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Hydrodynamics

Represent flow by streamlines

Types of flow

steady vs. non-steady (turbulent),
rotational vs. irrotational,
compressible vs. incompressible,
viscous vs. non-viscous

Continuity (conservation of mass)

$$ { {\normalsize \Delta m_1}\over{\normalsize \Delta t} } = { {\normalsize \Delta m_2}\over{\normalsize \Delta t} } $$ or $$ \rho_1 A_1 V_1 = \rho_2 A_2 V_2 $$

Bernoulli's equation (conservation of energy)

$$ p + {1\over2}\rho v^2 + \rho g y = \mbox{const} $$ i.e. lower pressure $p$ where speed $v$ is higher: lift of a wing, atomizer, $v_{\mbox{efflux}} = v_{\mbox{free~fall}}$ (Torricelli)

Viscous force $$ F = \frac{\eta A v}{y}, \quad [\eta]=\mbox{Pa}\cdot\mbox{s} $$

Viscous flow rate in a pipe (Poiseuille's Law)

$$ Q = { {\normalsize \pi r^4 \Delta p}\over {\normalsize 8\eta L} } $$

Viscosities, $\eta$, of various fluids, in N s m^-2

Honey	10
Glycerine at 20°C	1.5
10-wt motor oil at 30°C	0.250
Whole blood at 37°C	2.72x10^-3
Water at 0°C	1.79x10^-3
Water at 20°C	1.0055x10^-3
Water at 100°C	2.82x10^-4
Air at 20°C	1.82x10^-5

Reynolds' number ${\cal R}$ (dimensionless)

$$ {\cal R} = { \normalsize {2 v \rho r}\over{\normalsize \eta} } = { {\normalsize \mbox{inertial~forces}}\over{\normalsize \mbox{viscous~forces}} } $$

${\cal R} < 2000$: flow laminar; ${\cal R} > 2000$: flow turbulent