Chapter 4: Fluids

Why does a 100,000-ton steel ship float, while a tiny iron pebble sinks the moment you toss it in a pond?

It’s not about weight — the ship weighs millions of times more than the pebble. It’s about how the mass is distributed through space. The ship is mostly hollow, so its average mass-per-volume is low. The pebble is solid iron, so its mass-per-volume is high.

That mass-per-volume property is density, and it’s the foundation of every fluids problem on the MCAT — floating, sinking, pressure, buoyancy, even why ice floats on water (a fact essential to aquatic life).

Density

Density tells you how much mass is packed into a given amount of space.

The density of water is the single most important reference value in fluid mechanics — memorize it and you’re ready for almost any density problem.

Unit Conversions

The two density units you’ll encounter on the MCAT are kg/m³ and g/cm³. Convert between them by multiplying or dividing by 1000:

• 1 g/cm³ = 1000 kg/m³.
• g/cm³ → kg/m³: multiply by 1000.
• kg/m³ → g/cm³: divide by 1000.

Why 1000? There are 1000 g per 1 kg, and 1,000,000 cm³ per 1 m³ (100 cm per side, cubed). When you divide those out, the net factor is 1000. Don’t memorize the derivation — just remember the 1000.

Specific Gravity

Specific gravity (SG) compares an object’s density to the density of water. It’s a dimensionless ratio — no units.

Specific gravity gives you an instant float-or-sink prediction:

• SG < 1 → less dense than water → floats.
• SG = 1 → same density as water → neutrally buoyant (suspended at any depth).
• SG > 1 → denser than water → sinks.

Why Ice Floats

Ice has a density of 917 kg/m³ (SG = 0.917), making it less dense than liquid water. This is unusual — most solids are denser than their liquid form (think of how candle wax sinks when you drop a chunk into melted wax).

Water is weird because of hydrogen bonding. When water freezes, the H-bonds lock molecules into an open hexagonal lattice with more empty space than the disordered liquid arrangement. So 1 mL of liquid water becomes more than 1 mL of ice (about 9% bigger), and the ice floats.

This anomaly is critical for life on Earth: ice floats on top of lakes and oceans, insulating the liquid water below and allowing aquatic life to survive winter. If ice were denser than water, lakes would freeze from the bottom up, killing fish and disrupting global climate.

Applying Density to the MCAT

Most MCAT density problems come in one of three flavors:

1. Calculate density from mass and volume (or rearrange to find mass or volume).
2. Predict floating or sinking from specific gravity.
3. Compare densities of layered liquids — the densest fluid always settles to the bottom.

When a passage shows you layered liquids, denser ones are always lower. Oil floats on water (oil’s SG ≈ 0.9). Mercury sits below water (SG = 13.6 — way denser). A common test setup: oil/water/mercury layers in a single tube.

Worked Example

A statue has a mass of 5400 g and a volume of 600 cm³. What’s its density? Will it float in water? In mercury?

• $\rho = m/V = 5400/600 = 9.0$ g/cm³.
• SG = 9.0 → much denser than water → sinks in water (it’s likely brass or copper).
• Compare to mercury (SG = 13.6) → the statue is less dense than mercury → floats in mercury.

That last result feels strange but it’s real — gold (SG ≈ 19) sinks in mercury, but most metals (steel, copper, brass) actually float on it.

Lying on a bed of nails — somehow — doesn’t hurt. Hundreds of nail tips share your weight, so each one bears only a tiny fraction. Now stand on a single nail. Same total weight, all concentrated on one tiny area. Excruciating.

The number that captures this difference is pressure: force divided by the area over which the force acts. Same force can be gentle (spread over big area, low pressure) or devastating (focused on a tiny point, high pressure). Snowshoes use this principle to keep you on top of snow; stiletto heels use it to leave dents in hardwood floors. Both are about pressure, not weight.

Defining Pressure

Pressure is a scalar, not a vector. At any point in a fluid, pressure pushes equally in every direction — up, down, sideways, all the same. This is fundamentally different from force, which has a specific direction. A force pushes; pressure squeezes.

Pressure Units

The MCAT uses four pressure units interchangeably. You have to be fluent in all of them.

Key relationships:

• 1 atm = 101,325 Pa ≈ $10^5$ Pa (use the approximation on the MCAT — much faster).
• 1 atm = 760 mmHg = 760 torr (mmHg and torr are exactly the same thing, different name).
• 1 atm ≈ 14.7 psi (rarely tested, but might appear in a passage).

Atmospheric Pressure

The atmosphere is a column of air about 100 km tall. That air has weight, and it presses down on everything below it. At sea level, that pressure is 1 atm = 101,325 Pa.

Atmospheric pressure decreases with altitude because there’s less air above you the higher you go:

• Sea level: 1 atm.
• Denver (~1.6 km): 0.83 atm.
• Top of Mt. Everest (~8.8 km): ~0.33 atm — about a third of sea-level pressure.
• Cruising altitude of a commercial plane (~10 km): ~0.25 atm (which is why cabins are pressurized — you’d black out otherwise).

This is why hikers feel short of breath at high altitudes: each lungful contains ~⅓ as many oxygen molecules at the top of Everest as at sea level.

Gauge Pressure vs. Absolute Pressure

This distinction trips up many students. There are two ways to report pressure:

• Absolute pressure is the total pressure at a point, including the atmospheric pressure pushing on everything. Always positive.
• Gauge pressure is the pressure above (or below) atmospheric pressure. A flat tire reads 0 on a tire gauge, but the air inside is still at 1 atm — gauge zero just means “same as atmospheric.”

Blood pressure readings ($\frac{120}{80}$ mmHg) are gauge pressures. They tell you how much the blood pressure exceeds atmospheric, not the absolute pressure in your arteries (which is actually 120 + 760 = 880 mmHg during systole).

Pressure in Fluids vs. Solids

In a solid, force can act in a specific direction over a specific area — a nail pressing into wood, a hammer striking a chisel. In a fluid, pressure acts equally in every direction at any given point.

That’s a direct consequence of fluids being unable to resist shear stress: if the pressure in a fluid were higher in one direction than another, the fluid would simply flow until the imbalance disappeared. So at every microscopic point, the fluid is squeezed equally from all sides.

Worked Example

A 70 kg person stands on one foot. The bottom of their shoe has area 0.02 m². What pressure does their foot exert on the floor? ($g = 10$ m/s².)

• Force = $mg = 700$ N.
• Pressure = $F/A = 700/0.02 = 35{,}000$ Pa = 35 kPa.

That’s about a third of an atmosphere — substantial, but spread over enough area that the floor doesn’t notice. Now switch to a stiletto heel with area 0.0001 m² instead: pressure = $700/0.0001 = 7{,}000{,}000$ Pa = 7 MPa. That’s why stilettos make dents and regular shoes don’t.

Ever notice that your ears hurt when you swim down to the bottom of a pool, but not while standing in the shallow end? That pain is hydrostatic pressure — the weight of all the water sitting above you, pressing on your eardrums.

The deeper you go, the more water sits above you, the bigger the pressure. The relationship is dead simple: pressure increases linearly with depth, period. Same idea explains why dams are thicker at the bottom (more pressure to hold back), why deep-sea creatures have specialized anatomy (they live under hundreds of atmospheres of pressure), and why your blood pressure in the veins of your feet is higher than in your head when you’re standing up.

Hydrostatic Pressure

In a stationary fluid, pressure increases linearly with depth.

Three insights pop out of this equation:

1. Pressure depends only on depth, not on the shape or width of the container. A narrow tube and a giant lake at the same depth have the same pressure at the bottom. This sounds wrong (the lake has vastly more water above you!) — but the extra water also spreads over a much larger area at the bottom. Pressure is force per area, and the two cancel exactly.

2. Pressure is the same at all points at the same depth (within a single connected fluid). Move horizontally — pressure doesn’t change. Only vertical position matters.

3. Layered fluids: add each layer’s contribution. If you have oil floating on water, the pressure at the bottom isn’t $\rho_{water} g (h_{total})$. It’s $\rho_{oil} g h_{oil} + \rho_{water} g h_{water}$. Each layer above contributes independently.

Worked Example

A diver descends to 10 m in the ocean ($\rho_{seawater} = 1025$ kg/m³). What’s the absolute pressure at that depth?

• $P_{abs} = P_{atm} + \rho g h = 101{,}325 + (1025)(9.8)(10) = 101{,}325 + 100{,}450 \approx 201{,}775$ Pa.
• That’s about 2 atm.

So at just 10 m down, the diver experiences twice the surface pressure. At 20 m, ~3 atm. At 30 m, ~4 atm. Each 10 m of water adds another atmosphere — that’s the rule of thumb every scuba diver learns.

Pascal’s Law

Pascal’s law says that pressure applied to an enclosed fluid is transmitted equally and undiminished to every point in the fluid and to the walls of the container.

Plain English: push on one part of a confined fluid, and the entire fluid feels the same pressure increase.

Hydraulic Systems

Pascal’s law is the principle behind hydraulic lifts, hydraulic brakes, hydraulic presses, and even the hydraulic systems that move heavy excavator arms. A small force applied to a small piston produces a pressure increase that’s transmitted to a large piston, where the same pressure × bigger area = much bigger output force.

Worked Example: Hydraulic Lift

A mechanic pushes with 100 N on a piston of area 0.01 m². The car sits on a piston of area 1 m². What force lifts the car?

• $P_1 = F_1/A_1 = 100/0.01 = 10{,}000$ Pa.
• $F_2 = P \times A_2 = 10{,}000 \times 1 = 10{,}000$ N.

The mechanic’s 100 N becomes 10,000 N at the lifting piston — enough to easily lift a small car. The area ratio (100×) is the force multiplier.

The catch (Pascal’s law isn’t free): if the small piston gets pushed down 1 meter, the large piston only rises 0.01 m (1 cm). Same volume of fluid moves, but spread over a much larger piston. Work done is identical on both sides: $100 \times 1 = 10{,}000 \times 0.01 = 100$ J. Conservation of energy strikes again — there’s no free lunch in physics.

Step into a swimming pool and you immediately feel lighter. Try to push a beach ball underwater and it pushes back with surprising force.

Both experiences are buoyancy — an upward force that every fluid exerts on every object that’s submerged (or partly submerged) in it. The amount of upward push depends on how much fluid the object displaces, not on the object itself.

Archimedes worked this out 2200 years ago — supposedly while taking a bath, then running naked through Syracuse shouting “Eureka!” The legend may be embellished, but the physics he discovered is exactly what the MCAT still tests. Buoyancy is one of the most reliably tested topics in fluids.

Test your prediction below. Set the two densities, press Drop, and watch where the block settles. Compare the red weight arrow with the blue buoyant force arrow: the block stops sinking exactly when the two arrows match, and if the buoyant force maxes out (fully submerged) before it can match the weight, the block sinks.

Archimedes’ Principle

The critical detail: buoyant force depends on the fluid’s density (not the object’s), and on the volume displaced (which equals the volume of the object that’s actually submerged).

Floating vs. Sinking

Whether an object floats or sinks depends on how its density compares to the fluid:

• $\rho_{object} < \rho_{fluid}$ → floats (buoyant force can match the object’s weight before it’s fully submerged).
• $\rho_{object} = \rho_{fluid}$ → neutrally buoyant (hovers wherever you put it).
• $\rho_{object} > \rho_{fluid}$ → sinks (even fully submerged, buoyancy can’t match the weight).

For a floating object, the buoyant force exactly equals the object’s weight:

$F_b = mg \;\Rightarrow\; \rho_{fluid} \cdot V_{submerged} \cdot g = \rho_{object} \cdot V_{total} \cdot g$

This rearranges to a powerful and famous result:

Apparent Weight

When you weigh an object while it’s submerged in a fluid, the scale reads less than the object’s true weight, because the buoyant force pushes up and partially supports it.

Why Steel Ships Float

A solid steel block ($\rho \approx 7800$ kg/m³) sinks the moment it touches water. So how does a steel ship float?

The trick is shape. A ship isn’t a solid block — it’s a hollow shell that encloses an enormous volume of air. The relevant quantity is the average density of the entire ship (steel hull + air-filled interior), and that average density is much less than water.

Same way: $V_{displaced}$ for a ship is huge — the hull pushes aside a lot of water. The weight of that displaced water (the buoyant force) exceeds the ship’s total weight. As long as the average density of the ship-plus-its-internal-air is less than water, it floats. Crush the same ship into a solid steel cube and it sinks.

Buoyancy in Different Fluids

An object that sinks in one fluid might float in another. A swimmer who barely floats in a freshwater pool ($\rho = 1000$ kg/m³) bobs effortlessly in the Dead Sea ($\rho \approx 1240$ kg/m³) because the saltier, denser water provides more buoyant force per unit volume displaced.

The same logic applies in gases: a helium balloon rises in air because helium ($\rho \approx 0.16$ kg/m³) is much less dense than air ($\rho \approx 1.2$ kg/m³). The buoyant force from the displaced air exceeds the weight of the helium plus the balloon material. Same physics — just less dramatic, because gas densities are tiny.

Common MCAT Buoyancy Traps

• “The heavier object experiences more buoyancy.” Wrong. Buoyant force depends on volume displaced and fluid density, not on the object’s weight or mass.
• “The buoyant force on a sinking object is zero.” Wrong. A sinking object still experiences a buoyant force — it just isn’t big enough to prevent sinking. $F_b < mg$ for any sinking object, but $F_b > 0$.
• Confusing $\rho_{fluid}$ with $\rho_{object}$. The buoyant force formula uses the fluid’s density. The object’s density only matters for the object’s weight.

Worked Example

A 5 kg object with volume 0.004 m³ is fully submerged in water. Find the buoyant force and the apparent weight. ($g = 9.8$ m/s².)

• $F_b = \rho_{water} \cdot V \cdot g = 1000 \times 0.004 \times 9.8 = 39.2$ N.
• Actual weight: $mg = 5 \times 9.8 = 49$ N.
• Apparent weight: $49 - 39.2 = 9.8$ N.

Object density: $\rho = m/V = 5/0.004 = 1250$ kg/m³ > 1000 kg/m³ → it does sink. But underwater it feels only $\frac{1}{5}$ as heavy as it does on land.

Pull a rubber band: it stretches a lot. Hang the same weight from a steel wire: it stretches almost imperceptibly. Both deform — the question is how much and whether they snap back.

Every solid material deforms under force. Some return to their original shape (elastic), others stay deformed (plastic), and pushing too hard breaks them entirely. This section gives you the language — stress, strain, Young’s modulus, elastic limit — to describe all of that quantitatively.

The MCAT doesn’t lean heavily on this topic, but it’s on the AAMC content list and shows up in passages about bones, tendons, materials engineering, and biomechanics. It also marks the natural boundary between solid mechanics and the fluid mechanics in the rest of this chapter.

Stress: Force per Area

Stress measures how much force is applied per unit cross-sectional area. The formula looks identical to pressure — but stress applies specifically to solids being deformed.

There are three kinds of stress:

• Tensile stress — pulling/stretching forces (a rubber band being stretched, a tendon being loaded).
• Compressive stress — squeezing/crushing forces (stepping on a soda can, your femur supporting your weight).
• Shear stress — forces parallel to a surface in opposite directions (sliding the top of a deck of cards sideways while holding the bottom).

Strain: The Response to Stress

Strain measures how much the material actually deforms in response to stress. It’s a dimensionless ratio.

Young’s Modulus: Stiffness

Young’s modulus (also called the elastic modulus) measures stiffness — how much stress is needed to produce a given strain.

The Elastic Limit

Below the elastic limit (a.k.a. yield point), a material returns to its original shape when force is removed — like a stretched rubber band snapping back. Above the elastic limit, the material undergoes plastic (permanent) deformation — it stays bent or stretched even after you let go. Push even further and you reach the fracture point, where the material breaks.

You’ve seen this every time you’ve bent a paperclip a little (it springs back) versus bent it a lot (it stays bent) versus bent it back-and-forth too many times (it snaps).

Why This Matters for Fluids

This section sits inside the fluids chapter because it marks the boundary between solid and fluid behavior. The defining property of a fluid is that it cannot resist shear stress — apply any shearing force and the fluid simply flows in response. Solids, in contrast, can hold a fixed deformation against shear (within their elastic range).

That single distinction — solids resist shear, fluids don’t — is the reason fluid mechanics exists as its own branch of physics. Once you can’t assume things hold their shape, all the equations change. Everything in the rest of this chapter (Bernoulli, continuity, viscosity, Pascal’s law) flows (literally) from this property.

Worked Example

A bone of cross-sectional area $4 \times 10^{-4}$ m² is loaded with a 2000 N compressive force (about 200 kg pushing down). If the bone is 0.4 m long and Young’s modulus for bone is $1.8 \times 10^{10}$ Pa, how much does it shorten?

• Stress: $\sigma = F/A = 2000 / (4 \times 10^{-4}) = 5 \times 10^6$ Pa.
• Strain: $\varepsilon = \sigma/E = (5 \times 10^6)/(1.8 \times 10^{10}) \approx 2.78 \times 10^{-4}$.
• $\Delta L = \varepsilon \cdot L_0 = 2.78 \times 10^{-4} \times 0.4 \approx 1.1 \times 10^{-4}$ m = 0.11 mm.

So a 200 kg load shortens the bone by only about a tenth of a millimeter. Bone is impressively stiff — which is why your skeleton can support you without obviously deforming.

Try drinking honey through a straw. Now try the same straw with water. The water flows easily; the honey barely moves. Same straw, same suction — wildly different flow.

The difference is viscosity: a fluid’s internal resistance to flow. Viscosity is to fluids what friction is to solids — and it has huge implications for everything from blood circulation, to IV drip rates, to why your motor oil gets a “winter” rating.

This section also introduces Poiseuille’s law, an equation with a single, dramatic feature: flow depends on the fourth power of the tube radius. That $r^4$ explains atherosclerosis, why thin needles drip slowly, and why a stuffy nose feels so much worse than the slight swelling actually warrants.

Viscosity: Fluid Friction

Viscosity ($\eta$, Greek “eta”) measures a fluid’s resistance to flowing. High-viscosity fluids (honey, syrup, motor oil) flow slowly. Low-viscosity fluids (water, alcohol, air) flow easily.

The SI unit of viscosity is the pascal-second (Pa·s). Blood is about 3–4 × 10⁻³ Pa·s — roughly 3 to 4 times more viscous than water (the cells in suspension cause the extra resistance).

Viscosity generally decreases as temperature increases. Warm honey pours faster than cold honey. This is also why blood flows more easily at body temperature than at room temperature.

Laminar vs. Turbulent Flow

Fluid flow comes in two fundamentally different patterns:

• Laminar flow is smooth, orderly, layered flow. Each layer slides past its neighbor without mixing. In a pipe, the velocity profile is parabolic — fastest at the center, dropping to zero right at the walls (where friction with the pipe halts the fluid).
• Turbulent flow is chaotic — eddies, vortices, swirling, mixing. It happens at high speeds, in wide pipes, and in low-viscosity fluids.

Poiseuille’s Law

This is the single most important equation in MCAT fluid dynamics. It describes the volume flow rate of a viscous fluid through a cylindrical tube — a blood vessel, an IV line, a pipe.

The $r^4$ Dependence: THE Key Insight

Of all the variables, the radius matters far more than anything else because it appears as the fourth power. This is the single most important takeaway from the entire fluid-dynamics section.

Putting numbers to it:

Effects of the Other Variables

While radius dominates, the other variables still matter:

• Pressure difference ($\Delta P$): Flow is directly proportional. Double the pressure gradient → double the flow. This is why the heart pumps harder when arteries narrow — it has to overcome more resistance.
• Viscosity ($\eta$): Flow is inversely proportional. Thicker blood (dehydration, polycythemia) flows more slowly. Dehydration also raises clotting risk for the same reason — sluggish flow.
• Length ($L$): Flow is inversely proportional. A longer tube = more resistance. Rarely changes in blood vessels (they don’t get longer overnight), but matters for IV lines and catheters.

Resistance to Flow

Poiseuille’s law can be rewritten in a form that looks just like Ohm’s law for circuits ($V = IR$):

This analogy isn’t a coincidence. Many physical systems share the structure “driving force = (rate of flow) × resistance” — fluid in pipes, current in wires, heat in walls, even gas through membranes.

Worked Example

A blood vessel with radius $r$ has its radius cut in half by atherosclerotic plaque. By what factor must the heart raise blood pressure to maintain the original flow rate?

• Original flow: $Q_0 \propto r^4$.
• New flow with half-radius: $(r/2)^4 = r^4/16$ — only $\frac{1}{16}$ of original.
• To restore flow to $Q_0$, the pressure difference must rise by a factor of 16.

That’s why severe atherosclerosis is so dangerous: not only does it slow flow drastically, it forces the heart to work enormously harder, leading to hypertension, heart strain, and eventually failure.

Put your thumb partly over the end of a garden hose. The water that was streaming out gently now sprays out fast. You didn’t change the faucet, the pressure, or the water supply — you just made the opening smaller. Yet the water is way faster.

Why? The same amount of water still has to come out per second. If it has less area to go through, it has to move faster to get the same volume out per unit time.

That intuition is the continuity equation, and it underlies everything from garden hoses, to plumbing, to why blood slows down in your capillaries (which is critical for letting oxygen exchange happen).

Conservation of Mass in Fluids

For an incompressible fluid (one whose density doesn’t change — like water or blood), whatever flows into a pipe has to flow out. Mass can’t accumulate inside a rigid tube or vanish into thin air. That conservation of mass leads directly to the continuity equation.

The key insight: when the cross-sectional area shrinks, velocity grows (and vice versa). Inversely proportional.

• Halve the area → double the velocity.
• Quarter the area → quadruple the velocity.

The Importance of Total Cross-Sectional Area

Here’s where most students trip up: the continuity equation uses the total cross-sectional area at a given level of the system, not the area of a single tube.

The circulatory system is the canonical example. The aorta is one big tube. It branches into arteries, then arterioles, then billions of capillaries. Each individual capillary is microscopic — but with billions of them in parallel, the total cross-sectional area of all the capillaries combined is roughly 600 times the area of the aorta.

Notice the pattern: blood speeds up again in the veins and vena cavae as the total cross-section shrinks on the return trip. The continuity equation works in both directions — area and velocity always trade.

When Continuity Does NOT Apply

The continuity equation assumes:

1. Incompressible fluid — density stays constant. Valid for liquids; not valid for gases at varying pressures. (For the MCAT, blood and water are always incompressible.)
2. Steady flow — flow rate doesn’t change with time. Real heartbeats are pulsatile, but MCAT problems typically assume steady-state.
3. No leaks — no fluid enters or leaves the system between the two points. If a pipe has a hole or a side branch, you have to account for it separately.

Worked Example

A garden hose has a diameter of 2 cm. You attach a nozzle that narrows the opening to 0.5 cm diameter. If water flows through the hose at 1 m/s, what’s the speed coming out of the nozzle?

• $A \propto d^2$, so the area ratio is $(2/0.5)^2 = 16$.
• Continuity: $A_1 v_1 = A_2 v_2 \Rightarrow v_2 = v_1 \times (A_1/A_2) = 1 \times 16 = 16$ m/s.

So the water comes out 16× faster than it was moving inside the hose — about 36 mph. That’s exactly why nozzles let you spray water across a yard.

Stand in a hot shower with the water running hard. Watch the curtain. Within a few seconds, it creeps inward and clings to your legs.

You might think the water is somehow pulling the curtain — but the water never touches the curtain. What’s actually happening: the air inside the shower is moving (rushing upward with the steam, swirling around). That moving air has lower pressure than the still air outside the curtain. The higher outside pressure pushes the curtain in.

That counterintuitive idea — fast-moving fluid has lower pressure than slow-moving fluid — is the punch of Bernoulli’s equation, one of the most elegant ideas in fluid dynamics. Once you internalize it, you can explain shower curtains, airplane lift, why a window slams shut when wind blows past, and how a doctor uses a venturi nebulizer to spray medication.

Bernoulli’s Equation: Energy Conservation for Fluids

Bernoulli’s equation is just conservation of energy applied to a flowing fluid. The same way a roller coaster trades PE for KE on the way down, a moving fluid trades among three forms of energy: pressure energy, kinetic energy, and gravitational potential energy.

Between any two points along a streamline:

$P_1 + \tfrac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \tfrac{1}{2}\rho v_2^2 + \rho g h_2$

The Big Takeaway: Speed Up → Pressure Down

The most-tested consequence of Bernoulli’s equation is dead simple:

When fluid velocity increases, pressure decreases. And vice versa.

Often this is the only thing you need to know for an MCAT question. The exam describes a situation where fluid speeds up and asks what happens to the pressure — the answer is essentially always “it decreases.”

Watch the tracer particles speed up as they squeeze through the throat, and keep an eye on the two standpipe gauges: P₁ sits over the wide section and P₂ over the constriction. Tighten the constriction or raise the flow rate and watch P₂ fall. The drop scales with $v^2$, so small changes in speed produce large changes in pressure.

Assumptions (When Bernoulli’s Applies)

Bernoulli’s holds for an ideal fluid:

1. Incompressible — constant density (true for liquids).
2. Non-viscous — no internal friction (an approximation; works well for water and air at moderate speeds).
3. Laminar flow — smooth, non-turbulent.
4. Steady flow — flow rate constant in time.

When viscosity matters (narrow blood vessels, thick fluids), Bernoulli’s alone isn’t enough — you need Poiseuille’s law (§4.6) to capture the viscous losses.

Special Cases

Case 1: Fluid at rest ($v_1 = v_2 = 0$). Bernoulli reduces to $P_1 + \rho g h_1 = P_2 + \rho g h_2$, or $\Delta P = \rho g \Delta h$ — exactly the hydrostatic pressure equation from §4.3. No coincidence: hydrostatics is just Bernoulli with no flow.

Case 2: Same height ($h_1 = h_2$). Bernoulli reduces to $P_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \tfrac{1}{2}\rho v_2^2$. Pure horizontal-flow case. This is the version that shows up in 90% of MCAT problems. Speed up → pressure drops; slow down → pressure rises.

Case 3: Open tank draining (Torricelli’s theorem). A tank with a hole at depth $h$ below the water surface drains at $v = \sqrt{2gh}$ — the same speed an object would have after falling from height $h$. Pure energy conservation: gravitational PE → KE.

Applications

• Airplane lift. Air moves faster over the curved upper surface of a wing than the flatter lower surface. Faster air → lower pressure on top → net upward force = lift. (Real aerodynamics is more complex — angle of attack and circulation matter too — but the Bernoulli explanation is the one the MCAT expects.)
• Shower curtain effect. Fast-moving air inside the shower has lower pressure than the still air outside → net inward force on the curtain.
• Chimney draft. Hot air rises fast in the chimney, creating low pressure at the base → fresh air gets sucked in from the room to feed the fire.
• Carburetors and atomizers. Fast air over a small tube of liquid creates low pressure that pulls the liquid up, where it gets sprayed into a fine mist (think nebulizers, perfume sprayers, paint sprayers).
• Wind blowing past a window. Fast outside air lowers outside pressure → indoor pressure pushes the window outward (or the curtains inward) — sometimes hard enough to slam doors shut.

Worked Example

Water flows through a horizontal pipe that narrows from 4 cm² to 1 cm². The pressure in the wide section is 200,000 Pa and the velocity is 1 m/s. Find the pressure in the narrow section. ($\rho_{water} = 1000$ kg/m³.)

• Step 1 — Continuity. $A_1 v_1 = A_2 v_2 \Rightarrow v_2 = (4/1)(1) = 4$ m/s.
• Step 2 — Bernoulli (same height, so $h$ terms cancel). $P_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \tfrac{1}{2}\rho v_2^2$.
• $200{,}000 + \tfrac{1}{2}(1000)(1)^2 = P_2 + \tfrac{1}{2}(1000)(16)$.
• $200{,}500 = P_2 + 8000 \Rightarrow P_2 = 192{,}500$ Pa.

Pressure dropped by 7500 Pa as the fluid sped up — exactly as Bernoulli predicts.

Hold two sheets of paper a couple centimeters apart in front of your face. Now blow hard between them. Most people expect the sheets to fly apart — but they actually pull together.

That’s a parlor trick, but the physics is real and important. The fast-moving air between the sheets has lower pressure than the still air outside the sheets. Higher outside pressure pushes the sheets inward.

This is the Venturi effect — the predictable consequence of fluid speeding up through a constriction (and, by Bernoulli, dropping in pressure). Once you see this pattern, you’ll see it everywhere: airplane lift, perfume sprayers, asthma attacks, atherosclerosis, the Pitot tube on the wing of every commercial aircraft.

The Venturi Effect

The Venturi effect is what happens when fluid passes through a constriction (a narrow section) in a pipe. It’s a direct combination of the continuity equation (§4.7) and Bernoulli’s equation (§4.8).

The logic chain:

1. Continuity: the same volume of fluid must pass through the narrow section per second as the wide section. So fluid velocity increases in the constriction.
2. Bernoulli: as velocity increases, pressure decreases.
3. Result: the narrow section has higher velocity and lower pressure than the wide sections on either side.

The Venturi Tube

A Venturi tube is a pipe with a deliberately narrowed middle section (called the throat). Pressure gauges at the wide and narrow sections show a measurable pressure difference — lower in the throat where the fluid is moving fastest.

Practical uses of the Venturi tube:

• Measuring flow rate. Measure the pressure difference between the wide and narrow sections, plug into Bernoulli, and you get the flow rate. Industrial flow meters do exactly this.
• Aspirators and atomizers. The low-pressure zone in the throat draws in a secondary fluid through a side tube. This is how perfume sprayers, paint sprayers, and old-school carburetors all work.
• Medical nebulizers. Use the Venturi effect to draw liquid medication into a fast airstream and break it into a fine mist for inhalation.

The Pitot Tube

A pitot tube measures fluid velocity (most commonly airspeed on an aircraft). It works by comparing two pressure measurements:

1. Total (stagnation) pressure — measured by a tube pointing directly into the oncoming flow. The fluid is brought to a halt at the tube opening, converting all its KE into pressure: $P_{total} = P_{static} + \tfrac{1}{2}\rho v^2$.
2. Static pressure — measured by a port flush with the surface, perpendicular to the flow.

The difference between total and static pressure is the dynamic pressure ($\tfrac{1}{2}\rho v^2$), and from that you back out velocity:

$v = \sqrt{\dfrac{2(P_{total} - P_{static})}{\rho}}$

Venturi Effect in the Body

The Venturi effect is responsible for several biological phenomena:

Airway narrowing in asthma. When bronchi constrict, air velocity in the narrowed airways increases (continuity). The resulting lower pressure can cause further collapse of the flexible airway walls, worsening the obstruction. This is a positive-feedback loop: narrowing → faster air → lower internal pressure → more collapse → more narrowing.

Atherosclerotic vessels. Blood flowing past a partial blockage speeds up. The Venturi-effect low-pressure zone at the narrowing can pull the vessel walls inward, worsening the obstruction over time.

Worked Example

Air flows through a horizontal Venturi tube. The wide section has area 20 cm² and the throat has area 5 cm². If air moves at 4 m/s in the wide section, what’s the air velocity in the throat?

• Continuity: $A_1 v_1 = A_2 v_2 \Rightarrow v_2 = (20/5)(4) = 16$ m/s.

Air moves 4× faster through the throat (because area dropped by 4×). And by Bernoulli, the pressure in the throat is lower than in the wide section — that’s the suction effect that makes Venturi devices useful.

A water strider walks on a pond without sinking. A steel needle, carefully placed flat, floats on water despite being denser than water. A paper towel sucks up a spill, pulling water upward against gravity.

None of this can be explained by buoyancy or pressure. They all rely on surface tension and capillary action — forces that come from the molecular interactions at fluid surfaces and interfaces. These effects matter especially at small scales: thin tubes, narrow gaps, fine fibers. They’re also crucial in biology — surfactant in your lungs prevents the alveoli from collapsing every time you exhale.

Surface Tension

Molecules in the interior of a liquid are pulled equally in all directions by neighboring molecules. The forces cancel, and the molecule doesn’t experience any net pull.

Molecules at the surface are different. They have neighbors below them and to the sides — but no neighbors above. So they get pulled inward and sideways but not outward. That uneven pull creates surface tension, which makes the surface behave like a stretched elastic membrane.

Surface tension ($\gamma$) is measured in N/m (force per unit length of surface). Water has a relatively high surface tension (~0.073 N/m at 20°C) because of strong hydrogen bonds between water molecules.

Things that change surface tension:

• Temperature. Higher temperature → lower surface tension (faster molecules → weaker effective attractions).
• Surfactants. Substances like soap dramatically reduce surface tension by inserting themselves into the surface and breaking up the hydrogen-bond network. This is why soapy water “wets” things better — it spreads and penetrates fabric far more easily than plain water.
• Intermolecular forces. Stronger intermolecular forces → higher surface tension. Water (H-bonds) > ethanol (weaker H-bonds) > nonpolar liquids.

Adhesion vs. Cohesion

Two types of intermolecular attraction matter for how fluids interact with surfaces:

• Cohesion = attraction between molecules of the same substance (water-to-water). Cohesion is what creates surface tension.
• Adhesion = attraction between molecules of different substances (water-to-glass). Adhesion is what makes water cling to the inside of a glass.

The Meniscus

Look at water in a thin glass tube — the surface curves upward at the edges. That curved surface is the meniscus, and its shape tells you which force is winning:

• Concave meniscus (curves up at the edges) = adhesion > cohesion. Fluid is more attracted to the container walls than to itself. Example: water in glass.
• Convex meniscus (curves down at the edges) = cohesion > adhesion. Fluid is more attracted to itself than to the walls. Example: mercury in glass.

Capillary Action

Capillary action is the ability of a liquid to flow into narrow spaces against gravity, driven by adhesion and surface tension. Dip a thin glass tube into water:

1. Water molecules adhere to the glass walls and start climbing up.
2. Surface tension pulls the rest of the water surface upward along with them.
3. The water column rises until the upward adhesive/surface-tension forces balance the downward weight of the water column.

The narrower the tube, the higher the water rises. Why? A narrower tube has a larger surface-area-to-volume ratio, so the surface forces (adhesion, surface tension) win out over gravity (which pulls on the volume).

Mercury: The Opposite Case

Mercury in a glass tube does the opposite of water. Mercury atoms are strongly attracted to each other (high cohesion — that’s why mercury forms beads) but only weakly attracted to glass (low adhesion). So mercury in glass:

• Forms a convex meniscus (curves down at the edges).
• Shows capillary depression — mercury inside a thin tube sits lower than mercury outside.
• Doesn’t wet glass surfaces — it beads up instead of spreading.

This is why mercury thermometers (when they were common) had cleanly readable, non-clinging mercury columns: the mercury didn’t smear on the glass, just rolled cleanly up and down.

Every equation in this chapter was derived by physicists studying water in pipes. But the MCAT’s favorite application of fluid mechanics is not plumbing — it’s the human circulatory system. Blood vessels are pipes. The heart is a pump. Blood is a viscous fluid. The physics translates directly, and the exam exploits that connection relentlessly.

This section pulls together everything from the previous ten sections — continuity, Poiseuille, Bernoulli, hydrostatic pressure — and applies it all to the body. It also explains some surprising clinical phenomena: why aneurysms keep growing, why a 50% blockage drops flow by 94%, why your blood pressure differs by 100+ mmHg between your head and your feet, and why standing up too fast makes you dizzy.

The Circulatory System as a Fluid Network

The cardiovascular system is a closed-loop fluid circuit. The heart generates a pressure difference ($\Delta P$). Blood flows through a branching network of vessels. The flow obeys exactly the same equations you’ve been working with throughout this chapter:

Continuity and Capillary Blood Flow

Blood leaves the heart through the aorta at ~40 cm/s. By the time it reaches the capillaries, it has slowed to ~0.03 cm/s — about a thousand times slower. Why?

Continuity equation + total cross-sectional area. The aorta is a single tube (cross-section ~4 cm²). It branches into thousands of arteries, millions of arterioles, and billions of capillaries. The combined cross-sectional area of all the capillaries is ~3000–6000 cm² — over a thousand times the area of the aorta.

$A_1 v_1 = A_2 v_2 \Rightarrow$ if total $A$ rises by 1000×, $v$ drops by 1000×.

Poiseuille’s Law and Atherosclerosis

Atherosclerosis (plaque buildup in arterial walls) narrows the vessel radius. Poiseuille’s law reveals why even moderate narrowing has catastrophic effects on flow.

Because flow scales with $r^4$:

• 25% radius reduction → flow = $(0.75)^4 \approx$ 32% of normal.
• 50% radius reduction → flow = $(0.5)^4 =$ 6.25% of normal.
• 75% radius reduction → flow = $(0.25)^4 \approx$ 0.4% of normal.

A “50% blocked artery” sounds moderate. The actual flow is 6% of normal — enough to cause ischemia (tissue death from insufficient blood supply) and trigger a heart attack or stroke.

The body compensates for arterial narrowing in two ways:

1. Increase $\Delta P$ — the heart pumps harder (raising blood pressure). This works in the short term but overworks the heart and eventually causes heart failure.
2. Vasodilate other vessels — nitroglycerin and similar drugs widen coronary arteries. Because of the $r^4$ dependence, even a small radius increase restores a lot of flow. (5% wider radius → ~22% more flow. 25% wider → 144% more flow.)

Bernoulli’s Equation and Aneurysms

An aneurysm is a localized balloon-like bulge in an artery wall. What does the physics say about flow through one?

Apply continuity, then Bernoulli:

1. The aneurysm has a larger cross-sectional area than the normal vessel.
2. By continuity, blood slows down in the bulge.
3. By Bernoulli, slower blood = higher static pressure in the bulge.
4. That extra pressure pushes outward on the already-weakened wall, causing the aneurysm to grow larger.

This is a self-reinforcing positive-feedback loop: bigger aneurysm → slower blood → more pressure on the wall → bigger aneurysm. Eventually it can rupture, which in the aorta is often fatal. This is why doctors monitor known aneurysms carefully and intervene surgically before they reach a critical size.

Stenosis: The Opposite of Aneurysm

A stenosis is a narrowing of a vessel, usually from plaque. Same physics, opposite direction:

1. Smaller cross-section → blood speeds up (continuity).
2. Faster blood → lower lateral pressure (Bernoulli).
3. Low pressure can pull flexible vessel walls inward, worsening the narrowing.

So stenosis also has a positive-feedback loop, just running the other way. And it combines with Poiseuille’s $r^4$ effect to drastically reduce downstream flow.

Blood Pressure Measurement

When blood pressure is measured with a sphygmomanometer (the inflatable arm cuff), the underlying physics is fluid flow:

1. The cuff inflates above systolic pressure, completely compressing the brachial artery → flow stops.
2. The cuff slowly deflates. When the cuff pressure drops just below systolic pressure, blood squirts through the partially compressed artery — but only briefly during each systole.
3. That intermittent, turbulent flow produces audible Korotkoff sounds (heard through a stethoscope).
4. As cuff pressure drops below diastolic pressure, the artery is fully open all the time, flow becomes laminar, and the sounds disappear.

So systolic = the cuff pressure where you first hear Korotkoff sounds; diastolic = the pressure where the sounds vanish. The whole technique works because turbulent flow is audible and laminar flow is silent.

Hydrostatic Pressure and Posture

Blood pressure isn’t constant throughout the body — gravity pulls blood downward, so there’s a hydrostatic column ($P = \rho g h$) on top of whatever pressure the heart provides.

Standing upright, blood pressure in your feet is higher than at your heart by $\rho g h$, where $h$ is the vertical distance heart-to-feet (~1.3 m). With $\rho_{blood} \approx 1060$ kg/m³:

$\Delta P = (1060)(9.8)(1.3) \approx 13{,}500 \text{ Pa} \approx 100 \text{ mmHg}$

So if heart-level BP is $\frac{120}{80}$, foot-level BP is roughly $\frac{220}{180}$. Brain-level BP (~0.4 m above the heart) is roughly $\frac{90}{50}$.