Level 2 — RecallFluid Mechanics

Fluid Mechanics

⏱ 30 minutes40 marksprintable — key stays hidden on paper

Level: 2 (Recall / Standard textbook problems / Short derivations) Time Limit: 30 minutes Total Marks: 40

Use $...$ or $...$ for all mathematics. Take $g = 9.8\ \mathrm{m/s^2}$ and water density $\rho = 1000\ \mathrm{kg/m^3}$ unless stated otherwise.

Q1. Define a fluid in terms of its response to shear stress, and state why fluids have no fixed shape. (3 marks)

Q2. The specific gravity of mercury is $13.6$ . Compute its density. A dynamic viscosity of mercury is $\mu = 1.55\times10^{-3}\ \mathrm{Pa\cdot s}$ ; find its kinematic viscosity $\nu$ . (4 marks)

Q3. Starting from a force balance on a fluid column, derive the hydrostatic pressure relation $p = \rho g h$ . (4 marks)

Q4. State the Young–Laplace equation for a spherical droplet and for a soap bubble (two surfaces). A water droplet has radius $1.0\ \mathrm{mm}$ and surface tension $0.072\ \mathrm{N/m}$ . Compute the excess (gauge) pressure inside. (5 marks)

Q5. State Archimedes' principle. A solid block of volume $2.0\times10^{-3}\ \mathrm{m^3}$ is fully submerged in water. Find the buoyant force acting on it. (4 marks)

Q6. State the continuity equation $\rho A v = \text{const}$ and its physical basis. Water flows through a pipe that narrows from diameter $0.10\ \mathrm{m}$ to $0.05\ \mathrm{m}$ . If the speed in the wide section is $2.0\ \mathrm{m/s}$ , find the speed in the narrow section. (4 marks)

Q7. State Bernoulli's equation and list its four key assumptions. (5 marks)

Q8. Water flows from a large open tank through an orifice located $5.0\ \mathrm{m}$ below the free surface. Using Torricelli's theorem (a consequence of Bernoulli), find the exit speed. (3 marks)

Q9. Define the Reynolds number and give its expression $Re = \rho v L/\mu$ . For water ( $\rho = 1000\ \mathrm{kg/m^3}$ , $\mu = 1.0\times10^{-3}\ \mathrm{Pa\cdot s}$ ) flowing at $0.5\ \mathrm{m/s}$ in a pipe of diameter $0.02\ \mathrm{m}$ , compute $Re$ and state whether the flow is laminar or turbulent (transition $\approx 2300$ ). (4 marks)

Q10. State the Kutta–Joukowski theorem for lift per unit span, defining each symbol. (4 marks)

Answer keyMark scheme & solutions

Q1. (3 marks)

A fluid is a substance that deforms continuously (flows) under the action of any applied shear stress, however small — it cannot resist shear in static equilibrium. (2)
Because it cannot sustain a shear stress at rest, a fluid conforms to the shape of its container, having no fixed shape of its own. (1)

Q2. (4 marks)

Density: $\rho = SG \times \rho_{water} = 13.6 \times 1000 = 13600\ \mathrm{kg/m^3}$ . (2) — specific gravity is density relative to water.
Kinematic viscosity: $\nu = \dfrac{\mu}{\rho} = \dfrac{1.55\times10^{-3}}{13600} = 1.14\times10^{-7}\ \mathrm{m^2/s}$ . (2) — definition $\nu = \mu/\rho$ .

Q3. (4 marks)

Consider a fluid element of cross-sectional area $A$ and height $dz$ at depth. Forces: pressure $p$ at top, $p+dp$ at bottom, weight $\rho g A\,dz$ . (1)
Vertical equilibrium (measuring $h$ downward): $(p+dp)A = pA + \rho g A\,dz \Rightarrow \dfrac{dp}{dz} = \rho g$ . (2) — balance of pressure difference against weight.
Integrating from surface ( $p_0$ ) to depth $h$ : $p - p_0 = \rho g h$ ; taking gauge pressure $p = \rho g h$ . (1)

Q4. (5 marks)

Spherical droplet (one surface): $\Delta p = \dfrac{2\sigma}{R}$ . (1.5)
Soap bubble (two surfaces): $\Delta p = \dfrac{4\sigma}{R}$ . (1.5)
Numeric (droplet): $\Delta p = \dfrac{2(0.072)}{1.0\times10^{-3}} = 144\ \mathrm{Pa}$ . (2)

Q5. (4 marks)

Archimedes' principle: a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. (2)
$F_B = \rho_{water}\, V\, g = 1000 \times 2.0\times10^{-3} \times 9.8 = 19.6\ \mathrm{N}$ . (2)

Q6. (4 marks)

Continuity: $\rho A v = \text{const}$ ; physical basis is conservation of mass — mass entering equals mass leaving a control volume in steady flow. (2)
For incompressible flow $A_1 v_1 = A_2 v_2 \Rightarrow v_2 = v_1 (D_1/D_2)^2 = 2.0 \times (0.10/0.05)^2 = 2.0 \times 4 = 8.0\ \mathrm{m/s}$ . (2)

Q7. (5 marks)

Bernoulli: $p + \tfrac{1}{2}\rho v^2 + \rho g z = \text{const}$ along a streamline. (1)
Assumptions (1 each, max 4): steady flow, inviscid (no viscosity), incompressible, along a single streamline (also acceptable: no shaft work/heat transfer). (4)

Q8. (3 marks)

Torricelli: $v = \sqrt{2gh}$ . (1)
$v = \sqrt{2 \times 9.8 \times 5.0} = \sqrt{98} = 9.9\ \mathrm{m/s}$ . (2)

Q9. (4 marks)

$Re = \dfrac{\rho v L}{\mu}$ ; ratio of inertial to viscous forces. (1)
$Re = \dfrac{1000 \times 0.5 \times 0.02}{1.0\times10^{-3}} = \dfrac{10}{10^{-3}} = 10000$ . (2)
$Re = 10000 > 2300 \Rightarrow$ turbulent flow. (1)

Q10. (4 marks)

Kutta–Joukowski: lift per unit span $L' = \rho V_\infty \Gamma$ . (2)
Symbols: $\rho$ = fluid density, $V_\infty$ = free-stream velocity, $\Gamma$ = circulation around the body. (2)

[
  {"claim":"Mercury density = 13600 kg/m3 and nu = 1.14e-7 m2/s", "code":"rho=13.6*1000; nu=1.55e-3/rho; result = (rho==13600) and (abs(nu-1.14e-7)<1e-9)"},
  {"claim":"Droplet excess pressure = 144 Pa", "code":"dp=2*0.072/1.0e-3; result = abs(dp-144)<1e-6"},
  {"claim":"Buoyant force = 19.6 N", "code":"F=1000*2.0e-3*9.8; result = abs(F-19.6)<1e-9"},
  {"claim":"Narrow section speed = 8.0 m/s", "code":"v2=2.0*(0.10/0.05)**2; result = abs(v2-8.0)<1e-9"},
  {"claim":"Torricelli exit speed approx 9.9 m/s", "code":"v=sqrt(2*9.8*5.0); result = abs(float(v)-9.899)<0.01"},
  {"claim":"Reynolds number = 10000 (turbulent)", "code":"Re=1000*0.5*0.02/1.0e-3; result = (abs(Re-10000)<1e-6) and (Re>2300)"}
]