Foundations — Stream function, velocity potential

Before you can read the parent note 2.2.11 Stream function, velocity potential, you must own every symbol it throws at you. Below, each symbol gets three things: plain words, the picture, and why the topic needs it. They are ordered so each one leans only on the ones above it.

1. A point in the plane: $x$ and $y$

The picture: a flat sheet of graph paper. Every dot on it has an address $(x,y)$.

Why the topic needs it: the fluid fills a flat region (we study 2D flow — a thin sheet, like a shallow river seen from above). To talk about "the water here versus there" we must be able to name spots. That is all $(x,y)$ does.

2. Speed with a direction: the velocity vector $\vec{V}=(u,v)$

The picture: at each spot on the sheet, draw a tiny arrow showing where the water is heading and how fast. That arrow's shadow on the horizontal axis is $u$; its shadow on the vertical axis is $v$.

Why the topic needs it: this is the villain of the story. At every one of the infinitely many points we have two numbers, $u$ and $v$. That is the "two functions" the parent note wants to shrink down to one. If $u$ or $v$ can be negative, that just means the flow points left or down there — every sign is allowed.

3. The slope of a hill in one direction: the partial derivative $\frac{\partial f}{\partial x}$

Here is the first tool. Meet it slowly.

The curly $\partial$ (say "partial dee") is used instead of the straight $d$ precisely to remind you: other variables are held still. $\partial f/\partial y$ is the same idea facing north.

Why the topic needs it: every single formula in the parent — $u=\partial \varphi /\partial x$, $v=-\partial \psi /\partial x$, continuity, Laplace — is built from partial derivatives. They are how we say "the clever scalar changes this fast in this direction."

We will also use a shorthand: $f_x$ means $\partial f/\partial x$, and $f_{xy}$ means "differentiate by $x$, then by $y$." Same thing, fewer symbols.

4. The uphill arrow: the gradient $\nabla f$

The picture: stand on the hill $f$. Water poured out would run down the slope; $\nabla f$ is the arrow pointing the opposite way, up the slope, perpendicular to the flat contour lines.

Why the topic needs it: the velocity potential is defined by $\vec{V}=\nabla \varphi$ — "the flow is the uphill arrow of some hill $\varphi$." You cannot read that line until $\nabla$ means something. Notice: $\nabla f$ turns one scalar $f$ into two numbers (an arrow) — exactly the "unpack one into two" move the topic exploits.

5. Two jobs for del: divergence $\nabla \cdot \vec{V}$ and curl $\nabla \times \vec{V}$

The same del symbol does two more jobs, this time eating an arrow-field and spitting out a rate.

Why the topic needs both:

• Incompressible (fluid can't be squished, mass conserved) means $\nabla \cdot \vec{V}=0$ — nothing created or destroyed in any box. This is the Continuity Equation, and it is what lets the stream function $\psi$ exist.
• Irrotational (no local spinning) means $\nabla \times \vec{V}=0$ — see Vorticity and Irrotational Flow. This is what lets the velocity potential $\varphi$ exist.

Each condition ties $u$ and $v$ together with one equation, which is precisely how we drop from two unknowns to one.

6. The two star scalars: $\varphi$ and $\psi$

Now the payoff symbols can finally be named.

When both exist, matching their definitions of $u$ and $v$ gives the Cauchy-Riemann Equations, and each scalar obeys the Laplace Equation ${\nabla }^2\varphi =0,{\nabla }^2\psi =0$. The orthogonal grid they draw is a flow net. The potential idea mirrors the Electrostatic Potential $\vec{E}=-\nabla V$, and once the flow is solved, pressures follow from the Bernoulli Equation.

7. How it all feeds together

Equipment checklist

Cover the right side and see if you can answer each before revealing.