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

2.2.19 · Physics › Fluid Mechanics

YEH HAI KYA?

Dimensionless kyun? Kyunki yeh do forces compare karta hai same units ke saath, toh units cancel ho jaate hain. Isse $R e$ ek universal number ban jaata hai — ek toy pipe aur ek oil pipeline same behave karte hain agar unka $R e$ match kare.

YEH LAMINAR VS TURBULENT PREDICT KYUN KARTA HAI? (First principles se derive karo)

Yaad mat karo — build karo. Flow mein do competing effects hote hain.

1. Inertial force — fluid ki tendency hai ki woh apne momentum ke saath aage badhta rahe. Force = rate of change of momentum. Area $A = L^{2}$ se momentum flux, mass-flow $\times$ velocity hai: $F_{inertia} \sim (ρ v A) \cdot v = ρ v^{2} L^{2}$

2. Viscous force — internal friction jo shear resist karta hai. Newton's law of viscosity se, shear stress $τ = μ \frac{d v}{d y}$ . Velocity length $\sim L$ ke across $\sim v$ ke barabar change hoti hai, toh: $τ \sim μ \frac{v}{L}, F_{visc} = τ \cdot A \sim μ \frac{v}{L} L^{2} = μv L$

3. Ratio lo: $R e = \frac{F _{inertia}}{F _{visc}} = \frac{ρ v ^{2} L ^{2}}{μv L} = \frac{ρ v L}{μ} ✓$

CRITERION KAISE USE KAREIN (pipe flow, $L =$ diameter)

$R e$ range	Regime
$R e < 2000$	Laminar (smooth, layered)
$2000 < R e < 4000$	Transitional (unstable, intermittent)
$R e > 4000$	Turbulent (chaotic, mixing)

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

Worked Examples

Steel-manned Mistakes

Active Recall

Recall Test karne ke liye click karo

Q: $R e$ kaun si do forces compare karta hai? → inertial / viscous.
Q: Viscosity upar hai ya neeche? → neeche (denominator).
Q: Bada $R e$ matlab? → turbulent.
Q: $R e$ ke units? → koi nahi (dimensionless).
Q: $F_{inertia} \sim ρ v^{2} L^{2}$ derive karo. → $\overset{m}{˙} = ρ v L^{2}$ , times $v$ .

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek crowd ek hallway se guzar rahi hai. Agar woh dheere aur politely chalein, toh sab neat lines mein rehte hain — woh laminar hai. Agar woh sprint karein, toh log bump karte hain, swirl karte hain, aur yeh ek chaotic scramble ban jaata hai — woh turbulent hai. Reynolds number bas yeh hai: "woh kitna rush karna chahte hain" (speed, weight) divided by "woh kitna ek doosre ko haath pakad ke slow karte hain" (stickiness/viscosity). Bada number = chaos. Chhota number = neat lines. Honey super sticky hai, toh woh hamesha neat rehti hai. Paani tezi se jaaye toh pagal ho jaata hai.

Flashcards

Reynolds number physically kya represent karta hai?

Flowing fluid mein inertial forces aur viscous forces ka ratio.

Reynolds number formula likho.

R e = ρ v L / μ = v L / ν

Kya Re dimensionless hai?

Haan — yeh ek pure number hai (do forces ka ratio).

Inertial force scale derive karo.

Mass flow

\overset{m}{˙} = ρ v L^{2}

, velocity

v

carry karta hai, toh

F_{in er t ia} \sim ρ v^{2} L^{2}

Viscous force scale derive karo.

τ = μ d v / d y \approx μv / L

; area

L^{2}

se multiply karne par

F_{v i sc} \sim μv L

milta hai.

Pipe flow ke liye approx critical Re (laminar→turbulent)?

~2000 ke neeche laminar, ~4000 ke upar turbulent.

Pipe flow ke liye kaun sa length L use hota hai?

Pipe diameter D.

Zyada viscosity flow ko zyada turbulent banati hai ya kam?

Kam — μ denominator mein hai, toh Re kam hota hai (zyada laminar).

Critical speed formula Re se?

v_{c} = R e μ / (ρ L)

Honey almost hamesha laminar kyun hoti hai?

Uska huge μ Re ko tiny bana deta hai, toh viscosity dominate karta hai aur disturbances damp ho jaate hain.

Connections

Viscosity and Newton's law of viscosity — $μ$ aur $τ = μ d v / d y$ define karta hai jo derivation mein use hote hain.
Poiseuille's law — sirf laminar ( $R e < 2000$ ) regime mein valid hai.
Stokes' law and terminal velocity — ek sphere par low- $R e$ drag.
Bernoulli's principle — ideal (inviscid) flow assume karta hai, yeh $R e \to \infty$ idealisation hai.
Dimensional analysis — kyun $R e$ jaise dimensionless numbers universal hote hain.
Drag force and drag coefficient — $C_{d}$ , $R e$ ka ek function hai.