Question bank — Streamlines, pathlines, streaklines

False — ek crossing par fluid ko ek saath do alag velocity directions ki zaroorat hogi, lekin $\vec{v}$ ka ek point par exactly ek hi value hota hai. Crossings sirf wahan allowed hain jahan $\vec{v}=0$ ho (stagnation points), jahan "direction" undefined hai.

True — agar field time ke saath kabhi nahi badlata, toh ek particle har point par jo direction "dekhta" hai wahi hoti hai jo streamline ne draw ki thi, toh wo kabhi alag nahi ho sakti. Ye parent note par boxed golden rule hai: steady flow ⇒ streamlines = pathlines = streaklines.

Zaroori nahi — wo us instant par tangent hoti hain (dono local $\vec{v}$ follow karti hain), lekin generally baad mein diverge ho jaati hain. Wo coincidence se kisi stretch par coincide bhi kar sakti hain; unsteadiness sirf difference ko allow karti hai, ye har jagah ise force nahi karti.

True — wo particle jiski release time current instant ke barabar hai ($\tau =t$) abhi bhi source $(x_p,y_p)$ par baitha hai, toh source point har moment streakline ka ek end hota hai.

False — construction se velocity streamline ke tangent hoti hai, toh line ke across uska component zero hota hai. Exactly isi liye streamlines flow reasoning mein impermeable "walls" ki tarah act karti hain.

True — unsteady flow mein poora field har instant khud ko redraw karta hai, toh snapshot badal jaata hai. Steady flow mein pattern hamesha ke liye frozen rehta hai.

True — alag particles same point se alag times par guzar sakte hain. Pathlines time-parametrised trails hain, toh ek spatial crossing ka matlab sirf "same jagah, alag clock reading" hai, jo bilkul theek hai.

False — general rule ye hai ki ek regular point ($\vec{v}\ne 0$) par har streamline ko wahan $\vec{v}$ ki single well-defined direction follow karni hogi, toh us point ko share karne wali do curves usi tangent aur usi continuation ko share karengi, unhe identical streamline banane par majboor karengi. Alag streamlines ki tangency isliye sirf ek stagnation point ($\vec{v}=0$) par possible hai, jahan direction undefined hai.

Error — streamline ek instant par defined hoti hai, toh jo geometry ye capture karti hai woh hai "is frozen moment par arrows kis taraf point karte hain." Mathematically iska matlab hai ki field values $u(x,y,t)$ aur $v(x,y,t)$ ko ek fixed $t$ par evaluate karna hoga: equation sirf space variables $x,y$ mein ek ODE hai, jisme $t$ ek constant parameter ke roop mein aata hai. Agar aap integrate karte waqt $t$ ko advance karte hain, toh aap $dx$ ko $dy$ se ek snapshot par relate nahi kar rahe — aapne field ke time-changes smuggle kar liye hain aur na streamline na pathline produce ki hai.

Error — ek plume ek streakline hai: ye bahut saare alag parcels ko jodta hai jo bahut saare alag times $\tau$ par release hue, sab abhi dekhe gaye. Ek parcel ka actual route uski pathline hai, jo generally unsteady wind mein ek alag curve hoti hai.

Error — ye streamline ODE hai (ek spatial relation). Pathline ke liye time ODEs $\frac{dx}{dt}=u,\frac{dy}{dt}=v$ chahiye jo $t_0$ par $(x_0,y_0)$ se forward integrate honi chahiye. Time driving variable hai, frozen label nahi.

Error — ye sirf steady flow mein coincide hoti hain. Ise generally assume karna silently woh "history" drop kar deta hai jo unsteady dye patterns ko single-particle trails se alag banati hai. Pehle $\partial \vec{v}/\partial t=0$ check karein.

Error — ye har jagah instantaneous direction dikhata hai. Unsteady flow mein, jab tak particle agli point par pahunchta hai field wahan badal chuka hota hai, toh particle ka real route (uski pathline) us streamline se alag ho jaata hai jis par wo tangent tha.

Half-error — circular streamlines ka matlab hai snapshot circular hai; kya particles poore circles trace karte hain ye flow ke steady hone par depend karta hai. $\vec{v}=(-y,x)$ ke liye flow steady hai, toh yahan theek hai — lekin reasoning "streamline shape = particle path" sirf steadiness ki wajah se valid hai, automatically nahi.

Kyunki aap ek spot par continuously marker inject karte hain aur usse photograph karte hain — woh visible ribbon by definition saare particles hain jo source se guzre, yaani ek streakline. Dekhen Flow visualization techniques (dye, smoke, PIV). Iske bajaaye streamlines paane ke liye ek instant par poore velocity field ko measure karna zaroori hai.

Ek pathline ek particle ko time ke through follow karti hai, toh time woh cheez hai jo advance ho rahi hai; ek streamline ek single-instant snapshot hai, toh time fixed rakha jaata hai aur sirf space ($x,y$) vary karta hai. $t$ ka alag role donon ke beech sabse gehri structural difference hai.

$\partial \vec{v}/\partial t=0$ ke saath field ko kab ka koi memory nahi hota — har particle jo ek point par aata hai wahi direction follow karta hai jo field ne hamesha rakhi, toh snapshot, single trail, aur many-particle streak sab ek hi locus trace karte hain. Ye golden rule hai.

Kyunki ek streamline ek instant par saari space par defined hoti hai, jabki ek particle ka route time par defined hota hai. Unsteady flow mein field moving particle ke neeche badal jaata hai, toh uski trail (pathline) generally us streamline se alag ho jaati hai jis par wo tangent thi. Dono alag sawaalon ke jawab deti hain.

Kyunki "saare particles jo source se guzre" ka matlab hai ek particle per release time $\tau$; har ek apni pathline obey karta hai, aur hum unhe current $t$ par ek saath photograph karte hain. $t$ freeze karke aur $\tau$ sweep karke unhe ek visible curve mein thread kiya jaata hai.

Ek flow tube (streamtube) streamlines ka ek bundle hai jo ek closed surface banata hai — imagine karein ek hose bani neighbouring streamlines se, figure mein shaded band ke roop mein sketch ki gayi hai. Kyunki koi fluid streamline cross nahi karta, koi fluid tube ki walls se leak nahi kar sakta, toh jo bhi mass ek end mein enter karta hai woh dusre se exit karna zaroori hai — woh sealed-wall bookkeeping hi tube par apply ki gayi continuity equation hai.

Ye undefined ho jaata hai — tangent hone ke liye koi direction nahi hai, toh multiple streamlines wahan mil sakti hain. Stagnation points hi wo ek jagah hain jahan streamlines legitimately intersect kar sakti hain.

Ek single point — zero velocity ke saath ye kabhi move nahi karta, toh uska "trail" sirf stagnation point hi hai. Uski streakline (agar woh point ek source hai) bhi sirf woh point hai.

Haan — $\vec{v}$ mein koi explicit $t$ nahi hai, toh $\partial \vec{v}/\partial t=0$, flow steady hai, aur golden rule pathlines = streamlines = circles banata hai. Particles sach mein origin ke around orbit karte hain. Dekhen Steady vs unsteady flow.

Sirf source point — kisi bhi particle ke paas abhi tak door jaane ka time nahi tha, toh streak ki length zero hai aur time ke saath baahir ki taraf badhti hai.

Teeno identical horizontal straight lines hain — steadiness aur constant field ka matlab hai ki snapshot, single trail, aur dye streak sab ek hi set of parallel lines hain. Golden rule ka ek clean sanity check.

Nahi — kisi bhi frozen instant par har point ek direction share karta hai, toh streamlines parallel straight lines hain. Lekin pathlines curve kar sakti hain, kyunki ek particle move karte waqt time ke saath direction ko badalta hua experience karta hai. Ye unsteadiness ko difference ka akela cause isolate karta hai.