A pipe of cross-sectional area $A_1 = 8\text{ cm}^2$ carries fluid at velocity $v_1 = 50\text{ cm/s}$. The pipe narrows to $A_2 = 2\text{ cm}^2$. The flow rate through the pipe is:

Ideal fluid: incompressible, non-viscous, irrotational (no turbulence). Real fluids: have viscosity, compressibility at high speeds. Streamline: curve tangent to velocity vector at each point; no two streamlines cross. Stream tube: bundle of streamlines. Stagnation point: where fluid velocity = 0; full $\frac{1}{2}\rho v^2$ converted to pressure ($P_0 = P + \frac{1}{2}\rho v^2$). Pitot tube: measures stagnation pressure to find fluid speed.

$P + \frac{1}{2}\rho v^2 + \rho gh = $ const along streamline. Applications: Aeroplane wings: faster air flow over curved upper surface → lower pressure above wing than below → net upward force (lift). Spinning ball (Magnus effect): cricket/football spin causes curved trajectory due to pressure difference from asymmetric airflow. Chimney effect: natural ventilation by warm rising air. Bunsen burner: gas flow entrains air for combustion.

Viscous flow in pipe (Poiseuille\'s law): $Q = \frac{\pi r^4 \Delta P}{8\eta L}$ (flow rate proportional to $r^4$, inverse to viscosity $\eta$ and length $L$). Doubling radius → 16× flow rate. This is why arterial narrowing (atherosclerosis) dramatically reduces blood flow. Blood flow: blood is non-Newtonian fluid (viscosity depends on shear rate). Stokes drag: $F = 6\pi\eta rv$ for sphere of radius $r$ at speed $v$. Terminal velocity: $v_t = 2r^2g(\rho_{sphere}-\rho_{fluid})/9\eta$.

Surface tension $T$: force per unit length = $F/l$ (N/m). Energy per unit area: $T$ (J/m$^2$). Pressure excess inside bubble: soap bubble = $4T/r$ (two surfaces); liquid drop = $2T/r$; air bubble in liquid = $2T/r$. Capillary rise: $h = 2T\cos\theta/\rho g r$. Meniscus: concave (wetting liquids, $\theta<90°$, like water in glass). Convex: non-wetting ($\theta>90°$, mercury in glass). Detergents reduce surface tension → easier cleaning.

Pressure at depth h below a free surface is $P = P_0 + \rho gh$, and it depends only on the depth, not on the shape or total volume of the container — the hydrostatic paradox. Pascal's law follows: pressure applied to an enclosed fluid is transmitted undiminished throughout, which is the working principle of the hydraulic lift, where a small force on a small piston produces a large force on a large one in the ratio of their areas.

A body immersed in a fluid experiences an upward buoyant force equal to the weight of fluid displaced. It floats when its average density is less than the fluid's, and the fraction submerged equals the ratio of the two densities — which is why roughly nine-tenths of an iceberg lies below water. Apparent weight is the true weight minus the buoyant force, and a body of exactly the fluid's density hangs suspended at any depth.

For an incompressible fluid in steady flow, $A_1v_1 = A_2v_2$, so the flow speeds up where the pipe narrows. This is conservation of mass rather than energy, and it is usually the first equation to write in a flow problem — it supplies the relation between the two speeds that Bernoulli's equation then needs. It also explains why a stream of water from a tap narrows as it falls: gravity increases v, so A must decrease.