Detailed Notes
Every notes page is a complete treatment: the derivation carried out in full, the reasoning behind each step, worked examples, and links to the calculators and formulas it connects to. No summaries pretending to be notes.
What Is a Fluid? The Continuum Hypothesis
Detailed notes on the definition of a fluid, shear response, and why the continuum hypothesis lets us use calculus on fluids — with the Knudsen number criterion.
Viscosity and Newton's Law of Viscosity
Detailed notes on dynamic and kinematic viscosity, Newton's law of viscosity τ = μ du/dy, its physical reasoning, and Newtonian vs non-Newtonian fluids.
Hydrostatic Pressure: Full Derivation
Derivation of the hydrostatic equation dp/dz = −ρg from a force balance on a fluid element, with P = ρgh, Pascal's law and a worked manometer example.
The Material Derivative
Detailed notes on the material derivative D/Dt: Lagrangian vs Eulerian descriptions, the chain-rule derivation, local vs convective acceleration, with an example.
The Continuity Equation: Derivation
Derivation of mass conservation for fluids: the differential continuity equation from a control volume, the incompressible form ∇·u = 0, and the 1-D A₁v₁ = A₂v₂ result.
Bernoulli's Equation: Derivation and Limits
Full derivation of Bernoulli's equation from the work–energy theorem along a streamline, its four assumptions, pressure-head-velocity trade-offs, and failure modes.
Reynolds Number and Flow Regimes
Why Re = ρvD/μ decides laminar versus turbulent flow: nondimensionalisation of Navier–Stokes, Reynolds' 1883 experiment, regime thresholds and worked examples.
Hagen–Poiseuille Law: Full Derivation
Complete derivation of laminar pipe flow: the shear balance on a fluid cylinder, the parabolic velocity profile, integration to Q = πΔPr⁴/8μL and the r⁴ scaling.
Prandtl's Boundary-Layer Concept
Notes on the 1904 boundary-layer idea: why inviscid theory failed at walls, the scaling argument, the boundary-layer equations, and displacement thickness.
The Blasius Flat-Plate Solution
The Blasius boundary layer worked in detail: stream function, similarity variable, reduction to f‴ + ½ff″ = 0, the numbers f″(0) = 0.
MHD: The Lorentz Force in Fluid Flow
How magnetohydrodynamics enters the momentum equation: the J×B body force, Ohm's law for moving conductors, the low-magnetic-Reynolds simplification and the Hartmann number.
Similarity Transformations: The Stretching Sheet
Step-by-step similarity reduction of the MHD stretching-sheet boundary layer: choosing η and f, collapsing the PDEs to f‴+ff″−f′²−Mf′=0, and the exact solution.
Nanofluid and Hybrid Nanofluid Models
The Tiwari–Das and Buongiorno frameworks for nanofluid flow, effective-property correlations for density, viscosity and conductivity, and the hybrid extension.
Shooting Method and bvp4c
How boundary-layer ODEs are solved in practice: converting BVPs to IVPs by shooting, Newton iteration on the missing slope, collocation with bvp4c, and validation practice.
Heat Transfer over a Stretching Sheet
The thermal boundary layer on a stretching sheet: reduction of the energy equation to θ″ + Pr fθ′ = 0, the roles of Pr and M, and the Nusselt number.
Completing the Square
Detailed notes on completing the square: the geometric picture, the algebraic algorithm, vertex form, and how it proves the quadratic formula.
The Power Rule: Statement and Proof
Detailed notes proving d/dx xⁿ = nxⁿ⁻¹ from the limit definition via the binomial theorem, with the linearity extension and worked examples.