.

 Written by:- Ms. Ojimadu David

 
✍️ *FLASHPEE EDUCATIONAL TEAM*