The Ocean of Fluids

NODE 01

What is a Fluid?

• Liquids and Gases.
• Ability to flow, continuous deformation.
• Shape determined by container.

NODE 02

Liquids vs Gases

• Liquids: Definite volume, low compressibility.
• Gases: No fixed volume, highly compressible.

NODE 03

Density

m

V

• ρ = m/V. SI unit: kg/m³.
• First major quantity of the fluid world.
• Crucial for floating and fluid selection.

NODE 04

Relative Density

ρ SUBSTANCE

ρ WATER

• Density of substance / Density of reference.
• Water is common reference for liquids/solids.
• Dimensionless quantity.

NODE 05

Pressure

• P = F/A. The central language of fluids.
• Force applied normally over an area.
• SI unit: Pascal (Pa).

NODE 06

Everyday Pressure

vs

• Sharp knives, needles (small A, high P).
• Snow shoes, broad foundations (large A, low P).

NODE 07

Pressure in Liquid

• P = P₀ + ρgh.
• Increases with depth (h) and density (ρ).
• Gauge vs Absolute pressure.

NODE 08

Hydrostatic Pressure

• Fluid at rest.
• At the same depth in connected fluid, pressure is equal.

NODE 09

Atmospheric Press

• Air has mass and produces pressure.
• Changes with altitude.
• Sea-level atmospheric pressure.

NODE 10

Barometer

VAC

• Measures atmospheric pressure.
• Mercury column supported by air pressure.
• Vacuum above column (ideal model).

NODE 11

Manometer

• U-tube measuring pressure difference.
• Compares unknown pressure to reference (gauge).

NODE 12

Pascal's Principle

• Confined fluid at rest:
• Applied pressure change is transmitted undiminished throughout.

NODE 13

Hydraulic Machines

• F₁/A₁ = F₂/A₂ → F₂ = F₁(A₂/A₁).
• Force multiplication (lifts, brakes).
• Does NOT mean energy is created.

NODE 14

Archimedes' Princ

• Upward buoyant force = weight of displaced fluid.
• Fᵦ = ρ_fluid * g * V_displaced.

NODE 15

Buoyancy

• The complete force picture: Weight ↓, Buoyancy ↑.
• Sink (W > Fᵦ), Float (W = Fᵦ), Rise (W < Fᵦ).

NODE 16

Floatation

• Floating body: Fᵦ = mg.
• Weight of body = Weight of displaced fluid.
• Ice in water, average density.

NODE 17

Why Ships Float

• Steel sinks, but a steel ship floats.
• Shape allows it to displace sufficient water volume.
• Buoyant force balances its weight.

NODE 18

Apparent Weight

APP WT =ACTUAL WT- BUOYANCY

• Immersed object appears lighter.
• Connects Realm 07 (Gravity) with fluids.

NODE 19

Fluid Flow

• The fluid begins moving.
• Steady vs Unsteady, Laminar vs Turbulent.

NODE 20

Streamlines

• Gives local direction of fluid motion (steady flow).
• Never cross. Dense lines = stronger flow.

NODE 21

Volume Flow Rate

Q=A·v

• Q = ΔV/Δt.
• For uniform flow: Q = Area × velocity.
• Units: m³/s. Prepares for continuity.

NODE 22

Continuity Equation

• For steady incompressible flow: A₁v₁ = A₂v₂.
• Conservation of mass.
• Narrow pipe → greater speed.

NODE 23

Mass Flow Rate

ṁ=ρ·A·v

• Mass per unit time.
• Connects fluid mechanics to conservation laws.

NODE 24

Bernoulli's Princ

P +½ρv² +ρgh = C

• Energy framework of flowing fluids.
• Pressure E + Kinetic E + Gravitational PE = Const.
• Energy conservation for ideal fluid flow.

NODE 25

Bernoulli Apps

🌬️✈️💨

• Venturi effect, atomizers, sprayers.
• Pitot tubes, qualitative lift explanations.

NODE 26

Venturi Effect

• Fluid through narrow section: area ↓ → speed ↑.
• Speed ↑ → static pressure ↓.
• Flow meters, aspirators.

NODE 27

Torricelli's Law

• Ideal efflux from a tank: v = √(2gh).
• Fluid escaping through an opening.
• Connects to Realm 07 free-fall speed.

NODE 28

Viscosity

• Internal resistance to fluid flow/deformation.
• Water vs Oil vs Honey.
• Internal friction between fluid layers.

NODE 29

Newton's Viscosity

τ =η(dv/dy)

• τ = η (dv/dy).
• Shear stress, velocity gradient, dynamic viscosity.
• Connects to stress from Realm 08.

NODE 30

Laminar Flow

• Smooth layers, ordered motion.
• Limited mixing between layers.
• Stable flow patterns.

NODE 31

Turbulent Flow

• Irregular fluctuations, eddies, strong mixing.
• Energy dissipation.
• Reynolds number: Re = ρvL/η.

NODE 32

Stokes' Law

F = 6πηrv

• Small sphere moving slowly through viscous fluid.
• Depends on radius, viscosity, speed.
• Particle settling.

NODE 33

Terminal Velocity

• Falling object: initially Weight > Drag (accelerates).
• Eventually: Weight = Buoyancy + Drag.
• Net force = 0, constant terminal speed reached.

NODE 34

Surface Tension

• Intermolecular forces minimize surface area.
• Surface molecules, cohesion, surface energy.
• Force per unit length (N/m).

NODE 35

Surface Phenom

💧🫧🦟

• Water droplets, soap bubbles.
• Small insects on water.
• Liquid films.

NODE 36

Capillarity

• Capillary rise/fall in narrow tubes.
• Adhesion, cohesion, contact angle, meniscus.
• Plants, soil, ink, wicks.

NODE 37

Angle of Contact

• Liquid-solid interface, wetting.
• Hydrophilic vs hydrophobic behavior.
• Determines if liquid rises or falls in tube.

NODE 38

Excess Pressure

DROP2T/r

BUBBLE4T/r

• Pressure differences caused by curved surfaces.
• Soap bubble has TWO surfaces (4T/r).

NODE 39

Energy Conserv

P

K

U

TOTAL E

• Pressure E + Kinetic E + Grav E = Total Mech E.
• Connects directly back to Realm 06.

NODE 40

Fluid Machines

🚰⚙️💉

• Dams, pipelines, pumps, hydraulic lifts.
• Water turbines, sprayers, medical flow systems.

NODE 41

Atmos & Geo Fluids

• Atmosphere, ocean currents, winds, storms.
• Convection, river flow, erosion.

NODE 42

Biological Fluids

♥

• Blood flow, airflow in lungs, capillaries.
• Circulatory pressure, viscosity, medical pumps.

NODE 43

Experiment Lab

ρ

Fᵦ

η

P

• Measure density, Archimedes principle.
• Flow rate, viscosity, capillary rise.

NODE 44

Graph Chamber

• Pressure vs depth, Flow vs P-diff.
• Extract gradient, proportionality.

NODE 45

Misconceptions

PRESSURE = FORCE

X

• "Pressure = Force" → No. P = F/A.
• "Heavier sinks" → No. Average density matters.
• "Viscosity = Density" → No.

NODE 46

Solving Engine

• At Rest? → P = P₀+ρgh / Pascal / Buoyancy.
• Moving? → A₁v₁ = A₂v₂ (Mass) / Bernoulli (Energy).
• Viscosity? Surface Effect?

NODE 47

The Grand Synthesis

PASCALBERNOULLISTOKES

• Gravity → Pressure. Force → Stress.
• Energy → Flow. Conservation → Continuity.
• Density → Pressure → Buoyancy → Flow → Viscosity.