Chapter 12: Fluid Mechanics – Study Notes

Fluid Mechanics

Introduction to Fluids

Fluid mechanics is the study of substances that can flow, such as liquids and gases. Unlike solids, fluids do not maintain a fixed shape and conform to the boundaries of their containers.

• Fluids include air, water, and other substances that flow.

• Fluids adapt to the shape of their container and do not retain a fixed form.

Learning Goals

• Understand the concepts of density and pressure in fluids and how they are measured.

• Calculate the buoyant force exerted by a fluid on an immersed object.

• Apply Bernoulli’s equation to relate pressure and flow speed in fluid dynamics.

• Distinguish between laminar and turbulent flow, and understand how tube size affects flow speed.

• Compare viscous and turbulent flow to ideal (non-viscous) flow.

Density

Definition and Properties

Density is a fundamental property of matter, defined as mass per unit volume. It is denoted by the Greek letter rho (ρ).

• For a homogeneous material: $\rho = \frac{m}{V}$ where $m$ is mass and $V$ is volume.

• The SI unit of density is the kilogram per cubic meter (kg/m3).

• Density is constant throughout a homogeneous material.

Example: Both a steel wrench and a steel nail have the same density, even though their masses and volumes differ, because they are made of the same material.

Densities of Common Substances

Material	Density (kg/m3)
Air (1 atm, 20°C)	1.20
Ice	0.92 × 103
Water	1.00 × 103
Blood	1.06 × 103
Aluminum	2.7 × 103
Lead	11.3 × 103
Gold	19.3 × 103

Pressure in Fluids

Definition and Units

Pressure is defined as the perpendicular force per unit area exerted on a surface:

• $p = \frac{F_{\perp}}{A}$

• The SI unit of pressure is the pascal (Pa): 1 Pa = 1 N/m2.

• Pressure is a scalar quantity; it has magnitude but no direction.

Pressure at Depth

The pressure at a depth h in a fluid of uniform density is greater than the pressure at the surface:

• $p = p_0 + \rho g h$

• $p_0$ = pressure at the surface (often atmospheric pressure)

• $\rho$ = density of the fluid

• $g$ = acceleration due to gravity

• $h$ = depth below the surface

Key Point: The pressure at a given depth is the same regardless of the shape of the container.

Hydrostatic Pressure and Gauge Pressure

• Hydrostatic pressure is the pressure due to the weight of the fluid above a certain point.

• Gauge pressure is the pressure relative to atmospheric pressure: $p_{\text{gauge}} = p - p_{\text{atm}}$

• Absolute pressure is the total pressure, including atmospheric pressure.

Experimental Facts about Fluids at Rest

• Fluids exert pressure in all directions.

• At any point at rest, the pressure is the same in all directions.

• Pressure increases with depth.

• The force due to fluid pressure always acts perpendicular to any surface in contact with the fluid.

Example Problem: Submarine Hatch

Problem: What force must be applied to a hatch (1.2 m × 0.6 m) at a depth of 10 m in seawater (density 1025 kg/m3)?

• Area, $A = 1.2 \times 0.6 = 0.72$ m2

• Pressure, $p = \rho g h = 1025 \times 9.8 \times 10 = 100,450$ Pa

• Force, $F = pA = 100,450 \times 0.72 = 72,324$ N

Atmospheric Pressure

• Atmospheric pressure is due to the weight of the air above us.

• At sea level: $p_{\text{atm}} = 1.01 \times 10^5$ Pa = 1 bar = 14.7 lb/in2

• Atmospheric pressure decreases with altitude.

Pascal’s Law

Statement and Applications

Pascal’s Law: A change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid and the walls of its container.

• Mathematically: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

• Basis for hydraulic systems, where a small force applied to a small area can produce a large force on a larger area.

Example: Hydraulic car lifts use Pascal’s law to raise heavy vehicles with relatively little input force.

Consequences of Hydrostatic Pressure & Pascal’s Law

• Hydraulics: Small force on a small surface can be converted into a large force on a large surface.

• Air pressure decreases with height; water pressure increases with depth.

• Pressure gauges can measure both relative (gauge) and absolute pressure.

Archimedes’ Principle and Buoyancy

Archimedes’ Principle

Any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.

• $F_b = \rho_f V_{\text{disp}} g$

• $\rho_f$ = density of fluid

• $V_{\text{disp}}$ = volume of fluid displaced

• $g$ = acceleration due to gravity

Proof Outline: The pressure at the bottom of a submerged object is greater than at the top, resulting in a net upward force.

Flotation and Conditions for Floating

• An object floats if the buoyant force equals its weight.

• Condition for floating: $\rho_{\text{object}} \leq \rho_{\text{fluid}}$

• Objects denser than the fluid sink; less dense objects float.

Example: Ships float because their average density (including air inside) is less than that of water.

Summary Table: Densities of Common Substances

Material	Density (kg/m3)
Air (1 atm, 20°C)	1.20
Ice	0.92 × 103
Water	1.00 × 103
Blood	1.06 × 103
Aluminum	2.7 × 103
Lead	11.3 × 103
Gold	19.3 × 103

Key Equations

• Density: $\rho = \frac{m}{V}$

• Pressure: $p = \frac{F_{\perp}}{A}$

• Pressure at depth: $p = p_0 + \rho g h$

• Buoyant force: $F_b = \rho_f V_{\text{disp}} g$

• Pascal’s Law: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

Additional info: This summary covers the first half of Chapter 12 (Fluid Mechanics), focusing on fluids at rest, density, pressure, hydrostatics, Pascal’s law, and Archimedes’ principle. Later sections (not included in these slides) would cover fluid dynamics, Bernoulli’s equation, viscosity, and turbulence.