2.2.18 · D1Navier-Stokes equations — derivation from Newton's second la
Navier–Stokes equation kuch aur nahi balki ==Newton ka F = m a vocabulary hai — arrows, chhote symbols, slopes — jo aapko chahiye tab jab woh ek sentence samajh mein aaye.
Is page par yeh assume kiya gaya hai ki aap kuch nahi jaante. Hum har symbol ko earn karenge jo parent note (the parent topic ) mein use hota hai, ek aisi order mein jahan har cheez sirf pehle wali cheez par lean karti hai. Hum pehle picture banate hain, phir coordinates , phir quantities , aur tabhi woh operators jo unhe combine karte hain.
Intuition Hum fluid ko cubes mein kyun kaatte hainEk river mein bahut zyada molecules hain jinhe ek ek karke track karna possible nahi. Toh hum fluid ko tiny cubes ke ek bade grid mein slice karte hain. Har cube itna chhota hota hai ki fluid ki properties uske andar almost constant hoti hain, lekin itna bada hota hai ki usme trillions of molecules hon (taaki woh smooth "stuff" ki tarah behave kare, na ki individual bouncing dots). Hum Newton ka law ek aisi cube par apply karte hain, phir cube ko ek point tak shrink hone dete hain.
Figure mein jo cube hai woh hamara hero hai. Jo kuch bhi aata hai — coordinates, quantities, operators — woh isliye banaya gaya hai taaki hum is cube ke liye F = m a
Definition Teen axis directionsx y z − z u x u x Picture: teen arrows ek room ke corner par milte hain, right angles par.
Definition Position vectorr = ( x , y , z ) kahan hai: x y z Picture: origin (room ka corner) se us point tak ek arrow.
d x , d y , d z d x tiny step hai x d y y d z z Picture: upar figure mein cube ke teen coloured edge-lengths.
Ab jab hamare paas directions aur unke along tiny steps hain, hum finally cube ko size aur weigh kar sakte hain.
Definition Volume elementd V d V = d x d y d z length × width × height ek infinitesimally small box ka (abhi define kiye gaye tiny steps use karke).
Topic ko yeh isliye chahiye kyunki forces aur mass per cube measure hote hain, aur hum end mein d V mass density chahiye, jo hum Section 5 mein milenge.)
Intuition Do tarah ki quantitiesKuch quantities sirf ek number hoti hain (ek scalar ): temperature, for example. Doosri quantities ko ek number AUR direction dono chahiye (ek vector ): ek dhakka, ya ek velocity. Hum ek letter ke upar ek chhota arrow draw karte hain yeh kehne ke liye ki "yeh wala ek vector hai."
a a jiske upar arrow ho (u F P ρ ∣ u ∣ u Picture: ek scalar ek colour shade hai jo space mein fill hoti hai; ek vector ek arrow hai jo har point par draw hota hai.
Reveal-check:
Kya density ρ Ek scalar — har point par sirf ek number, koi direction nahi.
Kya velocity u Ek vector — yeh ek speed aur ek direction dono carry karta hai (upar arrow note karein).
Definition Velocity fieldu ( r , t ) vector (Section 3 se ek scalar-with-arrow) position r t . Yeh dono space aur time ka function likha jaata hai.
Picture: neeche arrow-forest — flow ke har point par ek arrow draw kiya gaya. Jahan arrows lambe hain, fluid fast hai; jahan chhote, slow.
Intuition Velocity DO cheezein par kyun depend karti haiHawa ke arrows ka ek weather map hour to hour badal sakta hai (time dependence), aur ek single instant mein bhi arrows jagah jagah alag hote hain (space dependence). Ek beh raha patta dono effects ek saath feel karta hai — yahi double dependence hume baad mein material derivative invent karne par majboor karti hai.
ρ ρ kg / m 3 scalar hai. Ek cube ka mass tab d m = ρ d V Picture: fluid kitna "heavy per box" hai — mercury ek dense (crowded) fluid hai, air ek thin hai.
P P = area of that surface force pushing on a surface pascals mein measure hota hai, Pa = N / m 2 scalar hai: har point par yeh ek single number hai, har direction mein same squeeze.
Picture: fluid hamare cube ke har face par press karta hai. P force milta hai — yahi exact wajah hai ki P difference cube ke across baad mein ek net force per volume − ∇ P
μ μ sticky / gooey hai — ek moving layer apne neighbour ko kitni strongly drag karti hai. Pascal-seconds mein measure hota hai, Pa ⋅ s μ ≈ 0.001 Pa ⋅ s Picture: fluid ki do stacked sheets ek doosre ke past slide karti hain; μ Pa ⋅ s Pa 1/ s Viscosity & Newton's law of viscosity mein.
kya hoti hai, zero se
Ek derivative jawaab deti hai "agar main input ko thoda nudge karun, output kitna badlega?" Yeh ek graph ki steepness hai — rise over run jab run almost nothing tak shrink ho jaaye.
Definition Ordinary derivatived x df f x f Picture: ek curve ko hug karne wali tangent line; uski tilt hi derivative hai.
Definition Partial derivative∂ x ∂ f f x , y , z , t partial derivative ka matlab hai "sirf x ∂ Picture: ek hilly surface ko ek direction ke along slice karo aur us slice ka slope padhna.
Topic ko derivatives kyun chahiye: forces differences se aati hain — ek cube ke across pressure difference, layers ke beech velocity difference — aur difference-per-distance hi ek derivative hai.
∇ ∇ packaging hai:
∇ = ( ∂ x ∂ , ∂ y ∂ , ∂ z ∂ ) instruction hai jo kuch act karne ke liye wait kar rahi hai.
Definition Scalar ka gradient:∇ P ∇ P = ( ∂ x ∂ P , ∂ y ∂ P , ∂ z ∂ P ) vector jo us direction mein point karta hai jahan pressure sabse tezi se badhta hai, jiska length = kitni steeply badhta hai.
Picture: ek pahaad par, gradient arrow seedha upar ki taraf point karta hai. Fluid pressure pahaad se neeche push hota hai, isliye pressure force per volume − ∇ P P
∇ ⋅ u ∇ dot ek scalar deta hai: ∇ ⋅ u = ∂ x ∂ u x + ∂ y ∂ u y + ∂ z ∂ u z bahar spread ho raha hai (ek source) ya pile in ho raha hai (ek sink).
Picture: arrows sab ek spot se bahar ki taraf point kar rahe hain → positive divergence. Ek incompressible fluid ke liye kuch pile up nahi hota, toh ∇ ⋅ u = 0 Continuity equation .
Definition Advection operator( u ⋅ ∇ ) u ⋅ ∇ = u x ∂ x ∂ + u y ∂ y ∂ + u z ∂ z ∂ jab aap flow ke saath chalte ho ." Ise u ( u ⋅ ∇ ) u Material derivative .
Picture: downstream ride karte hue aur apne around ke arrows ko badlte feel karte hue.
∇ 2 ∇ 2 = ∇ ⋅ ∇ = ∂ x 2 ∂ 2 + ∂ y 2 ∂ 2 + ∂ z 2 ∂ 2 second derivative. Yeh ek point ki value ko uske neighbours ke average se compare karta hai.
Picture: ek bump jo apne surroundings se upar hai uska negative ∇ 2 ∇ 2 u difference of a difference hai (sliding-rate mein change cube ke across).
∇ 2 vector u ∇ 2 scalar ke liye define kiya gaya tha. Ek vector par yeh component by component act karta hai:
∇ 2 u = ( ∇ 2 u x , ∇ 2 u y , ∇ 2 u z ) alag alag teen parts u x , u y , u z μ ∇ 2 u
∇ P ∇ ⋅ u Kyun sahi lagta hai: dono ∇ Kami: ∇ ek scalar par ek vector banata hai (gradient); ∇ ⋅ ek vector par ek scalar banata hai (divergence). Chhota sa dot sab kuch badal deta hai.
Fix: neighbours padho: koi dot nahi + scalar → gradient (vector out); dot + vector → divergence (scalar out).
Definition Gravitational accelerationg g 9.81 m/s 2 g = ( 0 , 0 , − g ) = ρ g d V Picture: seedha neeche point karta ek arrow, har jagah same.
F F = m a F m a
Ise ek short chain ki tarah padho, spider-web ki tarah nahi: picture → coordinates → quantities → operators → the equation .
operators grad div laplacian
Har foundation final sentence mein ek word ban jaata hai: acceleration (D u / D t − ∇ P μ ∇ 2 u ρ g
Khud test karo — tum derivation ke liye tabhi ready ho jab tum bina dekhhe har cheez ka jawaab de sako.
d x , d y , d z d V x y z d V = d x d y d z
Cube d V d m = ρ d V ρ
Scalar aur vector mein kya difference hai, fluid ka ek ek example ke saath? Scalar bas ek number hai (pressure
P ); vector ki magnitude aur direction dono hoti hain (velocity
u , arrow ke saath marked).
Velocity field u ( r , t ) r t Flow jagah jagah alag ho sakta hai (space) aur time ke saath badal sakta hai (time); ek moving particle dono feel karta hai.
Pressure P Force per unit area, P = force / area Pa = N / m 2
μ Viscosity — fluid layers ke beech stickiness/drag, pascal-seconds Pa ⋅ s
d x df ∂ x ∂ f Ordinary derivative single input ko vary karta hai; partial derivative ek variable ko wiggle karta hai baaki sab ko fixed rakhte hue.
∇ P Ek vector, us direction mein point karta hai jahan pressure sabse tezi se badhta hai (fluid opposite taraf push hota hai, isliye − ∇ P
∇ ⋅ u = 0 Fluid incompressible hai — kisi bhi point par koi net spreading-out ya piling-up nahi (continuity).
∇ 2 u Component by component:
∇ 2 u = ( ∇ 2 u x , ∇ 2 u y , ∇ 2 u z ) .
Viscosity second derivative kyun use karti hai, first nahi? Net viscous force cube ke across stresses ka ek difference hai; stress already ek derivative hai, toh net force ko ek aur chahiye.
Ek fluid cube par teen forces kaun si hain, aur kaun sa equation unhe acceleration se jodta hai? Pressure (
− ∇ P ), viscosity (
μ ∇ 2 u ), gravity (
ρ g ); milake yeh
ρ D u / D t ke barabar hain (Newton's
F = m a ).
