Fluid Mechanics

Syllabus

Fluid properties; fluid statics, forces on submerged bodies, stability of floating bodies; control volume analysis of mass, momentum and energy; fluid acceleration; differential equations of continuity and momentum; Bernoulli's equation; dimensional analysis; viscous flow of incompressible fluids, boundary layer, elementary turbulent flow, flow through pipes, head losses in pipes, bends and fittings.

Key concepts and formulas

Properties. Newton's law of viscosity τ = μ du/dy; kinematic viscosity ν = μ/ρ. Capillary rise h = 4σ cos θ/(ρgd). Bulk modulus K = −dp/(dV/V).

Statics and buoyancy. p = ρgh. Force on a plane surface F = ρgAh̄. A floating body is stable if its metacentric height GM > 0, where BM = I/V (I = second moment of the waterline area, V = displaced volume) and GM = BM − BG.

Kinematics. Continuity for incompressible flow: ∂u/∂x + ∂v/∂y = 0. A flow is irrotational if ∂v/∂x − ∂u/∂y = 0. Stream function: u = ∂ψ/∂y, v = −∂ψ/∂x.

Bernoulli and momentum.

• p/ρg + V²/2g + z = constant along a streamline (steady, inviscid, incompressible).
• Venturimeter: Q = C_dA₁A₂√(2gh)/√(A₁² − A₂²).
• Momentum equation: ΣF = ṁ(V_out − V_in). Force of a jet on a fixed flat plate normal to it: F = ρAV².

Viscous flow.

• Flow between fixed parallel plates and in pipes (Hagen-Poiseuille): Δp = 32μVL/D², f = 64/Re, V_max = 2V_avg in a pipe and 1.5V_avg between plates.
• Darcy-Weisbach h_f = fLV²/(2gD); minor losses K V²/(2g); sudden expansion loss (V₁ − V₂)²/(2g).

Boundary layer. Laminar (Blasius): δ/x = 5/√Re_x; the transition on a flat plate is at about Re_x = 5 × 10⁵. Drag force F_D = ½C_DρAV².

Dimensional analysis. Reynolds number (inertia/viscous), Froude (inertia/gravity), Weber (inertia/surface tension), Mach (inertia/compressibility), Euler (pressure/inertia).

Worked example

Water flows through a venturimeter with inlet diameter 200 mm and throat 100 mm; the manometer shows a head difference of 0.5 m of water and C_d = 0.98. Find Q. Then find the force of a 50 mm water jet at 20 m/s on a fixed plate normal to it, and the boundary layer thickness at 1 m on a plate in air at 5 m/s (ν = 1.5 × 10⁻⁵ m²/s).

import math

g = 9.81
A1, A2 = math.pi * 0.2 ** 2 / 4, math.pi * 0.1 ** 2 / 4
Q = 0.98 * A1 * A2 * math.sqrt(2 * g * 0.5) / math.sqrt(A1 ** 2 - A2 ** 2)
print(f"venturimeter Q = {Q * 1000:.3f} L/s")
A, V = math.pi * 0.05 ** 2 / 4, 20
print(f"jet force on plate = {1000 * A * V ** 2:.2f} N")
Re = 5 * 1 / 1.5e-5
print(f"Re_x = {Re:.0f}, delta = {5 * 1 / math.sqrt(Re) * 1000:.2f} mm")

venturimeter Q = 24.898 L/s
jet force on plate = 785.40 N
Re_x = 333333, delta = 8.66 mm

Practice questions