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

2.2.3 · D1 · Physics › Fluid Mechanics › Viscosity — dynamic μ, kinematic ν = μ - ρ; Newtonian vs non

Parent note padne se pehle, tumhe har woh symbol khud se samajh lena hai jo woh tumhare saamne throw karta hai. Neeche, har cheez ko zero se build kiya gaya hai: plain meaning → the picture → topic ko uski zaroorat kyun hai. Inhe is tarah order kiya gaya hai ki har ek sirf upar waalon par lean kare.

1 — Fluid ki layers (cards ka deck)

Plain words: Ek liquid ko bahut patli sheets ke stack ki tarah imagine karo, jaise flat rakha hua cards ka deck. Har sheet apni speed par sideways move kar sakti hai.

The picture: Figure 1 dekho. Bottom card table par chipki hui hai (speed $0$). Top card ko right side mein drag kiya ja raha hai. Beech wale cards apne neighbours se drag hote hain, har ek neeche wale se thoda zyada fast. Arrows ka yeh staircase viscosity ki poori kahani hai.

Topic ko iski zaroorat kyun hai: Viscosity in layers ke beech jo hota hai usse define hoti hai. Agar layers ka picture tumhare dimag mein nahi hai, toh koi bhi symbol kuch nahi maanta.

2 — $u$ : ek layer ki speed

Plain words: $u$ simply ek fluid layer ki sideways speed hai, metres per second (m/s) mein measure ki gayi.

The picture: Figure 1 mein, $u$ har card par blue arrow ki length hai. Lamba arrow = fast layer; koi arrow nahi = frozen layer.

3 — $y$ aur velocity gradient $\frac{du}{dy}$

Plain words: $y$ woh height hai jo gap ke across measure ki jaati hai, bottom plate ($y=0$) se top plate ($y=h$) tak. Velocity gradient $\frac{du}{dy}$ yeh poochta hai: "jab main thoda upar chadhta hoon, toh layer kitni zyada fast move karti hai?"

The picture: Figure 2 mein, speed $u$ ko sideways aur height $y$ ko upar draw karo. Layers ek straight slanted line trace karti hain. Us line ki steepness hi $\frac{du}{dy}$ hai. Gentle slope = layers barely differ hoti hain = weak shearing. Steep slope = neighbours ek-doosre ke past race karte hain = strong shearing.

Topic ko iski zaroorat kyun hai: $\frac{du}{dy}$ Newton's law of viscosity ka single input hai. Parent page par har viscous force yahin se shuru hoti hai.

4 — Shear rate $\dot{\gamma }$ (gradient ka ek nickname)

Plain words: $\dot{\gamma }$ (Greek "gamma", upar ek dot ke saath) velocity gradient ka bas ek aur naam hai: $\dot{\gamma }=\frac{du}{dy}$. Dot ka matlab hai "rate" — shape ko per second kitni tezi se shear kiya ja raha hai.

The picture: Figure 2 jaisi hi slanted line. Simple straight-line profile ke liye, $\dot{\gamma }$ = (top speed $U$) ÷ (gap $h$), kyunki straight line ki steepness har jagah constant hoti hai.

Topic ko iski zaroorat kyun hai: Jab parent Newtonian vs non-Newtonian fluids classify karne ke liye $\tau$-versus-$\dot{\gamma }$ graphs draw karta hai, horizontal axis hi $\dot{\gamma }$ hoti hai.

5 — Shear stress $\tau$ (sideways rub, per area)

Plain words: $\tau$ (Greek "tau") woh sideways dragging force hai jo ek layer ke face ke har square metre par spread hoti hai. Yeh ek aisi pressure hai jo seedha nahi, sideways push karti hai.

The picture: Figure 1 mein, $\tau$ woh red arrow hai jo do cards ke beech ke face par act karta hai — yeh sliding direction ke along point karta hai, use resist karta hai, upar wali layer aur neeche wali layer dono par.

Topic ko iski zaroorat kyun hai: $\tau$ Newton's law ka output hai — woh cheez jo viscosity ek gradient ke response mein produce karti hai.

6 — Dynamic viscosity $\mu$ (the muscle)

Plain words: $\mu$ (Greek "mu") woh number hai jo batata hai ki ek given gradient ke liye kitna stress milta hai. Yeh fluid ki personal "stickiness rating" hai.

The picture: $\tau$-vs-$\dot{\gamma }$ graph par (Figure 3, blue line), $\mu$ literally slope hai. Steep line = high $\mu$ (honey). Flat line = low $\mu$ (paani).

Topic ko iski zaroorat kyun hai: $\mu$ poore chapter ka star hai — yeh wahi hai jo "viscosity" usually ek word mein matlab rakhta hai.

7 — Density $\rho$ (deck kitni heavy hai)

Plain words: $\rho$ (Greek "rho") woh mass hai jo har cubic metre mein packed hai — fluid ka ek chunk kitna bhaari hai.

The picture: Do identical boxes: ek honey se bhara (heavy, high $\rho$), ek air se (light, tiny $\rho$). Same size, wildly different mass.

Topic ko iski zaroorat kyun hai: Newton's second law kehta hai acceleration = force ÷ mass. Yeh predict karne ke liye ki viscosity actually fluid ko kitni tezi se move karti hai, tumhe viscous force ko fluid ki inertia se divide karna hoga — aur inertia $\rho$ se set hoti hai. Woh division agla symbol deta hai.

8 — Kinematic viscosity $\nu$ (the nimbleness)

Plain words: $\nu$ (Greek "nu" — "v" jaisa dikhta hai, letter v nahi hai) dynamic viscosity ko density se divide karna hai. Yeh ek alag sawaal ka jawaab deta hai: "motion fluid ke through kitni tezi se spread hota hai?"

9 — Apparent viscosity ${\mu }_{app}$ aur power-law $n$, $K$

Plain words: Un fluids ke liye jinki stickiness shear karne par badal jaati hai, koi single $\mu$ nahi hota. Iske bajaaye hum har shear rate par ratio measure karte hain aur use apparent viscosity kehte hain: ${\mu }_{app}=\frac{\tau }{\dot{\gamma }}$

Power-law model bending curve describe karta hai: $\tau =K{\dot{\gamma }}^n$

• $K$ = "consistency" — overall kitna sticky hai ($\mu$ ka stand-in ki tarah).
• $n$ = bending exponent. $n=1$ → straight line → Newtonian. $n<1$ → curve neeche bend karta hai → shear-thinning. $n>1$ → curve upar bend karta hai → shear-thickening.

The picture: Figure 3 mein, blue line ($n=1$) straight hai. Green curve ($n<1$) steep se shuru hoti hai phir flatten ho jaati hai — shear karna aasaan hota jaata hai. Red curve ($n>1$) gentle se shuru hoti hai phir steep ho jaati hai — jitna tezi se push karo, utna zyada fight back karta hai.

Topic ko iski zaroorat kyun hai: Yahi teen curves parent note ke dil mein "Newtonian vs non-Newtonian" classification hain.

Prerequisite map

Equipment checklist

Khud ko test karo — right side cover karo aur zor se jawab do.

Ready? Toh parent topic par wapas jao — wahan har symbol ab tumhara hai.