Question bank — Navier-Stokes equations — derivation from Newton's second law for fluid

Galat — ek particle phir bhi convective term $(\vec{u}\cdot \nabla )\vec{u}$ ke through accelerate kar sakta hai, jaise ek patta narrow hoti nadi mein drift karte waqt speed up karta hai, bhale hi har fixed point pe flow kabhi nahi badalti. (Convective-acceleration figure dekho.)

Galat — uniform pressure sabhi faces pe equally push karta hai aur cancel ho jata hai; sirf gradient $-\nabla P$ (blob ke across ek difference) net force density deta hai, isliye $P$ ki value motion ke liye irrelevant hai.

Galat — yeh tab vanish hoti hai jab velocity field linear ya uniform ho (zero curvature ${\nabla }^2\vec{u}=0$), speed se regardless; tez uniform flow mein koi viscous force nahi, jabki slow curved flow mein badi ho sakti hai.

Galat — $\mu =0$ deta hai Euler's equation ($\rho D\vec{u}/Dt=-\nabla P+\rho \vec{g}$); hydrostatics tabhi milti hai jab additionally $\vec{u}=0$ bhi set karo.

Galat — yeh nonlinear hai (velocity apne aap ke gradient ko multiply karti hai); yahi ek term turbulence aur fluid dynamics ki zyaadatar mushkil ka kaaran hai.

Galat — dono generally nonzero hoti hain aur balance karti hain: $\nabla P=\rho \vec{g}$ ($\hat{z}$ upar ke saath, yeh padha jata hai $\partial P/\partial z=-\rho g$). Zero net force ka matlab nahi ki individual force densities vanish ho jaayein.

Galat — viscosity momentum ko diffuse karti hai; yeh utni hi aasani se ek slow layer ko tez kar sakti hai usse faster layer ke saath drag karke. Yeh velocity differences remove karti hai, speed per se nahi.

Galat — yeh momentum conservation hai, yani $\vec{F}=m\vec{a}$ per unit volume. Energy ek alag (derivable) balance hai.

Galat — $\nabla \cdot \vec{u}=0$ incompressibility hai. Ek gas jo compress ya expand ho rahi hai uska $\nabla \cdot \vec{u}\ne 0$ hoga; density $\rho$ phir space aur time mein vary karti hai, aur yahi sound waves ko exist karne deta hai.

Error — ek field ke liye particle ek naye location pe move karta hai jahan $\vec{u}$ alag hota hai, isliye tumhe convective term add karna padta hai: sahi acceleration hai $D\vec{u}/Dt=\partial \vec{u}/\partial t+(\vec{u}\cdot \nabla )\vec{u}$.

Error — dimensionally aur physically galat; force per volume ke liye blob ke across pressure ka difference chahiye, jo gradient $-\nabla P$ hai (units $\text{N}/{\text{m}}^3$). Ek single value $P$ (units $\text{N}/{\text{m}}^2$) force density nahi ho sakti.

Error — $\tau =\mu \partial u_x/\partial y$ sirf poore viscous stress tensor ${\tau }_{ij}=\mu (\partial u_i/\partial x_j+\partial u_j/\partial x_i)$ ka ek component hai, jo yahan valid hai kyunki humne simple shear choose ki. Saath hi $\tau$ ek stress hai (force per area); net force density element ke across stress ka difference hai, jo ek aur derivative add karta hai, aur incompressible flow ke liye $\mu {\nabla }^2\vec{u}$ deta hai.

Error — hum $dV$ isliye drop karte hain kyunki $\vec{F}=m\vec{a}$ likhne ke baad yeh dono sides ka common factor hai; chhotaapan fields ko differentiable treat karne ko justify karta hai, cancellation ko nahi.

Error — gravity ek body force hai jo $\vec{F}=m\vec{a}$ ke force side pe baithti hai; material derivative sirf acceleration ($\vec{a}$) side se belong karta hai.

Error — $\nabla \cdot \vec{u}=0$ incompressibility constraint hai jo viscous term ko simplify karne ke liye use hoti hai; compressible gases ke liye yeh fail hoti hai, $\rho$ vary karta hai, aur ek extra term $(\mu /3+\zeta )\nabla (\nabla \cdot \vec{u})$ bulk viscosity $\zeta$ ko involve karte hue appear hota hai. Yeh optional bookkeeping nahi hai.

Error — pehla hai (velocity dotted into the gradient) $\vec{u}$ pe acting (ek vector, convective acceleration); doosra hai ($\vec{u}$ ka divergence, ek scalar) $\vec{u}$ se times. Bilkul alag operations hain.

Kyunki $\vec{F}=m\vec{a}$ ek matter ke baare mein law hai (usi blob of mass ke baare mein), aur woh blob move karta hai; ek fixed point dekhne se alag alag particles us se guzarte hain. Isliye material derivative $D/Dt$.

Ek face pe stress pehla derivative hai $\mu \partial u_x/\partial y$; net force density blob ke across us stress ka difference hai, jo ek derivative ka derivative hai — ek second derivative $\mu {\nabla }^2\vec{u}$. (Viscous-diffusion figure dekho: force profile ki curvature se aati hai.)

Fluid high pressure se low pressure ki taraf push hota hai; $\nabla P$ increasing $P$ ki taraf point karta hai, isliye force density $-\nabla P$ hai, decreasing pressure ki taraf pointing. (Pressure-gradient figure dekho.)

Steady, unidirectional flow jisme pressure gradient $\partial P/\partial x=-G$ hai, $x$-equation reduce hoti hai boundary-value ODE $\mu \frac{d^2{u}_x}{d{y}^2}=-G$ $0\le y\le h$ pe no-slip $u_x(0)=u_x(h)=0$ ke saath. Ek baar integrate karo: $u_x^{\prime }=-\frac{G}{\mu }y+C_1$. Dobara integrate karo: $u_x=-\frac{G}{2\mu }{y}^2+C_{1}y+C_2$. $u_x(0)=0$ apply karo $\Rightarrow C_2=0$; $u_x(h)=0$ apply karo $\Rightarrow C_1=\frac{Gh}{2\mu }$. Isse downward parabola milta hai $u_x(y)=\frac{G}{2\mu }y(h-y)$, dono walls pe zero aur centre pe maximum.

Har term ki size estimate karo typical speed $U$ aur length $L$ use karke. Inertial term scale karta hai $\rho (\vec{u}\cdot \nabla )\vec{u}\sim \rho U^2/L$; viscous term scale karta hai $\mu {\nabla }^2\vec{u}\sim \mu U/L^2$. Unka ratio hai $\frac{\rho U^2/L}{\mu U/L^2}=\frac{\rho UL}{\mu }=\frac{UL}{\nu }\equiv \text{Re}$. Bada $\text{Re}$ → inertia (aur turbulence) jeetta hai; chhota $\text{Re}$ → viscosity sab smooth kar deti hai. Note karo density $\rho$ essential hai; isse drop karna galat dimensions deta hai.

Continuum picture mein blob ka mass $\rho dV$ hai, isliye uska weight $\rho \vec{g}dV$ hai; volume se divide karne par body-force density $\rho \vec{g}$ milti hai (units $\text{N}/{\text{m}}^3$) uniformly.

Viscosity remove karna sirf friction term delete karta hai; fluid phir bhi move aur accelerate kar sakta hai, isliye inertia aur convective terms survive karte hain — yahi precisely inviscid flow equation hai, rest equation nahi.

${\nabla }^2\vec{u}=0$, isliye koi viscous force nahi — koi velocity difference nahi matlab layer-on-layer drag nahi, bhale hi fluid move kar raha ho.

Shear stress (first derivative) wahan zero hai kyunki velocity maximum hai, lekin viscous force density $\mu d^2{u}_x/d{y}^2$ nonzero aur everywhere constant hai — force supply karne wali curvature hai, slope nahi.

Velocity wall pe zero hai, isliye koi acceleration nahi aur koi inertia nahi; pressure gradient puri tarah viscous term se balance hoti hai, isliye wall friction wahan sabse zyada hoti hai jahan profile sabse steep hoti hai.

Sirf $0=-\nabla P+\rho \vec{g}$ bachta hai — har velocity-dependent term (unsteady, convective, viscous) vanish ho jaata hai, hydrostatics $dP/dz=-\rho g$ recover hoti hai ($\hat{z}$ upar ke saath).

Viscous term $\mu {\nabla }^2\vec{u}$ inertia ke comparison mein chhota ho jaata hai, isliye flow approximately Euler flow jaisi behave karti hai — siwaaye walls ke paas thin boundary layers ke jahan viscosity hamesha matter karti hai.

Nahi — $\partial \vec{u}/\partial t=0$ lekin $\vec{u}$ ka direction path ke saath change hota hai, isliye $(\vec{u}\cdot \nabla )\vec{u}$ ek centripetal-style acceleration provide karta hai; steady ka matlab kabhi static nahi hota.

Dono inertia side aur gravity term vanish ho jaate hain, chhod ke $0=-\nabla P+\mu {\nabla }^2\vec{u}$ — ek degenerate massless balance; physically unrealistic, yeh signal karta hai ki density woh "matter" carry karta hai jo Newton's law ko chahiye. (Isse $\nu =\mu /\rho$ bhi blow up ho jaata hai.)

Haan — sound density $\rho$ aur pressure $P$ ka ek chhota oscillation hai. Iske liye $\nabla \cdot \vec{u}\ne 0$ chahiye (fluid locally compress/expand hota hai), isliye yeh incompressible model ke bahar rehta hai; momentum equation ka varying-$\rho$ continuity equation ke saath coupling hi wave ko carry karta hai.

Nahi — bulk-viscosity term $\nabla (\nabla \cdot \vec{u})$ ke proportional hai, aur incompressibility $\nabla \cdot \vec{u}=0$ force karti hai, isliye yeh bilkul drop ho jaata hai. Yeh sirf compressible motions jaise shock waves aur sound absorption ke liye matter karta hai.