Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations: Foundations

This page assumes you have seen none of the notation. We introduce each symbol from a picture, in an order where each one leans only on the ones already defined. Read top to bottom.

0. The scene we keep drawing

Picture a pipe with gas flowing left-to-right, and one spot where the gas is forced to a dead stop — a wall, or the nose of a probe. That stopping point is where "static" turns into "total". We will name and picture every quantity on this figure, one at a time, starting from the flow speed.

Figure — Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations

Everything below labels one piece of this picture.

1. $V$ — the speed of the bulk flow

$V$ is the ordered motion — every molecule drifting the same way. Contrast that with temperature (next), which is disordered jiggle.

2. $T$ — static temperature (disordered energy)

3. $P$ and $ρ$ — static pressure and density

Recall Why "static" if the gas is moving?

"Static" ::: means "measured in the fluid's own frame", i.e. what you'd read moving alongside it — NOT that the gas is at rest.

These three, $T, P, ρ$ , are tied together by a single equation of state, but that equation needs one more symbol — a gas-specific constant — before we can write it. That is the very next section.

4. $R$ — the specific gas constant, and the ideal-gas law

5. Heat and internal energy — $Q$ and $u$

Before we can talk about energy conservation we need names for the two energies that can enter or hide inside the gas.

Definition

$Q$ — heat transfer (with a sign convention) Plain words: energy that crosses the boundary of a gas lump because of a temperature difference — the flame-under-the-pan kind of energy. Measured in joules (J); the dot form $\dot{Q}$ means "joules per second" (a rate of heat flow). Sign convention (used throughout): $Q > 0$ means heat enters the gas; $Q < 0$ means heat leaves it. So "adiabatic" ( $Q = 0$ ) means neither — nothing crosses the boundary in either direction. The picture: wavy red arrows pointing inward through the pipe walls count as positive; arrows pointing outward are negative. If the walls are perfectly insulated, $\dot{Q} = 0$ and there are no arrows at all. Why the topic needs it: the stagnation derivation assumes $\dot{Q} = 0$ (no leak either way); to say that cleanly, we first need both the symbol $Q$ and its sign rule.

6. Internal energy vs enthalpy — $c_{v}$ , $c_{p}$ , $h$

To do energy conservation we need the right kind of energy. Two flavours:

7. $γ$ — the specific-heat ratio

Recall Why does

$c_{p} = \frac{γ R}{γ - 1}$ ? From $c_{p} - c_{v} = R$ (Mayer) and $γ = c_{p} / c_{v}$ ::: substitute $c_{v} = c_{p} / γ$ into Mayer: $c_{p} - c_{p} / γ = R \Rightarrow c_{p} (1 - 1/ γ) = R \Rightarrow c_{p} = γ R / (γ - 1)$ .

8. $a$ — the speed of sound

Intuition WHY

$a = γ R T$ — the physical reason A sound wave is a chain of tiny squeeze-and-release pulses. Two things set how fast the pulse hands itself on:

Stiffness — how strongly the gas pushes back when squeezed. For a fast pulse the squeeze happens too quickly for heat to escape, so it is adiabatic, and the relevant stiffness is $γ P$ (the springiness dial $γ$ times pressure). Stiffer gas → the "push-back" news travels faster → $a$ larger, so $a^{2} \propto γ P$ .
Inertia — how heavy the gas is. Heavier gas is sluggish and passes the pulse on more slowly, so $a^{2} \propto 1/ ρ$ .

Putting them together, $a^{2} = γ P / ρ$ . Now use the ideal-gas law $P / ρ = R T$ from §4: $a^{2} = γ \frac{P}{ρ} = γ R T ⟹ a = γ R T .$ Notice $a$ depends only on $T$ (with $γ, R$ fixed): hotter gas carries sound faster because its molecules already jiggle harder and pass the pulse along sooner. That is why the "stopped" temperature also controls the sound speed everywhere in the flow.

Speed strand (left): $V$ and $T$ combine — via the sound speed $a$ — into the Mach number $M$ .
Energy strand (middle): heat $Q$ and internal energy $u$ build enthalpy $h$ , which powers the energy balance that yields the stopped temperature $T_{0}$ .
Order strand (right): constant entropy $s$ gives the isentropic law, which lifts $T_{0}$ into the stopped pressure and density $P_{0}, ρ_{0}$ .

Related destinations you'll reach with these tools: Isentropic flow relations, Bernoulli's equation (incompressible limit), Steady-flow energy equation (First Law for control volumes), Pitot tube measurement, and the parent Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations.

Equipment checklist

Self-test: cover the right side and see if you can state each from memory.

$V$ ::: bulk flow speed — the ordered motion that gets scrambled when the flow stops.
$T$ ::: static temperature — disordered molecular jiggle in the fluid's own frame.
$P, ρ$ ::: static pressure and density, linked by $P = ρR T$ .
$R$ ::: specific gas constant, $287 J/ (kg \cdot K)$ for air.
$Q$ ::: heat crossing the boundary; $Q > 0$ enters, $Q < 0$ leaves, $Q = 0$ is adiabatic.
$u$ ::: specific internal energy, $u = c_{v} T$ for an ideal gas.
$c_{p}$ vs $c_{v}$ ::: heat per kg per K at constant pressure vs volume; $c_{p} > c_{v}$ because the gas also does push-work.
Mayer ::: $c_{p} - c_{v} = R$ — the extra push-work per K is exactly $R$ .
$h$ ::: enthalpy $= u + P / ρ = c_{p} T$ ; the correct energy for flowing gas because it includes flow-work $P / ρ$ .
$γ$ ::: $c_{p} / c_{v} = 1.4$ for air; sets the stagnation exponents $\frac{γ}{γ - 1}$ and $\frac{1}{γ - 1}$ .
$a$ ::: local sound speed $γ R T$ — from stiffness $γ P$ over inertia $ρ$ ; how fast pressure news travels.
$M$ ::: Mach number $V / a$ — the true measure of compressibility; every ratio is written in it.
subscript $0$ ::: value the gas reaches when smoothly brought to rest.
$s$ ::: entropy — a one-way "scramble counter"; constant only in reversible flow.
adiabatic ::: no heat crosses the boundary ( $Q = 0$ ) → conserves $T_{0}$ .
isentropic ::: adiabatic + reversible ( $s$ constant, $P / ρ^{γ} =$ const) → conserves $P_{0}$ too.

Foundations — Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations