2.2.10
Sochो ek pond pe wind se patte ud rahe hain. Flow ko "draw" karne ke teen alag tarike hain, aur wo tabhi same dikhte hain jab flow steady ho (time ke saath change na ho).
Ek streamline ek snapshot hai — "abhi is waqt sab cheez KAHAN point kar rahi hai?"
Ek pathline ek particle ka trail hai time ke saath — "YE patta kahan kahan gaya?"
Ek streakline smoke plume hai — "wo saare particles jo kabhi bhi EK point se guzre."
WHY care? Kyunki experiments tumhe streaklines dete hain (dye/smoke), theory tumhe streamlines deti hai (velocity field), aur tracking pathlines deti hai. Inhe confuse karna unsteady flow mein galat answers deta hai.
Definition Velocity fieldEk fluid flow ko space aur time ke har point pe ek velocity vector se describe kiya jaata hai:
v ( r , t ) = ( u ( x , y , t ) , v ( x , y , t ) ) MATLAB kya hai: ( x , y ) t v
Ek flow steady ∂ t ∂ v = 0 bhale hi individual particles move karte rahein .
Intuition Streamline KYA hai?Ek aisi curve jo is tarah draw ki ho ki ek instant mein, uski tangent har jagah local velocity ke parallel ho. Koi bhi fluid streamline ko cross nahi karta (construction se hi velocity uske saath saath point karti hai).
Worked example Worked: rotating fieldv = ( − y , x ) Step 1 Equation likho: − y d x = x d y Kyun? u = − y , v = x u d x = v d y Step 2 Cross-multiply karo: x d x = − y d y Kyun? Bas ratios ko rearrange karna hai variables separate karne ke liye.
Step 3 Integrate karo: 2 x 2 = − 2 y 2 + C ⇒ x 2 + y 2 = R 2 Kyun? Dono sides exact differentials hain. Result: streamlines concentric circles hain — samajh aata hai, field swirl karta hai.
Actual trajectory jo ek fluid particle trace karta hai jab time aage badhta hai. MATLAB: ek single particle ko tag karo aur use film karo; film ka trail hi pathline hai.
Worked example Worked: unsteady fieldu = 1 , v = t Step 1 d t d x = 1 ⇒ x = x 0 + ( t − t 0 ) Kyun? Constant horizontal speed integrate karo.
Step 2 d t d y = t ⇒ y = y 0 + 2 1 ( t 2 − t 0 2 ) Kyun? t t Step 3 t t = t 0 + ( x − x 0 ) y Result: released particle ke liye ek parabola milti hai.
Fixed time t un saare particles ka locus jo ek fixed point ( x p , y p ) τ ≤ t MATLAB: ek point par continuous dye inject karo; jo visible streak dikhti hai wahi streakline hai.
Worked example Same fieldu = 1 , v = t Step 1 τ x = ( t − τ ) y = 2 1 ( t 2 − τ 2 ) τ Kyun? τ x 0 = y 0 = 0 Step 2 Observation time fix karo, maan lo t = 2 τ ≤ 2 ( x , y ) Result: pathline se alag ek curve milti hai — yahi prove karta hai ki unsteady flow mein teeno alag hote hain.
Steady flow ⇒ streamlines = pathlines = streaklines WHY: agar v
Common mistake Common errors ko steel-man karoMistake 1: "Streamline dikhata hai ki particle kahan jaata hai."
Kyun sahi lagta hai: particle actually local velocity ke saath move karta hai. Flaw: jab tak particle agle point tak jaata hai, unsteady flow mein wahaan ka field change ho chuka hota hai, isliye uska path original streamline se alag ho jaata hai. Fix: streamline = instantaneous snapshot; pathline = time-integrated trajectory.
Mistake 2: "Streaklines aur pathlines same cheez hain."
Kyun sahi lagta hai: dye injection se lagta hai ek particle trace ho raha hai. Flaw: streakline alag alag times par release hue alag alag particles ko connect karti hai, sabko abhi dekha jaata hai. Pathline ek particle ko saare time ke saath dikhata hai. Fix: yaad rakho "streak = kaafi particles, ek source, ek instant."
Mistake 3: streamline ODE mein t
Fix: u d x = v d y t
Recall Feynman: 12-saal ke bachche ko samjhaoEk river imagine karo jisme patte hain aur ek smoke machine hai.
Streamline: Ek photo lo. Chhote chhote arrows banao jo dikhayein paani har jagah kahan push kar raha hai. Arrows ko connect karo — wahi streamline hai. Ye "abhi ka" hai.Pathline: EK patta chhordo aur dekho wo poore din kahan baha. Jo wet trail wo chhod ke jaata hai wahi uski pathline hai. Ye "ek patte ki poori kahani" hai.Streakline: SAME chimney se smoke lagaatar phunkna band mat karo. Jo smoke hawa mein dikhrahi hai wahi streakline hai — ek hi jagah se saare smoke puffs, ek saath dekhe gaye.
Agar river kabhi apna mood nahi badalta (steady), to teeno drawings same ho jaati hain!
Mnemonic Teeno yaad rakho"SNAP, TRAIL, SMOKE"
S treamline = SNAP shot (abhi, pura field)P athline = ek particle ka TRAIL (ek particle, poora time)StreaK line = ek chimney ka SMOKE (kaafi particles, ek spot, abhi)
Streamline kya hai? Ek aisi curve jis ki tangent har jagah instantaneous velocity field ke parallel ho (ek snapshot); koi flow ise cross nahi karta.
Pathline kya hai? Ek single fluid particle ki actual trajectory jo time ke saath trace hoti hai.
Streakline kya hai? Ek fixed instant par, un saare particles ka locus jo ek fixed point se guzre hain.
2D mein Streamline ODE? u d x = v d y t
Pathline equations? d t d x = u ( x , y , t ) , d t d y = v ( x , y , t )
Teeno kab coincide karte hain? Steady flow mein (
∂ v / ∂ t = 0 ).
Streamline equation mein time variable hai ya constant? Ek constant (frozen) parameter.
v = ( − y , x ) Concentric circles x 2 + y 2 =
Continuous dye injection wala experiment kaunsi curve dikhata hai? Ek streakline.
Streakline aur pathline mein key difference? Streakline = kaafi particles (ek source) ek instant mein; pathline = ek particle poore time mein.
Velocity field and material derivative Steady vs unsteady flow Continuity equation Stream function ψ Lagrangian vs Eulerian description Flow visualization techniques (dye, smoke, PIV)
all particles through one point
