Batch 3: Fluid Dynamics and Surface Phenomena
1. Fluid Flow and Basic Concepts
Fluid dynamics deals with fluids (liquids and gases) in motion. Understanding how fluids behave under motion is essential in engineering, aviation, medicine, and environmental science.
Types of Flow
Steady Flow: Properties at a point do not change with time
Unsteady Flow: Flow variables vary with time
Laminar Flow: Smooth, orderly motion (layered flow)
Turbulent Flow: Irregular, chaotic motion
2. Continuity Equation (Conservation of Mass)
For an incompressible fluid, mass is conserved as it flows.
A
1
v
1
=A
2
v
2
Where:
A = cross-sectional area
v = velocity
Key Idea:
When a pipe narrows → velocity increases
When a pipe widens → velocity decreases
3. Bernoulli’s Equation
Bernoulli’s principle expresses conservation of energy in a moving fluid.
P+
2
1
ρv
2
+ρgh=constant
Meaning of Terms
P → pressure energy
2
1
ρv
2
→ kinetic energy per unit volume
ρgh → potential energy per unit volume
Key Assumptions
Fluid is incompressible
Flow is steady
No viscosity (no energy loss)
Motion along a streamline
Important Insight
High velocity → low pressure
Low velocity → high pressure
This is known as the Venturi effect.
Worked Example (Simplified)
A fluid flows in a horizontal pipe:
v
1
=3.5m/s
v
2
=0.35m/s
ρ=1000kg/m
3
P
1
=2000Pa
Since height is constant:
P
2
=P
1
+
2
1
ρ(v
1
2
−v
2
2
)
P
2
=2000+500(12.25−0.1225)
P
2
≈8063.75Pa
Answer: P
2
≈8.06kPa
Applications (Exam Focus)
Aircraft lift (pressure difference on wings)
Venturi meter (measuring flow rate)
Blood flow in arteries
Spray systems and carburetors
4. Surface Tension
Surface tension explains why liquids behave as if their surface is a stretched membrane.
Definition
Surface tension is the force per unit length acting along the surface of a liquid.
Origin
Molecules inside a liquid experience equal attraction in all directions
Molecules at the surface experience net inward force
5. Cohesion and Adhesion
Cohesion: Attraction between similar molecules
Adhesion: Attraction between different substances
Effects
Strong cohesion → droplets form (e.g., water beads)
Strong adhesion → liquid spreads (e.g., water on glass)
6. Capillarity (Capillary Action)
Capillary action is the rise or fall of liquid in a narrow tube.
h=
ρgr
2γcosθ
Where:
γ = surface tension
θ = contact angle
r = tube radius
Key Observations
Narrower tube → higher rise
Depends on balance between adhesion and cohesion
Example Insight
Water rises in glass because:
Adhesion > cohesion → concave meniscus
Mercury falls in glass because:
Cohesion > adhesion → convex meniscus
7. Viscosity
Viscosity measures resistance to flow.
High viscosity → thick fluids (e.g., oil)
Low viscosity → thin fluids (e.g., water)
Physical Meaning
It represents internal friction between fluid layers.
Practical Insight
Temperature increase → viscosity decreases (liquids)
Important in lubrication, blood flow, and industrial processes
8. Drops and Bubbles
Surface tension explains the shape and pressure of droplets and bubbles.
Pressure Difference (Laplace Law)
For a droplet:
ΔP=
R
2γ
For a soap bubble:
ΔP=
R
4γ
Key Concept
Smaller radius → higher internal pressure
This is why tiny droplets are more stable
9. Meniscus Formation
The curved surface of a liquid in a container is called a meniscus.
Concave (water): climbs walls
Convex (mercury): falls away from walls
10. Floating Due to Surface Tension
Some objects float not because of buoyancy but due to surface tension.
Examples:
Needle on water
Water-walking insects
SUMMARY FOR EXAMS AND TESTS
Continuity equation ensures mass conservation
Faster fluid → lower pressure (Bernoulli principle)
Surface tension arises from molecular imbalance
Capillary rise increases as radius decreases
Viscosity measures resistance to flow
Droplets are spherical due to minimum surface area
Pressure inside bubbles depends on radius
Adhesion vs cohesion determines liquid behavior