Fluid kinematics — Eulerian vs Lagrangian description: Question bank

2.2.9 · D5 · Physics › Fluid Mechanics › Fluid kinematics — Eulerian vs Lagrangian description

Shuru karne se pehle, yeh do pictures apne dimag mein rakh lo (poora topic inhi par tika hai):

Machli — tum fluid ke ek lump ko tag karte ho aur uski journey follow karte ho. Yeh Lagrangian view hai; iske saath jaane wala derivative hai $\frac{D}{D t}$ (particle ka name tag fixed rakhta hai).
Pul — tum ek fixed jagah par khade ho aur jo bhi fluid guzarta hai use dekhte ho. Yeh Eulerian view hai; iske saath jaane wala derivative hai $\frac{\partial}{\partial t}$ (position fixed rakhta hai).

Dono ko jodne wala number convective term $(v \cdot \nabla) ϕ$ hai, jo measure karta hai "field ki value kitni badlti hai sirf isliye kyunki particle nai jagah par move ho gaya."

True ya false — justify karo

"Steady" flow ka matlab hai ki har fluid particle constant speed se move karta hai.

False. Steady ka matlab hai ki har fixed point par field time ke saath nahi badalta (

\partial / \partial t = 0

), na ki kisi particle ki speed constant hai. Ek particle phir bhi speed up kar sakta hai ek tez region mein jaake — yahi convective term

(v \cdot \nabla) v

hai.

Steady flow mein fluid particle ki acceleration hamesha zero hoti hai.

False. Sirf local part

\partial v / \partial t

vanish hota hai. Convective part

(v \cdot \nabla) v

bahut bada ho sakta hai — jaise ki fluid ek nozzle throat mein accelerate karta hai jabki flow pattern khud kabhi nahi badalta.

\frac{D ϕ}{D t}

aur

\frac{\partial ϕ}{\partial t}

ek hi derivative ke bas do notations hain.

False.

\partial / \partial t

position fixed rakhta hai (bridge ki reading);

D / D t

particle label fixed rakhta hai (machli ki reading). Dono mein exactly

(v \cdot \nabla) ϕ

ka antar hai, aur yeh sirf tab agree karte hain jab woh term zero ho.

Agar flow unsteady hai (

\partial v / \partial t \neq = 0

), toh har particle accelerate kar raha hoga.

Generally False. Local aur convective terms cancel ho sakti hain:

\frac{\partial v}{\partial t} + (v \cdot \nabla) v

kisi point par zero sum ho sakta hai chahe har piece nonzero ho, jis se wahan particle acceleration zero milti hai.

Ek hi flow ki Lagrangian velocity aur Eulerian velocity numerically alag quantities hain.

False — yeh same velocity hai, bas alag tarike se label ki gayi hai. Lagrangian ise particle tag

a

aur

t

ki function ke roop mein likhta hai; Eulerian same value ko current position

x

aur

t

ki function ke roop mein likhta hai. Same particle/jagah/time par evaluate karne par dono agree karte hain.

Material derivative sirf scalar (jaise temperature) par apply hoti hai, vector par nahi.

False. Yeh particle ki kisi bhi property par apply hoti hai. Scalar ke liye ek equation milti hai; velocity jaise vector ke liye ise component-by-component apply karte hain, jis se acceleration

\frac{D v}{D t}

milti hai.

Agar

\nabla ϕ = 0

har jagah ho, toh material derivative local derivative ban jaati hai.

True. Convective term

(v \cdot \nabla) ϕ

zero ho jaata hai jab

ϕ

mein koi spatial variation nahi hoti, isliye particle sirf field ke time-change ko feel karta hai — move karne par koi "alag jagah" nahi milti.

Convective term hi fluid dynamics ko mathematically mushkil banata hai.

True (velocity ke liye).

(v \cdot \nabla) v

velocity ko uske apne gradient se multiply karta hai, jis se equations nonlinear ho jaati hain — yahi nonlinearity turbulence ki jad hai aur isliye closed-form solutions rare hain.

Error dhundho

"Flow steady hai, isliye drift karne wale parcel ke liye

D T / D t = 0

hoga."

Galat. Steady sirf

\partial T / \partial t

ko khatam karta hai. Agar temperature space mein vary karti hai, toh parcel phir bhi

(v \cdot \nabla) T \neq = 0

feel karta hai jab woh garam ya thande fluid mein drift karta hai. Steady

\neq =

zero material rate.

"Ek student ne path

x = a e^{t}

u = a e^{t}

derive kiya aur ise Eulerian velocity field bataya."

Galat — ismein abhi bhi particle label

a

hai, jo ek Eulerian field mein nahi ho sakta. Path invert karo (

a = x e^{- t}

) aur substitute karo taaki

u (x, t) = x

mile, jo sirf current position aur time ki function hai.

"Kyunki instruments fixed points par baithe hain, Newton ka doosra niyam naturally Eulerian frame mein likha jaata hai."

Galat.

F = ma

ek specific particle ki acceleration ke baare mein statement hai, isliye yeh naturally Lagrangian hai. Hume Lagrangian acceleration ko Eulerian language mein convert karna padta hai

\frac{D v}{D t} = \frac{\partial v}{\partial t} + (v \cdot \nabla) v

use karke, phir ise fixed instrument points par apply karte hain.

(v \cdot \nabla) ϕ

ka matlab hai

v

aur

ϕ

ka dot product lena."

Galat.

ϕ

(ek scalar) ki koi direction nahi hoti jisse dot liya ja sake. Operator

v \cdot \nabla = u \partial_{x} + v \partial_{y} + w \partial_{z}

ϕ

par act karta hai; pehle operator

v \cdot \nabla

banao, phir use

ϕ

differentiate karne do.

"Lagrangian description mein

x

ek independent variable hai jise main freely vary kar sakta hoon."

Galat. Lagrangian bookkeeping mein label

a

independent variable hota hai;

x = x (a, t)

ek dependent output hai jo batata hai ki woh tagged particle kahan gaya. Eulerian description mein

x

independent hota hai.

"Ek fixed thermometer aur hawaon ke saath drift karta balloon temperature change ki same rate report karenge."

Galat. Thermometer

\partial T / \partial t

padhta hai (sirf local); balloon

D T / D t = \partial T / \partial t + (v \cdot \nabla) T

padhta hai, jo naye temperatures mein ride karne se aane wala change add karta hai. Yeh tab hi match karte hain jab flow speed ya spatial temperature gradient zero ho.

Kyun wale sawaal

Hum do descriptions kyun rakhte hain ek choose karne ki bajaye?

Kyunki physics (Newton ka niyam) naturally Lagrangian hai — particles ke baare mein — jabki hamare measurements Eulerian hain — fixed points par. Material derivative woh bridge hai jo hume particle ke niyam field variables mein likhne deta hai jo hum actually measure kar sakte hain.

Convective term aata hi kyun hai — particle ka rate field ka time rate kyun nahi hota?

Kyunki particle move karta hai. Do instants ke beech woh ek nai location par hota hai jahan field ki already alag value ho sakti hai. Convective term us "relocation ki wajah se change" ka hisaab rakhta hai, jise

\partial / \partial t

(fixed spot) bilkul miss kar deta hai.

Acceleration term

(v \cdot \nabla) v

ko nonlinear kyun kaha jaata hai?

Kyunki velocity, velocity ke derivative se multiply hoti hai — unknown

v

uske apne spatial gradient se.

v

ko double karne par yeh term zyada se zyada double ho jaata hai, isliye equation

v

mein linear nahi hai.

Eulerian field mein convert karte waqt particle label kyun hatana zaroori hai?

Kyunki Eulerian field ka jawab hai "yahan, abhi kya value hai?" — yeh depend nahi kar sakta ki kaun sa particle guzar raha hai. Label

a

sirf Lagrangian information hai, isliye ise path equation ke zariye

(x, t)

se replace karna padta hai.

Chain rule material derivative mein exactly do tarah ke terms kyun produce karta hai?

Kyunki

ϕ (x (t), y (t), z (t), t)

mein time dependence ke do sources hain: explicit

t

slot (local term deta hai) aur moving coordinates

x (t), y (t), z (t)

(teen convective pieces dete hain). Chain rule inhe saaf alag kar deta hai.

Pipe wall par laga pressure gauge seedha particle ki material rate report kyun nahi kar sakta?

Kyunki woh ek location par chipka hua hai aur wahan

\partial p / \partial t

measure karta hai. Uske paas koi tarika nahi hai ki woh

(v \cdot \nabla) p

contribution sense kare jo ek moving parcel feel karta hai — uske liye parcel ko follow karna padta hai, jo ek fixed gauge nahi kar sakta.

Edge cases

Agar fluid rest par hai (

v = 0

) har jagah, toh material derivative kya ban jaata hai?

Yeh local derivative

\partial ϕ / \partial t

par collapse ho jaata hai, kyunki convective term

(v \cdot \nabla) ϕ

zero velocity se multiply hota hai. Koi motion nahi toh Lagrangian aur Eulerian rates coincide karte hain.

Ek uniform field (har jagah same value) ke liye jo time mein vary karta hai, ek moving particle kya feel karta hai?

Sirf local rate

\partial ϕ / \partial t

. Spatial gradient zero hai isliye convective term vanish ho jaata hai — move karna matter nahi karta jab har location par same value ho.

Socho ek particle momentarily rest par hai (

v = 0

ek instant par) ek unsteady flow mein. Kya uski material acceleration zero hai?

Zaroor nahi. Us instant convective term zero hai, lekin local term

\partial v / \partial t

nonzero ho sakta hai, isliye particle phir bhi rest se accelerate kar sakta hai.

Perfectly steady flow mein ek streamline ke along jahan speed bhi har jagah constant ho, particle ki acceleration kya hai?

Flow direction mein zero — dono terms vanish hote hain (local steadiness se, convective kyunki speed mein koi spatial change nahi). Lekin agar streamline curved hai, toh convective term constant speed par bhi centripetal acceleration deta hai.

Bahut slow flow aur strong time-dependence ki limit mein dono descriptions ka distinction kya hoga (jaise slowly seeping fluid under a fast-changing tap)?

Convective term (chhota

v

) local term ke muqable mein negligible ho jaata hai, isliye

D / D t \approx \partial / \partial t

. Eulerian aur Lagrangian rates lagbhag coincide karte hain kyunki relocation bahut kam contribute karta hai.

Agar do alag particles same point

x

par do alag times par hain, kya unki wahan same velocity hogi?

Unsteady flow mein, nahi — Eulerian value

v (x, t)

t

par depend karti hai, isliye har ek ek alag local velocity mein aata hai. Steady flow mein, haan —

x

par field time-independent hai, isliye us point se guzarne wale har particle ki wahan same velocity hoti hai.

Recall Jaane se pehle self-check karo

$\frac{D ϕ}{D t}$ ka kaun sa term steady flow mein bachta hai? ::: Convective term $(v \cdot \nabla) ϕ$ ; local term $\partial ϕ / \partial t$ zero hota hai. Kaun si ek quantity Eulerian time rate ko Lagrangian mein convert karti hai? ::: Convective term $(v \cdot \nabla) ϕ$ , $\partial ϕ / \partial t$ mein add hoke. Lagrangian velocity ko Eulerian field banane ke liye kya hatana padta hai? ::: Particle label $a$ , jo path equation $x = x (a, t)$ ke zariye eliminate hota hai.