2.2.13Fluid Mechanics
Reynolds transport theorem
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WHY do we even need it?
In particle mechanics you track a fixed mass forever. In fluids, the same fluid particles are constantly moving — tracking them (the Lagrangian view) is hopeless. We prefer to nail down a window in space, the control volume (CV), and watch fluid pass through (the Eulerian view).
The problem: conservation laws are Lagrangian statements.
So we need a machine that converts into control-volume language. That machine is the RTT.
WHAT are the objects?
HOW to derive it from scratch
Take a system that, at time , exactly coincides with the control volume. A short time later the system has drifted: part of it flowed out (region III), part of the CV is now occupied by new fluid that flowed in (region I), and the bulk overlaps (region II).
Step 1 — Bookkeeping at and . At time : the system occupies the CV exactly, so Why this step? We deliberately chose them to coincide, so we can swap them at this instant.
At time : the system has left region I but entered region III: Why? The CV at contains the new inflow (I) which is not system, and is missing the outflow (III) which is system — so correct it.
Step 2 — Form the derivative.
=\underbrace{\frac{d}{dt}\!\int_{CV}\!\rho b\,dV}_{\text{storage}} +\underbrace{\lim_{dt\to0}\frac{B_{III}-B_{I}}{dt}}_{\text{net efflux}}$$ *Why?* The two CV terms become $\frac{d}{dt}B_{CV}$; the I and III terms become surface fluxes. **Step 3 — Turn flux into a surface integral.** In time $dt$, fluid near a patch $dA$ of the surface sweeps a slanted cylinder of volume $$dV=(\vec v\cdot\hat n)\,dt\,dA,$$ where $\hat n$ is the **outward** normal. So the property carried = $\rho b\,(\vec v\cdot\hat n)\,dt\,dA$. *Why $\vec v\cdot\hat n$?* Only the velocity **component along the normal** actually crosses the surface; tangential flow slides along it and crosses nothing. Dividing by $dt$ and integrating over the **whole** closed surface (outflow gives $\vec v\cdot\hat n>0$, inflow gives $<0$, automatically handling I and III): $$\frac{B_{III}-B_{I}}{dt}\to\oint_{CS}\rho b\,(\vec v\cdot\hat n)\,dA.$$ > [!formula] Reynolds Transport Theorem (fixed CV) > $$\boxed{\;\frac{dB_{sys}}{dt}=\frac{d}{dt}\int_{CV}\rho\,b\,dV+\oint_{CS}\rho\,b\,(\vec v\cdot\hat n)\,dA\;}$$ > *(Left = "following the matter". Right = "**unsteady storage** inside" + "**net outflow** through the surface".)* **Moving CV?** Replace $\vec v$ in the surface term by the **relative** velocity $\vec v_r=\vec v-\vec v_{CS}$, because only motion *relative to the surface* actually crosses it. --- ## The laws fall out instantly > [!example] Continuity (mass): $b=1$ > $\dfrac{dm_{sys}}{dt}=0$ so > $$0=\frac{d}{dt}\int_{CV}\rho\,dV+\oint_{CS}\rho(\vec v\cdot\hat n)\,dA.$$ > **Why this step?** Setting the left side to zero just encodes "mass is conserved." Steady incompressible flow ⇒ $\sum(\rho A v)_{out}=\sum(\rho A v)_{in}$. > [!example] Momentum: $b=\vec v$ > $$\sum\vec F=\frac{d}{dt}\int_{CV}\rho\vec v\,dV+\oint_{CS}\rho\vec v(\vec v\cdot\hat n)\,dA.$$ > **Why?** $B=m\vec v$ ⇒ $b=\vec v$, and $\frac{d(m\vec v)_{sys}}{dt}=\sum\vec F$. This is the basis of jet/thrust force problems. > [!example] Number plug-in (steady jet on a plate) > Water jet, $\rho=1000$, $A=2\times10^{-3}\,\text{m}^2$, $v=10\,\text{m/s}$, hits a wall and spreads sideways (loses all x-momentum). > Mass flow: $\dot m=\rho A v=1000(2\times10^{-3})(10)=20\,\text{kg/s}$. > Steady ⇒ storage term $=0$. x-momentum flux out $=0$, in $=\dot m v=200$. > $$F_x=\oint\rho v_x(\vec v\cdot\hat n)dA = 0-(-\dot m v)=200\ \text{N (force on jet from wall)}.$$ > **Why the sign?** Inflow has $\vec v\cdot\hat n<0$; that minus sign flips, giving a positive reaction. The plate pushes back $200\,$N. --- > [!mistake] Steel-manning the classic errors > **(1) "Use the actual speed $\vec v$ for a moving CV."** *Feels right* because $\vec v$ is the only velocity you measured. **Fix:** crossing the surface depends on motion *relative to the surface*, so use $\vec v_r=\vec v-\vec v_{CS}$. A surface moving *with* the fluid has zero flux even if $\vec v\ne0$. > **(2) "Drop the $\frac{d}{dt}\int_{CV}$ term whenever flow exists."** *Feels right* because something is clearly flowing. **Fix:** that storage term is about **unsteadiness in time** ($\partial/\partial t$), not about flow. It vanishes only for **steady** flow. > **(3) Using full speed instead of the normal component.** *Feels right* — "the fluid is fast!" **Fix:** only $\vec v\cdot\hat n$ pierces the surface; tangential flow contributes nothing. > **(4) Forgetting $\rho b$ vs $b$.** RTT integrates $\rho b\,dV$, not $b\,dV$, because the stored amount is property-per-mass times mass, and $dm=\rho\,dV$. --- > [!recall]- Feynman: explain to a 12-year-old > Imagine a window in a river. You want to count fish (the "stuff"). The number of fish **inside the window box** can change two ways: fish are **born or vanish inside** (the storage term — only matters if things change with time), or fish **swim across the window edges** (the surface term — out the right, in the left). RTT just says: *change for the original gang of fish = change you see in the box + net fish that swam out.* That's it. > [!mnemonic] Remember the structure > **"System = Storage + Surface."** (Three S's.) Left of equals = following the *System*; right = *Storage* inside + flux through the *Surface*. --- ## Active recall #flashcards/physics What does RTT translate between? ::: The system (Lagrangian) rate of change and the control-volume (Eulerian) description. General RTT formula? ::: $\frac{dB_{sys}}{dt}=\frac{d}{dt}\int_{CV}\rho b\,dV+\oint_{CS}\rho b(\vec v\cdot\hat n)\,dA$ What is $b$? ::: The intensive property, $b=dB/dm$ (property per unit mass). $b$ for mass, momentum, energy? ::: $1$, $\vec v$, $e$ (specific energy). Why does the surface term use $\vec v\cdot\hat n$? ::: Only the velocity component along the outward normal actually crosses the surface; tangential flow slides along it. When does the storage term $\frac{d}{dt}\int_{CV}$ vanish? ::: For steady flow (no time dependence inside the CV). What replaces $\vec v$ in the surface term for a moving CV? ::: The relative velocity $\vec v_r=\vec v-\vec v_{CS}$. RTT with $b=1$ gives which law? ::: Continuity / conservation of mass. RTT with $b=\vec v$ gives which law? ::: Linear momentum equation, $\sum\vec F=\dots$. Why coincide system and CV at time $t$? ::: So $B_{sys}(t)=B_{CV}(t)$, allowing the swap that starts the derivation. Sign of $\vec v\cdot\hat n$ for inflow vs outflow? ::: Inflow negative, outflow positive (outward normal convention). --- ## Connections - [[Continuity equation]] — the $b=1$ special case. - [[Momentum equation (control volume)]] — the $b=\vec v$ special case; jet thrust. - [[Eulerian vs Lagrangian description]] — the two viewpoints RTT bridges. - [[Material derivative]] — the differential ("point") analogue of RTT. - [[Divergence theorem]] — converts the surface flux into a volume integral, yielding differential conservation laws. - [[Bernoulli equation]] — energy form, $b=e$, under restrictions. ## 🖼️ Concept Map ```mermaid flowchart TD CL[Conservation laws are Lagrangian] SYS[System - fixed mass] CV[Control volume - fixed region] RTT[Reynolds Transport Theorem] B[Extensive property B] b[Intensive property b equals dB/dm] BCV[B stored in CV integral rho b dV] COIN[System coincides with CV at time t] STOR[Storage term d/dt B_CV] FLUX[Net efflux term] SURF[Surface integral rho b v dot n dA] CL -->|written for| SYS SYS -->|hard to track| CV CV -->|needs translation| RTT RTT -->|converts| CL B -->|per unit mass gives| b B -->|stored as| BCV b -->|used in| BCV RTT -->|derived from| COIN COIN -->|yields| STOR COIN -->|yields| FLUX FLUX -->|swept volume gives| SURF BCV -->|becomes| STOR ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, Physics ke saare conservation laws (mass, momentum, energy) ek **fixed lump of matter** ke liye likhe gaye hain — yaani ek "system" jisme hamesha same molecules hote hain. Lekin fluid mein toh particles continuously move karte rehte hain, unhe chase karna impossible hai. Isliye hum ek **control volume (CV)** fix kar lete hain — ek window space mein — aur dekhte hain fluid usme se kaise guzarta hai. Problem yeh hai ki laws system ke liye hain, par hame CV ki language chahiye. **Reynolds Transport Theorem yahi translation karta hai.** > > Core idea bahut simple hai: jis "stuff" $B$ ko track kar rahe ho (mass, momentum...), uski system-wali rate of change = (CV ke andar jo store ho raha hai uska time-rate) + (jo CV ki surface se net bahar nikal raha hai). Bas teen S yaad rakho: **System = Storage + Surface.** Surface wale term mein $\vec v\cdot\hat n$ aata hai kyunki sirf normal direction wala velocity component hi surface ko cross karta hai — tangential flow toh bas surface ke saath slide karta hai, kuch cross nahi karta. > > Do important traps: (1) Storage term $\frac{d}{dt}\int_{CV}$ tab zero hota hai jab flow **steady** ho — flow hone se nahi, *time-change* na hone se. (2) Agar CV khud move kar raha hai, toh surface term mein full velocity nahi, **relative velocity** $\vec v_r = \vec v - \vec v_{CS}$ daalo, kyunki crossing toh surface ke relative hota hai. > > Iska practical fayda? $b=1$ daalo toh continuity equation, $b=\vec v$ daalo toh momentum equation milti hai — jisse aap jet thrust, rocket, pipe bend pe force sab nikaal sakte ho. Ek hi theorem, aur saare CV problems khul jaate hain. JEE/GATE mein yeh foundation hai. ![[audio/2.2.13-Reynolds-transport-theorem.mp3]]