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{net}=\big(P{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{0} = incident/ambient pressure at surface (not always 1\ \text{atm}; e.g., pressure cookers have elevated P{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{0} → higher boiling T → faster cooking, retains moisture).

Gauge Pressure

  • Pressure measured by a typical gauge (e.g., tire gauge): P{gauge}=P - P{atm}=P{0}+\rho g z - P{atm}
    • If P{0}=P{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{abs}=P{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.