Fluid Dynamics Exhaustive Study Notes

Fluid dynamics is the study of the movement of liquids and gases in a certain direction.
The study considers fluids at rest as a specific case of motion with zero velocity ( $v = 0$ ).
The fundamental components of fluid dynamics include:
- Fluid flow.
- Viscosity.
- Bernoulli's principles.
Fluid mechanics is essential to life and the physical world:
- The human body is approximately $65\%$ water.
- Roughly $2/3$ of the Earth's surface is covered by water.
- The Earth's atmosphere extends to a height of $17\,km$ above the surface.
Fluid dynamics has historically shaped several aspects of civilization:
- Geomorphology (the study of physical features of the surface of the earth).
- Human migration and the development of civilization.
- Modern scientific and mathematical theories and methods.
- Warfare.
- It affects every part of daily lives, from climate to technology.

The study of fluid dynamics dates back to Ancient Greece with the investigation of fluid statics.
Key contributors to the field include:
- Archimedes (c. $287-212\,\text{BC}$ ): Known for investigating fluid statics.
- Isaac Newton ( $1642-1727$ ).
- Gottfried Wilhelm Leibniz ( $1646-1716$ ).
- Daniel Bernoulli ( $1667-1748$ ).
- Leonhard Euler ( $1707-1783$ ).
- Claude-Louis Navier ( $1785-1836$ ).
- George Gabriel Stokes ( $1819-1903$ ).
- Osborne Reynolds ( $1842-1912$ ).
- Ludwig Prandtl ( $1875-1953$ ).
- Geoffrey Ingram Taylor ( $1886-1975$ ).

A fluid is a substance in either the liquid or gas phase.
Factors affecting fluid flow include:
- The layers of the fluid.
- The streamline of the flow.
- The density of the substance.
- The flow area (flow rate depends on area).
Liquids vs. Gases:
- Liquids: Groups of molecules move relative to each other. They maintain a relatively constant volume due to strong cohesive forces. Liquids take the shape of their container and form a "free surface" in a gravitational field when the container is larger than the volume.
- Gases: Molecules are widely spaced with very small cohesive forces. A gas expands to fill the entire available space of its container and does not form a free surface.
Atomic Arrangement by Phase:
- Solid: Molecules are arranged in a pattern repeated throughout and remain in relatively fixed positions. Intermolecular bonds are at their strongest.
- Liquid: Molecules can rotate and translate freely about each other.
- Gas: Molecules move about at random and are far apart; molecular ordering is nonexistent. Intermolecular bonds are at their weakest.
Gas and Vapor:
- Gas: The vapor phase of a substance is called a gas when it exists above the critical temperature.
- Vapor: This term implies the current phase is not far from a state of condensation.

Pressure Measurement: On a microscopic scale, pressure is determined by the interaction of individual gas molecules. On a macroscopic scale, it is measured using a pressure gauge.
Macroscopic (Classical) Approach: Does not require knowledge of individual molecular behavior. It provides a direct and easy method for analyzing engineering and practical problems.
Microscopic (Statistical) Approach: Analyzes problems based on the average behavior of large groups of individual molecules.

Types of Flow:
- Steady (Laminar) flow.
- Turbulent flow.
Internal vs. External Flow:
- Internal Flow: The flow of a fluid in a pipe or duct where it is completely bounded by solid surfaces (e.g., water in a pipe).
- External Flow: The flow of an unbounded fluid over a surface (e.g., airflow over a tennis ball or over a wire). External flows often feature a "turbulent wake region" behind the object.
- Open-channel Flow: A specific type of internal flow where a duct is only partially filled, allowing for a free surface (e.g., water in a drainage canal).

Environment: River hydraulics and air pollution studies.
Atmospheric Phenomena: Analysis of weather and climate, including tornadoes, thunderstorms, and hurricanes.
Transportation: Design and performance of aircraft, spacecraft, surface ships, submarines, automobiles, and high-speed rail.
Energy and Industry: Wind turbines, power plants, piping, and plumbing systems.
Sports and Recreation: Water sports, auto racing, cycling, surfing, and offshore racing.

Fluid mechanics governs processes within the human body, such as gas exchange ( $CO_2$ and $O_2$ ).
Medical Devices:
- Fluid dynamics is crucial in the design of artificial hearts, blood pumps, and Ventricular Assist Devices (VAD).
- Insulin Pumps: Used for patients with Type 1 diabetes who cannot produce sufficient insulin. The pump resides near the belly button and delivers liquid insulin into the bloodstream via an "infusion set" to regulate glucose for energy.
Treatments:
- Chemotherapy: Used to control tumors.
- Blood Thinners (e.g., Heparin): Used to maintain blood flow and prevent the formation of blood clots. This fall under hydrodynamics/aerodynamics study as it involves calculating medicine circulation throughout the body.

First presented by Daniel Bernoulli in his book Hydrodynamica in $1738$ .
Fundamental Relationship: An increase in fluid speed occurs simultaneously with a decrease in internal pressure or a decrease in the fluid's potential energy.
Conservation Law of Energy: Bernoulli's principle relates Pressure, Potential Energy, and Kinetic Energy. The sum of these values in a unit volume remains constant.
Bernoulli's Equation:
- $P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$
- Where:
- $P$ = Pressure Energy per unit volume.
- $\frac{1}{2}\rho v^2$ = Kinetic Energy per unit volume.
- $\rho gh$ = Potential Energy per unit volume.
The Bernoulli Effect: When fluid speed increases (v_2 > v_1), the internal pressure decreases (P_2 < P_1) provided height remains constant.
Application - LIFT:
- In flight, air moves faster over the top of a wing (creating lower pressure).
- Slower air moves under the wing (creating higher pressure).
- The pressure differential results in an upward force called lift.

The relationship between flow, pressure, and resistance is defined by:
- $\text{FLOW} = \frac{\Delta P}{R}$
- Where $\Delta P$ is the change in pressure and $R$ is the resistance.
Blood Flow Characteristics:
- Blood flow involves "Shear Stress" on endothelial cells.
- Pressure reduces as blood moves from the Left Ventricle (LV) to the Right Atrium (RA).
- Healthy flow typically presents as laminar flow.