Assumptions in Bernoulli — steady, inviscid, incompressible, along streamline

2.2.15 · Physics › Fluid Mechanics

Chaar assumptions, wahan se derive kiye jahan se aate hain

Hum Bernoulli ko Newton ke 2nd law se derive karte hain ek chote fluid parcel par jo ek streamline ke saath slide kar raha hai. Har assumption ek term ki kimat hai jise hum chhor dete hain.

Derivation setup karna (first principles)

Ek tiny cylindrical fluid element lo jisi ki length $d s$ hai, cross-section $A$ hai, aur yeh ek streamline $s$ ke saath lie kar raha hai. Forces along $s$ :

Pressure pichhe wali face par push karta hai $P A$ , aage wali face par $- (P + d P) A$ . Net pressure force $= - A d P$ .
Gravity ka component along $s$ : weight $= ρ A d s g$ , streamline ke saath component $= - ρ A d s g \frac{d h}{d s}$ jahan $h$ height hai.

Newton ka 2nd law: $mass \times acceleration = net force$ . Mass $= ρ A d s$ .

Parcel ki acceleration material derivative hai: $a = \frac{D v}{D t} = unsteady \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial s}$

Sab kuch jodne par ( $A$ se divide karo): $ρ d s (\frac{\partial v}{\partial t} + v \frac{\partial v}{\partial s}) = - d P - ρ g d h$

Ab assumptions ek ek karke apply karo

(1) Steady $\Rightarrow \frac{\partial v}{\partial t} = 0$ . Hum unsteady term drop kar dete hain: $ρ v d v = - d P - ρ g d h$

(2) Inviscid — humne kabhi koi friction force nahi likha. Viscosity nahi hone ki assumption se humne silently ek viscous shear term drop kar diya jo $- τ d A_{side}$ ke roop mein appear hota. Viscosity ke saath, energy heat mein convert hoti hai aur "constant" downstream decay karta hai.

(3) Incompressible $\Rightarrow ρ = const$ , toh hum $d P / ρ$ ko saaf saaf integrate kar sakte hain. Streamline ke saath integrate karo: $\int ρ v d v + \int d P + \int ρ g d h = 0$ $\frac{1}{2} ρ v^{2} + P + ρ g h = const$

(4) Along a streamline — har step $s$ ke saath integrate kiya. Constant alag alag streamlines par alag ho sakta hai jab tak flow irrotational bhi na ho.

Figure — Assumptions in Bernoulli — steady, inviscid, incompressible, along streamline

Worked examples

Common mistakes (steel-manned)

Flashcards

Bernoulli ki equation ke chaar assumptions kya hain?

Steady, inviscid, incompressible, aur ek streamline ke saath apply kiya.

"Steady" assumption kaun sa acceleration term khatam karta hai?

Material derivative mein local/unsteady term

\partial v / \partial t

Bernoulli ko ek streamline ke saath kyun apply karna chahiye?

Kyunki constant sirf ek streamline ke saath guaranteed hai; alag streamlines ke alag constants ho sakte hain jab tak flow irrotational na ho.

Viscosity Bernoulli balance se kaun si physical quantity hataati hai, aur kismein?

Mechanical (pressure+kinetic+potential) energy, irreversibly heat mein convert hoti hai — isliye sum constant rehne ke liye "inviscid" assumption chahiye.

Kaun sa integral messy ho jaata hai agar fluid compressible ho?

\int d P / ρ

— variable

ρ

ke saath yeh

P / ρ

ke barabar nahi rehta, isliye cleanly integrate karne ke liye incompressible assumption chahiye.

Gas flow ko incompressible treat karne ka rule of thumb kya hai?

Mach number

< 0.3

Ek lambi rough horizontal constant-area pipe mein, kaun si assumption fail hoti hai aur symptom kya hai?

Inviscid fail hoti hai; pressure pipe ke saath drop hota hai chahe

v

aur

h

unchanged hon.

Ek fluid parcel ki streamline ke saath material (total) acceleration kya hai?

\frac{D v}{D t} = \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial s}

Recall Feynman: ek 12 saal ke bachche ko explain karo

Socho tum ek smooth waterslide par slide kar rahe ho. Paani ki ek tube teen "coins" ka ek "energy budget" maintain karti hai: push (pressure), speed, aur height. Yeh coins swap kar sakti hai — height ko speed se trade kar sakti hai — lekin total same rehta hai. LEKIN yeh tabhi kaam karta hai jab: slide tum ride karte waqt change na ho (steady), yeh bilkul slippery ho taaki koi energy heat ke roop mein na rube (inviscid), paani squish na kare (incompressible), aur tum apni khud ki slide lane par raho (along a streamline). Koi bhi rule todo aur tumhare teen coins ka total same nahi rahega.

Connections

Bernoulli's Equation — woh result jo yeh assumptions license karte hain.
Continuity Equation — Bernoulli ke saath pair hoti hai; area change hone par $v$ deti hai.
Viscosity and Poiseuille Flow — tab kya hota hai jab inviscid assumption drop ho jaati hai.
Boundary Layer — woh region jahan walls ke bilkul paas inviscid fail hoti hai.
Material Derivative — steady vs unsteady distinction ka source.
Mach Number and Compressibility — incompressible assumption ki limit set karta hai.
Irrotational Flow — woh extra condition jo Bernoulli ke constant ko streamlines ke across global banati hai.