Fluids: Density & Pressure Fundamentals

Underwater Lakes Beneath the Mediterranean

• Three hypersaline “lakes” exist >$4000\ \text{m}$ below sea level; brine is $5\text{–}10$ times saltier than over-lying seawater.
  • Extreme density prevents mixing, creating a sharp interface analogous to oil/water in salad dressing.
  • Behave like surface lakes: measurable tides, shorelines, beach ridges, swash zones.
  • Deep-sea submersibles that land on the brine bob and generate circular ripples just like a stone in a pond.
• Serves as a striking real-world illustration of the physics of fluids (density differences, surface phenomena, fluid/solid interactions).

Scope of the Chapter

• Review of foundational quantities: density and pressure.
• Hydrostatics: behavior of fluids at rest; buoyancy to be explored later.
• Fluid dynamics: Bernoulli’s equation, flight aerodynamics.
• Physiological applications: blood flow, air flow in respiration.

Definitions: Fluids vs. Solids

• Fluids = substances that flow and conform to container shapes (liquids & gases).
  • Examples: methane gas in home pipelines; air entering lungs, filling alveoli.
• Solids = retain independent shape; do not flow.

Shared Mechanical Characteristics

• Both fluids and solids exert normal (perpendicular) forces on surfaces.
• Only solids sustain shear (tangential) forces; fluids cannot.
• Large perpendicular forces from fluids can mimic solids (belly-flop from height can hurt like hitting concrete).

Density ($\rho$)

• Definition: $\rho = \dfrac{m}{V}$ (scalar, no direction).
• SI unit: $\text{kg}\,\text{m}^{-3}$.
  • Common MCAT units: $\text{g}\,\text{mL}^{-1}$ or $\text{g}\,\text{cm}^{-3}$.
  • Reminder: $1\ \text{mL} = 1\ \text{cm}^{3}$, but $1\ \text{L} \neq 1\ \text{m}^{3}$ (actually $1000\ \text{L} = 1\ \text{m}^{3}$).
• Water benchmark (at $1\ \text{atm},\ 4^{\circ}\text{C}$): $1\ \text{g}\,\text{cm}^{-3}=1000\ \text{kg}\,\text{m}^{-3}$.
• Weight of fluid/solid sample: $F_g = \rho V g$ (frequent in buoyancy problems).

Specific Gravity (SG)

• Compares density of substance to density of water:
  $SG = \dfrac{\rho}{1\ \text{g}\,\text{cm}^{-3}}$ (unitless, decimal form).
• Predicts sinking vs. floating in water (SG > 1 → sinks; SG < 1 → floats).
• Example: Benzene
  • $\rho_{benzene}=877\ \text{kg}\,\text{m}^{-3}$.
  • $SG = \dfrac{877}{1000}=0.877$ (will float on water).

Pressure ($P$)

• Definition: $P = \dfrac{F}{A}$ where $F$ is magnitude of normal force.
• SI unit: Pascal (Pa) $\big(1\ \text{Pa}=1\ \text{N}\,\text{m}^{-2}\big)$.
• Additional units & conversions:
  • $1.013\times10^{5}\ \text{Pa}=760\ \text{mmHg}=760\ \text{Torr}=1\ \text{atm}$.

Scalar Nature of Pressure

• Although tied to force (a vector), pressure itself is scalar—same in all directions at a point inside a fluid (neglecting gravity).
• In closed container with gas, random molecular motion yields equal pressure on every surface regardless of orientation.
• Unequal pressures across a surface produce a net force that is vectorial → drives airflow in lungs, bursts windows, inflates plastic over broken car windows.
• Gravity induces vertical pressure gradients → foundational for hydrostatics.

Example: Skyscraper Window

• Window dimensions: $2.0\ \text{m}\times3.5\ \text{m}$.
• Inside pressure $=1\ \text{atm}$; outside storm pressure $=0.997\ \text{atm}$.
• Net outward force:
  $F<em>{net}=\big(P</em>{in}-P_{out}\big)A$
  $=\big(1-0.997\big)\text{atm}\times\dfrac{1.013\times10^{5}\ \text{Pa}}{1\ \text{atm}}\times(2.0)(3.5)\ \text{m}^{2}$
  $\approx2.1\times10^{3}\ \text{N}$ (actual $2128\ \text{N}$).

Absolute (Hydrostatic) Pressure

• Total pressure at depth $z$ in a fluid: $P = P_{0} + \rho g z$
  • $P<em>{0}$ = incident/ambient pressure at surface (not always $1\ \text{atm}$; e.g., pressure cookers have elevated $P</em>{0}$).
• Practical notes:
  • Atmospheric pressure varies with altitude (Denver $0.83\ \text{atm}$, Death Valley $1.01\ \text{atm}$).
  • Influences hemoglobin O$2$ affinity, liquid boiling points, cooking times (pressure cooker raises $P</em>{0}$ → higher boiling T → faster cooking, retains moisture).

Gauge Pressure

• Pressure measured by a typical gauge (e.g., tire gauge): $P<em>{gauge}=P - P</em>{atm}=P<em>{0}+\rho g z - P</em>{atm}$
  • If $P<em>{0}=P</em>{atm}$ (open fluid), then $P_{gauge}=\rho g z$.

Example: Diver 20 m Below Sea Surface

• Data: $z=20\ \text{m};\ \rho_{seawater}=1025\ \text{kg}\,\text{m}^{-3};\ g\approx9.8\ \text{m}\,\text{s}^{-2}$.
• Gauge pressure:
  $P_{g}=\rho g z\approx1025\times9.8\times20\approx2.0\times10^{5}\ \text{Pa}$ (actual $2.01\times10^{5}\ \text{Pa}$).
• Absolute pressure:
  $P<em>{abs}=P</em>{atm}+P_{g}=1.013\times10^{5}+2.01\times10^{5}\approx3.02\times10^{5}\ \text{Pa}$.

Connections & Implications

• Physics principles in this section underpin upcoming topics:
  • Buoyancy (Archimedes’ principle) relies on weight $\rho V g$ and pressure gradients $\rho g z$.
  • Bernoulli’s equation will expand on pressure–velocity relationships for moving fluids.
  • Human physiology: breathing (pressure differential across lungs), blood flow (gauge pressures within vasculature).
  • Engineering & everyday life: pressure cookers, airplane cabins, HVAC, weather phenomena (tornado window bursts).
• Ethical/practical: Understanding pressure/density crucial for safe deep-sea exploration, design of submersibles, mitigating barotrauma in divers.