Foundations — Drag — pressure (form) drag, skin friction drag

2.2.24 · D1 · Physics › Fluid Mechanics › Drag — pressure (form) drag, skin friction drag

Yeh page Drag topic ke liye toolbox hai. Isse pehle ki hum form drag ya skin friction ki baat karein, un formulas mein har letter aur symbol ka matlab kuch aisa hona chahiye jo tum imagine kar sako. Toh hum unhe ek ek karke, bilkul zero se banate hain. Neeche kuch bhi assume nahi kiya gaya ki tumne pehle yeh notation dekhi hai.

0 · Woh scene jo hum describe kar rahe hain

Ek solid object imagine karo — ek ball, ek plate, ek car — jo moving air ya water mein baitha hai. Ya object move kar raha hai aur fluid still hai; dono same hi hain, bas point of view badalta hai. Fluid surface ke saath slide karta hai aur, jahan bhi touch karta hai, press karta hai aur rub karta hai. Hamara poora kaam us interaction ke har piece ko naam dena hai.

1 · Speed aur direction — $v$ aur $\hat{x}$

Picture: Figure 1 mein lambe fluid-flow arrows ki length. Lamba arrow = bada $v$.

Topic ko yeh kyun chahiye: drag exist hi isliye karta hai kyunki fluid ko raste se hat'ta karna padta hai, aur tez hat'tana zyada costly hota hai. Drag $v^2$ (speed squared) par depend karega, isliye $v$ master dial hai.

Picture: body se door free stream par chipka ek chhota arrow, downstream point karta hua; poora $x$-axis uske saath lie karta hai.

Yeh kyun chahiye: drag specifically fluid ki push ka woh part hai jo downstream along $\hat{x}$ point karta hai. "$\hat{x}$ ke saath wala part extract" karne ke liye hum doosre arrows ko is ek se compare karenge (§6 dekho).

2 · Surface patch — $d{A}_s$ aur $\hat{n}$

Picture: Figure 1 mein single orange patch. Poori surface laakhon aise patches ki mosaic hai.

Picture: Figure 1 mein blue arrow, orange tile se flagpole ki tarah khada.

Dono kyun chahiye: pressure $\hat{n}$ ke saath push karta hai (seedha andar), aur uski strength patch size $d{A}_s$ se multiply hoti hai. Patch nahi toh add karne ke liye force nahi.

3 · Pressure $p$ aur ek patch par pressure-force $d{\vec{F}}_p$

Picture: ball ke front par packed fluid imagine karo, har tile par equally hard inward squeeze karta hua har taraf se. Squeeze jitna dense, $p$ utna bada.

Yeh kyun chahiye: body ka front high pressure mein hota hai (fluid pile up ho raha hai), back aksar low pressure mein (fluid gap fill nahi kar sakta). Woh front-minus-back imbalance do drags mein se ek hai — form drag. $d{\vec{F}}_p$ ke bina hum ise likh hi nahi sakte.

4 · Viscosity aur shear — $\mu$, $u(y)$, ${\tau }_w$, aur shear force $d{\vec{F}}_{\tau }$

Yeh do forces mein se trickier hai, isliye hum ise ek picture ke saath char steps mein banate hain.

4a · Velocity profile $u(y)$

Picture: Figure 2 mein horizontal arrows ki row — wall ke paas chhote, upar lamba. Badhte arrows ka woh fan velocity profile hai.

4b · Gradient $\frac{du}{dy}$

Sirf $u$ ki jagah derivative kyun? Viscous rubbing care nahi karta ki fluid kitni tez move kar raha hai — yeh care karta hai ki neighbouring layers ek doosre se kitna slide past karte hain. Woh sliding exactly steepness $du/dy$ hai. Gradient sahi tool hai kyunki yeh adjacent layers ke beech difference measure karta hai, jo rubbing feel karti hai.

4c · Viscosity $\mu$ aur wall shear stress ${\tau }_w$

4d · Ek patch par shear force $d{\vec{F}}_{\tau }$

${\tau }_w$ ek stress hai (force per area); ek tile par actual force paane ke liye hum tile ke area se multiply karte hain aur ise woh direction point karte hain jis taraf rub act karta hai. Hume woh direction chahiye jis taraf rub point karta hai — use $\hat{t}$ bulao (§6 mein carefully banaya gaya) — taaki hum vector force likh sakein:

Picture: Figure 1 mein green sideways arrow — woh tile par act karta $d{\vec{F}}_{\tau }$ hai.

5 · Fluid density $\rho$ aur dynamic pressure $q$

Picture: ek box mein bhare chhote fluid dots ki sankhya. Zyada dots = bada $\rho$ = raste se hat'tane ke liye bhaari cheez.

$\frac{1}{2}$ kahan se aata hai? Yeh kinetic-energy factor hai. Mass $m$ ke fluid parcel mein energy $\frac{1}{2}m{v}^2$ hoti hai; per unit volume ($m/\text{vol}=\rho$) woh $\frac{1}{2}\rho v^2$ hai. Toh $q$ literally flow ki kinetic energy density hai, aur $\frac{1}{2}$ seedha $\frac{1}{2}m{v}^2$ se aata hai. Dekho Bernoulli's Principle, jahan $q$ static pressure se trade karta hai.

Yeh kyun chahiye: drag fluid mass ko raste se hat'tane se aata hai, aur $q$ exactly "moving fluid kitna hard push karta hai" wala number hai. Har drag formula $q$ times area times shape number hoga.

6 · Dot product aur tangent $\hat{t}$ — drag part slice karna

Formal drag formula patch forces $d{\vec{F}}_p=-p\hat{n}d{A}_s$ (§3) aur $d{\vec{F}}_{\tau }={\tau }_w\hat{t}d{A}_s$ (§4d) ko unke along-flow part tak slice karta hai. Yeh slicing karne wala tool dot product hai — toh hum ise abhi earn karte hain, aur yeh bhi pin down karte hain ki exactly $\hat{t}$ kya hai.

Pressure cases ($-p\hat{n}$ use karke):

• Front (Figure 3, left): $\hat{n}$ partly into flow point karta hai, toh $\hat{n}\cdot \hat{x}<0$ — lekin $d{\vec{F}}_p$ minus sign carry karta hai, isliye yeh positive backward drag contribute karta hai. ✓
• Back (right): $\hat{n}$ flow ke saath point karta hai, $\hat{n}\cdot \hat{x}>0$. Equal back-pressure front cancel kar deta; low-pressure wake ise drop kar deta hai, isliye woh cancel nahi karta — net drag rehti hai.
• Top/bottom ($\hat{n}\perp \hat{x}$): $\hat{n}\cdot \hat{x}=0$ — wahan pressure se zero drag add hota hai (push purely sideways hai).

Shear cases (${\tau }_w\hat{t}$ use karke):

• Long flat sides (Figure 4): $\hat{t}$ seedha downstream point karta hai, toh $\hat{t}\cdot \hat{x}\approx 1$ — rub poori tarah backward hai, sab drag count hota hai. Isliye edge-on flat plate almost pure skin friction hai.
• Curved nose/tail par: $\hat{t}$ upar ya neeche tilt hota hai, toh $0<\hat{t}\cdot \hat{x}<1$ — sirf rub ka downstream component count hota hai.
• Woh patch jiska tangent crosswise run kare ($\hat{t}\perp \hat{x}$): $\hat{t}\cdot \hat{x}=0$ — woh rub sideways hai aur koi drag add nahi karta, exactly top/bottom pressure case mirror karta hua.
• Sign: ${\tau }_w>0$ hamesha (yeh fluid ko woh direction drag karta hai jis taraf fluid move karta hai), aur $\hat{t}$ downstream liya gaya hai, toh skin-friction drag hamesha $\ge 0$ hota hai — yeh kabhi tumhe aage nahi dhakelta.

7 · Total drag $D$, surface sum $\oint$, aur packaged law

Picture: poori body ke around walk karo; har tile par $d{\vec{F}}_p$ aur $d{\vec{F}}_{\tau }$ ka along-flow slice lo, bucket mein daalo; $\oint$ bucket ka total hai. Woh total $D$ hai: $D=\underset{\text{form drag}}{\underbrace{\oint (-p)(\hat{n}\cdot \hat{x})d{A}_s}}+\underset{\text{skin friction drag}}{\underbrace{\oint {\tau }_w(\hat{t}\cdot \hat{x})d{A}_s}}$

Surface sum se tidy formula tak

Woh exact integral ko har point par pressure aur shear chahiye — usually jaanna impossible. Toh hum ise repackage karte hain. Yahan reasoning hai, step by step:

1. Push ka scale kya set karta hai? Moving fluid ka kinetic push, yaani §5 se dynamic pressure $q=\frac{1}{2}\rho v^2$. Surface par har pressure aur shear $q$ ka koi multiple hai.
2. Area ka scale kya set karta hai? Ek single reference area $A_{\text{ref}}$ — bluff body ke liye yeh frontal (projected) area hai, woh silhouette jo flow "dekhta" hai. Yeh §2 ki wetted surface area $A_s$ se alag $A$ hai (wahan notation warning yaad karo); hum subscript $\text{ref}$ rakhte hain taaki kabhi blur na ho.
3. Baki sab — messy geometry (wake kitna bada hai, pressure actually kaise vary karta hai) — ek dimensionless number mein sweep ho jaata hai, drag coefficient $C_D$.

Push-scale ko area-scale se aur shape-number se multiply karke:

Yeh integral jaisi physics kyun hai: dono $\oint$ terms dono "(ek pressure) $\times$ (ek area) $\times$ (ek geometric factor)" form mein hain. Kyunki surface par har pressure dynamic pressure $q$ ke saath scale karta hai, aur har area reference area $A_{\text{ref}}$ ke saath, poora messy sum zaroor $q{A}_{\text{ref}}\times (\text{koi pure number})$ mein collapse hoga — aur woh bacha hua pure number, saari shape aur wake information carry karta hua jise hum haath se compute nahi kar sake, exactly $C_D$ hai. Toh packaged law naya law nahi hai: yeh surface sum $D=\oint (\dots )$ hai jisme har unknown ek experimentally measured coefficient mein quarantine hai. Yehi poora trick hai — hard geometry ko $C_D$ mein chhupaao, easy physics ($\rho$, $v$, $A_{\text{ref}}$) explicit rakho.

Agle pointers jo milenge: Reynolds Number decide karta hai kaise flow behave karta hai (aur kya $C_D$ roughly constant rehta hai), aur drag Terminal Velocity ko feed karta hai jab woh weight balance karta hai.

Prerequisite map

Upar har node is page (§1–§7) par define kiya gaya symbol hai, isliye map sirf woh reorganise karta hai jo tum pehle se jaante ho — koi naya term andar nahi aata.

Equipment checklist

Khud se test karo — har ek reveal karo sirf apna jawab zor se kehne ke baad.