Question bank — Potential flow — irrotational, inviscid; superposition of basic flows

False — yeh ek model hai. Real fluids mein viscosity hoti hai, lekin thin boundary layers ke bahar flow often nearly inviscid behave karta hai, isliye model lift aur pressure achhe se capture karta hai.

False — global circular paths aur local spin same nahi hain. Free vortex mein $\nabla \times \vec{v}=0$ har jagah hai except uske singular centre par, isliye yeh irrotational hai.

True — Laplace's equation linear hai, isliye solutions ka koi bhi sum ek solution hai; yahi cheez superposition ko legal banati hai.

False — pressure Bernoulli's equation se aata hai, jo speed mein nonlinear hai ($p+\frac{1}{2}\rho |\vec{v}{|}^2=$const). Pehle velocity fields add karo, phir total speed se ek baar pressure compute karo.

False — yeh zero net drag predict karta hai (d'Alembert's paradox) kyunki koi viscosity nahi hai aur koi wake nahi hai; yeh lift aur pressure ke liye trustworthy hai, drag ke liye nahi.

True — $u=\partial \psi /\partial y,v=-\partial \psi /\partial x$ ke saath, divergence ${\psi }_{xy}-{\psi }_{yx}=0$ identically ban jaata hai, isliye mass construction se conserved hoti hai.

True — Cauchy–Riemann equations force karti hain ki $\nabla \varphi$ aur $\nabla \psi$ perpendicular hon; geometrically $\nabla \varphi$ flow ke along point karta hai jabki streamlines flow ke saath run karti hain, isliye equipotentials streamlines ko squarely cut karti hain, ek orthogonal grid banati hain (figure s02 dekho).

True — stagnation streamline woh special streamline hai jo ek stagnation point se guzarti hai aur flow ko inside/outside mein split karti hai; kyunki koi fluid kisi bhi streamline ko cross nahi karta, yeh separatrix exactly ek rigid boundary ki tarah act karti hai. Isi tarah cylinder aur Rankine nose "appear" hote hain.

False — vortex $\psi =-\frac{\Gamma }{2\pi }\ln r$ $r=a$ par constant hai, isliye circle ek streamline bana rehta hai; yeh speeds aur isliye pressure change karta hai, lift produce karta hai, koi nayi shape nahi.

Sirf irrotational flow ke liye, aur tab bhi $\varphi$ globally single-valued sirf ek simply connected region mein hai; zero curl $\varphi$ ke liye locally precondition hai, lekin ek hole (vortex) phir bhi $\varphi$ ko multi-valued bana sakta hai.

Galat — centre ko enclosing karne wale loop ki circulation $\Gamma \ne 0$ hoti hai. Irrotationality sirf un loops ke liye zero circulation guarantee karta hai jo singular centre ko wrap nahi karte (region simply connected nahi hai).

Ulta hai — $|\sin \theta |$ top/bottom par sabse bada hota hai ($\theta =\pm 90^{\circ }$), wahan maximum speed $2U$ deta hai; stagnation points front aur back par hain ($\theta =0,\pi$) jahan $\sin \theta =0$ hai (figure s02 dekho).

Jaise $r\to 0$, $u_r\to \infty$: origin ek sachcha singularity hai, koi real physical point nahi. Model sirf usse door valid hai.

Ek single source uniform flow ke saath ek open half-body deta hai (ek rounded Rankine nose upstream jo kabhi downstream close nahi hoti). Body ko Rankine oval mein close karne ke liye equal strength ka matching sink chahiye.

Viscosity appear nahi hoti — Kutta–Joukowski theorem purely potential-flow result hai. Viscosity ka role sirf $\Gamma$ ki physical value select karna hai (Kutta condition ke zariye), lift expression mein enter karna nahi.

Irrotational deta hai $\vec{v}=\nabla \varphi$ (simply connected region mein); incompressible deta hai $\nabla \cdot \vec{v}=0$; substitute karne par ${\nabla }^2\varphi =0$ milta hai, isliye teeno velocity components ek harmonic scalar mein encode hain.

$\psi$ streamlines visible banata hai (uski constant lines flow paths hain) aur automatically continuity satisfy karta hai, jabki $\varphi$ ki constant lines flow ke perpendicular equipotentials hain.

Cylinder flow par ek vortex superpose karna ek side ko speed up aur doosri side ko slow down karta hai; Bernoulli phir pressure difference aur ek net sideways force deta hai — Magnus effect, $L^{\prime }=\rho U\Gamma$ se quantify kiya gaya.

Kyunki Laplace's equation linear hai, mushkil flows sirf kuch catalogued simple flows (uniform, source, vortex, doublet) ke sums hain, fluid dynamics ko Lego assembly mein badal deta hai.

Bernoulli speed mein quadratic hai, isliye pehle saare velocity contributions ko ek total $\vec{v}$ mein add karo, phir ek baar uski length square karo — ek sum ko square karna squares ke sum se alag hai.

$\nabla \varphi$ steepest potential rise ki direction mein point karta hai, jo flow direction $\vec{v}$ hai; streamlines $\vec{v}$ ke along run karti hain; isliye constant $\varphi$ ka curve ($\nabla \varphi$ ke perpendicular) streamline ke perpendicular hona chahiye — Cauchy–Riemann equations ise algebraically state karti hain.

Undefined/infinite — centre ek singularity hai jo saari circulation carry karta hai; baaki har jagah vorticity exactly zero hai.

Radial speed $m/(2\pi r)\to \infty$; model singular point par break down karta hai aur sirf $r>0$ ke liye meaningful hai.

$m<0$ ke saath source ek sink ban jaata hai aur uska inflow ab stream ko upstream (left, $\theta =\pi$) side par balance karta hai jaise ek source karta hai; on-axis balance $U-\frac{m}{2\pi x}=0$ phir bhi $x_s=m/(2\pi U)<0$ deta hai, yani upstream. (Formula $x_s=-m/(2\pi U)$ $\theta =\pi$ ke liye likha gaya hai jahan $x=-|x|$ hai.)

Stagnation points $\theta =0,\pi$ par jahan speed zero hai — Bernoulli se, minimum speed matlab maximum pressure.

Critical circulation ${\Gamma }_p=4\pi Ua$ par: do stagnation points neeche ($\theta =-\frac{\pi }{2}$) par merge ho jaate hain, aur $\Gamma >4\pi Ua$ ke liye ek single stagnation point cylinder ke neeche fluid mein hota hai.

Necessarily nahi — agar hole ke around circulation nonzero hai, toh $\varphi$ har loop mein $\Gamma$ se increase karta hai, multi-valued ban jaata hai chahe $\vec{v}$ single-valued rahe.

Exactly zero (d'Alembert's paradox) — no viscosity aur fore-aft symmetric pressure ka consequence, body shape se regardless.