Viscosity & Poiseuille

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.