Fluid Dynamics

February 2, 2023

Contents

0 Preliminary Introduction  3
1 Kinematics  4
1.1 Pathlines & Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Kinematic Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Stream Function for 2D Incompressible Flow . . . . . . . . . . . . . . . . 9
2 Dynamics of Inviscid Flow  12
2.1 Surface and Volume Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The Euler Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Bernoulli’s Equation for Steady Flow with Potential Forces . . . . . . . . 15
2.4 Hydrostatic Pressure and Archimedes Principle . . . . . . . . . . . . . . . 20
2.5 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Interpretation of ω as 2 × average location rate . . . . . . . . . . . 22

Lectures

Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Start of lecture 1

0 Preliminary Introduction

Continuum Mechanics:

• Fluid dynamics (liquids, gases)

• Solid mechanics (solids, elasticity, fracture)

• Other (complex fluids, soft matter, biomechanics) Average over irrelevant molecular detail to get a continuum description in terms of fields. For example

• velocity u(x, t)

• pressure p(x, t)

• density ρ(x, t) x = U(t) is uniform flow, u = U(t) + R(t) × x solid body motion. u(x) steady, u(x, t) unsteady. Simple physics - mass, momentum, Newton’s Laws, vector calculus, maths methods. Combine these to find the velocity u and the accompanying forces. Kinematics: velocities and trajectories Dynamic: forces and equations of motion. Forces: gravity, pressure, viscous stress (internal friction). We will consider inviscid flow good approximation for water / air (except on very small scales). We will also consider simple viscous flow. More to come in Part II Fluids & Waves and lots of courses in Part III. Applications Fluids are everywhere! Application for the environment: atmosphere / polar ice caps (climate change), pollution, ventilation, wind power. For biology: breathing, blood, flying, swimming, cells. Aerosols to astrophysics.