# Faculty of Mathematics

Please be aware that Fluid Dynamics in particular is an area where the Cambridge undergraduate course is much more advanced and specialised than corresponding courses at many other universities, including many in the UK. This is also reflected in the level of the Part III Fluids courses.

The notes below are intended to convey an idea of the previous knowledge, both in terms of mathematical methods and fluid dynamics, required, and how to prepare for Part III, if necessary. If you are unsure whether you have the required level of preparation in some of the areas, test yourself by having a go at some example sheet or past paper questions from the relevant undergraduate courses. For ecample sheets, we recommend you look only at the main questions, excluding any "extra", "additional" or "starred" questions. (Refer to the General Resources for example sheets and past papers.) Prepare yourself well by having a realistic idea of what awaits you in Part III!

### Prerequisite areas

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## Mathematical Methods

This section deals in particular with the crucial mathematical methods that all Part III students wanting to take courses in Continuum Mechanics should be very comfortable with. You are expected to have seen all this material already, but we are trying to help you remember it and to push you to practice until you are really confident and fluent in the use of these methods. (The material in this section is also useful for some courses in Relativity and Gravitation.)

Relevant undergraduate courses are (for relevant schedules, example sheets and exam questions, refer to the General Resources):

• Part IA Differential Equations (DE), Vectors & Matrices (VM), Vector Calculus (VC)
• Part IB Methods (M)
• [Part II Mathematical Biology (MB), Fluid Dynamics II (F2)]

### Reality check

• Solution of ODEs, including by series solution. (DE)
• Elementary linear algebra (eigenvectors and eigenvalues, â€¦) (VM)
• Index notation (or Einstein or suffix notation) and the summation convention. (VM)
• Cylindrical and spherical coordinates. (VC)
• Grad, Div, Curl (in Cartesian and polar coordinates). (VC)
• Integral theorems: Divergence Theorem, Stokes' Theorem. (VC)
• Tensors. (VC)
• Solution of PDEs by separation of variables. (M)

You can test whether you are at the correct level on the questions given in these notes on Index Notation and Tensors.

### Prerequisites

• Scaling analysis. Similarity solutions of PDEs. (DE, MB, F2)

You can use these notes and exercises on Scaling Analysis to check your current level and guide your study.

### Useful books and resources

• Lecture Notes for Part IA Vector Calculus by Professors Allanach and Evans.
• Lecture Notes on separation of variables (Part IB Methods) by Professor Josza.

## Fluid Mechanics

The material in this section is needed for all courses in the subject area except for Perturbation Methods. Not all courses will require knowledge of all these ideas, but will focus on different area of fluid mechanics (e.g. viscous or inviscid fluid dynamics).

Relevant undergraduate courses are (for schedules, example sheets and exam questions, refer to the General Resources):

• Part IB Fluid Dynamics (FD)
• Part II Fluid Dynamics II (F2)

### Reality check

• Navier-Stokes equations. Reynolds number: Euler and Stokes limits. (FD, F2)
• Boundary conditions at rigid and free surfaces (and free surfaces with surface tension). (FD, F2)
• Lagrangian and Eulerian descriptions. Convective (material) derivative. (FD)
• Streamfunctions. Potential flows. (FD)
• Inviscid flows: Bernoulli's theorem. (FD)
• Vorticity and the vorticity equation. (FD, F2)
• Simple viscous flows: Taylor and Couette flows, Stokes flow past a rigid sphere. (F2)

### Prerequisites

• Properties of Stokes flows: Linearity, Reversibility. Minimum Dissipation Theorem. (F2)
• Lubrication theory. (F2)
• Boundary layer equations. (F2)
• Stability of unidirectional inviscid flows. (F2)

### Useful books and resources

• Chapters 1-8 of Elementary Fluid Dynamics by D.J. Acheson (Oxford), as suggested in the formal prerequisites.
• Lectures notes for Part II Fluid Dynamics II by Professor Proctor.

## Theory of Complex Variables

The material in this section is needed for Perturbation Methods, but very useful for other courses in the subject area, too.

Relevant undergraduate courses are (for schedules, example sheets and exam questions, refer to the General Resources):

• Part IB Methods (M), Complex Methods (CM)
• [Part II Asymptotic Methods (AM)]

### Reality check

• Fourier and Laplace transforms. Fourier superposition. (M)
• Evaluation of integrals using complex variable techniques (e.g. residue calculus). (CM)

### Prerequisites

• Green's functions for simple PDEs on unbounded domains (e.g. diffusion equation). (M).
• Elementary asymptotic techniques. (AM)

### Useful books and resources

• Lecture Notes on Fourier transforms (Part IB Methods) by Professor Josza.
• Lecture Notes on Green's functions for PDEs (Part IB Methods) by Professor Josza.

## Dynamical Systems

The material in this section is needed for Convection and Magnetoconvection, but is useful for many courses in the subject area.

Relevant undergraduate courses are (for schedules, example sheets and exam questions, refer to the General Resources):

• Part IA Differential Equations (DE)
• Part II Mathematical Biology (MB), [Part II Dynamical Systems (DS)]

### Reality check

• Classification of stationary points of a dynamical system, and use for sketching of phase plots. (DE, MB)

### Prerequisites

• Elementary bifurcation theory. (DS)

## Electromagnetism

The material in this section is needed for Convection and Magnetoconvection and Biological Physics and Complex Fluids. Note that only a basic understanding of electromagnetism is required.

The relevant undergraduate course is (for schedules, example sheets and exam questions, refer to the General Resources):

• Part IB Electromagnetism (EM)

### Reality check

• Electrostatic potential. Charge conservation. (EM)
• Maxwell's equations (EM)

## Statistical Physics

The material in this section is needed for Theoretical Physics of Soft Condensed Matter, Quantum Fluids; a basic understanding of statistical physics is required for some sections of Biological Physics and Complex Fluids.

The relevant undergraduate course is (for schedules, example sheets and exam questions, refer to the General Resources):

• Part II Statistical Physics (SP)

### Reality check

• Statistical Ensembles: Partition Functions. (SP)
• Grand Canonical Ensemble. Chemical Potential. (SP)
• Boltzmann distribution. (SP)
• Laws of Classical Thermodynamics. (SP)

### Prerequisites

• Landau Theory of phase transitions. (SP)

## Quantum Mechanics

The material in this section is needed for Quantum Fluids, and no other courses in the subject area.

Relevant undergraduate courses are (for schedules, example sheets and exam questions, refer to the General Resources):

• Part IB Quantum Mechanics (QM)
• Part II Principles of Quantum Mechanics (PQM)

Prerequisites to be completed.