Viscosity — dynamic μ, kinematic ν = μ - ρ; Newtonian vs non-Newtonian

2.2.3 · Physics › Fluid Mechanics

WHY karte hain hum viscosity ki zaroorat?

WHAT problem solve kar rahe hain? Ek ideal fluid mein, layers freely slip hoti hain — paddle push karo aur woh kabhi slow nahin hota. Lekin real fluids drag karti hain. Chai stir karo aur woh eventually ruk jaati hai. Kuch cheez bulk motion ko heat mein convert kar rahi hai. Woh "kuch cheez" viscosity hai.

WHY drag exist karta hai? Molecules adjacent layers ke beech jump karte hain aur apna momentum carry karte hain. Ek fast layer slow layer ko momentum donate karti hai (use speed up karti hai); slow layer fast layer se momentum churaati hai (use slow down karti hai). Net effect: ek shear stress jo layers ke beech velocity difference ko oppose karta hai.

Toh viscosity fundamentally ek velocity gradient ke baare mein hai, velocity itself ke baare mein nahin. Uniform flow (saari layers same speed par) mein zero viscous stress hota hai, chahe speed kitni bhi zyada ho.

HOW karte hain formula scratch se build

Step 1 — Stress kya depend kar sakta hai? Experiment (Newton ka guess): force per area $τ$ jo top plate ko move karte rakhne ke liye chahiye woh hai

plate speed $U$ ke proportional (tez → zyada drag),
gap $h$ ke inversely proportional (moti cushion → kam drag).

Toh $τ \propto U / h$ . Why? Kyunki $U / h$ exactly velocity gradient $d u / d y$ hai straight-line profile ke liye. Proportionality local hai: stress local gradient par depend karta hai.

Step 2 — Constant ka naam rakho. Proportionality constant ko dynamic (absolute) viscosity $μ$ define karo:

Units, derived not memorized: $[μ] = \frac{[ τ ]}{[ d u / d y ]} = \frac{Pa}{s ^{- 1}} = Pa\cdots = \frac{kg}{m\cdots}$ (Purani CGS unit: 1 poise $= 0.1$ Pa·s; water ≈ 1 centipoise $= 1 0^{- 3}$ Pa·s.)

Kinematic viscosity ν — WHY divide karte hain ρ se?

m²/s kyun beautiful hai: yeh ek diffusion coefficient ki units hain. $ν$ literally batata hai ki momentum fluid mein kitni tez diffuse hota hai. Time $t$ mein, momentum $\sim ν t$ distance spread karta hai.

Newtonian vs Non-Newtonian

Ek handy power-law model: $τ = K \overset{γ}{˙}^{n}$ . $n = 1$ Newtonian; $n < 1$ shear-thinning; $n > 1$ shear-thickening.

Worked Examples

Common mistake Steel-manned misconceptions

"Viscous force velocity par depend karta hai, toh ek tez uniform stream mein bada viscous stress hoga." Yeh kyun sahi lagta hai: tez cheezein obviously daily life mein zyada drag experience karti hain. Fix: viscous stress velocity gradient $d u / d y$ par depend karta hai, $u$ par nahin. Ek poora river ek rigid block ki tarah 10 m/s par move kar raha hai toh zero internal viscous stress hai. Nature mein drag walls ke paas gradients se aata hai.

"Higher μ ka matlab hamesha higher ν hai, toh honey water se zyada slowly momentum diffuse karti hai." Yeh kyun sahi lagta hai: honey obviously "thicker" hai. Fix: $ν = μ / ρ$ density se bhi divide karta hai. Air vs water compare karo: air ka μ ~50× chhota hai phir bhi uska ν ~15× bada hai kyunki uska ρ ~800× chhota hai.

"Non-Newtonian ka matlab sirf bahut viscous hai." Fix: iska matlab hai ki μ shear rate ke saath constant nahin hai. Water ka fixed μ hai (Newtonian); ketchup ka μ squeeze karne par drop karta hai (non-Newtonian). Magnitude irrelevant hai; constancy test hai.

Recall Feynman: 12-saal ke bachche ko samjhao

Ek deck of cards imagine karo. Top card slide karo aur baaki thoda saath aate hain kyunki cards ke beech friction hoti hai. Viscosity woh friction hai liquid ke andar. Water ke cards slippery hain (asaani se slide hote hain); honey ke cards sticky hain. Ab: yeh tabhi matter karta hai jab cards alag-alag speeds par slide ho rahi hoon. Agar poora deck ek saath move kare, koi rubbing nahin, koi friction nahin. "Kinematic viscosity" sirf poochh rahi hai ki deck kitna bhari bhi hai — ek halka slippery deck (air) ek push ko bahut tez spread karne deta hai.

Flashcards

Newton's law of viscosity

τ = μ \frac{d u}{d y}

— shear stress = dynamic viscosity × velocity gradient.

Viscous stress kis physical quantity par depend karta hai (velocity ya uska gradient)?

Velocity gradient

d u / d y

par; uniform flow mein zero viscous stress hota hai.

Dynamic viscosity μ ki SI units

Pa·s = kg/(m·s).

Kinematic viscosity ν ki definition

ν = μ / ρ

, units m²/s.

μ ko ρ se divide kyun karte hain ν paane ke liye?

Viscous force ko inertia se compare karne ke liye; ν momentum ka diffusivity hai (diffusion coefficient ki units, m²/s).

Air ki kinematic viscosity water se badi kyun hai jab air "thinner" lagti hai?

Air ka μ chhota hai lekin uska ρ ~800× chhota hai, toh μ/ρ end mein bada ho jaata hai.

Newtonian fluid define karo

Jiska μ constant ho; τ vs shear-rate origin se ek straight line hai (water, air, glycerin).

Shear-thinning vs shear-thickening

Thinning: apparent μ drop hota hai jab shear-rate badhti hai (ketchup, blood); thickening: apparent μ badhta hai (cornstarch/oobleck).

Power-law model aur n kya indicate karta hai

τ = K \overset{γ}{˙}^{n}

; n=1 Newtonian, n<1 shear-thinning, n>1 shear-thickening.

Bingham plastic kya hai?

Flow hone se pehle ek yield stress τ₀ chahiye:

τ = τ_{0} + μ_{p} \overset{γ}{˙}

(toothpaste).

Couette flow gap h, plate speed U mein shear rate kya hai?

\overset{γ}{˙} = U / h

(linear profile).

Apparent viscosity formula

μ_{a pp} = τ / \overset{γ}{˙}

(jab μ vary kare tab use hota hai).

Connections

Reynolds Number — $ν$ use karta hai: $R e = U L / ν$ , inertia aur viscous forces ka ratio.
Boundary Layer — jahan velocity gradients (aur viscous stress) walls ke paas concentrate hote hain.
Poiseuille Flow — μ pressure ke under pipe mein flow rate set karta hai.
Stokes Drag — sphere par drag $F = 6 π μ r v$ , μ ka direct use.
Momentum Diffusion — $ν$ ek diffusion coefficient ki tarah; heat/mass diffusion se analogy.
Terminal Velocity — viscous drag aur gravity ka balance.