Foundations — Bernoulli's equation — derivation from F = ma along streamline

Before you can read the parent derivation, you need to see what each letter stands for. This page builds every one of them from nothing, in the order they first appear, so no symbol is used before it has a picture attached. If a term already feels obvious, treat its reveal line at the bottom as a quick self-test.

0 — What is a fluid parcel? (the object everything happens to)

Everything in Bernoulli happens to a parcel: an imaginary tiny box of liquid that we mentally paint a different colour and follow as it drifts along. It is not a real, separate thing — the water around it is identical — but by choosing a definite chunk we get a definite mass, and mass is what Newton's law needs.

1 — The streamline and $s$ (the road the parcel drives on)

A streamline is a curve that is everywhere tangent to the fluid's velocity — draw an arrow for the velocity at each point, join the arrows head-to-tail, and you have the streamline. The parcel rides along it like a car on a road.

We measure distance travelled along that road with the letter $s$. So $s$ is not left-right or up-down; it is "how far along the winding path have we gone."

Why the topic needs it. Newton's law is a vector statement. By measuring along the streamline we only ever deal with the component of force in the direction of motion — the only component that can change the parcel's speed. That is why the whole derivation collapses to one-dimensional algebra. See Equation of Continuity for another idea built on streamlines.

2 — Tiny increments: $ds$, $dy$, $dP$, $dv$ (the "$d$" means "a sliver of")

The little "$d$" in front of a quantity means "a very small piece of it." It is a slice so thin that across it everything except the change stays effectively constant.

• $ds$ — a tiny length of streamline the parcel occupies.
• $dy$ — the tiny rise in height across that length.
• $dP$ — the tiny change in pressure from the back face to the front face.
• $dv$ — the tiny change in speed the parcel gains crossing that slice.

3 — Density $\rho$ (how heavy the fluid is, per box)

Density $\rho$ (Greek letter "rho", rhymes with "row") is mass packed into each unit of volume: $\rho =\text{mass}/\text{volume}$. Water is about $\rho =1000{\text{kg/m}}^3$ — a cube of side one metre holds a tonne.

Why the topic needs it. Newton's law needs mass, and the parcel's mass is $dm=\rho \times dV$. Bernoulli assumes $\rho$ is constant (incompressible), which is exactly what lets us pull it outside the integral in Step 5 of the parent note.

4 — Area $A$, volume $dV$, mass $dm$ (turning "size" into "mass")

• Area $A$ — the size of a flat face, e.g. the circular end of the parcel-cylinder; units ${\text{m}}^2$.
• Volume $dV$ — how much space the parcel fills. For a cylinder of face $A$ and length $ds$: $dV=Ads$.
• Mass $dm$ — how much stuff is in it: $dm=\rho dV=\rho Ads$.

Why the topic needs it. These three turn the abstract "parcel" into a concrete number you can put into $F=ma$. Area $A$ also appears in Equation of Continuity as the throttle that sets speed.

5 — Pressure $P$ (a push spread over a face)

Pressure $P$ is force pressing on a surface, divided by that surface's area: $P=F/A$. Its units are pascals, $\text{Pa}={\text{N/m}}^2$. Crucially, a fluid pushes on any surface perpendicular to it, and it pushes on both faces of our parcel.

Why the topic needs it. Pressure differences are one of only two forces in the whole derivation. See Hydrostatic Pressure for how $P$ behaves when the fluid is not moving.

6 — Gravity $g$, height $y$, and the slope $\sin \theta$

• $g$ — the strength of gravity's pull, $\approx 9.8{\text{m/s}}^2$; it points straight down.
• $y$ — height, measured upward from any chosen floor. Higher parcel = larger $y$.
• $\theta$ — the angle the streamline makes with the horizontal, so $\sin \theta =dy/ds$ is the fraction of gravity that lies along the road.

Why the topic needs it. Gravity is the second (and last) force. Height differences are what make Torricelli's tank drain — see Torricelli's Law.

7 — Speed $v$ and its change with position $\frac{dv}{ds}$

$v$ is the parcel's speed along the streamline. In steady flow the speed at any fixed spot never changes — but different spots have different speeds, so a moving parcel still finds itself in a faster region a moment later.

8 — The chain rule and the derivative $\frac{dv}{dt}$ (why we can turn time into distance)

Acceleration is "change of speed per unit time", $a=\frac{dv}{dt}$. But our forces are naturally written in terms of distance $s$, not time. The chain rule — a rule for stacking two rates of change — lets us swap: $a=\frac{dv}{dt}=\underset{\text{speed change per metre}}{\underbrace{\frac{dv}{ds}}}\times \underset{=v}{\underbrace{\frac{ds}{dt}}}=v\frac{dv}{ds}$

Why this tool and not another? We want a distance-based acceleration because our forces (pressure, gravity) are laid out along distance $s$. The chain rule is the only clean way to convert the time-derivative Newton demands into the distance-derivative our geometry supplies. This one move is what produces the $\frac{1}{2}\rho v^2$ term in the final equation. Built on Newton's Second Law.

9 — Newton's law $F=ma$ and the work–energy view

• $F=ma$ — net force equals mass times acceleration. Push twice as hard, accelerate twice as fast; twice the mass, half the acceleration.
• Multiplying $F=ma$ through by a displacement turns it into the Work-Energy Theorem: net work done = change in kinetic energy. Bernoulli can be read either way — the parent note calls these "two readings of the same equation."

10 — The integral sign $\int$ (adding up all the slivers)

The tall S, $\int$, means "sum up." Each sliver contributed a tiny relation $dP+\rho gdy+\rho vdv=0$; integrating from point 1 to point 2 adds every sliver's contribution along the whole streamline: ${\int }_1^{2}dP+\rho g{\int }_1^{2}dy+\rho {\int }_1^{2}vdv=0.$

How these feed the topic

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, reread that section above before opening the parent note.