## MATH2625 Fluid Dynamics

### 15 creditsClass Size: 200

Module manager: Professor Serguei Komissarov
Email: S.S.Komissarov@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

(MATH1010 and MATH1012 and MATH2365) or (MATH1005 and MATH2365) or (MATH1050 and MATH1400 and MATH2365) or (PHYS1290 and PHYS1300 and PHYS2370) or (SOEE1301 and SOEE1311), or equivalent

Module replaces

MATH2620 Fluid Dynamics 1

This module is approved as a discovery module

### Module summary

Fluid dynamics is the science of the motion of materials that flow. Understanding fluid dynamics is a significant mathematical challenge with important implications in an enormous range of fields in science and engineering, from physiology, aerodynamics, climate, to astrophysics. This course gives an introduction to fundamental concepts of fluid dynamics. It includes a formal mathematical description of fluid flow and the derivation of their governing equations, using techniques from calculus and vector calculus. The theoretical background is applied to a series of simple flows (e.g. bath-plug vortex or stream past a sphere), giving the student a feel for how fluids behave, and experience in modelling everyday phenomena. A wide range of courses, addressing more advanced concepts in fluid dynamics, with a variety of applications (polymers, astrophysical and geophysical fluids, stability, and turbulence), follow naturally from this introductory course.

### Objectives

This course demonstrates the importance of fluid dynamics and how physical phenomena can be understood using rigorous mathematics. It introduces students to concepts of mathematical modelling. Students will learn how to use mathematical methods to derive approximate solutions and to critically assess the validity of these results.

Learning outcomes
Students will be able to:
1. Understand how vector calculus can be used to describe fluid flow.
2. Mathematically model fluid flow in a number of situations.
3. Solve simple problems involving fluid motion, pressure, vorticity and waves.
4. Describe the physical mechanisms giving rise to some everyday phenomena involving fluid motion.

### Syllabus

- Vector calculus as applied to fluid dynamics (grad, div, curl, directional derivative, surface and volume integrals). Concept of pressure, and Archimedes’ Principle. Concept of vorticity and circulation.

- Mathematical modelling of fluids. Elementary kinematics (particle paths and streamlines). Mass conservation. Incompressibility and streamfunctions. Kinematic boundary conditions.

- Potential flows. Irrotational flow and velocity potential. Flows as solutions of Laplace's equation; elementary singular flows. Method of images. Introduction to complex potential.

- Dynamics. Forces acting on fluids. Euler’s equation; vorticity equation. Shape of the free surface of rotating fluids. Bernoulli's invariant (steady). Forces exerted by fluids (e.g., walls, cylinders; drag and lift).

- Shallow water model. Introduction to linear waves: solutions in 1D (d’Alembert’s solution), and 2D by separation of variables.

### Teaching methods

 Delivery type Number Length hours Student hours Lectures 22 1.00 22.00 Tutorials 10 1.00 10.00 Private study hours 118.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 150.00

### Opportunities for Formative Feedback

Feedback on module worksheets used for in-course assessment. Publication of model solutions.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 85.00 Total percentage (Assessment Exams) 85.00

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