Physics · Ch 9 — Mechanical Properties of Fluids

Summary

Fluids can flow: unlike a solid, a fluid (liquid or gas) has no fixed shape of its own and takes the shape of its container. A liquid also has a fixed volume and a free surface at the top; a gas is compressible and expands to fill whatever container holds it.
Pressure in a fluid: Pressure at depth $h$ in a static fluid of density $\rho$ is $P = P_0 + \rho g h$ , where $P_0$ is the pressure at the free surface. Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container.
Units of pressure: the SI unit is the pascal (Pa) = 1 N m $^{-2}$ . Common non-SI units in everyday and scientific use: 1 atmosphere (atm) $\approx 1.01 \times 10^5$ Pa, 1 bar $= 10^5$ Pa, and 1 torr (also called 1 mm of mercury) $\approx 133$ Pa $= 0.133$ kPa.
Equation of continuity: For an incompressible fluid in steady flow, $A_1 v_1 = A_2 v_2$ , where $A$ is cross-sectional area and $v$ is flow speed. This expresses conservation of mass.
Bernoulli's principle: For an ideal fluid (incompressible, non-viscous, steady, irrotational flow), $P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$ along a streamline. It is a statement of conservation of energy per unit volume.
Applications of Bernoulli's equation: Explains lift on an aerofoil (faster air above, lower pressure) and the velocity of efflux from a tank (Torricelli's theorem: $v = \sqrt{2gh}$ ).
Viscosity: Internal friction in fluids. Newton's law of viscous flow: $\tau = \eta \frac{dv}{dy}$ , where $\tau$ is shear stress, $\eta$ is coefficient of viscosity, and $\frac{dv}{dy}$ is velocity gradient. SI unit: $\text{Pa·s}$ (poiseuille).
Stokes' law: For a small sphere moving slowly through a viscous fluid, drag force $F = 6 \pi \eta r v$ , where $r$ is radius and $v$ is terminal velocity.
Terminal velocity: When net force on a falling sphere becomes zero: $v_t = \frac{2}{9} \frac{(\rho_s - \rho_f) g r^2}{\eta}$ .
Surface tension: Force per unit length acting along the surface of a liquid, $\sigma = \frac{F}{L}$ . It arises from cohesive forces and causes spherical droplet shapes, capillary rise, and meniscus formation. …

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