Chapter 13: Fluids – Properties, Pressure, and Dynamics

Fluids and Density

States of Matter and Their Properties

Understanding the behavior of fluids begins with the classification of matter into solids, liquids, gases, and plasma. Each state has distinct molecular arrangements and properties:

• Solids: Definite volume and shape; molecules vibrate about fixed positions due to strong intermolecular forces.

• Liquids: Definite volume but no definite shape; molecules move more freely than in solids.

• Gases: No definite volume or shape; molecules are far apart and move randomly.

• Plasma: Ionized gas with free electrons; found in stars.

Crystalline and Amorphous Solids

• Crystalline solids: Atoms are arranged in an ordered structure (e.g., NaCl).

• Amorphous solids: Atoms are arranged almost randomly (e.g., glass).

Density

Density ($\rho$) is defined as mass per unit volume:

$\rho \equiv \frac{M}{V}$

• SI unit: kg/m3

• 1 g/cm3 = 1000 kg/m3

• Density of solids and liquids changes slightly with temperature and pressure; gases vary greatly.

Specific Gravity

Specific gravity (SG) is the ratio of the density of a substance to the density of water at 4°C. It is dimensionless:

$SG = \frac{\rho_{substance}}{\rho_{water}}$

• Density of water at 4°C: 1.000 g/cm3

• Example: Mercury's SG is 13.6.

Pressure in Fluids

Definition of Pressure

Pressure (P) is the force (F) applied perpendicular to a surface divided by the area (A):

$P \equiv \frac{F}{A}$

• SI unit: Pascal (Pa), where 1 Pa = 1 N/m2

• Other units: atm, bar, mmHg, psi

• 1 atm = $1.01325 \times 10^5$ Pa = 760 mmHg = 14.70 psi

Pressure Variation with Depth

In a fluid at rest, pressure increases with depth due to the weight of the fluid above:

$P = P_0 + \rho g h$

• $P_0$ is atmospheric pressure at the surface.

• $\rho$ is fluid density, $g$ is acceleration due to gravity, $h$ is depth.

Hydrostatic Equilibrium

All points at the same depth in a static fluid have the same pressure. If not, fluid would flow from high to low pressure until equilibrium is reached.

Pascal’s Principle

A change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid and to the walls of the container. This principle is the basis for hydraulic systems:

$\frac{F_1}{A_1} = \frac{F_2}{A_2}$

• A small force applied to a small area can produce a large force on a larger area.

Pressure Measurement Devices

• Manometer: Measures pressure relative to atmospheric pressure using a U-shaped tube.

• Barometer: Measures atmospheric pressure using a column of mercury.

• Sphygmomanometer: Measures blood pressure in mmHg.

Absolute vs. Gauge Pressure

• Absolute pressure: Measured relative to a vacuum.

• Gauge pressure: Measured relative to atmospheric pressure.

• Relationship: $P_{abs} = P_{atm} + P_{gauge}$

Buoyancy and Archimedes’ Principle

Archimedes’ Principle

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

$B = \rho_{fluid} V_{displaced} g$

• The buoyant force is due to the pressure difference between the top and bottom of the object.

Floating and Sinking

• If the object's density is less than the fluid's, it floats; if greater, it sinks.

• For floating objects, the weight of the displaced fluid equals the object's weight.

Fluids in Motion

Streamline (Laminar) and Turbulent Flow

• Streamline (Laminar) flow: Fluid particles follow smooth paths; velocity at any point is constant over time.

• Turbulent flow: Irregular, with mixing and eddy currents; occurs at high velocities or with abrupt changes in flow.

Equation of Continuity

For an incompressible fluid, the mass flow rate is constant throughout a pipe:

$A_1 v_1 = A_2 v_2$

• Where $A$ is cross-sectional area and $v$ is fluid speed.

• Speed increases in narrower sections of pipe.

Bernoulli’s Principle and Equation

Bernoulli’s equation relates pressure, kinetic energy per unit volume, and potential energy per unit volume along a streamline:

$P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

• As fluid speed increases, pressure decreases (and vice versa).

• Explains lift on airplane wings, atomizers, and other phenomena.

Applications of Bernoulli’s Principle

• Atomizer: Fast-moving air reduces pressure, drawing liquid up and dispersing it as a spray.

• Airplane wing: Faster airflow above the wing creates lower pressure, resulting in lift.

Viscosity and Poiseuille’s Equation

Viscosity

Viscosity is a measure of a fluid's resistance to flow, caused by internal friction between layers moving at different velocities. High viscosity fluids (like syrup) flow more slowly than low viscosity fluids (like water).

Summary Table: Key Fluid Properties and Principles

Concept	Definition/Formula	SI Unit
Density ($\rho$)	$\rho = \frac{M}{V}$	kg/m3
Specific Gravity (SG)	$SG = \frac{\rho_{substance}}{\rho_{water}}$	Dimensionless
Pressure (P)	$P = \frac{F}{A}$	Pa (N/m2)
Hydrostatic Pressure	$P = P_0 + \rho g h$	Pa
Buoyant Force (B)	$B = \rho_{fluid} V_{displaced} g$	N
Continuity Equation	$A_1 v_1 = A_2 v_2$	m3/s
Bernoulli’s Equation	$P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$	Pa