3.2 Fluid Dynamics: Viscosity, Continuity & Bernoulli's Equation

Fluid statics describes fluids at rest; fluid dynamics describes fluids in motion — blood surging through arteries, air rushing through airways, water moving through a narrowing pipe. The same handful of relationships explain all of it, and Content Category 4B expects you to move fluently between the equations and their biological applications.

Viscosity and Poiseuille Flow

Viscosity (η, eta) is a fluid's internal resistance to flow — essentially, friction between fluid layers sliding past one another. Honey is highly viscous; water is much less so. Blood is a non-Newtonian fluid in detail (viscosity depends on shear rate and hematocrit), but MCAT problems usually treat it as a viscous fluid with an effective viscosity several times that of water.

For smooth, layered (laminar) flow of a viscous fluid through a cylindrical tube, Poiseuille's Law gives the volumetric flow rate:

Q = (π × ΔP × r⁴) / (8ηL)

where Q is flow rate, ΔP is the pressure difference driving the flow, r is the tube's radius, η is viscosity, and L is the tube's length. The single most important feature of this equation for the MCAT is that flow rate scales with the fourth power of the radius. Doubling a vessel's radius increases flow sixteenfold (2⁴ = 16) at a constant pressure gradient; halving the radius drops flow to 1/16 of its original value. Equivalently, resistance R = 8ηL/(πr⁴) scales as 1/r⁴. This extreme sensitivity to radius is why small anatomical or pathological changes in vessel diameter — a partially clogged artery, a constricted airway, arteriolar smooth-muscle tone — produce dramatic changes in flow, a relationship explored further in the next section on the circulatory system.

Increasing viscosity (for example, polycythemia raising hematocrit) or increasing path length raises resistance and lowers Q for a given ΔP. Raising the driving pressure difference increases flow linearly — the heart can compensate for high resistance only by generating a larger pressure gradient.

The Continuity Equation and Turbulence

For an incompressible fluid (density doesn't change) flowing through a tube of varying cross-sectional area, conservation of mass requires that volume flow rate stay constant throughout the tube. This gives the continuity equation:

A1v1 = A2v2 (or Q = Av = constant)

where A is cross-sectional area and v is fluid speed. In words: fluid speeds up where the tube narrows and slows down where the tube widens. Since a cylindrical tube's cross-sectional area is A = πr², a small change in radius produces a much larger change in area (and therefore velocity) — cutting the radius in half cuts the area to one-fourth, which quadruples the fluid's speed at that point.

Important subtlety for biology: when many vessels run in parallel (the entire capillary bed), the relevant A is the sum of the individual cross-sections. Capillaries are each tiny, but their combined area is huge, so capillary blood velocity is slow — ideal for exchange.

Smooth, orderly laminar flow (fluid moving in parallel layers) can break down into chaotic, swirling turbulent flow once velocity is high enough. Qualitatively, turbulence is favored by high fluid velocity, large vessel diameter, low viscosity, and high fluid density — the same factors that raise the Reynolds number (Re ∝ ρvd/η). The MCAT rarely demands a numerical Re calculation; it tests the conceptual direction. Turbulent flow dissipates more energy than laminar flow (as heat and sound) and, in the body, can be heard as a bruit or heart murmur: turbulence downstream of a narrowed heart valve or partially blocked artery is a hallmark finding on physical exam.

Surface Tension

Surface tension arises because molecules at a liquid's surface feel a net inward cohesive pull from their neighbors (with no matching pull from the air above), causing the surface to behave like a stretched elastic membrane. Surface tension explains why water beads up, why insects can walk on water, and why a needle can float on a water surface despite being denser than water.

In the lungs, alveoli (tiny air sacs) are lined with a thin fluid film, and the surface tension of that fluid would cause alveoli to collapse if left unopposed. Surfactant, a phospholipid mixture secreted by type II alveolar cells, reduces this surface tension and keeps small alveoli from collapsing into larger neighboring ones — a direct application of the Law of Laplace (collapsing pressure P = 2T/r for a sphere), where higher surface tension T or a smaller radius r means a higher collapsing pressure. Premature infants who lack adequate surfactant develop respiratory distress syndrome precisely because high alveolar surface tension drives collapse, especially of the smallest alveoli. On the MCAT, connect surface tension → Laplace → surfactant → alveolar stability whenever a respiratory physiology passage appears in the Chem/Phys section.

Bernoulli's Equation

Bernoulli's equation is a statement of energy conservation for an ideal (non-viscous, incompressible) fluid flowing steadily along a streamline:

P + ½ρv² + ρgh = constant

The three terms represent static pressure energy density, kinetic energy per unit volume, and gravitational potential energy per unit volume. They trade off: if one rises along the streamline, another must fall so the sum stays constant. For flow through a horizontal tube (h constant), the equation simplifies to:

P1 + ½ρv1² = P2 + ½ρv2²

This means that as fluid speeds up, its pressure must drop to keep total mechanical energy density constant — the key insight tested repeatedly across this content category. Real blood is viscous, so Bernoulli is an approximation, but it still correctly predicts the direction of the pressure-velocity trade-off at a stenosis. Do not confuse Bernoulli (ideal, inviscid energy balance) with Poiseuille (viscous flow rate through a tube): passages sometimes test whether you pick the right model for the question asked.

The Venturi Effect and Pitot Tube

The Venturi effect is the direct combination of the continuity equation and Bernoulli's equation: at a constriction in a tube, fluid speeds up (continuity) and its pressure drops (Bernoulli). This principle drives carburetors and aspirators, and — physiologically — explains the local pressure drop across a stenotic (narrowed) blood vessel or heart valve. Low pressure at a severe constriction can even promote vessel wall collapse or contribute to further turbulence just past the narrowing.

A pitot tube works in the opposite direction, measuring fluid velocity by comparing the static pressure of moving fluid to the "stagnation pressure" measured at a point where the flow is brought to rest — exploiting the same pressure-velocity trade-off described by Bernoulli's equation. Pitot tubes measure airspeed in aircraft and can appear in MCAT passages about flow-measurement instruments. Mentally: high speed ↔ low static pressure; bringing flow to rest converts kinetic energy density back into pressure.

Worked Example: Continuity and Bernoulli's Equation Together

Blood flows through a healthy segment of an artery with radius 4 mm at a speed of 1 m/s. The vessel then narrows, due to atherosclerotic plaque, to a radius of 2 mm. Approximate the blood's density as ρ ≈ 1,000 kg/m³ for easy mental math, and assume the vessel is horizontal.

Step 1 — Find the new velocity using the continuity equation. Since area A = πr², halving the radius quarters the area:

A1v1 = A2v2 → v2 = v1 × (r1/r2)² = 1 m/s × (4/2)² = 1 m/s × 4 = 4 m/s

Step 2 — Find the pressure drop using Bernoulli's equation. With height constant:

P1 + ½ρv1² = P2 + ½ρv2²

ΔP = P1 − P2 = ½ρ(v2² − v1²) = ½(1,000)(4² − 1²) = 500 × (16 − 1) = 500 × 15 = 7,500 Pa

So the pressure at the narrowed segment drops by about 7,500 Pa (roughly 56 mmHg, since 1 mmHg ≈ 133 Pa) relative to the wider segment upstream, even though the fluid is squeezing through a plaque-narrowed vessel. This counterintuitive result — faster flow, lower local pressure — is exactly what the Venturi effect predicts, and it is part of the biophysical explanation for why severely stenotic vessels can be prone to further turbulence or collapse just past the constriction.

Step 3 — Radius sensitivity check with Poiseuille (optional contrast). If the entire vessel length were narrowed to half radius rather than a short stenosis, Q would fall by a factor of 16 for the same ΔP. A short stenosis is better treated with continuity + Bernoulli for local velocity and pressure; Poiseuille governs the overall viscous resistance of a long tube.

Common MCAT Traps

• Applying continuity and Bernoulli without checking assumptions — both treat the fluid as incompressible with steady flow; Bernoulli also assumes negligible viscosity. The MCAT will usually state or imply these conditions.
• Forgetting that Q ∝ r⁴ in Poiseuille's Law, not r² or r. A question describing a vessel radius change is testing whether you remember the exponent is 4, not the more intuitive area exponent of 2.
• Reversing the pressure-velocity relationship in Bernoulli's equation. Faster flow means lower pressure at that point, not higher — a common point of confusion for students who associate "more speed" with "more force/pressure."
• Confusing viscosity's role with Bernoulli's equation, which assumes an ideal, non-viscous fluid; Poiseuille's Law is the one that explicitly accounts for viscosity.
• Using individual capillary area instead of total parallel area when applying continuity to the microcirculation — collective area is what sets capillary velocity.