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Fluid Dynamics

Year 3 · 4 sections · 42 concepts · 68 practice questions

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What's covered

free1 · Kinematics11 concepts

Overview 1 · The plan of this section
Definition 1.1 · Velocity fields, pathlines and streamlines
Example 1.2 · Pathlines and streamlines can differ
Proposition 1.3 · The material derivative
Theorem 1.4 · The Reynolds transport theorem
Corollary 1.5 · Conservation of mass and incompressibility
Proposition 1.6 · The stream function
Definition 1.7 · Vorticity and irrotational flow
Proposition 1.8 · Local motion is translation, rotation and strain
Definition 1.9 · Circulation
Check 1.10 · Check: describing a flow

2 · Inviscid flow10 concepts

Overview 2 · The plan of this section
Theorem 2.1 · Euler's equations
Theorem 2.2 · Bernoulli's theorem
Example 2.3 · Applications of Bernoulli
Theorem 2.4 · Kelvin's circulation theorem
Proposition 2.5 · Potential flow
Definition 2.6 · Complex potentials
Example 2.7 · Flow past a cylinder
Theorem 2.8 · d'Alembert's paradox and the Kutta–Joukowski lift
Check 2.9 · Check: solving inviscid flow problems

3 · Viscous flow11 concepts

Overview 3 · The plan of this section
Definition 3.1 · Stress and Newtonian fluids
Theorem 3.2 · The Navier–Stokes equations
Definition 3.3 · Boundary conditions
Example 3.4 · Couette flow
Example 3.5 · Poiseuille flow
Proposition 3.6 · The Reynolds number and dynamic similarity
Proposition 3.7 · Energy dissipation
Remark 3.8 · Stokes drag and terminal velocity
Remark 3.9 · Boundary layers
Check 3.10 · Check: exact viscous solutions

4 · Water waves10 concepts

Overview 4 · The plan of this section
Definition 4.1 · The linearised water-wave problem
Theorem 4.2 · The dispersion relation
Corollary 4.3 · Deep and shallow water
Proposition 4.4 · Group velocity
Example 4.5 · Particle paths under waves
Proposition 4.6 · Wave energy
Example 4.7 · Ocean swell and shoaling
Remark 4.8 · Capillary waves
Check 4.9 · Check: wave calculations

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Proposition 1.3 · The material derivative

$D/Dt = \partial_t + \mathbf u\cdot\nabla$ : local change plus change from being carried along.

Proof

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