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Hydrodynamics
Represent flow by streamlines
Types of flow
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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
$$F = \frac{\eta A v}{y}$$, $$[\eta]=\mbox{Pa}\cdot\mbox{s}$$
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
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