Fluid Mechanics

Pressure is defined as the limit of force per unit area as the area approaches zero: $P = \lim_{\Delta S \rightarrow 0} \frac{F}{\Delta S}$

Key properties:

• It's a scalar quantity (magnitude only)
• For homogeneous and nonviscous fluids, pressure at a point is independent of the orientation of the area
• Measured in pascals (Pa) or N/m²
• Differs from force by being intensive (independent of size) while force is extensive
• Pressure acts perpendicular to any surface in contact with the fluid

Example: 1 Pa equals the pressure exerted by a force of 1 N spread over 1 m²

Derivation of Archimedes' principle:

1. Consider a body submerged in a fluid
2. Imagine replacing the body with the same fluid of equal volume
3. This substituted fluid would be in equilibrium
4. Forces on this substitute fluid:
   • Weight mg downward
   • Buoyant force B upward from surrounding fluid
5. For equilibrium: B = mg (where m is mass of displaced fluid)
6. Since the surrounding fluid exerts the same pressure on the original body as on the substituted fluid, the buoyant force on the body equals the weight of the displaced fluid

Archimedes' principle: A body partially or fully immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced.

Limitations/invalid scenarios:

• Non-inertial reference frames: If the fluid is accelerating, the principle requires modification
• Compressible fluids: When density varies significantly with depth
• Surface tension effects: Dominant at very small scales
• Non-homogeneous fluids: When density varies throughout the fluid
• Highly viscous flows: Where viscous forces are significant

Steady vs. Turbulent Flow:

Steady (Streamline) Flow:

• Fluid particles passing through any point follow the same path over time
• Velocity at each point remains constant with time
• Predictable, smooth flow patterns
• Occurs at lower velocities
• Can be analyzed with simpler mathematics

Turbulent Flow:

• Chaotic, irregular fluid motion
• Velocity at any point changes erratically with time
• Contains eddies and vortices
• Occurs at higher velocities
• Requires statistical methods for analysis

Streamline:

• A curve whose tangent at any point gives the direction of fluid velocity at that point
• All fluid particles passing through any point on a streamline follow that streamline
• Represents the path a fluid particle would take in steady flow

Why streamlines cannot cross:

• If two streamlines crossed, a fluid particle at the intersection would have two different velocity directions simultaneously
• This would create a logical contradiction since a particle can only move in one direction at a time
• At an intersection point, the fluid would need to follow two different paths, which is physically impossible

For two points at different depths in water:

1. Hydrostatic pressure equation: $P_2 - P_1 = \rho g h$

2. For water with density 1000 kg/m³ and points with 5 m vertical separation: $P_2 - P_1 = (1000 \text{ kg/m}^3)(9.8 \text{ m/s}^2)(5 \text{ m})$ $P_2 - P_1 = 49,000 \text{ Pa} = 49 \text{ kPa}$

3. Practical applications:

   • This is why your ears feel pressure when diving underwater
   • Water pressure increases by about 10 kPa per meter of depth
   • Deep-sea submersibles must withstand tremendous pressure differences
   • Dams are designed thicker at the base to withstand greater pressure at greater depths

Note: This equation assumes a fluid with constant density, which is a good approximation for liquids like water under normal conditions.

The work-energy theorem applies to fluid flow as follows:

1. For fluid moving from point 1 to point 2:

   • Work is done by pressure forces at both ends: $W_1 = P_1A_1v_1\Delta t$ and $W_2 = -P_2A_2v_2\Delta t$
   • Work is done by gravity: $W_g = \Delta m g(h_1 - h_2)$

2. This work causes a change in kinetic energy:

   • $\Delta KE = \frac{1}{2}\Delta m(v_2^2 - v_1^2)$

3. By conservation of mass (continuity equation):

   • $\Delta m = \rho A_1v_1\Delta t = \rho A_2v_2\Delta t$

4. Applying work-energy theorem ($W_{total} = \Delta KE$):

   • $P_1\frac{\Delta m}{\rho} - P_2\frac{\Delta m}{\rho} + \Delta m g(h_1 - h_2) = \frac{1}{2}\Delta m(v_2^2 - v_1^2)$

5. Simplifying yields Bernoulli's equation:

   • $\frac{P_1}{\rho} + gh_1 + \frac{1}{2}v_1^2 = \frac{P_2}{\rho} + gh_2 + \frac{1}{2}v_2^2$

In a Venturi tube:

• Pressure decreases at the constriction (P₂ < P₁)
• This follows directly from Bernoulli's principle
• The mathematical relationship is: P₁ + ½ρv₁² = P₂ + ½ρv₂²
• Since v₂ > v₁ (velocity increases at constriction), P₂ must decrease
• The pressure difference can be expressed as: P₁ - P₂ = ½ρ(v₂² - v₁²)

This demonstrates energy conservation: as kinetic energy increases at the constriction, pressure energy must decrease to maintain constant total energy along a streamline.

Real-world application: Carburetors use this principle to draw fuel into an air stream.

The derivation combines two key fluid principles:

1. Bernoulli's equation between the free surface (1) and the hole (2): $P_0 + \frac{1}{2}ρv_1^2 + ρgh = P_0 + \frac{1}{2}ρv_2^2$

2. The equation of continuity states that A₁v₁ = A₂v₂, which means v₁ = v₂(A₂/A₁)

When these are combined:

• The atmospheric pressure P₀ cancels out on both sides
• Substituting v₁ in terms of v₂ gives: $\frac{1}{2}ρ\left(\frac{A_2}{A_1}\right)^2v_2^2 + ρgh = \frac{1}{2}ρv_2^2$

This simplifies to: $\left[1-\left(\frac{A_2}{A_1}\right)^2\right]v_2^2 = 2gh$

When A₂ << A₁ (the hole is much smaller than the tank), the fraction becomes negligible, resulting in: $v_2 = \sqrt{2gh}$

This explains why the exit velocity depends only on the height of the fluid column and not on the density of the fluid.

The Magnus effect is a physical phenomenon where a spinning object moving through a fluid experiences a force perpendicular to both its direction of motion and axis of rotation.

Key mechanisms:

• When a ball spins while moving through air, it creates asymmetric airflow patterns
• On the side where the surface motion matches airflow direction, the relative air speed increases
• On the opposite side where surface motion opposes airflow, the relative air speed decreases
• According to Bernoulli's principle, faster-moving air creates lower pressure
• This pressure differential creates a force perpendicular to both the flight path and spin axis

Example: A cricket ball with vertical spin axis moving horizontally will experience a sideways force, causing it to "swing" left or right depending on spin direction.

Bernoulli's principle states that as fluid velocity increases, pressure decreases. For a spinning ball:

1. The spin creates different relative air velocities on opposite sides of the ball:

   • Higher velocity where ball surface and air move in the same direction
   • Lower velocity where ball surface and air move in opposite directions

2. This velocity difference creates a pressure gradient:

   • Lower pressure on the side with faster relative airflow
   • Higher pressure on the side with slower relative airflow

3. The resulting force (F) is:

   • Perpendicular to both the spin axis and velocity vector
   • Directed from the high-pressure side toward the low-pressure side
   • Proportional to both spin rate and forward velocity

The force direction follows the right-hand rule: if your fingers curl in the direction of spin, your thumb points in the approximate direction of the force.

Applications utilizing pressure reduction in fluid constrictions:

1. Aspirator pumps: Used for spraying liquids by drawing them up through a tube connected to a constriction
2. Venturi meters: Measure fluid flow rates by calculating velocity from pressure differences
3. Carburetors: Mix air and fuel in engines by drawing fuel into the airstream through a constriction
4. Atomizers: Create fine mists for perfumes, medical inhalers, and paint sprayers
5. Bunsen burners: Draw air into the gas flow to create a hotter flame
6. Vacuum systems: Laboratory vacuum pumps use water flow through constrictions to create suction
7. Injectors: Used in various industrial processes to introduce one fluid into another

All these applications work because when fluid velocity increases in the constriction, pressure decreases according to Bernoulli's principle.