This document discusses fluid dynamics and Bernoulli's theorem. It defines key terms like ideal fluid, streamline flow, turbulent flow, viscosity, and drag force. It presents the equation of continuity which states that the product of area and fluid speed is constant for incompressible flow. Bernoulli's theorem is then introduced, which states that the total energy per unit volume remains constant in steady ideal fluid flow. The properties of ideal fluids are outlined and Bernoulli's theorem is proved using the work-energy principle. Applications of Bernoulli's theorem to fluid flow are also mentioned.

2. FLUIDDYNAMICS
• Basic terms.
• Ideal Fluid.
• Equation of Continuity.
• Bernoulli'sTheorem.
• Application of Bernoulli’sTheorem.

3. FLUID:
• A fluid is a substance which can
flows.
• Such as liquids , gases and plasma.
• Example: Water, air etc…

4. Fluid Dynamics
• Study of fluid in motion.
Viscosity:
• The frictional effect b/w different layers
of a flowing fluid is the viscosity of the
fluid.
Drag Force:
• An object moving through a fluid
experiences a retarding force called a
drag force.

5. Fluid Flow
Streamline /Laminar Flow:
• Every particle of fluid during flow has
constant velocity, pressure , density
and having regularity.
Turbulent Flow:
• The irregular and non-steady fluid
flow is called turbulent flow.
• Velocity , pressure , and density
remain non – uniform.

6. IDEAL FLUID
Properties of Ideal Fluid:
1. Fluid is non-viscous (Internal Friction is neglected).
2. Fluid is incompressible (i.e. Constant Density).
3. Flow is Steady (Laminar).
4. Flow is irrotational (i.e. No angular momentum)

7. EQUATION OF CONTINUITY
𝐦𝐚𝐬𝐬 = 𝐝𝐞𝐧𝐢𝐬𝐭𝐲 × 𝐯𝐨𝐥𝐮𝐦𝐞
∆𝑚1 = 𝜌𝐴1∆𝑥1
∵ ∆𝑥1=𝑣1 × ∆𝑡
∆𝑚1 = 𝜌𝐴1 𝑣1 × ∆𝑡
Similarly
∆𝑚2 = 𝜌𝐴2∆𝑥2
∵ ∆𝑥1=𝑣1 × ∆𝑡
∆𝑚2 = 𝜌𝐴2 𝑣2 × ∆𝑡
“It states that the product of the area and the fluid speed at all points
along a pipe is constant for an incompressible fluid.”

8. • Because the fluid is incompressible and the flow is steady, then
∆𝑚1 = ∆𝑚2
𝜌𝐴1 𝑣1 × ∆𝑡 = 𝜌𝐴2 𝑣2 × ∆𝑡
𝐴1 𝑣1 = 𝐴2 𝑣2
So the product:
𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
It has Dimensions:
𝐴 ×
∆𝑥
∆𝑡
=
𝑉𝑜𝑙𝑢𝑚𝑒
𝑡𝑖𝑚𝑒
It is either called Volume Flux or Flow Rate

9. • The speed of water spraying from the end
of a garden hose increases as the size of
the opening is decreased with the thumb.

10. Bernoulli’sTheorem
• It is simply a statement of Law of conservation of energy applied to
liquid in motion.
This theorem states that:
“For the steady flow of an ideal fluid, the total energy (i.e., sum
of pressure, potential energy & kinetic energy) per unit volume
remains constant through the flow.”
𝑃 + 𝜌𝑔ℎ +
1
2
𝜌𝑣2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

11. Proof:
The force exerted on lower segment:
𝐹1 = 𝑃1 𝐴1
TheWork Done by force on this
segment is:
𝑊1 = 𝐹1∆𝑥1
𝑊1 = 𝑃1 𝐴1∆𝑥1
• Similarly on the upper segment:
𝑊2 = −𝑃2 𝐴2∆𝑥2
This work done is negative because
the it is against Fluid Fow

12. The Force 𝐹1 moves the Liquid a distance
∆𝑥1 & the liquid moves a distance ∆𝑥2
against the Force 𝐹2.
Therefore, the net work done on liquid is:
𝑊 = 𝑃1 𝐴1∆𝑥1 − 𝑃2 𝐴2∆𝑥2
𝑊 = 𝑃1(𝐴1∆𝑥1) − 𝑃2(𝐴2∆𝑥2)
∵ 𝐴1∆𝑥1= 𝐴2∆𝑥2= m/ρ
𝑊 = 𝑃1 − 𝑃2 V

13. Part of thisWork is utilized by the fluid in changing its Kinetic Energy & a
part is used in changing its Gravitational Potential Energy:
∆𝐾. 𝐸 =
1
2
𝑚𝑣2
2
−
1
2
𝑚𝑣1
2
∆𝑃. 𝐸 = 𝑚𝑔ℎ2 − 𝑚𝑔ℎ1
None of theWork Done on the liquid has been used to overcome the internal
friction because the liquid is non-viscous.
• According to Law of Conservation of Energy:
𝑊 = ∆𝐾. 𝐸 + ∆𝑃. 𝐸
𝑃1 − 𝑃2 V =
1
2
𝑚𝑣2
2
−
1
2
𝑚𝑣1
2
+ 𝑚𝑔ℎ2 − 𝑚𝑔ℎ1
𝑃1 − 𝑃2 m/ρ =
1
2
𝑚𝑣2
2
−
1
2
𝑚𝑣1
2
+ 𝑚𝑔ℎ2 − 𝑚𝑔ℎ1

14. 𝑃1 − 𝑃2 m/ρ = m(
1
2
𝑣2
2
−
1
2
𝑣1
2
+ 𝑔ℎ2 − 𝑔ℎ1)
𝑃1 − 𝑃2 =
1
2
ρ𝑣2
2
−
1
2
ρ𝑣1
2
+ ρ𝑔ℎ2 − ρ𝑔ℎ1
Rearranging the Eq :
𝑃1 +
1
2
ρ𝑣1
2
+ ρ𝑔ℎ1 = 𝑃2 +
1
2
ρ𝑣2
2
+ ρ𝑔ℎ2
This Equation is often expressed as:
𝑃 + 𝜌𝑔ℎ +
1
2
𝜌𝑣2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

15. Bernoulli’s equation shows that the pressure of a fluid decreases as
the speed of the fluid increases.