Fluids

Bernoullis Theorem
Pascal's Law of Fluid Pressures Buoyancy & Archimedes' Principle Surface Tension

Bernoullis Theorem


It was given by Swiss physicist Daniel Bernoulli (1700-1782). Bernoulli’s theorem is based on the principle of conservation of energy applied to a liquid in motion i.e. for incompressible, non-viscous, irrotational fluid.

Bernoullis Theorem states that :

"The pressure in a Fluid decreases as its velocity increases."



In the diagram , the same amount of fluid has to pass through the constriction during any given time as passes through the wider parts of the river, so the fluid velocity v2 in the constriction is larger than the velocity v1 outside it. (This phenomenon is easy to observe in any creek or river.) As a consequencethe pressure P2 is smaller than the pressure P1.



The same principle operates in the second diagram. During any given time interval the same volume has to pass through the narrow section A1 of the pipe with diameter 2h1 as through the wide section A2 (V1 = V2). Therefore the velocity v1 is larger than the velocity v2, and the pressure in the narrow part is smaller than in the wider part.

Consider tube of varying cross-section through which an ideal liquid is made to flow.















P1= pressure applied on the liquid at A, inlet
P2 = pressure at B, outlet
A1 = area of cross section of the tube at A
A2 = area of cross section of the tube at B

h1, h2 = height of section A and B from the reference level.
v1,v2 = velocity of the liquid flow at A and B respectively.
p = density of the liquid flowing through the tube.
P1 > P2 because the liquid flows from A1 to A2.

The mass m of the liquid crossing per second through any part of the tube is
a1v1p = a2v2p = m (according to the equation of continuity )

or As a1 > a2
Therefore, v2 > v1
Now force on the liquid at section A = P1a1
And force on the liquid at section B = P2a2
Work done/second on the liquid at section A = P1a1 v1 = P1V
Work done/second by the liquid at section B = P2a2 v2 = P2V
Net work done/ second on the liquid by the pressure energy in moving the liquid from
section A to B = P1V – P2V

When the mass m of the liquid flows in one second from A to B its height increases from h1 to h2 and its velocity increases from v1 to v2.
Therefore, increase in potential energy/second of the liquid from A to B = mgh2 – mgh1
Increase in kinetic energy /second of the liquid from A to B







According to work energy principle,

work done/second by the pressure energy = increase in P.E./second + increase in K.E./second












Dividing throughout by m we get






so, we get:






Now,

P / p = pressure energy per unit mass
gh = potential energy per unit mass
1/2v2 = kinetic energy per unit mass

Pressure energy per unit mass + P.E. per unit mass + K.E. per unit mass = constant







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Pascal's Law of Fluid Pressures

Also called Pascal's Principle.

Pascal's law — developed by French mathematician Blaise Pascal — states that when there is an increase in pressure at any point in a confined fluid,there is an equal increase at every other point in the container.

Definition of Pressure: If F is the magnitude of the normal force on the piston and A is the surface area of a piston, then the pressure, P, of the fluid at the level to which the device has been submerged as the ratio of the force to area.


P = F / A


Since the pressure is force per unit area, it has units of N/m2 in the SI system. Another name for the SI unit of pressure is Pascal (Pa).

1 Pa = 1 N/m2



An important application of Pascal's law is the Hydraulic Press.

A force F1 is applied to a small piston of area A1. The pressure is transmitted through a liquid to a larger piston of area A2. Since the pressure is the same on both sides, we see that P = F1/A1 = F2/A2. Therefore, the force F2 is larger than F1 by multiplying factor A2/A1.

Hydraulic brakes, car lifts, hydraulic jacks, and forklifts all make use of this principle.






















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Buoyancy & Archimedes' Principle


When a body is wholly or partially immersed in a fluid, the displaced fluid has a tendency to regain its original position, which exerts an upward force on the body. This upward force acting on the body immersed in a fluid is called Upward Thrust or Buoyant Force or Simply Buoyancy.


Archimedes’ principle
states that:

"When a body is partially or fully immersed in a fluid at rest, the fluid exerts an upward force of buoyancy equal to the weight of the displaced fluid".



















 

Condition of Floatation


The weight W of the body which acts downward and the Buoyant force acting upward. Buoyant force (say W1) may be greater than, equal to or less than W.


















When W > W1

The body sinks in the fluid because of higher downward pull.

When W = W1
The body just floats or is at rest in the fluid.

When W < W1
The body floats comfortably.

The body floats because the weight of the liquid displaced by the immersed part of the body is at least equal to or greater than the weight of the body.




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Surface Tension

This phenomenon is valid with liquids only and not gasses.


Surface tension basically due to the intermolecular attractions in the liquid surface due to this a membrane effect can be seen on the surface.

Or

The cohesive forces between liquid molecules are responsible for the phenomenon known as surface tension. The molecules at the surface do not have other like molecules on all sides of them and consequently they cohere more strongly to those directly associated with them on the surface. This forms a surface "film" which makes it more difficult to move an object through the surface than to move it when it is completely submersed.




















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