Visual walkthrough — Fluid definition — shear stress, no fixed shape

2.2.1 · D2 · Physics › Fluid Mechanics › Fluid definition — shear stress, no fixed shape

Yeh parent note ke peeche ki picture-story hai. Agar koi word unfamiliar lage, toh use use kiye jaane se pehle draw kiya jaayega.

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Step 1 — Ek force ko do seedhe tukdon mein baanto

KYA. Force bas ek push (ya pull) hota hai jisme ek direction hoti hai, jise arrow ki tarah draw karte hain. Jab woh arrow kisi flat surface pe pada, toh woh almost kabhi bhi seedha surface ke andar ki taraf point nahi karta. Toh hum us ek tedhe arrow ko do aaise arrows mein chop karte hain jinki directions clean hoti hain:

• ek jo seedha surface ke andar point karta hai — uski size $F_n$ (letter $n$ hai normal ke liye, matlab "perpendicular"),
• ek jo surface ke saath flat point karta hai — uski size $F_t$ (letter $t$ hai tangential ke liye, matlab "sideways sliding").

KYUN. Yeh do tukde bilkul alag kaam karte hain. Perpendicular piece $F_n$ surface ko squash karne ki koshish karta hai. Along-the-surface piece $F_t$ use sideways smear karne ki koshish karta hai. Ek fluid squashing aur smearing pe bilkul alag tarike se react karta hai, isliye inhein alag karna zaroori hai — tab hi hum kuch sach bol sakte hain.

PICTURE. Blue tedha arrow real force $\vec{F}$ hai. Orange arrow $F_t$ (sideways sliding) hai, green arrow $F_n$ (andar press) hai. Orange aur green arrows milke (tip-to-tail) blue arrow banaate hain.

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Step 2 — Force ko area se divide karke stress banao

KYA. Hum har force piece ko us surface ke area $A$ se divide karte hain jis par woh act karta hai. Isse do nayi quantities milti hain: $\underset{\text{normal stress}}{\underbrace{\sigma }}=\frac{F_n}{A},\underset{\text{shear stress}}{\underbrace{\tau }}=\frac{F_t}{A}.$

Yahan har symbol apni jagah justify karta hai:

• $\sigma$ (Greek letter "sigma") normal stress hai — surface ko kitna hard press kiya ja raha hai, per unit of area.
• $\tau$ (Greek letter "tau", "cow" se rhyme karta hai) shear stress hai — surface ko sideways kitna hard smear kiya ja raha hai, per unit of area.
• $A$ flat surface area hai jis par force land karta hai, square metres ${\text{m}}^2$ mein measure hota hai.
• $\sigma$ aur $\tau$ dono pascals mein measure hote hain: $1\text{Pa}=1{\text{N/m}}^2$ (ek newton ek square metre pe spread hua).

Area se KYUN divide karen? Kyunki koi material deform hota hai intensity se, raw force se nahi. Apna thumb table pe press karo: gentle. Wahi force ek needle tip se push karo: woh puncture kar deta hai. Same $F$, tiny $A$, enormous stress. Toh fluid kitna giita hai yeh baat karne ke liye, humein stress (force per area) mein baat karni hogi, sirf force mein nahi.

PICTURE. Wahi orange smearing force ek bade plate pe spread ho toh chhota $\tau$ (pale); chote plate pe squeeze ho toh bada $\tau$ (deep). Arrows apni length rakhte hain (same force) lekin smear ki density badal jaati hai.

Ab se hum sirf $\tau$, yaani shear stress, track karenge — kyunki shear woh cheez hai jise ek fluid resist nahi kar sakta.

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Step 3 — Experiment: do plates aur beech mein ek trapped film

KYA. Ek thin film of fluid ko do flat plates ke beech mein trap karo. Bottom plate bolted still hai. Top plate ko steadily sideways drag kiya jaata hai ek speed se jise hum $u$ kehte hain (speed, metres per second mein, $\text{m/s}$). Plates $y$ distance apart hain, seedha bottom se top tak measure kiya (metres mein).

KYUN yeh setup? Yeh pure shear apply karne ka sabse clean tarika hai — sirf sideways smearing, koi squashing nahi. Agar hum yahan measure kar sakein ki fluid kaise fight back karta hai, toh humne exactly woh property isolate kar li hai jo ek fluid ko define karti hai. Har doosra flow in simple shears ka mixture hota hai.

PICTURE. Bottom plate grey aur fixed (little hatching = "held"). Top plate right ki taraf move kar rahi hai blue speed arrow ke saath labelled $u$. Gap height $y$ ko ek vertical measuring bar se mark kiya gaya hai. Fluid gap bharta hai.

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Step 4 — No-slip rule ek velocity profile set karta hai

KYA. Plate ko touch karne wala fluid us plate ke saath move karta hai — woh us par slide nahi karta. Yeh no-slip condition hai. Toh:

• bottom (height $0$) par fluid speed $0$ se move karta hai,
• top (height $y$) par fluid speed $u$ se move karta hai,
• beech wala fluid kisi in-between speed se move karta hai jo smoothly badhti hai jab tum upar chadhte ho.

Height $y^{\prime }$ par baithe layer ki speed ko hum $u(y^{\prime })$ likhte hain — padho "$u$ of $y^{\prime }$", matlab "height $y^{\prime }$ par jo speed milti hai." $y^{\prime }$ par chhota prime bas yeh kehta hai ki "gap ke andar kaheen ek height", use total gap $y$ se alag rakhne ke liye.

KYUN yeh important hai? Kyunki yeh humein batata hai ki fluid ek lump ki tarah nahi move karta — woh layers mein move karta hai jo ek doosre ke upar slide karti hain, top par fast, bottom par slow. Woh layered sliding hi shearing hai. Heights ke hisaab se stack ki gayi speeds ki yeh picture velocity profile kehlati hai.

PICTURE. Horizontal arrows ka ek vertical stack: bottom par zero length, top par sabse lamba, evenly badhta hua. Unki tips ek straight slanted line trace karti hain — woh straight line simple fluid ke liye velocity profile hai.

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Step 5 — Film ko deform hote dekho: yahan "strain" ka matlab

KYA. Gap ke across paint kiya gaya ek tall thin rectangle of fluid freeze karo. Thoda time $dt$ (bahut chota sa moment) wait karo. Us moment mein top edge aage slide karti hai jabki bottom ruki rehti hai, toh rectangle parallelogram mein tilt ho jaata hai. Top aage ek chhoti distance se khisak jaata hai $\underset{\text{top kitna khisaka}}{\underbrace{dudt}},$ jahan $du$ woh extra speed hai jo top ki bottom ke upar hai, aur $dt$ thoda sa waqt hai.

Tilt ko ek chhote angle $d\gamma$ se measure kiya jaata hai ($\gamma$ Greek "gamma" hai). Thodi si tilt ke liye, angle sideways slide ÷ height ke barabar hota hai: $d\gamma =\frac{dudt}{dy},$ jahan $dy$ hamare fluid rectangle ki (chhoti) height hai.

KYUN angle? Kyunki "yeh kitna sheared hai?" naturally ek tilt hota hai. Upar se push ki gayi ek tall thin book ek angle se jhuk jaati hai — woh lean angle hi shear strain $\gamma$ hai. Slide-over-height use karna messy geometry ko ek clean number mein convert kar deta hai.

PICTURE. Green rectangle (pehle) orange parallelogram mein tilt ho jaata hai ($dt$ ke baad). Top-edge slide $dudt$ horizontally mark ki gayi hai; height $dy$ vertically; chhota lean angle $d\gamma$ corner mein hai.

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Step 6 — Crucial move: strain ka rate, strain nahi

KYA. Tilt equation lo aur dono sides ko time $dt$ se divide karo. Right side par $dt$ cancel ho jaata hai: $\frac{d\gamma }{dt}=\frac{du}{dy}.$ Left side ko padho: "tilt angle per second kitni tezi se badhta hai." Right side ko padho velocity gradient ke roop mein: speed kitni tezi se badlati hai jab tum upar chadhte ho — Step 4 ke velocity profile ki steepness.

KYUN time se divide karen? Yahi baat ek fluid ko fluid banati hai. Ek solid ko push karo aur woh ek fixed angle tak tilt ho jaata hai aur ruk jaata hai — toh solid ka stress angle $\gamma$ itself ki parwah karta hai. Ek fluid ko push karo aur woh kabhi nahi rukta — angle forever badhta rehta hai — toh "final angle kya hai?" poochha karna bekar hai. Sirf yeh poochh sakte ho "woh kitni tezi se tilt ho raha hai?" Isliye $dt$ se divide karte hain: hum useless total angle ko meaningful rate se replace kar lete hain.

PICTURE. Do side-by-side stories. Left: ek solid block ek baar jhukta hai aur freeze ho jaata hai (single tilted shape, ek "stop" sign). Right: ek fluid block baar baar aur zyada jhukta rehta hai har moment (ever-greater tilts ka ek fan, time arrows). Fluid ka jawab rate mein hai, marked $\dot{\gamma }=du/dy$.

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Step 7 — Experiment missing constant $\mu$ supply karta hai

KYA. Nature (lab mein measure kiya, derive nahi kiya) kehti hai: jo shear stress tum apply karte ho woh proportional hoti hai fluid ke smear hone ki rate se: $\tau \propto \frac{du}{dy}.$ "Proportional" ko "equals" mein badlo proportionality ka ek constant daalkaar, jise $\mu$ (Greek "mu") kehte hain: $\boxed{\tau =\mu \frac{du}{dy}}$

• $\tau$ = shear stress (Pa) — Step 2 se smear intensity.
• $\mu$ = dynamic viscosity (Pa·s) — fluid ki "thickness." Bada $\mu$ = honey (zyada fight karta hai), chhota $\mu$ = water (asaani se de deta hai), har fluid ke liye measure kiya jaata hai.
• $du/dy$ = velocity gradient (per second) — Step 6 se steepness.

KYUN formula nahi, ek constant? Kyunki $\mu$ ko geometry se reason out nahi kiya ja sakta — yeh substance par depend karta hai. Geometry ne humein law ki shape di ($\tau$ ∝ steepness); experiment ek particular fluid ke liye fit hone wala number $\mu$ supply karta hai. Jo fluids is straight-line law ko follow karte hain unhe Newtonian kehte hain.

PICTURE. Ek graph: horizontal axis gradient $du/dy$ hai, vertical axis stress $\tau$ hai. Har fluid origin se ek straight line hai; uska slope $\mu$ hai. Honey (steep, orange), water (shallow, blue). Origin se matlab: koi smear rate nahi ⇒ koi stress nahi.

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Step 8 — Degenerate case: fluid at rest ($du/dy=0$)

KYA. Ab motion band karo. Dono plates still hain, kuch slide nahi karta, toh har layer ki same (zero) speed hai: velocity gradient $du/dy=0$ hai. Use law mein daalo: $\tau =\mu \times 0=0.$

KYUN yeh case poori baat hai. Yeh single line hi fluid ki definition hai. Rest mein, ek fluid zero shear stress carry karta hai, chahe woh kitna bhi thick ho — kyunki $\mu$ finite hai aur woh zero se multiply karta hai. Ek solid aisa nahi karta: rest mein bhi ek solid apne stored tilt se ek leftover shear $\tau =G\gamma$ rakhta hai. Isliye ek fluid apni shape nahi rakh sakta (koi bhi tiny leftover shear use flow kara dega) aur bajaye iske container fill karne ke liye slump kar jaata hai — exactly woh "no fixed shape" fact jo parent note se shuru hota hai.

PICTURE. Velocity profile ek single vertical line mein collapse ho jaati hai (sab speeds zero). Stress readout $0$ par aa jaata hai. Fluid ka ek blob, koi bhi shear sustain karne mein unable, flat spread ho kar ek bowl fill kar leta hai.

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Ek-picture summary

Upar sab kuch, ek diagram mein compress: force → shear alag karo → per-area $\tau$ deta hai → no-slip ek velocity profile banata hai → tilting per second $du/dy$ deta hai → $\mu$ se multiply karo → aur rest mein sab zero switch off ho jaata hai.

Recall Poore walkthrough ki Feynman retelling

Ek force surface ko lagbhag hamesha lean karke hit karta hai, toh pehle hum use ek "seedha andar press" wale hisse aur ek "slide-along" wale hisse mein chop karte hain. Slide-along wala hissa, jis area ko woh touch karta hai us par share ho jaata hai, shear stress $\tau$ hai — ek smear per square metre. Yeh dekhne ke liye ki ek fluid smear ke saath kya karta hai, hum use do plates ke beech trap karte hain aur top wali ko drag karte hain. Kyunki fluid jo bhi touch karta hai us se chipak jaata hai (no-slip), bottom layer still rehti hai, top layer race karte hue jaati hai, aur beech ki layers ek smooth staircase of speeds banati hain — velocity profile. Ab fluid mein ek tall block paint karo aur palk jhapkao: top aage khiski hai, toh block parallelogram mein lean ho gaya hai. Ek solid ke liye woh lean kisi angle par ruk jaata aur freeze ho jaata, toh solid angle ki parwah karta hai. Ek fluid ke liye lean kabhi nahi rukta — woh sirf jhukta rehta hai — toh sirf ek fair sawaal hai kitni tezi se jhuk raha hai, jo speed-staircase ki steepness hai, $du/dy$. Tab experiment aakhri ingredient whisper karta hai: jis smear ko push karna padta hai woh us steepness ka ek number $\mu$ se multiple hai, fluid ki thickness. Yahi $\tau =\mu du/dy$ deta hai. Aur punchline: sab kuch rok do, staircase flat ho jaata hai, $du/dy$ zero ho jaata hai, toh $\tau$ zero ho jaata hai — rest mein ek fluid bilkul koi shear carry nahi karta, exactly isliye woh apni shape kabhi nahi rakh sakta aur hamesha cup fill karne ke liye slump kar jaata hai.

Recall Quick self-check

Shear force ko area se kyun divide karte hain? ::: Per unit area intensity paane ke liye; wahi force ek bade patch par gentle hota hai aur ek tiny patch par fierce. Tilt ko time se kyun divide karte hain (rate use karo, angle nahi)? ::: Kyunki ek fluid kabhi tilting band nahi karta, toh total angle ki koi fixed value nahi hoti — sirf rate $du/dy$ meaningful hai. $\mu$ kahan se aata hai — geometry se ya experiment se? ::: Experiment se; geometry sirf shape $\tau \propto du/dy$ deti hai, aur $\mu$ har fluid ke liye measured slope hai. Kisi bhi fluid ke liye rest mein $\tau =0$ kyun hota hai? ::: Rest mein $du/dy=0$, aur $\mu \times 0=0$ chahe $\mu$ kitna bhi bada ho.

Connections

• Velocity Profile and No-Slip Condition — Steps 3–4 yahan use ki gayi profile banate hain.
• Viscosity and Newtonian vs Non-Newtonian Fluids — constant $\mu$ aur Step 7 ka straight-line law.
• Stress and Strain in Solids — contrast law $\tau =G\gamma$ jis par Step 6 rely karta hai.
• Hydrostatics — Fluids at Rest — Step 8 ka $du/dy=0$ degenerate case.
• Pressure in Fluids — jahan Step 1 se normal piece $\sigma$ aage le jaata hai.