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

Before you can read the parent note Potential flow, you need to earn every symbol it throws at you. We build them one at a time, each on top of the last. Never do we use a mark on the page you haven't already seen defined and drawn.

1. Vectors and the velocity field $\vec{v}$

The little arrow over the letter ($\stackrel{⃗}{}$) is our reminder that this quantity has a direction, not just a size. A number with only size (like temperature) is a scalar; a quantity with size and direction (like velocity) is a vector.

In 2D we split the arrow into two pieces along the axes:

• $u$ = how much of the arrow points along the horizontal $x$-axis,
• $v$ = how much points along the vertical $y$-axis.

So writing $\vec{v}=(u,v)$ just means "the arrow is $u$ steps right and $v$ steps up."

2. The gradient $\nabla \varphi$ — slope of a hill

Suppose you have a scalar map: at every point a single number $\varphi$ (say, height of a landscape). The gradient answers the question "which way is steepest uphill, and how steep?"

Two new marks appear here — let's earn them.

• The symbol $\partial$ (a "curly d") means a partial derivative: the slope of $\varphi$ when you nudge only one coordinate and freeze the rest. $\frac{\partial \varphi }{\partial x}$ = "how much does $\varphi$ change if I step a tiny bit in $x$, keeping $y$ fixed?" It is the rate of change, our tool for measuring steepness — we use derivatives (not plain subtraction) because the slope can be different at every point and in every direction.
• The symbol $\nabla$ ("nabla" or "del") is a bookkeeping device: apply it to a scalar and it hands back the vector of all the partial slopes.

3. The dot product and divergence $\nabla \cdot \vec{v}$

To ask "is fluid piling up here or spreading out?" we need a way to add up the outward flow at a point. The tool is the dot product feeding into the divergence.

Now apply $\nabla$ as if it were the vector $\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y}\right)$ and "dot" it into $\vec{v}$:

4. The cross product and curl $\nabla \times \vec{v}$ — local spin

The last derivative measures a different question: "does a tiny paddle-wheel dropped here start to spin?"

The $\times$ ("cross") between $\nabla$ and $\vec{v}$ is a second way vectors combine; in flat 2D it collapses to the single difference above.

5. Putting curl-zero to work: the potential exists

Here is the pivot the whole subject balances on:

WHAT this says: no local spin $\Rightarrow$ the velocity is the slope of some height-map $\varphi$. WHY it's true (intuition): if walking around any closed loop always returned you to the same "height," a consistent height-map can be drawn — and zero curl is exactly the guarantee that loops don't accumulate a net twist. WHAT it buys us: three velocity numbers $(u,v)$ collapse into one scalar $\varphi$. One function to solve for instead of many.

Combine the two conditions from §3 and §5:

$\underset{\text{incompressible}}{\underbrace{\nabla \cdot \vec{v}=0}}\text{and}\underset{\text{irrotational}}{\underbrace{\vec{v}=\nabla \varphi }}⟹\nabla \cdot (\nabla \varphi )={\nabla }^2\varphi =0.$

That final object ${\nabla }^2\varphi$ — "divergence of the gradient" — is the Laplacian, and ${\nabla }^2\varphi =0$ is Laplace's equation. It is linear, which is the secret behind superposition.

6. Polar coordinates $(r,\theta )$ — because flows are round

The parent describes sources, vortices and cylinders — all naturally circular. So instead of $(x,y)$ we often use:

• $r$ = distance from the origin (how far out),
• $\theta$ = angle measured anticlockwise from the positive $x$-axis (which way round).

The matching velocity pieces are $u_r$ (moving outward/inward) and $u_{\theta }$ (moving around the circle). The conversions worth remembering: $x=r\cos \theta ,y=r\sin \theta ,r=\sqrt{x^2+y^2}.$

This is why the parent writes a source as $u_r=\frac{m}{2\pi r}$ (pure outward) and a vortex as $u_{\theta }=\frac{\Gamma }{2\pi r}$ (pure round-and-round).

7. The stream function $\psi$ and the constants $m,\Gamma ,\kappa ,U$

The remaining letters are just dial settings — numbers that say "how strong":

These feed the pressure story through Bernoulli's equation, the lift story through the Kutta–Joukowski theorem and Magnus effect, and the shape-bending story through Conformal mapping.

8. How it all fits together

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