Section 02
The Idea, Step by Step
Put your thumb over the end of a garden hose and the water suddenly shoots out faster and farther. You didn't turn up the tap — you just made the opening smaller. That one trick is the heart of moving fluids: squeeze the path and the flow speeds up. Fill a sink and a beach ball pops back to the surface while a coin drops straight down. Stir honey and it fights back, but water barely resists. Three everyday surprises — and three ideas that explain all of fluid mechanics.
Naming the pieces
Picture water moving through a pipe of cross-sectional area $A$ at speed $v$. The amount passing any point each second is the same everywhere, because the water can't pile up or vanish. That gives the continuity equation $A_1 v_1 = A_2 v_2$. Take this sim's defaults: the pipe narrows from $A_1=0.10\ \text{m}^2$ to $A_2=0.030\ \text{m}^2$ with an inlet speed of $2\ \text{m/s}$. The outlet speed is $v_2 = (A_1/A_2)\,v_1 = (0.10/0.030)\times 2 \approx 6.7\ \text{m/s}$ — more than triple, just from squeezing the pipe.
Where the fluid speeds up, something has to pay for that extra motion, and the currency is pressure. That trade-off is Bernoulli's principle: fast-moving fluid has lower pressure. The narrow throat where the water races is also where it pushes outward the least.
Saying it precisely
Bernoulli's equation is just energy conservation written per unit volume along a streamline for an ideal fluid (incompressible, no viscosity, steady flow): $P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}$. The three terms are static pressure, dynamic pressure $\tfrac{1}{2}\rho v^2$, and a height term $\rho g h$. Buoyancy is a separate idea: a submerged object feels an upward push equal to the weight of the fluid it displaces, $F_b = \rho_f V g$, so it floats whenever its density is below the fluid's, $\rho_\text{obj} < \rho_\text{fluid}$. And real fluids resist flow — Poiseuille's law $Q = \pi r^4 \Delta P / (8\eta L)$ shows flow rate depends on the fourth power of the radius, so halving a pipe's radius cuts the flow 16-fold. The sliders map straight onto these: $A_1$ and $A_2$ set the pipe widths, $v_1$ the inlet speed, $\rho$ the fluid density, $\rho_\text{obj}$ the object's density, and $\eta$ the viscosity.
Try this in the sim above
In Bernoulli mode, shrink the outlet area $A_2$ and watch $v_2$ climb while $\Delta P$ swings more negative — speed up, pressure down. Switch to Buoyancy mode and set $\rho_\text{obj}$ just below, then just above, the fluid density of $1000\ \text{kg/m}^3$ and watch the object flip between "Float" and "Sink". In Viscous mode, raise the viscosity $\eta$ and see the flow rate $Q$ collapse, the way cold honey crawls where water would pour.
Section 03
Equations & Derivation
Continuity Equation
$$A_1 v_1 = A_2 v_2 \quad \text{(incompressible flow)}$$
Bernoulli's Equation
$$P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}$$
Pressure & Buoyancy
$$P = P_0 + \rho g h,\quad F_b = \rho_f V g,\quad \tau = \eta \frac{dv}{dy}$$
Symbol Definitions
| Symbol | Quantity | SI Unit |
|---|
| $P$ | Pressure | Pa = N m⁻² |
| $\rho$ | Fluid density | kg m⁻³ |
| $v$ | Flow velocity | m s⁻¹ |
| $h$ | Height above reference | m |
| $A$ | Cross-sectional area | m² |
| $F_b$ | Buoyant force | N |
| $\eta$ | Dynamic viscosity | Pa·s |
1
Hydrostatic pressure. At depth $h$ in a fluid: $P=P_0+\rho g h$. Each layer supports the weight of the fluid above it.
2
Archimedes' Principle. Buoyant force $F_b=\rho_f V_\text{submerged}g$ — equals the weight of displaced fluid. An object floats when $F_b\geq mg$.
3
Continuity. Mass conservation in steady flow: $\rho_1A_1v_1=\rho_2A_2v_2$. For incompressible fluid: $A_1v_1=A_2v_2$. Smaller cross-section → higher speed.
4
Bernoulli's Equation. Derived from energy conservation for ideal (incompressible, inviscid, steady) flow: $P+\tfrac{1}{2}\rho v^2+\rho g h=\text{const}$. Higher speed → lower pressure.
5
Viscosity. Real fluids resist shear: $\tau=\eta(dv/dy)$ (Newton's law of viscosity). Viscous flow: Poiseuille's Law, $Q=\pi r^4\Delta P/(8\eta L)$.
Ref: Halliday, Resnick & Walker, 10th Ed., Ch. 14; Serway & Jewett, 8th Ed., Ch. 14; Munson, Young & Okiishi — Fundamentals of Fluid Mechanics.
Section 05
Common Misconceptions
❌ Pressure in a fluid depends on the shape of the container.
✅ Pressure at a given depth depends only on depth $h$, fluid density $\rho$, and $g$ — not on container shape or total volume. This is Pascal's principle: pressure at any point in a connected static fluid at the same depth is equal.
📖 HRW 10th Ed., §14-2: Fluids at Rest.
❌ Bernoulli's equation explains lift because faster air has less pressure.
✅ This is incomplete. The pressure difference due to Bernoulli accounts for some lift, but the wing also deflects air downward — the reaction force (Newton's 3rd Law) provides additional lift. The 'equal transit time' argument (air must arrive simultaneously at trailing edge) is physically incorrect.
📖 Serway & Jewett, 8th Ed., §14.6; also: Weltner & Ingelman-Sundberg, AJP 67 (1999) 64.
❌ Objects that are denser than water always sink completely.
✅ A dense object floats if it is hollow — like a steel ship. It can also be held in equilibrium at any depth (neutral buoyancy) if its average density equals the fluid. Submarines achieve this by adjusting ballast tanks.
📖 HRW 10th Ed., §14-4: Archimedes' Principle.
❌ Viscosity is the same thing as thickness or density.
✅ Viscosity measures resistance to shear (flow), not thickness or density. Honey is viscous but not especially dense. Water is thin but has low viscosity. Blood plasma has similar density to water but higher viscosity. Viscosity depends on molecular interactions and temperature.
📖 Serway & Jewett, 8th Ed., §14.7: Viscous Fluid Flow.
Misconception research: Arons — A Guide to Introductory Physics Teaching; Weltner & Ingelman-Sundberg, Am. J. Phys. 67 (1999) 64.