## MATH3620 Fluid Dynamics 2

### 15 creditsClass Size: 60

Module manager: Dr Cedric Beaume
Email: C.M.L.Beaume@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2019/20

### Pre-requisite qualifications

MATH2620 or equivalent. Some knowledge of complex analysis, as in MATH2016 or MATH3017 is useful but not required.

This module is approved as a discovery module

### Module summary

This module follows on from the second year fluid dynamics course (MATH2620) by introducing the effects of fluid viscosity, discussing wave motion and using complex variable methods to calculate the lift on an aerofoil.

### Objectives

At the end of this module students should be able to:
- Describe the physical significance of the terms in the Navier-Stokes equations and use dimensional analysis to calculate the Reynolds number;
- Solve one-dimensional viscous flow problems, such as flows in channels with moving walls;
- Use lubrication theory to solve flows in narrow gaps;
- Calculate the simple boundary layer flows involving the diffusion of vorticity;
- Use complex variable techniques to obtain the aerodynamic lift on a simple aerofoil;
- Obtain the dispersion equations for gravity and capillary waves.

### Syllabus

-Stress tensor and Velocity Gradient. What is fluid viscosity? Mathematical description of the deformation and internal forces within a fluid. Stress and strain-rate tensors;
- Navier Stokes Equations. Derivation of the equations of motion for a viscous fluid. Simple one-dimensional flow solutions such as flow along pipes and channels. Definition of the Reynolds number describing the relative importance of inertia and viscosity. Discussion of high and low Reynolds number approximations;
- Lubrication Theory. Flows in narrow gaps such as in a slider bearing or squeezing flows between two plates;
- Boundary layers. Flows near boundaries at high Reynolds numbers. The vorticity equation for a viscous fluid. The flow above impulsively moving plate and diffusion of a vortex sheet. Discussion of boundary layers and separation;
- Complex Potential. Description of planar inviscid flows using the complex potential. Using conformal transformations to find the flow around an ellipse or flat plate. Calculating forces and torques from Blasius' theorem. The lift on an aerofoil;
- Water Waves. Description of small amplitude surface waves. Dispersions relation, group and phase velocity. Transition from capillary to gravity waves.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material. Reading as directed. Completing of assignment and assessments.

### Opportunities for Formative Feedback

Regular example sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated