# Dynamics of Inviscid Fluids

Course category:
3rd year
Module code:
MAT00033H
Year:
2012/13
Term:
Autumn
Credits:
10

## Aims

The aim of the module is to...

• learn the most basic notions of continuous mechanics and fluid dynamics of an inviscid fluid. Each item of the syllabus represents a theoretical keystone of fluid dynamics and has numerous applications in engineering, biology, medicine, aviation, environmental sciences, oceanography, atmospheric physics, astrophysics, military, various branches of industry and technology, etc.
• give you the general information and personal advice about the places of graduate studies and the areas of employment in the broad variety of areas related to applied mathematics and fluid dynamics.
• serve as a solid basis for some final year projects.

## Learning objectives

At the end of the module you should be able to...

• use the mathematical tools for the description of motions of a continuous medium;
• formulate the governing equations of fluid dynamics and solve these equations in simple cases;
• have rational ideas about various fluid flows around us;
• think actively about interesting applications of fluid dynamics;
• make effective practical steps towards your future graduate studies or employment in the areas related to applied mathematics and fluid dynamics.

## Syllabus

1. Kinematics of Continuous Medium

• Eulerian and Lagrangian coordinates. Partial and material derivatives. Velocity and acceleration.
• Euler’s formula for Jacobian. The transport theorem. Continuity equation.
• Rate-of-strain tensor and vorticity. Potential of velocity. Streamlines, vortex lines and trajectories.

2. Governing Equations of Fluid Mechanics:

• Cauchy’s stress vector. Conservation of momentum and stress tensor.
• General form of equations of motion. Symmetry of the stress tensor.
• Perfect (or ideal or inviscid) fluid and barotropic fluid.
• Perfect gas and inviscid incompressible fluid. Euler’s equations.
• Plane motions, rotationally symmetric motions, and axisymmetric motions. Stream functions.
• Euler’s equations in cartesian, cylindrical, and spherical coordinates.

3. Main Topics in Dynamics:

• Linearization procedure and sound waves.
• Hydrostatic equilibrium and Archimedes’ theorem.
• Momentum transfer equation and the flux of momentum.
• Convection of vorticity: Helmholtz’s equation and Cauchy’s integral. Vortex lines as material lines. Increasing of vorticity via stretching of material lines.
• Persistence of irrotationality: Lagrange’s theorem.
• Gromeka-Lamb’ form of Euler’s equations. Bernoullian theorems.
• Kelvin’s Circulation Theorem. Bath-tube vortices, tornadoes, hurricanes, and cyclones.

4. Motion of a Solid in an Inviscid Incompressible Fluid.

• General properties of potential flows caused by motions of a solid. Cylindrical and spherical harmonics.
• Flows past circular cylinder. Virtual mass and d’Alembert’s paradox.
• Circulation of velocity. Kelvin’s Circulation Theorem.
• The lift (Kutta-Joukovsky’) force. Why do airplanes fly?
• Motion of three-dimensional solid. Motion of a rigid sphere.

## Recommended texts

• The course is based on the detailed lecture notes by Prof Vladimirov.
• G K Batchelor, An Introduction to Fluid Dynamics, Cambr. Univ. Press.
• L D Landau and E M Lifshitz, Fluid Mechanics, Butterworth-Heinemann.

## Teaching

• Autumn Term. 2 lectures and 1 seminar per week. Weekly office hours.
• The lectures are supplemented by 4 sets of homework. Solving the problems represents an essential part of the course. The seminar class will underpin the students’ work on their assignments, both reviewing completed assignments, and troubleshooting assignments in progress.

## Assessment

Two hour closed examination in week 1 Spring Term (100%).

## Elective information

This module is not available as an elective.

## Prerequisites

• Operations with vectors. Operators curl, grad, and div. (taught in Vector Calculus)
• The divergence theorem and Stokes' theorem.  (taught in Vector Calculus)
• Main properties of partial derivatives, including the chain rule.
• Newton's laws of Classical Mechanics. Conservation of momentum and conservation of energy.
• Solving of simple linear ordinary differential equations. (taught in Differential Equations)