Foundations — Assumptions in Bernoulli — steady, inviscid, incompressible, along streamline

This page assumes you know nothing. Every letter in $P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$ will be earned one at a time, each with a picture.

Once these are solid, the parent note Bernoulli's assumptions reads like a story instead of a wall of symbols.

0. What is a "fluid parcel"?

Before any symbol, meet the star of the show.

Picture a single grain of dye dropped into a river. It doesn't teleport; it drifts along a path. That path is what we build everything on.

Figure 1 — a fluid parcel follows a definite path.

Why we need it: Newton's laws are written for objects. To use physics we must first decide which object — and that object is this parcel.

1. The streamline — the road the parcel drives on

Look at Figure 2 below. The cyan curve is a streamline. The amber arrows are the fluid velocity at three spots — each arrow lies along the curve. That is the whole definition: the curve is always tangent to the flow.

Figure 2 — a streamline is tangent to the velocity everywhere.

Why the topic needs it: Bernoulli's promise ("the sum stays constant") is only guaranteed along one streamline. So the streamline is the track on which we do all our accounting. Related idea: Irrotational Flow is the special case where the constant becomes the same on every streamline.

We will measure distance along this track with the letter $s$.

2. $\rho$ — density: how much stuff is packed in

Picture two identical boxes: one full of packed marbles (high $\rho$), one with a few marbles floating in air (low $\rho$). Water is about $1000$; air about $1.2$.

Why the topic needs it: our parcel's mass is $\rho \times (\text{its volume})$. We will pin down the parcel's exact shape and volume in §4; once we do, its mass becomes $\rho Ads$ — the thing we plug into Newton's law (defined in §5b). The assumption "incompressible" is exactly the statement $\rho$ never changes, which is why the parent note can pull $\rho$ out of an integral.

3. $P$ — pressure: the parcel's sideways "push"

Picture the parcel squeezed on all sides by its neighbours, like a person in a crowd. If the crowd pushes harder on the back than the front, there is a net forward shove. That difference is what drives the parcel.

We will meet the face area the pressure pushes on, called $A$, in §4; here just watch the difference between the two faces.

Figure 3 — net pressure force is back push minus front push, $=-AdP$.

In Figure 3 the back face feels pressure $P$ pushing forward (a force $+PA$) and the front face feels a slightly different pressure $P+dP$ pushing back (a force $-(P+dP)A$). The little $dP$ is the change in pressure over the step $ds$.

Why the topic needs it: pressure difference is one of only two forces that push the parcel along its track (the other is gravity). See Bernoulli's Equation for the full force balance.

4. $A$ and $ds$ — the parcel's shape

Picture a coin: its flat face is $A$, its thickness is $ds$. Multiply to get its volume.

Why the topic needs it: $A$ lets pressure (force per area) become an actual force ($P\cdot A$), and $Ads$ lets density become an actual mass. Without these the symbols can't enter Newton's law. This is exactly the $A$ we previewed in §2 and §3 — now it is earned.

5. $v$ — speed, and $\frac{1}{2}\rho v^2$ — the energy of motion

Why the $\frac{1}{2}$ and the square? Double the speed and the energy quadruples (it's $v^2$), because stopping something moving twice as fast takes four times the work. The $\frac{1}{2}$ falls out of adding up that work from rest — it is not a fudge factor.

Why the topic needs it: this is the "motion" term in Bernoulli's sum. When a parcel speeds up (bigger $v$), this term grows, so something else in the sum (usually $P$) must shrink.

5b. $F$, $a$, and Newton's second law — the engine of the whole derivation

Before we combine forces, we need the law that turns "forces on the parcel" into "the parcel changes speed".

Why the topic needs it: Bernoulli is derived by writing $F=ma$ for the parcel, where $F$ is the pressure force $-AdP$ plus the gravity force (§7), $m$ is the mass $\rho Ads$, and $a$ is the material acceleration (§8). Every later step is just this one law unpacked.

6. $g$, $h$, and $\rho gh$ — the energy of height

Picture raising a bucket of water: the higher you lift it, the more energy it can release by falling. That stored "willingness to fall" is $\rho gh$.

Why the topic needs it: it is the third and last term in the sum. A parcel climbing a pipe trades motion or pressure energy for height energy.

7. $dh/ds$ — how steeply the streamline climbs, and where the minus sign comes from

Figure 4 — uphill: $dh/ds>0$, gravity opposes forward motion (hence the minus sign).

Why we need a ratio here: gravity pulls straight down, but the parcel moves along the (possibly tilted) track. Only the part of gravity that lines up with the track can speed the parcel up or slow it down. That fraction is exactly the slope $dh/ds$.

Look at Figure 4: the amber arrow $ds$ climbs the track, the white arrow $dh$ is the height gained. Their ratio is $dh/ds$, and because climbing means fighting gravity, the force it produces carries the minus.

Why the topic needs it: this is the exact term the parent note writes for gravity along the streamline. Getting the sign right is what lets Bernoulli correctly trade height for speed and pressure in both uphill and downhill flow.

8. $\partial v/\partial t$ and $Dv/Dt$ — the two reasons a parcel speeds up

Here two new notations appear. Take them slowly.

Why the topic needs it: the "steady" assumption is precisely $\partial v/\partial t=0$ — it deletes the first term. Understanding these two pieces is why the parent can throw one away and keep the other.

9. The integral sign $\int$ — adding up along the track

Why the topic needs it: Newton's law gives us a relation between tiny pieces ($\rho vdv$, $dP$, $\rho gdh$). To get Bernoulli's whole-path statement we must add all the tiny pieces along the streamline — that is the integration step, and the leftover total is exactly the "$=\text{constant}$" from the top of this page. And it is here that "incompressible" ($\rho$ constant) matters: only then can $\rho$ slide outside the integral cleanly.

How the foundations feed the topic

The map below is a reading order, top to bottom. Start at the fluid parcel. From it we get a mass (via density $\rho$ and the shape $Ads$). Separately, pressure $P$ over the area $A$, plus gravity $g$ scaled by the slope $dh/ds$, give the net force along the track. The parcel's acceleration comes from the material derivative $Dv/Dt$. Newton's law $F=ma$ locks force, mass and acceleration together into one equation. Finally the integral sign $\int$ adds up that equation along the whole streamline, and out drops Bernoulli's constant sum $P+\frac{1}{2}\rho v^2+\rho gh$. If a picture won't render for you, this paragraph is the map in words.

Equipment checklist

Cover the right side and answer aloud; reveal to check.