Bernoulli's equation — derivation from F = ma along streamline: Visual walkthrough

2.2.14 · D2 · Physics › Fluid Mechanics › Bernoulli's equation — derivation from F = ma along streamli

Step 1 — Streamline kya hoti hai, aur "parcel" kya hota hai?

KYA HAI. Steadily behta paani invisible highways trace karta hai jinhein streamlines kehte hain — ek aisi curve jo har point par exactly us direction mein point karti hai jis taraf wahan fluid move kar raha hota hai. Hum ek aisi highway par baithe fluid ke ek tiny blob ko pick karte hain: ek chhota sa납작한 cylinder. Uske flat faces flow ke along point karte hain; uski cross-section ka area $A$ hai, aur woh $d s$ lamba hai. Us jagah fluid streamline ke along ek certain speed se move karta hai, jise hum $v$ kehte hain.

KYUN. Newton ka law, $F=ma$, ek definite chunk of matter ke baare mein baat karta hai. Fluid ek continuous smear hota hai, isliye pehle humein ek definite chunk — yeh cylinder — kaat lena hoga — tab hi hum "force = mass × acceleration" keh sakte hain.

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Figure — Bernoulli's equation — derivation from F = ma along streamline

Step 2 — Pressure ki push

KYA HAI. Pressure $P$ yeh hai ki surrounding fluid har square metre of face par kitni zyada force se dabata hai. Yeh back face par dabaata hai blob ko aage push karte hue, aur front face par dabaata hai blob ko peeche push karte hue. Agar back par pressure $P$ hai aur front par $P + d P$ (thoda zyada ya kam), toh hum dono pushes ko add karte hain.

KYUN. Sirf pressure ka farq blob par net force chhod jaata hai. Agar dono faces equally squeeze hoti, toh pushes cancel ho jaate — jaise do log ek door ko dono sides se equally dhakelte hain. Isliye humein $d P$ chahiye, yaani $d s$ length par pressure mein change.

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Step 3 — Slope ke along gravity ki pull

KYA HAI. Gravity blob ko seedha neeche $d m g$ force se kheenchti hai. Lekin us pull ka sirf woh hissa jo streamline ke along point karta hai blob ko speed up ya slow down kar sakta hai. Agar streamline $d s$ length chalte hue height $d y$ se upar jaati hai, toh woh horizontal se $θ$ angle banati hai, aur streamline ki steepness $sin θ = d y / d s$ se capture hoti hai.

KYUN. Motion ke sideways force speed nahi badal sakti — woh sirf path ko bend karti hai. Isliye hum sirf gravity ka slope-component rakhte hain. Hum $sin θ$ isliye use karte hain kyunki "rise $d y$ ", "streamline ke along run $d s$ " se bane chhote right-triangle mein, (opposite over hypotenuse) ka ratio exactly yahi hai ki gravity ka kitna fraction flow ke along act karta hai.

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Step 4 — "Steady" flow phir bhi kyun accelerate karta hai

KYA HAI. Steady flow mein $v$ har fixed spot par time ke saath kabhi nahi badlata. Phir bhi blob speed up hota hai — kyunki woh ek naye spot par travel karta hai jahan fluid naturally faster hai (jaise narrow pipe mein). Uska acceleration hai $a = v \frac{d v}{d s}$ .

YEH TOOL KYUN — chain rule. Hum $a = d v / d t$ chahte hain, lekin hum directly nahi jaante ki $v$ time par kaise depend karta hai; hum jaante hain ki $v$ position $s$ par kaise depend karta hai (narrow part mein fast, wide part mein slow). Chain rule hume swap karne deta hai:

a = \frac{d v}{d t} = how v changes with place \frac{d v}{d s} \cdot = v, the speed \frac{d s}{d t} = v \frac{d v}{d s} .

Ise convective acceleration kehte hain — blob "khud ko" ek faster zone mein le jaata hai.

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Step 5 — $F = ma$ assemble karo

KYA HAI. Dono forces (Steps 2, 3) add karo aur unhe mass × acceleration (Steps 1, 4) ke barabar set karo.

KYUN. Yahi hamare blob ke liye Newton's law hai. Upar sab kuch sirf ingredients ko honestly measure karna tha.

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Step 6 — Tiny changes add karo (integrate karo)

KYA HAI. Hum streamline ke along point 1 se point 2 tak chalte hain, har chhote $d P$ , $d y$ , aur $v d v$ ko add karte hue. "Infinitely many tiny pieces add karna" exactly wohi hai jo integral sign $\int$ ka matlab hai, isliye hum har term ke liye $\int_{1}^{2}$ likhte hain.

KYUN. Ek tiny slice hume local rule batata hai; saare slices add karne se do door points ko connect karne wala rule milta hai. Kyunki $ρ$ constant hai, woh integral ke bahar slide kar jaata hai.

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Step 7 — Edge cases: har corner check karo

KYA HAI. Jis derivation par tum trust karte ho woh extreme inputs mein survive karni chahiye. Chalao unhe.

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Ek-picture summary

Recall Feynman retelling — puri walkthrough seedhe shabdon mein

Hum behte paani se paani ka ek tiny cylinder kaat lete hain (Step 1). Do cheezein use uske path ke along push kar sakti hain: peeche ka paani aage ke paani se zyada press kare (Step 2), aur gravity, lekin sirf gravity ka woh hissa jo slope ke along point karta hai (Step 3). Halanki stream unchanging dikhti hai, hamara blob phir bhi speed up hota hai — kyunki woh ek aisi jagah slide karta hai jahan paani naturally faster move karta hai, aur chain rule ise $v d v / d s$ mein badal deta hai (Step 4). Hum "push equals mass times how-fast-it-speeds-up" likhte hain, area aur length cancel ho jaate hain, aur ek tidy line of tiny changes nikalta hai (Step 5). Hum woh tiny changes shuru se ant tak add karte hain, aur $v d v$ ka sum $\frac{1}{2} v^{2}$ ban jaata hai (Step 6). Result: pressure, plus speed-energy, plus height-energy hamesha ek streamline ke along same total mein add hote hain. Corners test karo — flat pipe venturi deta hai, still water hydrostatics deta hai, khula hole $2 g h$ deta hai, blocked tube pitot reading deta hai — aur har corner behave karta hai (Step 7).

Recall Quick self-check

Final equation se $A$ kyun gayab ho jaata hai? ::: Yeh har force term ko equally multiply karta hai (pressure aur gravity dono face area ke saath scale karte hain), isliye divide karne par cancel ho jaata hai — Bernoulli blob size se independent hai. $\frac{1}{2} ρ v^{2}$ mein $\frac{1}{2}$ kahan se aata hai? ::: $v d v$ integrate karne se, exactly jaise $\int x d x = \frac{1}{2} x^{2}$ . Flat pipe mein, agar speed double ho jaaye toh dynamic pressure ka kya hota hai? ::: Yeh chaar guna ho jaata hai (yeh $v^{2}$ ke saath jaata hai), isliye static-pressure drop chaar guna bada hota hai.