Fluid Dynamics

Fluids — liquids and gases — flow and transmit pressure. For the MCAT Chemical and Physical Foundations section, master density and pressure, hydrostatics (P = ρgh), Pascal's and Archimedes' principles, the continuity equation, Bernoulli's equation, and viscous (Poiseuille) flow. The exam loves to place these in biological contexts: blood pressure, circulation, IV drips, and gas exchange.

Density, Specific Gravity & Pressure

Density ρ = m/V (SI: kg/m³). Water is 1000 kg/m³ (= 1 g/cm³). Specific gravity is the ratio of a substance's density to that of water; it is dimensionless. An object with specific gravity < 1 floats in water; > 1 sinks.

Pressure P = F/A is force per unit area normal to a surface (SI: pascal, Pa = N/m²). Pressure in a fluid acts equally in all directions at a given depth. Useful conversions: 1 atm = 1.013 × 10⁵ Pa = 760 mmHg (torr) = 760 mm of Hg.

Hydrostatic pressure — gauge vs absolute

The pressure due to a column of fluid of depth h is the gauge pressure:

P_gauge = ρgh

The total or absolute pressure adds the pressure pushing down on the fluid surface (usually atmospheric):

P_abs = P₀ + ρgh

Key point: hydrostatic pressure depends only on depth, fluid density, and g — not on the shape or total volume of the container (the "hydrostatic paradox"). A blood-pressure reading of 120/80 mmHg is a gauge pressure (above atmospheric).

Pascal's Principle

A pressure change applied to an enclosed incompressible fluid is transmitted undiminished to every point in the fluid and to the walls of the container. This is the basis of the hydraulic lift:

P₁ = P₂ ⇒ F₁/A₁ = F₂/A₂

A small force on a small piston produces a large force on a large piston (multiplied by A₂/A₁). Energy is conserved: the large piston moves a proportionally smaller distance, so F·d is unchanged.

Buoyancy & Archimedes' Principle

Any object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces:

F_B = ρ_fluid V_displaced g

• Floating: F_B = weight of object; the object displaces its own weight of fluid. The fraction submerged = ρ_object/ρ_fluid.
• Sinking: if ρ_object > ρ_fluid, weight exceeds the maximum buoyant force and the object sinks.
• Apparent weight of a submerged object = true weight − F_B.

Equation of Continuity

For an incompressible, non-viscous fluid flowing steadily through a pipe of varying cross-section, the mass flowing per second past every point is the same. Mathematically:

A₁v₁ = A₂v₂    (Av = constant)

Where A is the cross-sectional area and v is the speed of the fluid. This is a direct consequence of conservation of mass. Wherever the pipe narrows, the fluid must speed up; wherever it widens, the fluid slows down. The product Av is called the volume flow rate (m³/s).

Worked example

Water flows through a pipe whose radius narrows from 4 cm to 2 cm. If v₁ = 2 m/s, then v₂ = (A₁/A₂)v₁ = (16/4)(2) = 8 m/s. Speed quadruples because area scales with r².

Bernoulli's Equation

For an ideal (incompressible, non-viscous, streamline) fluid, the sum of pressure energy, kinetic energy and potential energy per unit volume is constant along a streamline:

P + ½ρv² + ρgh = constant

P

Static pressure of the fluid (Pa).

½ρv²

Dynamic pressure due to fluid motion.

ρgh

Hydrostatic pressure due to height h above a reference level.

Direct consequence: where fluid speed is high, pressure is low (for flow at constant height). Combined with continuity (narrow → faster → lower pressure), this explains many MCAT scenarios.

Applications (MCAT-relevant)

• Arterial stenosis / plaque: blood speeds up through a narrowed artery, so local pressure drops — the vessel can be squeezed further shut, worsening the blockage.
• Aneurysm: where a vessel widens, flow slows and pressure rises, pushing the weakened wall outward.
• Venturi meter / flow measurement: flow rate is found from the pressure drop across a constriction.
• Atomizer / perfume sprayer: fast air across a tube tip lowers the pressure there, drawing liquid up.
• Airplane lift & pitot tubes: classic Bernoulli demonstrations of the speed–pressure trade-off.

Fluid Flow (Laminar, Turbulent)

Two distinct flow regimes occur in real fluids:

Fluid Drag

When a body moves through a fluid, it experiences a retarding force called drag. For a small spherical object moving slowly through a viscous fluid, Stokes' law gives the drag:

F = 6πηrv

where η is the coefficient of viscosity (Pa·s), r is the radius of the sphere, and v is its speed relative to the fluid. Drag rises linearly with speed at low Re, but for fast/large objects (turbulent regime) drag rises with v².

Coefficient of viscosity (η)

Viscosity is the internal friction of a fluid — resistance to shear. SI unit: Pa·s (or N·s/m²). For liquids viscosity decreases with temperature (e.g. honey thins on heating); for gases it increases with temperature.

Poiseuille's Law (Viscous Flow in Tubes)

For steady laminar flow of a viscous fluid through a rigid cylindrical tube, the volume flow rate Q depends strongly on the radius:

Q = πr⁴ΔP / (8ηL)

where ΔP is the pressure difference across the tube, r the radius, L the length, and η the viscosity. The dominant feature is the r⁴ dependence.

• Halving a vessel's radius cuts flow to (1/2)⁴ = 1/16 of its former value (at the same pressure).
• Biological context: small changes in arteriole radius (vasoconstriction / vasodilation) produce huge changes in blood flow and are the body's main way of regulating flow and blood pressure. This also explains why atherosclerotic narrowing dramatically raises resistance.
• Analogy to Ohm's law: ΔP ↔ V, Q ↔ I, so vascular resistance R = 8ηL/(πr⁴).

Surface Tension

Surface tension (γ, units N/m) arises because molecules at a liquid surface are pulled inward by cohesive forces, making the surface behave like an elastic membrane. It causes water droplets to bead, insects to walk on water, and capillary rise.

For a spherical bubble/droplet, Laplace's law gives the excess internal pressure ΔP = 2γ/r (droplet) or 4γ/r (soap bubble, two surfaces). Smaller radius → higher pressure.

Terminal Velocity

A body falling through a viscous fluid eventually reaches a constant speed when the net force on it is zero. At this point the weight is balanced by buoyancy plus the viscous drag. For a sphere of radius r, density ρ in a fluid of density σ:

v_t = 2r²(ρ − σ)g / (9η)

Key features:

• v_t increases with r² — larger droplets fall much faster.
• v_t increases with the density difference (ρ − σ).
• v_t decreases as viscosity η increases.
• If ρ = σ the body floats; if ρ < σ it rises (e.g. air bubble in water).

Worked MCQs

Five MCQs that capture the high-yield testing patterns for this chapter.

Quick Recap

• Density ρ = m/V; specific gravity = ρ/ρ_water (dimensionless).
• Hydrostatic: P_gauge = ρgh; P_abs = P₀ + ρgh. 1 atm = 760 mmHg = 1.013 × 10⁵ Pa.
• Pascal: F₁/A₁ = F₂/A₂ (hydraulic lift).
• Buoyancy: F_B = ρ_fluidV_dispg; floating fraction submerged = ρ_obj/ρ_fluid.
• Continuity: A₁v₁ = A₂v₂ (mass). Bernoulli: P + ½ρv² + ρgh = const (energy).
• Higher speed → lower pressure (stenosis, aneurysm, Venturi).
• Poiseuille: Q = πr⁴ΔP/(8ηL) — r⁴ dependence dominates blood flow.
• Surface tension γ (N/m); surfactant lowers alveolar collapse pressure.
• Terminal velocity v_t = 2r²(ρ − σ)g/(9η); Stokes' drag F = 6πηrv (low Re).