 # Mathematics and transport phenomena

## Presentation

Programme (detailed contents):

Mathematics:

Numerical series, function sequences and series, power series, Fourier series, separation of variables for the heat equation. Initiation to Matlab, initiation to structured programming.

Transport phenomena: Continuum media, concept of shear stress. Statics of fluids. Kinematics of fluids.

Principles of conservation of mass, momentum and energy. Writing local equations and the global balances of mass, momentum and energy. Rheological laws (examples of Newtonian fluid and some non-Newtonian fluids).

Boundary conditions.

Exact solutions of some model problems. Analogies of transfers (Newton, Fourier, Fick).

Organisation:

Mathematics:

v  Lectures with lecture notes.

v  Exercise sessions.

v  Computer session with Matlab.

v  Personal work.

Transport phenomena:

First the concepts of scale are introduced to enable the writing of the conservation equations, presented successively for mass, momentum and energy.

Then, the application of balances of momentum and matter are developed.

Lectures and tutorials are complemented by the completion of two lab works:

Thermal conductivity of a material and

rheological behavior of a fluid.

Main difficulties for students:

Mathematics:

Frequent confusion between sequences and series,for the convergence of numerical series difficulty to choosethe relevant method for each case,for the functions sequences and series difficulty not toconfuse the different types of convergence.Difficulties to get familiar with Matlab.

Transport phenomena:

• Determination of the relevant scale for writing and solving a specific problem.

Solving common mathematical equations.

## Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

Mathematics:

The concept of series, the different types of convergence, the power series and Fourier series, the solution of the heat equation, the use of Matlab.

Transport phenomena:

- The basics of continuum mechanics

- The concept of balance and the different scales of application

The student will be able to:

Mathematics:

Study the convergence of numerical series or function series, solve a differential equation with power series, compute a Fourier series, solve the mono-dimensional heat equation.

Apply the mathematical knowledge via the development of simple Matlab programs allowing to illustrate these concepts in examples taken from physics, biology and chemistry.

Transport phenomena:

- Define a system and its contours, and calculate inlet, outlet and transformation fluxes

- Write local and macroscopic balances of mass, energy and momentum

- Solve analytically simple problems of Newtonian fluid mechanics

## Needed prerequisite

First year mathematics, first year computing.

Unit « Thermodynamics -  fundamentals and applications » (# I1ANTH11) of the INSA first year curriculum or equivalent

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...