2.2.18
Intuition Ek-sentence ideaNavier–Stokes equation basically ==F = m a pressure , viscous friction , aur gravity , aur acceleration ko moving fluid ke saath track kiya jaata hai (material derivative).
F = m a particles ke baare mein hai. Fluid ek continuum hai infinitely many particles ka jo flow karte hain, deform karte hain, aur ek doosre se slide karte hain. Hum har molecule ko track nahi kar sakte. Isliye hum F = m a infinitesimal fluid element (control volume of mass ρ d V hi Navier–Stokes hai.
Definition Material (substantial) derivativeVelocity field u ( r , t ) dono time aur us position par jo usne move karke haasil ki hai. Uska sahi acceleration hai:
D t D u = ∂ t ∂ u + ( u ⋅ ∇ ) u local (unsteady) acceleration ; doosra hai convective acceleration (aap sirf ek faster region mein move karke speed up ho jaate ho).
Intuition Convective term kyun exist karta haiEk river mein khade ho jo narrow hoti hai. Bhale hi flow steady ho (∂ u / ∂ t = 0 faster paani mein carry ho raha hai. Aapke fixed point par speed constant hai, lekin particle ki speed change hoti hai. Woh change ( u ⋅ ∇ ) u
KAISE derive karte hain (chain rule, first principles):
Ek particle r ( t ) u ( r ( t ) , t ) d t d u = ∂ t ∂ u + ∂ x ∂ u d t d x + ∂ y ∂ u d t d y + ∂ z ∂ u d t d z d t d x = u x ( u ⋅ ∇ ) u
Ek fluid ka cube lo, volume d V = d x d y d z d m = ρ d V
d F body = ρ g d V
gradient kyun deta hai, sirf ek value nahi
Pressure har face par andar ki taraf push karta hai. Agar pressure uniform hota, toh left face par push right face ke push ko cancel kar deta — net zero. Net force ke liye cube mein pressure ka difference chahiye, yaani ek gradient.
KAISE (derivation): Left x = P ( x ) d y d z + x = − P ( x + d x ) d y d z − x d F x = [ P ( x ) − P ( x + d x )] d y d z = − ∂ x ∂ P d x d y d z d F pressure = − ∇ P d V
Intuition Viscosity = fluid layers ke beech friction kyun haiAgar paas-paas ke fluid layers alag-alag speed se move kar rahe hain, toh molecules ek doosre ko drag karte hain — momentum sideways diffuse hota hai. Newton's law of viscosity: shear stress τ = μ ∂ y ∂ u net viscous force iske across stress mein difference se aati hai — ek spatial derivative ka aur derivative, isliye ek second derivative.
KAISE (incompressible Newtonian fluid ke liye derivation):
Ek face par stress = μ ∂ u x / ∂ y y x [ μ ∂ y ∂ u x y + d y − μ ∂ y ∂ u x y ] d x d z = μ ∂ y 2 ∂ 2 u x d V x , y , z ∇ ⋅ u = 0 d F visc = μ ∇ 2 u d V
d V D u / D t
Intuition Ise ek sentence ki tarah padho(mass density × blob ka acceleration) = (pressure push) + (viscous drag) + (weight). Har term literally un teen forces mein se ek hai jo humne build kiye.
Worked example Euler's equation recover karoForecast: ek ideal fluid mein viscosity nahi hoti. μ = 0 Verify: ρ D t D u = − ∇ P + ρ g Euler's equation . ✓ Yeh step kyun? Ise inviscid case par reduce hona chahiye, jo humara known limit hai.
Worked example Hydrostatics recover karoForecast: fluid at rest, u = 0 Verify: 0 = − ∇ P + ρ g ∇ P = ρ g d z d P = − ρ g P = P 0 + ρ g h h Yeh step kyun? Koi motion nahi ⟹ pressure ko gravity balance karna chahiye, jo humara sabse basic fluid result hai.
Worked example Plane Poiseuille flow (plates ke beech steady flow)Flow u x ( y ) y = 0 , h ∂ P / ∂ x = − G y , z x 0 = G + μ d y 2 d 2 u x ⇒ d y 2 d 2 u x = − μ G u x ( 0 ) = u x ( h ) = 0 u x ( y ) = 2 μ G y ( h − y ) Yeh step kyun? Parabola viscous pressure-driven flow ka fingerprint hai — walls par zero, middle mein max.
Common mistake "Acceleration bas∂ u / ∂ t Kyun sahi lagta hai: particle mechanics mein, a = d v / d t Flaw: ek field ke liye, particle ek naye jagah move karta hai jahan u ( u ⋅ ∇ ) u Fix: "fluid ka acceleration" ke liye hamesha material derivative D / D t
Common mistake "Pressure khud flow drive karta hai, toh term− P Kyun sahi lagta hai: bada pressure = bada push.
Flaw: uniform pressure saari sides se equally push karta hai → zero net force. Sirf differences matter karte hain.
Fix: force per volume − ∇ P
Common mistake "Viscous termμ ∇ u Kyun sahi lagta hai: Newton's law of viscosity mein ek first derivative hai, τ = μ ∂ u / ∂ y Flaw: τ stress hai; net force blob ke across stress mein difference hai → ek aur derivative.
Fix: viscous force per volume = μ ∇ 2 u
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho paani ka ek chota sa floating cube. Teen cheezein ise push karti hain: uske aas-paas ka paani ek side se doosri side se zyada squeeze kar raha hai (pressure), paas-paas ka paani slide karke ise rub kar raha hai (stickiness/viscosity), aur gravity ise neeche kheench rahi hai. Newton ne kaha push = mass × speed-kitni-tezi-se-badal-rahi-hai. Tricky part yeh hai: cube speed up hota hai dono isliye kyunki river change ho rahi hai aur isliye kyunki cube ek faster jagah float karta hai. Saari pushes add karo, unhe mass times speed-change ke barabar set karo, aur woh poora sentence hi Navier–Stokes equation hai.
Mnemonic Structure yaad karo"My Pal Viscous Goes" → M ass·accel = P ressure + V iscous + G ravity.
Acceleration = "Local + Carried" (∂/∂t + u ⋅ ∇
Material derivative likho aur dono terms ko naam do.
Pressure force − ∇ P − P
Viscosity ke liye second derivative kyun?
NS kya ban jaata hai agar μ = 0 u = 0
Material derivative D u / D t ∂ u / ∂ t + ( u ⋅ ∇ ) u — local (unsteady) plus convective acceleration.
Convective term ( u ⋅ ∇ ) u Ek fluid particle velocity change karta hai different velocity ke region mein move karke, steady flow mein bhi.
Fluid element par net pressure force per unit volume? − ∇ P differences net force dete hain).
Pressure force gradient kyun hai, sirf P Uniform pressure saare faces par equally push karta hai → cancel ho jaata hai; sirf element ke across difference net force deta hai.
Viscous force per unit volume (incompressible Newtonian)? Viscosity second spatial derivative kyun hai? Stress μ ∂ u / ∂ y stress mein difference hai → ek aur derivative.
Full incompressible Navier–Stokes equation? ρ ( ∂ t u + ( u ⋅ ∇ ) u ) = − ∇ P + μ ∇ 2 u + ρ g .
Incompressible NS ke saath kaun sa constraint aata hai? NS with μ = 0 Euler's equation:
ρ D u / D t = − ∇ P + ρ g .
NS with u = 0 Hydrostatics:
∇ P = ρ g , yaani
d P / d z = − ρ g .
Plane Poiseuille flow ka velocity profile? u x ( y ) = 2 μ G y ( h − y )
NS kis physical principle se derive hoti hai? Newton's second law
F = m a jo ek infinitesimal fluid element par per unit volume apply ki gayi hai.
Infinitesimal element mass rho dV
Material derivative Du/Dt
Chain rule on u of r and t
