2.2.10 · D1Streamlines, pathlines, streaklines
Ek fluid flow kuch nahi hai sirf space ke har point par ek arrow jo batata hai ki fluid kidhar move kar raha hai aur kitni tezi se. Streamlines, pathlines, aur streaklines sirf teen alag-alag tarike hain un arrows ke field se pictures banane ke — toh in teen ko alag karne se pehle, hume ek single arrow padhna aana chahiye.
Yeh page kuch bhi assume nahi karta . Agar parent note Streamlines, pathlines, streaklines ne koi symbol throw kiya aur tum blink kar gaye, toh woh yahan milega. Hum har notation piece list karenge, batayenge ki woh plain words mein kya matlab rakhta hai , picture dikhayenge jo woh represent karta hai, aur explain karenge kyun yeh topic uske bina nahi chal sakta .
( x , y ) ek location ko pin karte hain. x y origin ( 0 , 0 )
Picture: graph paper par ek dot. x y Topic ko kyun chahiye: fluid space bharta hai, toh har sawaal — "yeh leaf kahan hai?", "mein dye kahan inject karun?" — ek pair ( x , y )
Hum sirf 2D mein kaam karenge (ek flat sheet) taaki har picture draw ho sake. Real flows mein ek teesra number z
τ doosra time-label jo sirf streaklines ke liye use hota hai: woh moment jab kisi chimney se ek particular smoke puff nikla.
Picture: smoke ka har puff ek name-tag pehne hota hai jis par likha hai "Main τ = 1.3 s Kyun: ek streakline mein bahut saare alag-alag τ abhi time t t τ
t τ t present moment hai (jab tum photo lete ho). τ birth moment hai. Ek streakline mein tum t freeze karte ho aur τ τ ≤ t sweep karte ho.
Intuition Vector = ek arrowEk vector ek arrow hai: uski ek length hoti hai (kitna bada) aur ek direction hoti hai (kidhar). Do numbers ise describe karte hain — x y
v = ( u , v )
u x v y Picture: ek arrow jisme ek horizontal leg u v Kyun: velocity mein dono speed aur heading hoti hai, toh ek plain number usse hold nahi kar sakta. Ek arrow kar sakta hai.
Common mistake Do cheezein dono "v v poora velocity vector hai. Plain v vertical part hai. Parent note dono use karta hai. Doubt ho toh chhota arrow dhundho.
Ab hum arrows ko har point par chipkate hain.
Definition Velocity fieldv ( r , t ) = ( u ( x , y , t ) , v ( x , y , t ) ) r = ( x , y ) t
r = ( x , y ) u ( x , y , t ) depend karti hai tum kahan ho aur kab ho." v Picture: ek poori sheet jisme chhote arrows bichhe hain, ek per point — arrows ka ek "field". Time-slider drag karo aur har arrow swing kar sakta hai.
Kyunki field of view ke har point par ek value (ek arrow) hai, jaise ek room mein har jagah temperature hoti hai. Yeh Eulerian picture hai — fixed points watch karo, fixed particles nahi (dekho Lagrangian vs Eulerian description ).
Kisi bhi equation se pehle, yahan bataya gaya hai ki teen drawings mein se har ek kya hai — ek sentence each, koi symbol nahi siwaay us arrow-field ke jo ab tum samajhte ho.
Definition Streamline (snapshot)Ek ==streamline ek aisi curve hai jo ek fixed instant t
Picture: time freeze karo, phir arrows ke through ek smooth line thread karo taaki line hamesha us arrow ki taraf point kare jo uske neeche hai.Kyun: yeh capture karta hai "ABHI SAB KUCH KIDHAR POINT KAR RAHA HAI" — ek moment par poore field ki picture.
Definition Pathline (trail)Ek pathline ek tagged fluid particle ki actual trajectory hai jo time aage badhne par sweeps out hoti hai. Ek particle, saare times ke across follow kiya.
Picture: ek single dyed leaf drop karo aur uski poori journey film karo; woh wet trail jo woh chhodta hai woh pathline hai.Kyun: yeh jawab deta hai "YEH particle actually kahan gaya", arrows ko integrate karta hai jo particle moment by moment milta hai.
Definition Streakline (plume)Ek ==streakline, ek fixed instant t
Picture: ek chimney se continuously smoke phunkne karo aur hanging plume ki photo lo — har puff, alag τ abhi ek saath dikh rahe hain.Kyun: yeh exactly wahi hai jo ek dye ya smoke experiment dikhata hai, toh hume pata hona chahiye ki yeh dono se kaise alag hai.
Is page ka baaki hissa un symbols ko build karta hai jin par yeh teen definitions secretly rely karti hain.
d x , d y d x tiny step. d y tiny step. "Tiny" ka matlab hai itna chhota ki curve us step ke upar straight dikhti hai.
Picture: kisi bhi curve mein zoom karo jab tak woh ek straight micro-segment na lag jaye; d x d y d r = ( d x , d y ) curve ke along .Kyun: ek streamline ek curve hai, aur "curve velocity ki tarah same direction mein point karta hai" ek statement hai uske tiny direction arrow d r
Definition Scaling factorλ λ λ ∈ R stretch ya shrink karta hai. Yeh kehna ki d r = λ v d r v λ
Picture: tiny step arrow d r v λ
Common mistake Division by zero — jabu = 0 v = 0 u d x = v d y dono components nonzero hain . Agar u = 0 u nahi kar sakte; balki directly d x = λ u = 0 vertically chalti hai (wahan x v = 0 horizontal streamline deta hai (y dono u = 0 v = 0 stagnation point — jahan koi single direction exist nahi karta aur streamline branch ho sakti hai. Pure-horizontal, pure-vertical, aur zero cases ko hamesha seedha d x = λ u , d y = λ v
Definition Ek moving particle ka position vectorr ( t ) = ( x ( t ) , y ( t )) time ke functions ban jaate hain: x ( t ) , y ( t ) r ( t ) = ( x ( t ) , y ( t ))
Picture: ek arrow jo origin se rooted hai jiska tip particle ke saath ride karta hai, jaise t Kyun: pathline hai r ( t )
d t d x d t x d x rightward speed hai.
Picture: x t Topic ko kyun chahiye: particle ki velocity uski position vector ki time-derivative hai, d t d r = ( d t d x , d t d y ) d t d x = u , d t d y = v
Common mistake Streamline ratio vs pathline derivative — key confusionu d x = v d y t d t d x = u t
Definition Integrating (tiny steps ko add karna)Derivative ka ulta: rate diya ho, toh poori journey ko saare tiny steps sum karke rebuild karo. Symbol ∫
Picture: bahut saare tiny d x
x 0 t 0 C t 0 initial time , woh instant jab tum tracking shuru karte ho (clock reading jab particle apne start point par baitha hai). x 0 initial position , toh x ( t 0 ) = x 0
Integral se bachne wala ==constant C starting information hai jo rate akele recover nahi kar sakta; ise x ( t 0 ) = x 0 kaunsi curve par ho.
Kyun: u d x = v d y d t d x = u matlab hai integrate karna. Alag ( x 0 , t 0 ) C
Worked example Tiny warm-upLo d t d x = 1 x 0 t 0 x ( t 0 ) = x 0 t 0 t x = x 0 + ∫ t 0 t 1 d t ′ = x 0 + ( t − t 0 ) . C x 0 x ( t 0 ) = x 0
∂ t ∂ v ∂ position fixed rakho, sirf time wiggle karo, aur pucho ki us fixed spot par arrow change hota hai ya nahi.
Picture: ek dot par khade raho, uska arrow dekho, aur clock watch karo. Agar arrow kabhi nahi hila, yeh quantity zero hai.Kyun: parent ka golden rule isi par hinge karta hai. Agar ∂ t ∂ v = 0 steady hai aur teeno curves ek saath coincide karti hain; warna woh alag ho jaati hain (dekho Steady vs unsteady flow ).
Intuition Steady ka matlab phir bhi motion hai!Steady ka matlab "kuch nahi hilta" nahi hai. Particles abhi bhi tezi se daud rahe hain. Iska matlab hai ki arrows ka pattern frozen hai — har fixed point hamesha same arrow dikhata hai, toh naye aane wale sab same route trace karte hain.
Tiny step d r along a curve
Streamline ratio dx/u = dy/v
Integration and constant C
Partial dv/dt steady test
Streamlines pathlines streaklines
Test karo khud ko — right side cover karo.
( x , y ) Haan: x y ( 0 , 0 )
t τ t τ t τ ≤ t
v v Chhota arrow
matlab full vector
( u , v ) ; bare
v sirf
y -component hai.
Words mein bata sakta hoon ki streamline, pathline, aur streakline kya hain Streamline = ek instant t
Velocity field ko har point par ek arrow ki tarah picture kar sakta hoon Ek sheet jisme arrows bichhe hain, har location par ek, possibly time ke saath swing karte hue.
Jaanta hoon ki d r = ( d x , d y ) along tiny arrow hai Zoom karo jab tak curve ek straight micro-segment na ban jaye; uske legs d x , d y
Jaanta hoon λ λ ∈ R d x = λ u , d y = λ v u d x = v d y
Jaanta hoon jab u = 0 v = 0 Divide mat karo; d x = λ u , d y = λ v u = 0 v = 0
Jaanta hoon r ( t ) = ( x ( t ) , y ( t )) Origin-rooted arrow jiska tip particle ke saath ride karta hai aur pathline sweep karta hai.
d t d x x t x
Jaanta hoon constant C x 0 , t 0 Integration start recover nahi kar sakta; C x ( t 0 ) = x 0
Jaanta hoon ∂ t ∂ v = 0 Har fixed point par arrow kabhi nahi badla; steady =
Velocity field and material derivative Steady vs unsteady flow Lagrangian vs Eulerian description Stream function ψ Continuity equation Flow visualization techniques (dye, smoke, PIV) 
