fluid mechanics

Overview of Fluid Mechanics

• Fluid Mechanics is divided into two primary sub-topics:
  • Fluid Statics: Deals with fluids at rest. In this course, it has a smaller quantum (volume of content).
  • Fluid Dynamics: Deals with fluids in motion. This topic carries a higher quantum in the syllabus.
• Significance and Syllabus:
  • In the National Book Foundation (Federal Board) textbook, this is titled "Fluid Mechanics."
  • In most other boards (except Sindh), it is titled "Fluid Dynamics," with the Statics portion typically taught in grades 9 and 10.
  • For Sindh Board, Statics and Dynamics are separate chapters.
  • Current MD-CAT and NUMS syllabi do not include this chapter, though it is part of the Aga Khan University syllabus.

Definition and Properties of Fluids

• Definition: A fluid is a combination of liquids and gases. Technically, it is anything that flows.
• Flow Property: If you leave a liquid or gas in one corner of a room, it will flow to the other side (e.g., perfume scent or water on a slope).
• Key Properties Comparison:
  • Shape:
    • Fluids: Conform to the shape of their container.
    • Gases: Occupy the entire available volume of a closed container. If moved from a $3\,dm^3$ container to a $300\,dm^3$ container, gas will expand to fill the entire $300\,dm^3$. This is due to minimal intermolecular forces of attraction.
    • Liquids: Maintain a fixed volume but occupy the shape of the container at the bottom level.
    • Solids: Have fixed shapes and volumes.
  • Compressibility:
    • Gases: Highly compressible because molecules are far apart.
    • Liquids: Slightly compressible. In this chapter, we focus on Ideal Liquids, which are considered completely incompressible.
  • Density ($\rho$):
    • Defined as mass per unit volume: $\rho = \frac{m}{V}$.
    • Gases have low density because they occupy large volumes for a small mass.
  • Viscosity:
    • Known as Fluid Friction. It is the resistance to flow.
    • Fluids slide through layers. When the top layer moves forward, the bottom layer (in contact with a surface) remains at rest due to inertia.
    • Relative motion between layers creates friction: the bottom layer pulls the top layer back, and the top layer pulls the bottom layer forward. This opposition is viscosity.
  • Surface Tension:
    • Molecules at the surface of a liquid are stretched like a membrane.
    • Gases do not have a defined surface and thus lack surface tension, whereas liquids do.

Pascal's Law

• Statement: Pressure applied to any point of an enclosed static fluid is transmitted perfectly and equally to all portions of the fluid and the walls of the container.
• Ideal Fluid Assumptions:
  • Zero fluid friction (non-viscous).
  • Incompressible (does not compress under pressure).
• Pressure Calculation: $P = \frac{F}{A}$. If $10\,N/m^2$ of pressure is applied at one piston, the same $10\,N/m^2$ is transmitted everywhere.

Application: The Hydraulic Lift

• Principle: Uses unequal cross-sectional areas to amplify force.
• Mechanism:
  • Applying a small force ($F_1$) on a small area ($A_1$) creates a pressure $P_1$.
  • This pressure transmits as $P_2$ to a larger area ($A_2$).
  • Since $P_1 = P_2$, then $\frac{F_1}{A_1} = \frac{F_2}{A_2}$.
  • Because $A_2$ is much larger than $A_1$, the output force $F_2$ is much larger than $F_1$.
• Mechanical Advantage ($MA$):
  • $MA = \frac{F_2}{F_1}$.
  • It defines how many times the machine multiplies the input force.
• Numerical Case Study:
  • Cylinder A ($r_1 = 5\,cm$) and Cylinder B ($r_2 = 10\,cm$).
  • Input force $F_1 = 200\,N$ applied to Cylinder A.
  • $A_1 = \pi \times (5 \times 10^{-2})^2 = 25 \times 10^{-4} \pi\,m^2$.
  • $A_2 = \pi \times (10 \times 10^{-2})^2 = 100 \times 10^{-4} \pi\,m^2$.
  • Result: $F_2 = F_1 \times \frac{A_2}{A_1} = 200 \times \frac{100 \pi \times 10^{-4}}{25 \pi \times 10^{-4}} = 200 \times 4 = 800\,N$.

Upthrust and Archimedes' Principle

• Upthrust (Buoyant Force): An upward force exerted by a fluid on an object immersed in it.
• Cause: When an object enters a fluid, it displaces fluid to make space. The displaced fluid molecules push back.
• Magnitude: Equal to the weight of the liquid displaced.
  • $F_{up} = m_{liquid} \times g$
  • Since $m = \rho \times V$, then $F_{up} = \rho_{fluid} \times V_{displaced} \times g$.
• Archimedes' Principle: When an object is immersed in a liquid, it experiences an upward thrust equal to the weight of the fluid it displaces.
• Apparent Weight: Objects feel lighter in water.
  • $\text{Apparent Weight} = \text{Actual Weight} - \text{Upthrust}$.
• Principles of Flotation:
  1. Weight > Max Upthrust: The object sinks (e.g., a stone). This happens when $\rho_{object} > \rho_{fluid}$.
  2. Weight = Max Upthrust: The object floats completely submerged at any level (like a fish).
  3. Weight < Max Upthrust: The object rises to the surface and floats partially submerged (like a ship). It sinks only until the weight of the displaced water equals the weight of the ship and cargo.

Surface Tension Details

• Molecular Basis: Interior molecules are pulled in all directions (net force zero). Surface molecules have no molecules above them; they are pulled sideways and downwards, creating a "stretched membrane."
• Mathematical Definition: Force per unit length ($\gamma$) acting on an imaginary line on the liquid surface.
  • $\gamma = \frac{F}{L}$.
• Direction: Tangential to the surface and perpendicular to the imaginary line.
• Units and Dimensions:
  • Units: $N/m$ or $dynes/cm$.
  • Dimensions: $[M L T^{-2}] / [L] = [M T^{-2}]$ or $[M^1 L^0 T^{-2}]$.
• Applications: Causes raindrops to be spherical (minimal surface area for a given volume) and allows small needles to float on water surfaces.

Fluid Dynamics: Drag Force and Stokes' Law

• Drag Force ($F_d$): A retarding force experienced by an object moving through a fluid. It is a type of fluid friction.
• Stokes' Law: For a sphere of radius $r$ moving slowly at speed $v$ through a fluid with viscosity $\eta$:
  • $F_d = 6\pi \eta r v$.
• Viscosity Dimensional Analysis ($\eta$):
  • From $F / (6\pi r v)$, dimensions are: $[M L T^{-2}] / ([L][L T^{-1}]) = [M L^{-1} T^{-1}]$.
  • Units: $kg\,m^{-1}\,s^{-1}$.

Terminal Velocity ($v_t$)

• Concept: When a sphere falls through a fluid, its velocity increases due to gravity, causing the drag force ($F_d$) to increase ($F_d \propto v$).
• Condition for Terminal Velocity: When Drag Force equals Weight ($F_d = W$), the net force is zero ($a = 0$), and the object falls with constant speed.
• Derivation:
  • $6\pi \eta r v_t = m g$
  • $m = \rho \times V = \rho \times (\frac{4}{3}\pi r^3)$
  • $6\pi \eta r v_t = (\rho \times \frac{4}{3}\pi r^3) g$
  • $v_t = \frac{2gr^2\rho}{9\eta}$.
• Example Problem: A water droplet ($r = 0.01\,cm = 10^{-4}\,m$) in air ($\eta = 19 \times 10^{-6}\,kg\,m^{-1}\,s^{-1}$, $\rho = 1000\,kg/m^3$).
  • Calculated result is approximately $1.1\,m/s$.
• Paratrooper Example: Opening a parachute increases the surface area, significantly increasing drag force to reach a low, safe terminal velocity (Dynamic Equilibrium).

Types of Fluid Flow

• Streamline (Laminar) Flow: Steady flow where every particle passing a point follows the exact same path/velocity as the previous particle. Streamlines never cross.
• Turbulent Flow: Irregular, unsteady flow with varying velocities and cross-paths.
• Reynolds Number ($Re$): A dimensionless number used to predict flow patterns.
  • $Re < 2300$: Laminar flow.
  • $2300 < Re < 4000$: Transition phase.
  • $Re > 4000$: Turbulent flow.

Equation of Continuity

• Basis: Law of Conservation of Mass.
• Statement: The mass flow rate remains constant throughout a pipe.
• Mass Flow Rate: $\frac{\Delta m}{\Delta t} = \rho A v$.
• Incompressible Fluids: Since $\rho$ is constant, the volume flow rate ($Av$) must be constant.
  • $A_1 v_1 = A_2 v_2$.
• Implication: Velocity is inversely proportional to cross-sectional area. Where a pipe narrows, the fluid speeds up (e.g., putting a thumb over a garden hose or blood moving from the aorta to narrower arteries).

Bernoulli's Principle

• Basis: Law of Conservation of Energy applied to fluids.
• Statement: For an ideal fluid, the sum of pressure energy, gravitational potential energy, and kinetic energy per unit volume is constant.
• Equation: $P + \rho g h + \frac{1}{2} \rho v^2 = \text{constant}$.
• Horizontal Pipe Case: If $h$ is constant, $P + \frac{1}{2} \rho v^2 = \text{constant}$. This means where velocity increases, pressure decreases.

Applications of Bernoulli's Principle

• Torricelli's Theorem (Velocity of Efflux): Calculating speed of water leaking from a hole in a tank.
  • Applying Bernoulli at the top (1) and orifice (2):
  • $P_1 = P_2 = \text{Atmospheric Pressure}$.
  • $v_1 \approx 0$ (Top surface area is huge compared to the hole).
  • $\rho g h_1 = \rho g h_2 + \frac{1}{2} \rho v_2^2$
  • $v_2 = \sqrt{2g(h_1 - h_2)}$.
• Venturi Meter: Measures speed by creating a constriction. High speed in the constriction creates low pressure, and the pressure difference identifies the flow rate.

Physics of Blood Flow

• Blood Pressure Definition: Force per unit area exerted by blood against the walls of blood vessels (arteries/aorta).
• Heart Cycles:
  • Systole (High Side): Left ventricle contracts, pushing blood into the aorta. High pressure $\approx 120\,mmHg$.
  • Diastole (Low Side): Left ventricle relaxes and fills. Low pressure $\approx 80\,mmHg$.
• Measurement (Sphygmomanometer):
  • An external cuff is inflated until pressure exceeds $120\,mmHg$, collapsing the artery and stopping flow (no sound in stethoscope).
  • Pressure is released; when external pressure match systolic, blood shoots through (turbulent flow), creating a "gurgling sound."
  • Pressure is released further; sounds vanish when flow becomes laminar (laminar flow occurs when external pressure drops below the diastolic level, and the artery never collapses).