Viscous flow — Poiseuille flow, velocity profile in pipe: Foundations

2.2.17 · D1 · Physics › Fluid Mechanics › Viscous flow — Poiseuille flow, velocity profile in pipe

Yeh page assume karta hai ki aapko kuch nahi pata. Pehle hum parent note Poiseuille flow topic padh saken, uske liye har letter kamaana padega. Upar se neeche padho — har block sirf upar wale blocks par rely karta hai.

0. "Pipe cross-section" actually dikhti kaise hai

Sab kuch ek seedhi round tube ke andar hota hai. Use seedha kaat ke dekho toh ek circle nazar aata hai. Use lengthways kaat ke dekho toh fluid ka ek lamba rectangle dikhta hai. Hum inhi do views ke beech switch karte rahenge, isliye pehle inhe apne dimag mein fix karo.

Circle (barrel ke andar se dekhne par) wahan hai jahan hum maapte hain ki fluid ka ek tukda centre se kitna door hai.
Lamba slice (side se dekhne par) wahan hai jahan hum fluid ko tube ke saath move karte dekhte hain.

1. $r$ — centre line se doori

Picture: circle view par, bullseye se bahar ki taraf ek arrow kheencho. Uski length $r$ hai.

Topic ko iske zaroori hone ki wajah: fluid ki speed centre se alag-alag doooriyon par alag-alag hoti hai. "Yahan speed vs wahan speed" describe karne ke liye "yahan" ka ek label chahiye — woh label $r$ hai.

2. $R$ — pipe ka radius (wall)

Picture: usi circle par, $R$ woh arrow hai jo bilkul edge ko touch karta hai. Koi bhi $r$ satisfy karta hai $0 \leq r \leq R$ .

3. $L$ — pipe ki length

Picture: side view par, $L$ rectangle ka horizontal span hai.

Topic ko iske zaroori hone ki wajah: push ko poori length ke saath friction overcome karna padta hai. Lamba pipe = zyada wall se fight = same flow ke liye bada push chahiye.

4. $v$ — fluid ki speed, aur $v (r)$

Picture: har distance $r$ par pipe ke saath ek chota arrow kheencho; lamba arrow = tez. Saare arrows ko stack karo aur unke tips ek curve trace karte hain — velocity profile.

Notation $v (r)$ kyun? Yeh ek machine hai: distance daalo, speed niklo. Yeh poore topic ka sabse important object hai — parabola $v (r) = \frac{Δ P}{4 η L} (R^{2} - r^{2})$ bas uss machine ka ek rule hai.

5. $η$ — viscosity (stickiness number)

Picture: fluid ki do sheets ek dusre ke saath slide karti hain. Bada $η$ = woh ek dusre par kheenchti hain (honey). Chota $η$ = woh freely slide karti hain (paani).

Topic ko iske zaroori hone ki wajah: stickiness hi woh cheez hai jo flow ko resist karti hai. Zyada $η$ matlab same result ke liye zyada push karna padega — poori law ke liye Viscosity and Newton's law of viscosity dekho.

6. $\frac{d v}{d r}$ — velocity gradient (calculus kyun chahiye)

Speed bahar jaane par change hoti hai. Lekin kitni tez change hoti hai? "Speed ka distance ke saath change ka rate" yahi $\frac{d v}{d r}$ ka matlab hai.

Picture: velocity-profile curve par, $\frac{d v}{d r}$ ek point par curve ka slope hai — beech mein flat (speed muskil se change ho rahi hai), wall ke paas steep (speed tezi se gir rahi hai).

Negative kyun? Bahar jaane par ( $r$ badhne par) speed girh jaati hai, toh slope negative hai: $d v / d r < 0$ . Parent derivation mein minus signs bas yahi fact hain.

7. $τ$ — shear stress (drag force per area)

Words mein padho: drag ( $τ$ ) = stickiness ( $η$ ) × kitni-tez-speed-change-hoti-hai ( $d v / d r$ ). Sticky fluid aur steep speed change dono rubbing ko badha dete hain.

"Per area" kyun? Ek badi surface ek choti se zyada rubbing karti hai, isliye hum drag per square metre quote karte hain aur baad mein real area se multiply karte hain. Parent note mein woh area ek cylinder ka curved side hai, $2 π r L$ .

8. $P$ , $P_{1}$ , $P_{2}$ aur $Δ P$ — pressure aur push

Picture: left end par high pressure squeeze ho raha hai, right par lower pressure — fluid high se low ki taraf nikalta hai, jaise squeeze ki gayi tube se toothpaste.

Topic ko iske zaroori hone ki wajah: pressure difference ke bina, viscosity fluid ko ruk jaane par majboor kar deti. $Δ P$ ki wajah se kuch bhi move hota hai.

9. $π r^{2}$ , $2 π r L$ , aur $Q$ — areas aur flow rate

Picture: imagine karo ki ek second mein pipe se nikla poora fluid ek bucket mein pakad lo — bucket mein jo volume hai woh $Q$ hai.

Dono areas kyun? Push flat end par act karta hai ( $π r^{2}$ ); friction side par act karta hai ( $2 π r L$ ). Inhi dono ko balance karna poori derivation hai. $Q$ alag-alag pipe widths ko kaise link karta hai uske liye Equation of continuity dekho.

Sab kuch kaise fit hota hai

Har arrow ka matlab hai "banane ke liye zaroori hai". Upar se loose pieces se $Q$ tak, jo payoff hai, padho.

Equipment checklist

Cover the right side and say each answer out loud before revealing.

r

kya measure karta hai, aur uski sabse choti value kya hai?

Distance from the pipe's centre line; sabse choti value

0

hai (centre par).

R

kya hai aur yeh

r

se kaise alag hai?

R

fixed wall radius hai (ek constant);

r

ek moving pointer hai,

0 \leq r \leq R

v (r)

ka words mein kya matlab hai?

Ek machine: distance

r

daalo, wahan fluid ki speed niklo.

No-slip condition ko equation mein likho.

v (R) = 0

— wall ko touch karne wala fluid stuck hai.

η

kya describe karta hai, aur honey ka

η

bada hai ya chota?

Stickiness / viscosity; honey ka bada hai, paani ka chota.

Plain words mein,

\frac{d v}{d r}

kya hai?

Speed curve ki steepness — bahar ki taraf ek tiny step lene par speed kitni change hoti hai.

Pipe mein

\frac{d v}{d r}

negative kyun hai?

Bahar jaane par speed girh jaati hai, isliye slope negative hai.

Newton's law of viscosity likho.

τ = η \frac{d v}{d r}

Δ P

kya hai aur yeh kya karta hai?

Pressure difference

P_{1} - P_{2}

; yeh woh push hai jo flow drive karta hai.

Pressure kis area par act karta hai, drag kis par?

Pressure flat end

π r^{2}

par; drag curved side

2 π r L

par.

Q

kya measure karta hai?

Fluid ka volume jo per second ek cross-section se guzarta hai (

m^{3} / s

Connections

Viscosity and Newton's law of viscosity — $η$ , $τ$ aur $d v / d r$ ka source.
Equation of continuity — $Q$ = area × speed kahan se aata hai.
Reynolds number and turbulence — yeh batata hai ki yeh laminar picture kab valid hai.
Bernoulli's principle — frictionless cousin (koi viscosity nahi).
Stokes' law and terminal velocity — pipe ki jagah sphere par viscous drag.
Blood flow and circulatory system — arteries ke liye $R^{4}$ rule kyun matter karta hai.