12.4 Viscosity and Laminar Flow; Poiseuille’s Law

Fluid Dynamics

Fluid dynamics studies how liquids and gases move through spaces. In this unit, the focus narrows to two practical questions: What makes flow smooth versus chaotic? And how do pipe dimensions and fluid properties determine flow rate? These ideas show up constantly in biological and medical contexts, from blood moving through arteries to IV drips delivering medication.

Laminar vs. Turbulent Flow

Laminar flow is smooth, orderly motion where fluid moves in parallel layers that don't mix with each other. Picture honey pouring slowly off a spoon. Each layer slides over the next in a predictable way, and you can trace the path of any fluid particle as a clean, parallel streamline.

Turbulent flow is the opposite: chaotic, irregular motion where layers mix and form eddies and vortices. Think of a fast-moving river churning around rocks. Streamlines become tangled and unpredictable.

What determines which type you get? Three main factors:

• Velocity: Higher speeds push flow toward turbulence
• Viscosity: Higher viscosity (thicker fluid) favors laminar flow
• Pipe diameter: Larger diameters make turbulence more likely

These factors combine into a single number called the Reynolds number ($Re$). It acts as a threshold indicator:

• $Re < 2300$: Flow is laminar
• $2300 < Re < 4000$: Transitional flow (a mix of both behaviors)
• $Re > 4000$: Flow is turbulent

For this course, most problems deal with laminar flow, since that's where Poiseuille's Law applies.

Viscosity and Fluid Behavior

Viscosity ($\eta$) measures a fluid's internal resistance to flow. It comes from friction between fluid layers as they slide past each other. A high-viscosity fluid like molasses resists motion and flows slowly. A low-viscosity fluid like water flows much more freely.

Temperature matters here, and the effect depends on the type of fluid:

• Liquids: Viscosity decreases as temperature rises (warm honey pours faster than cold honey)
• Gases: Viscosity increases as temperature rises

Two types of viscosity come up in problems:

• Dynamic viscosity ($\eta$): The ratio of shear stress to shear rate. This is the one that appears in Poiseuille's Law.
• Kinematic viscosity ($\nu$): Dynamic viscosity divided by fluid density ($\nu = \eta / \rho$). Used in Reynolds number calculations.

Newtonian fluids like water and air have a constant viscosity regardless of how fast you push them. Non-Newtonian fluids like blood and ketchup change viscosity depending on the applied shear stress. Blood, for instance, becomes less viscous at higher shear rates, which helps it flow through narrow capillaries.

Shear Rate and Viscosity

Shear rate is the rate at which fluid velocity changes from one layer to the next. In a pipe with laminar flow, the fluid at the center moves fastest while the fluid touching the pipe wall is essentially stationary (the "no-slip condition"). This means:

• Shear rate is highest near the pipe walls, where velocity changes rapidly over a short distance
• Shear rate is lowest at the center, where neighboring layers move at nearly the same speed

For non-Newtonian fluids, this variation in shear rate across the pipe cross-section means viscosity isn't uniform. Blood near the vessel wall experiences different viscosity than blood at the center, which is one reason blood flow modeling gets complicated.

Poiseuille's Law

Poiseuille's Law gives you the volume flow rate of a viscous fluid through a cylindrical pipe, assuming laminar flow of a Newtonian fluid. The equation is:

$Q = \frac{\pi r^4 \Delta P}{8 \eta L}$

where:

• $Q$ = volume flow rate
• $r$ = pipe radius
• $\Delta P$ = pressure difference between the two ends
• $\eta$ = dynamic viscosity of the fluid
• $L$ = length of the pipe

The most striking feature of this equation is the $r^4$ term. Radius has an enormous effect on flow rate. If you double the radius, flow rate increases by a factor of $2^4 = 16$. This is why even small changes in blood vessel diameter (from plaque buildup or vasodilation) have dramatic effects on blood flow.

You can also define flow resistance as:

$R = \frac{8 \eta L}{\pi r^4}$

This lets you write the relationship in a form that mirrors Ohm's Law in circuits: $\Delta P = Q \cdot R$. Pressure difference drives flow, resistance opposes it.

Pressure Changes in Pipes

In a pipe with steady laminar flow, pressure drops linearly along the length. The total pressure drop is:

$\Delta P = Q \cdot R = \frac{8 \eta L Q}{\pi r^4}$

Several factors increase the pressure drop:

• Higher flow rate (more fluid pushed through per second)
• Longer pipe (more surface area creating friction)
• Smaller radius (dramatically, because of the $r^4$ dependence)
• Higher viscosity (more internal friction)

In biological systems, the heart acts as the pump that overcomes these pressure drops to keep blood circulating. In medical applications like IV lines, clinicians choose tube diameter and fluid height (which sets $\Delta P$) to control the drip rate. In engineering, long-distance pipelines use large-diameter, smooth-walled pipes to keep resistance low and reduce the pumping power needed.