Noesis
2.2.19Fluid Mechanics

Reynolds number Re = ρvL - μ — laminar vs turbulent criterion

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🇮🇳 Yeh note Hinglish mein padho →

WHAT is it?

Why dimensionless? Because it compares two forces of the same units, the units cancel. That makes ReRe a universal number — a toy pipe and an oil pipeline behave alike if their ReRe matches.

WHY does it predict laminar vs turbulent? (Derive from first principles)

Don't memorise — build it. Flow has two competing effects.

1. Inertial force — the fluid's tendency to barrel ahead with its momentum. Force = rate of change of momentum. Momentum flux through area A=L2A=L^2 is mass-flow ×\times velocity: Finertia∼(ρ v A)⋅v=ρv2L2F_{\text{inertia}} \sim (\rho \, v \, A)\cdot v = \rho v^2 L^2

2. Viscous force — internal friction resisting shear. From Newton's law of viscosity, shear stress τ=μdvdy\tau = \mu \dfrac{dv}{dy}. The velocity changes by ∼v\sim v across length ∼L\sim L, so: τ∼μvL,Fvisc=τ⋅A∼μvLL2=μvL\tau \sim \mu \frac{v}{L}, \qquad F_{\text{visc}} = \tau \cdot A \sim \mu \frac{v}{L} L^2 = \mu v L

3. Take the ratio: Re=FinertiaFvisc=ρv2L2μvL=ρvLμ ✓Re = \frac{F_{\text{inertia}}}{F_{\text{visc}}} = \frac{\rho v^2 L^2}{\mu v L} = \frac{\rho v L}{\mu}\ \checkmark

HOW to use the criterion (pipe flow, L=L = diameter)

ReRe range Regime
Re<2000Re < 2000 Laminar (smooth, layered)
2000<Re<40002000 < Re < 4000 Transitional (unstable, intermittent)
Re>4000Re > 4000 Turbulent (chaotic, mixing)
Figure — Reynolds number Re = ρvL - μ — laminar vs turbulent criterion

Worked Examples

Steel-manned Mistakes

Active Recall

Recall Click to test yourself
  • Q: What two forces does ReRe compare? → inertial / viscous.
  • Q: Where is viscosity, top or bottom? → bottom (denominator).
  • Q: Large ReRe means? → turbulent.
  • Q: Units of ReRe? → none (dimensionless).
  • Q: Derive Finertia∼ρv2L2F_{\text{inertia}}\sim\rho v^2 L^2. → m˙=ρvL2\dot m = \rho v L^2, times vv.
Recall Feynman: explain to a 12-year-old

Imagine a crowd walking through a hallway. If they walk slowly and politely, everyone stays in neat lines — that's laminar. If they sprint, people bump, swirl, and it becomes a chaotic scramble — that's turbulent. The Reynolds number is just: "how much do they want to rush" (speed, weight) divided by "how much they're holding hands and slowing each other down" (stickiness/viscosity). Big number = chaos. Small number = neat lines. Honey is super sticky, so it always stays neat. Water rushing fast goes crazy.

Flashcards

What does the Reynolds number physically represent?
The ratio of inertial forces to viscous forces in a flowing fluid.
Write the Reynolds number formula.
Re=ρvL/μ=vL/νRe = \rho v L / \mu = vL/\nu
Is Re dimensionless?
Yes — it is a pure number (a ratio of two forces).
Derive the inertial force scale.
Mass flow m˙=ρvL2\dot m=\rho vL^2, carries velocity vv, so Finertia∼ρv2L2F_{inertia}\sim\rho v^2L^2.
Derive the viscous force scale.
τ=μ dv/dy≈μv/L\tau=\mu\,dv/dy\approx\mu v/L; times area L2L^2 gives Fvisc∼μvLF_{visc}\sim\mu vL.
Approx critical Re for pipe flow (laminar→turbulent)?
Laminar below ~2000, turbulent above ~4000.
For pipe flow, what length L is used?
The pipe diameter D.
Does higher viscosity make flow more or less turbulent?
Less — μ is in the denominator, so it lowers Re (more laminar).
Critical speed formula from Re?
vc=Re μ/(ρL)v_c = Re\,\mu/(\rho L).
Why is honey almost always laminar?
Its huge μ makes Re tiny, so viscosity dominates and damps disturbances.

Connections

  • Viscosity and Newton's law of viscosity — defines μ\mu and τ=μ dv/dy\tau=\mu\,dv/dy used in the derivation.
  • Poiseuille's law — valid only in the laminar (Re<2000Re<2000) regime.
  • Stokes' law and terminal velocity — low-ReRe drag on a sphere.
  • Bernoulli's principle — assumes ideal (inviscid) flow, the Re→∞Re\to\infty idealisation.
  • Dimensional analysis — why dimensionless numbers like ReRe are universal.
  • Drag force and drag coefficient — CdC_d is a function of ReRe.

Concept Map

numerator

denominator

equals ρvL over μ

same units cancel

Re below 2000

2000 to 4000

Re above 4000

dominates gives

dominates gives

Inertial force ρv²L²

Viscous force μvL

Reynolds number Re

Dimensionless universal

Laminar smooth layers

Turbulent chaotic mixing

Transitional unstable

Ratio of forces

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Reynolds number ek simple ratio hai: fluid ki "bhaagne ki zid" (inertia) divided by uski "chipchipahat jo rok rahi hai" (viscosity). Formula hai Re=ρvL/μRe = \rho v L/\mu. Upar inertia (ρv2L2\rho v^2 L^2 se aata hai) aur neeche viscosity (μvL\mu v L). Jab inertia jeet jaaye, yaani ReRe bada ho, toh flow chaotic ho jaata hai — isko turbulent kehte hain. Jab viscosity strong ho, ReRe chhota, toh flow smooth layers me chalta hai — laminar.

Pipe flow ke liye rule of thumb: Re<2000Re < 2000 matlab laminar, Re>4000Re > 4000 matlab turbulent, beech me transition zone. Yaad rakho — viscosity neeche hai, toh zyada chipchipa fluid (jaise honey) ka ReRe chhota hota hai aur wo hamesha laminar rehta hai. Yeh bahut log ulta samajh lete hain!

Important baat: ReRe dimensionless hai, koi units nahi. Isliye yeh universal hai — chhoti lab pipe aur badi oil pipeline same ReRe par same behaviour dikhayegi. Isi liye engineers chhote model bana ke test karte hain. Aur haan, pipe me LL ka matlab diameter hota hai, radius nahi — warna answer aadha aa jayega.

Critical speed nikaalna ho toh formula palat do: vc=Re μ/(ρL)v_c = Re\,\mu/(\rho L). Bas yeh tin cheezein pakdo — ratio of forces, viscosity neeche, 2000 ka thumb rule — aur poora topic clear ho jayega.

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Connections

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