Foundations — Viscous flow — Poiseuille flow, velocity profile in pipe

This page assumes you know nothing. Before we can read the parent note Poiseuille flow topic, we must earn every letter it uses. Read top to bottom — each block relies only on the ones above it.

0. What a "pipe cross-section" even looks like

Everything happens inside a straight round tube. Cut it straight across and you see a circle. Cut it lengthways and you see a long rectangle of fluid. We will keep switching between these two views, so fix them in your head first.

• The circle (looking down the barrel) is where we measure how far from the centre a bit of fluid sits.
• The long slice (looking from the side) is where we watch fluid move along the tube.

1. $r$ — distance from the centre line

Picture: on the circle view, draw an arrow from the bullseye outward. Its length is $r$.

Why the topic needs it: the speed of the fluid is different at different distances from the centre. To describe "speed here vs speed there" we need a label for "here" — that label is $r$.

2. $R$ — the pipe's radius (the wall)

Picture: on the same circle, $R$ is the arrow that just touches the edge. Any $r$ satisfies $0\le r\le R$.

3. $L$ — the length of the pipe

Picture: on the side view, $L$ is the horizontal span of the rectangle.

Why the topic needs it: the push has to overcome friction along the whole length. A longer pipe = more wall to fight = you need a bigger push for the same flow.

4. $v$ — the speed of the fluid, and $v(r)$

Picture: at each distance $r$ draw a little arrow pointing along the pipe; longer arrow = faster. Stack all those arrows and their tips trace a curve — the velocity profile.

Why the notation $v(r)$? It is a machine: feed in a distance, out comes a speed. This is the single most important object in the whole topic — the parabola $v(r)=\frac{\Delta P}{4\eta L}(R^2-r^2)$ is just a rule for that machine.

5. $\eta$ — viscosity (the stickiness number)

Picture: two sheets of fluid sliding past each other. Big $\eta$ = they drag hard on each other (honey). Small $\eta$ = they slide freely (water).

Why the topic needs it: stickiness is what resists the flow. More $\eta$ means you must push harder for the same result — see Viscosity and Newton's law of viscosity for the full law.

6. $\frac{dv}{dr}$ — the velocity gradient (why we need calculus)

The speed changes as you move outward. But how fast does it change? That "rate of change of speed with distance" is what $\frac{dv}{dr}$ means.

Picture: on the velocity-profile curve, $\frac{dv}{dr}$ is the slope of the curve at a point — flat in the middle (speed barely changing), steep near the wall (speed dropping fast).

Why negative? Moving outward ($r$ increasing) the speed drops, so the slope is negative: $dv/dr<0$. The minus signs in the parent derivation are just this fact.

7. $\tau$ — shear stress (the drag force per area)

Reading it in words: drag ($\tau$) = stickiness ($\eta$) × how-fast-speed-changes ($dv/dr$). Sticky fluid and a steep speed change both make the rubbing worse.

Why "per area"? A big surface rubs more than a small one, so we quote the drag per square metre and multiply by the real area later. In the parent note that area is the curved side of a cylinder, $2\pi rL$.

8. $P$, $P_1$, $P_2$ and $\Delta P$ — pressure and the push

Picture: high pressure squeezing at the left end, lower pressure at the right — fluid squirts from high to low, like toothpaste from a squeezed tube.

Why the topic needs it: without a pressure difference, viscosity would grind the fluid to a halt. $\Delta P$ is the reason anything moves at all.

9. $\pi r^2$, $2\pi rL$, and $Q$ — areas and flow rate

Picture: imagine catching all the fluid leaving the pipe in one second into a bucket — the volume in the bucket is $Q$.

Why both areas? The push acts on the flat end ($\pi r^2$); the friction acts on the side ($2\pi rL$). Balancing those two is the entire derivation. See Equation of continuity for how $Q$ links different pipe widths.

How it all fits together

Every arrow means "is needed to build". Read from the top loose pieces down to $Q$, the payoff.

Equipment checklist

Cover the right side and say each answer out loud before revealing.

Connections

• Viscosity and Newton's law of viscosity — the source of $\eta$, $\tau$ and $dv/dr$.
• Equation of continuity — where $Q$ = area × speed comes from.
• Reynolds number and turbulence — tells you when this laminar picture is valid.
• Bernoulli's principle — the frictionless cousin (no viscosity).
• Stokes' law and terminal velocity — viscous drag on a sphere instead of a pipe.
• Blood flow and circulatory system — why the $R^4$ rule matters for arteries.