Potential flow — irrotational, inviscid; superposition of basic flows

1. The two assumptions — WHAT and WHY

WHY irrotational matters: A theorem from vector calculus says if the curl of a field is zero (in a simply-connected region), that field is the gradient of a scalar. So:

$\nabla \times \vec{v} = 0 \quad\Longrightarrow\quad \vec{v} = \nabla \phi$

where $\phi$ is the velocity potential. One scalar replaces three velocity components.

2. Deriving Laplace's equation from scratch

HOW we combine the two ideas. We have:

Irrotational $\Rightarrow \vec{v} = \nabla\phi$
Incompressible $\Rightarrow \nabla\cdot\vec{v} = 0$

Substitute the first into the second:

$\nabla \cdot (\nabla \phi) = 0 \quad\Longrightarrow\quad \boxed{\nabla^2 \phi = 0}$

Why this step? Divergence of a gradient is the Laplacian by definition. So potential flow = solving Laplace's equation, the most studied linear PDE in physics.

3. The stream function (2D companion)

WHY define it: continuity is satisfied automatically: $\nabla\cdot\vec v=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=\frac{\partial^2\psi}{\partial x\partial y}-\frac{\partial^2\psi}{\partial y\partial x}=0.$ And irrotationality forces $\nabla^2\psi = 0$ too. The pair $(\phi,\psi)$ are conjugate harmonic functions: $\phi=$ const and $\psi=$ const lines are everywhere perpendicular (Cauchy–Riemann relations).

$\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y},\qquad \frac{\partial\phi}{\partial y}=-\frac{\partial\psi}{\partial x}$

4. The basic building-block flows

Derive each by writing the simplest $\phi$ / $\psi$ and reading off velocities.

Figure — Potential flow — irrotational, inviscid; superposition of basic flows

5. Superposition examples

Common mistake Steel-manning the common errors

"Irrotational means the fluid doesn't go in circles." Why it feels right: "rotation" sounds like circular motion. Fix: the free vortex moves in perfect circles yet is irrotational ( $\nabla\times\vec v=0$ ) everywhere except the singular center. Irrotational refers to local spin of a fluid element (vorticity), not the global path shape.

"Superposition adds pressures too." Why it feels right: velocities add, so why not pressure? Fix: Bernoulli is nonlinear in $v$ ( $p+\tfrac12\rho v^2=$ const). Add the velocity fields, then compute pressure once at the end from the total velocity. Never add pressures.

"Potential flow predicts drag." Fix: for steady flow past a body it predicts zero drag (d'Alembert's paradox) because there's no viscosity/wake. It's great for lift and pressure distribution, useless for drag.

Recall Feynman: explain to a 12-year-old

Imagine syrup with no stickiness flowing so smoothly that no little piece ever spins like a top. Then the whole flow can be described by one "height map" — fluid always rolls "downhill" on this map. Because the map's rule is simple (just adds up), you can take a "straight river" map, a "spreading fountain" map, and a "swirl" map and stack them on top of each other to invent the flow around a ball or a wing. Stack a swirl onto the ball-flow and you've explained why a spinning ball curves!

Flashcards

What two assumptions define potential flow?

Inviscid (

\mu=0

) and irrotational (

\nabla\times\vec v=0

Why does irrotationality let us define a velocity potential?

Zero curl ⇒ field is a gradient:

\vec v=\nabla\phi

Derive the governing equation of potential flow.

Incompressible

\nabla\cdot\vec v=0

plus

\vec v=\nabla\phi

gives

\nabla^2\phi=0

(Laplace).

Why is superposition valid here?

Laplace's equation is linear, so sums of solutions are solutions; velocities add.

Define the stream function and its key property.

u=\partial\psi/\partial y,\ v=-\partial\psi/\partial x

; lines

\psi=

const are streamlines; satisfies continuity automatically.

\phi

and

\psi

for a source of strength

m

\phi=\frac{m}{2\pi}\ln r,\ \psi=\frac{m}{2\pi}\theta

, with

u_r=m/(2\pi r)

u_\theta

for a free vortex of circulation

\Gamma

u_\theta=\Gamma/(2\pi r)

; irrotational except at center.

How do you build flow past a cylinder of radius

a

Uniform flow + doublet with

a^2=\kappa/(2\pi U)

; the circle

r=a

becomes a streamline.

Surface speed on a non-spinning cylinder?

u_\theta=-2U\sin\theta

; max

2U

at top/bottom, stagnation at front/back.

Stagnation point of uniform+source (Rankine half-body)?

x_s=-m/(2\pi U)

on the axis, where source push cancels the stream.

State Kutta–Joukowski lift.

L'=\rho U\Gamma

per unit span; lift from added circulation.

What is d'Alembert's paradox?

Steady potential flow past a body gives zero drag (no viscosity, no wake).

Common error: do pressures superpose?

No — add velocity fields, then apply (nonlinear) Bernoulli once.

Connections

Laplace's equation — the linear PDE that makes superposition work
Bernoulli's equation — converts the superposed velocity field into pressure
Vorticity and circulation — defines irrotationality and the vortex
Kutta–Joukowski theorem — lift from circulation
Cauchy–Riemann equations — link $\phi$ and $\psi$ as conjugate harmonics
Magnus effect — spinning cylinder/ball lift
Conformal mapping — extends these flows to airfoil shapes

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, potential flow ka pura jaadu do assumptions par tika hai: fluid me viscosity nahi (inviscid) aur har fluid element apne axis par ghumta nahi (irrotational, yaani vorticity zero). Jab curl zero hota hai, to vector calculus kehta hai ki velocity kisi scalar ka gradient ban jaata hai — usko hum velocity potential $\phi$ kehte hain. Ab teen velocity components ki jagah sirf ek scalar! Isme incompressibility ( $\nabla\cdot v=0$ ) daalo to seedha Laplace equation $\nabla^2\phi=0$ mil jaata hai.

Laplace equation linear hai — yahi sabse important baat hai. Linear matlab agar $\phi_1$ aur $\phi_2$ dono solution hain, to inka sum bhi solution hai. Isliye hum chote-chote simple flows — uniform flow, source/sink, vortex, doublet — ko bas add karke complicated flow bana sakte hain. Jaise uniform flow + doublet = ek cylinder ke aaspaas ka flow, kyunki ek particular circle streamline ban jaati hai jo solid body jaisa behave karti hai.

Ek bahut common galti: log sochte hain irrotational matlab fluid circle me nahi ghoomega. Galat! Free vortex perfect circles me ghoomta hai par phir bhi irrotational hai (center chhod ke). Irrotational ka matlab hai local spin zero, path ka shape nahi. Doosri galti — velocities add hoti hain par pressure add nahi hoti, kyunki Bernoulli nonlinear hai. Pehle total velocity nikaalo, phir ek hi baar Bernoulli laga ke pressure nikaalo.

Practical importance: cylinder + vortex add karo to ek side speed badhti hai, doosri ghatti hai, pressure difference se lift milta hai — $L'=\rho U\Gamma$ (Kutta-Joukowski). Yahi spinning ball ke curve aur airplane wing ke lift ka basic physics hai. Bas yaad rakho — potential flow drag predict nahi karta (d'Alembert paradox), lekin lift aur pressure distribution ke liye zabardast tool hai.

Potential flow — irrotational, inviscid; superposition of basic flows

1. The two assumptions — WHAT and WHY

2. Deriving Laplace's equation from scratch

3. The stream function (2D companion)

4. The basic building-block flows

5. Superposition examples

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Fluid Mechanics

Connections