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

2.2.15 · D1 · Physics › Fluid Mechanics › Assumptions in Bernoulli — steady, inviscid, incompressible,

Is page pe assume kiya gaya hai ki aap kuch nahi jaante. Har letter jo $P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$ mein hai, ek-ek karke earn kiya jayega, har ek ke saath ek picture.

Jab yeh solid ho jayein, toh parent note Bernoulli's assumptions symbols ki wall ki jagah ek story ki tarah padhega.

0. "Fluid parcel" kya hota hai?

Kisi bhi symbol se pehle, show ke star se milo.

Ek single grain of dye ka imagine karo jo ek river mein dala gaya. Woh teleport nahi karta; woh ek path ke saath drift karta hai. Wahi path hai jis par hum sab kuch build karte hain.

Figure 1 — ek fluid parcel ek definite path follow karta hai.

Hum ise kyun chahiye: Newton's laws objects ke liye likhe jaate hain. Physics use karne ke liye pehle humein decide karna hoga kaun sa object — aur woh object yahi parcel hai.

1. Streamline — woh road jis par parcel drive karta hai

Neeche Figure 2 dekho. Cyan curve ek streamline hai. Amber arrows teen jagahon par fluid velocity hain — har arrow curve ke along hai. Yahi poori definition hai: curve hamesha flow ke tangent hoti hai.

Figure 2 — ek streamline har jagah velocity ke tangent hoti hai.

Topic ko ise kyun chahiye: Bernoulli ka promise ("sum constant rehta hai") sirf ek streamline ke along guarantee hai. Toh streamline woh track hai jis par hum apna saara accounting karte hain. Related idea: Irrotational Flow woh special case hai jahan constant har streamline par same ho jaata hai.

Hum is track ke along distance ko letter $s$ se measure karenge.

2. $\rho$ — density: kitna stuff packed hai

Do identical boxes ka imagine karo: ek packed marbles se bhara (high $\rho$), ek mein kuch marbles air mein float kar rahe hain (low $\rho$). Paani lagbhag $1000$ hai; air lagbhag $1.2$.

Topic ko ise kyun chahiye: hamare parcel ki mass $\rho \times (\text{uska volume})$ hai. Hum §4 mein parcel ki exact shape aur volume pin down karenge; jab hum karte hain, uski mass $\rho Ads$ ho jaati hai — woh cheez jo hum Newton's law mein plug karte hain (§5b mein defined). "Incompressible" assumption bilkul yeh statement hai ki $\rho$ kabhi nahi badalti, yahi wajah hai ki parent note $\rho$ ko ek integral se bahar nikal sakta hai.

3. $P$ — pressure: parcel ka sideways "push"

Parcel ko imagine karo jo apne neighbours se har side se squeeze ho raha hai, jaise ek crowd mein koi insaan. Agar crowd peeche se zyada aur aage se kam push kare, toh ek net forward shove hota hai. Wahi difference parcel ko drive karta hai.

Hum §4 mein wo face area miliinge jis par pressure push karta hai, jise $A$ kehte hain; yahan bas do faces ke beech ka difference dekho.

Figure 3 — net pressure force back push minus front push hai, $=-AdP$.

Figure 3 mein back face pressure $P$ feel karta hai jo aage push karta hai (ek force $+PA$) aur front face thoda alag pressure $P+dP$ feel karta hai jo peeche push karta hai (ek force $-(P+dP)A$). Chhota $dP$ step $ds$ ke upar pressure mein change hai.

Topic ko ise kyun chahiye: pressure difference un do forces mein se ek hai jo parcel ko uske track ke along push karte hain (doosra gravity hai). Poore force balance ke liye Bernoulli's Equation dekho.

4. $A$ aur $ds$ — parcel ki shape

Ek coin ka imagine karo: uski flat face $A$ hai, uski thickness $ds$ hai. Multiply karo toh uska volume milta hai.

Topic ko ise kyun chahiye: $A$ pressure (force per area) ko ek actual force ($P\cdot A$) banne deta hai, aur $Ads$ density ko ek actual mass banne deta hai. Inke bina symbols Newton's law mein enter nahi kar sakte. Yahi woh $A$ hai jo humne §2 aur §3 mein preview kiya tha — ab yeh earn ho gaya.

5. $v$ — speed, aur $\frac{1}{2}\rho v^2$ — motion ki energy

$\frac{1}{2}$ aur square kyun? Speed double karo aur energy quadruple ho jaati hai (yeh $v^2$ hai), kyunki do baar tezi se chalti cheez ko rokne mein chaar baar kaam lagta hai. $\frac{1}{2}$ us kaam ko rest se add karne se nikalta hai — yeh koi fudge factor nahi hai.

Topic ko ise kyun chahiye: yeh Bernoulli ke sum mein "motion" term hai. Jab parcel speed up hota hai (bada $v$), yeh term badhti hai, toh sum mein koi aur cheez (usually $P$) shrink hoti hai.

5b. $F$, $a$, aur Newton's second law — poori derivation ka engine

Forces combine karne se pehle, hum us law ki zaroorat hai jo "parcel par forces" ko "parcel speed change karta hai" mein convert kare.

Topic ko ise kyun chahiye: Bernoulli parcel ke liye $F=ma$ likhne se derive hota hai, jahan $F$ pressure force $-AdP$ plus gravity force (§7) hai, $m$ mass $\rho Ads$ hai, aur $a$ material acceleration (§8) hai. Baad ke har step mein bas yahi ek law unpack hoti hai.

6. $g$, $h$, aur $\rho gh$ — height ki energy

Paani ki ek bucket uthane ka imagine karo: jitna upar uthao, utni zyada energy woh gir kar release kar sakti hai. Woh stored "girne ki willingness" $\rho gh$ hai.

Topic ko ise kyun chahiye: yeh sum ka teesra aur aakhiri term hai. Ek parcel jo pipe mein climb karta hai woh height energy ke liye motion ya pressure energy trade karta hai.

7. $dh/ds$ — streamline kitni steeply climb karti hai, aur minus sign kahan se aata hai

Figure 4 — uphill: $dh/ds>0$, gravity forward motion ka virodh karta hai (isliye minus sign hai).

Yahan ek ratio kyun chahiye: gravity seedha neeche khichti hai, lekin parcel track ke along (jo shayad tilted ho) move karta hai. Gravity ka sirf woh part jo track ke saath align hota hai parcel ko speed up ya slow down kar sakta hai. Woh fraction bilkul slope $dh/ds$ hai.

Figure 4 dekho: amber arrow $ds$ track pe climb karta hai, white arrow $dh$ gained height hai. Unka ratio $dh/ds$ hai, aur kyunki climbing ka matlab gravity se ladna hai, woh force jo produce karta hai minus carry karta hai.

Topic ko ise kyun chahiye: yahi woh exact term hai jo parent note streamline ke along gravity ke liye likhta hai. Sign theek karna woh hai jo Bernoulli ko dono uphill aur downhill flow mein height ko speed aur pressure ke saath theek se trade karne deta hai.

8. $\partial v/\partial t$ aur $Dv/Dt$ — do reasons jis se parcel speed up hota hai

Yahan do naye notations aate hain. Inhe slowly lo.

Topic ko ise kyun chahiye: "steady" assumption bilkul $\partial v/\partial t=0$ hai — yeh pehla term delete karta hai. In do pieces ko samajhna yahi wajah hai ki parent ek ko throw away kar sakta hai aur doosre ko rakh sakta hai.

9. Integral sign $\int$ — track ke along add karna

Topic ko ise kyun chahiye: Newton's law humein tiny pieces ke beech relation deta hai ($\rho vdv$, $dP$, $\rho gdh$). Bernoulli ka whole-path statement paane ke liye humein streamline ke along saare tiny pieces add karne honge — woh integration step hai, aur leftover total exactly woh "$=\text{constant}$" hai jo is page ke top pe tha. Aur yahan "incompressible" ($\rho$ constant) matter karta hai: sirf tabhi $\rho$ cleanly integral ke bahar slide kar sakta hai.

Foundations topic ko kaise feed karte hain

Neeche ka map ek reading order hai, top to bottom. Fluid parcel se shuru karo. Isse humein mass milti hai (density $\rho$ aur shape $Ads$ ke zariye). Alag se, pressure $P$ area $A$ ke upar, plus gravity $g$ slope $dh/ds$ se scale ki, track ke along net force dete hain. Parcel ka acceleration material derivative $Dv/Dt$ se aata hai. Newton's law $F=ma$ force, mass aur acceleration ko ek equation mein lock karta hai. Aakhir mein integral sign $\int$ us equation ko poori streamline ke along add karta hai, aur Bernoulli ka constant sum $P+\frac{1}{2}\rho v^2+\rho gh$ nikal aata hai. Agar koi picture tumhare liye render nahi ho rahi, toh yeh paragraph words mein wahi map hai.

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