NBPhO 2026

A liquid with density $\rho$ flows in a pipe with diameter $D$. At low mean speeds $v$, the flow is laminar (uniform and without vortices); the drag force per unit area $F_L = \mu\,du/dr$ on the liquid comes from wall friction, where $\mu$ (unit $\text{Pa}\cdot\text{s}$) is the dynamic viscosity, $u$ is the local speed, and $r$ is the distance from the axis. When $v$ is large enough, the flow becomes turbulent, filled with vortices whose velocity fluctuations are of the order of $v$ itself. The drag $F_T$ still comes from wall friction $\mu\,du/dr$, but the vortices squeeze the near-wall layer (across which the flow speed drops from $v$ to zero) to a thickness $\delta \ll D$ where the drag force per unit area $\mu v/\delta$ is of the order of the dynamic pressure $\rho v^2$ carried by the vortices.

i) (1.5 points) At any given flow speed, only one of the two mechanisms actually dominates the drag — but the dimensionless ratio of the two characteristic drag scales, $R = F_T/F_L$, can be computed regardless and tells us which regime we are in. This ratio is the Reynolds number. Express $R$ as a product of powers of $\rho$, $v$, $D$, and $\mu$.

A water pump with output power $P = 250\,\text{W}$ is used for watering a garden; it draws water from a depth $h = 20\,\text{m}$ directly into a hose of length $s = 40\,\text{m}$ and inner diameter $d = 13\,\text{mm}$. Water leaves the end of the hose at a volumetric rate $Q = 25\,\text{L}\cdot\text{min}^{-1}$. Density of water is $\rho_w = 1000\,\text{kg}\cdot\text{m}^{-3}$, viscosity $\mu_w = 1.1 \times 10^{-3}\,\text{Pa}\cdot\text{s}$ and gravitational acceleration is $g = 9.8\,\text{N}\cdot\text{kg}^{-1}$.

ii) (0.5 points) Determine the flow type in the hose. It is known that flow is turbulent when $R > 2500$ and laminar otherwise.

iii) (3 points) The gardener is cleaning tools by directing the water jet onto dirty surfaces. Where the jet meets the surface, the pressure rises above atmospheric; this excess pressure is what removes dirt. To make cleaning more efficient, the gardener attaches a spray nozzle set to narrow-jet mode, so that the exit cross-section of the nozzle is $f = 15\,\%$ of the hose’s cross-section. By what factor does the volumetric flow rate $Q$ change? By what factor does the excess pressure at the dirty surface change? Find both within $1\,\%$ accuracy. You are allowed to use numerical methods.