Find here the latest pdf version of my lecture notes, with elements of solution of most exercises and problems.

In these lecture notes, some equations and figures are empty and should be filled or realized by the students themselves: `DIY' means `Do it yourself' ! A `full' version of the lecture notes, without blank spaces and with the complete figures, may be asked by mail to Emmanuel Plaut.

On this dynamic web page, I sketch the planning and the program of the module. Further informations will be published regarding the assessment - grading methods, using, in particular, homeworks.

Starting from session 2, the students will use Mathematica on their laptop to plot fields, perform formal (symbolic) and numerical computations. A Mathematica licence specific to Mines Nancy will be used: you should receive before session 1 a mail from Wolfram Research, who produces and distributes Mathematica.
For session 1, a pocket calculator will be sufficient.

All lectures and the exam will take place in room P318.

  Natural thermoconvection, Rayleigh - Bénard thermoconvection: setup of the linear stability analysis with slip boundary conditions

Session 2 Thursday October 6

  Rayleigh - Bénard thermoconvection with slip boundary conditions: linear stability analysis

Preparatory work: theory

     Try to solve by yourself the beginning of ex. 1.1: find the coefficient of degree 1 and the constant coefficient of the characteristic equation (1.32).

Preparatory work: computing

     Install and configure Mathematica on your laptop, if this has not been done last week. Learn how to use it with tutorials, and by trying to do a few things by yourself; for instance look at

1. ce notebook interactif par Olivier Deck (que je remercie);
2. this Wolfram's fast introduction for programmers;
3. this Wolfram's tutorial on algebraic calculations;
4. this Wolfram's tutorial on symbolic calculations;
5. this Princeton's tutorial on symbolic computation...

Session 3 Thursday October 13

  Rayleigh - Bénard thermoconvection: linear and weakly nonlinear analysis, supercritical patterning bifurcation, elements on further transitions and on the case of no-slip boundary conditions; Lorenz model & chaos; confined geometry - flow reversals...

Preparatory work:

     Terminate ex. 1.1.

  Transition in open shear flows: generalities, the case of plane Poiseuille flow

Session 5 Thursday October 27

  Transition in plane Poiseuille flow: linear stability analysis, Tollmienn - Schlichting waves

Session 6 Thursday November 10

  Transition in plane Poiseuille flow: weakly nonlinear stability analysis, subcritical bifurcation to Tollmienn - Schlichting waves, subcritical & saddle-node bifurcations. Transition in open shear flows...

Preparatory work:

  Advance pb. 2.1 as far as possible, starting with PPF-LinAn.m.

  This time slot corresponds to the standard exam of duration 1 hour 30 mins... plus 30 mins for one student who has a bonus of 1/3 of the duration. All students will start at 9h30 and should bring their laptop for Mathematica and ARCHE, nothing else, plus the lecture notes given at the beginning of the module.

  The homework will be realized in groups. A hand-written solution will be requested. Some plots realized preferentially with Mathematica could be added, moreover, a notebook could be sent by email.

1. Ex. 1.2 General linear stability analysis of slip RBT
   ... for 3 students, after session 2.
   Scoring scale: 1: 4 points | 2: 6 points
   Total: 10 points + max 1 point of bonus for good english

2. Ex. 1.11 Stability analysis of a fluid layer with different thermal stratifications
   ... for 4 students, after session 2.
   Scoring scale: 1: 1 point | 2: 1.5 points | 3: 3 points | 4: 0.5 point | 5: 3.5 points | 6: 0.5 point
   Total: 10 points + max 1 point of bonus for good english

3. Ex. 2.7 3D linear stability analysis of 2D viscous open shear flows: Squire's transformation and theorem
   ... for 4 students, after session 5.
   Scoring scale: 1: 0.75 point | 2: 0.75 point | 3: 1.75 points | 4: 1 point | 5: 3.5 points | 6: 2.25 points
   Total: 10 points + max 1 point of bonus for good english

• The number of + got during the whole session, to measure your participation, yields a score N_P over 15 = the maximum of + over the 11 students .
• Your homework score multiplied by 2 yields a score over 20 points, N_H .
• Your examination score will be, over 20 points, N_E .
• Your module score will be something like (the coefficients might be slightly modified)
  N_M = 0.2 N_P + 0.4 N_H + 0.6 N_E .

• A grade G = A, B, C, D, E, F or FX will be deduced from N_M. I will give some infos (histograms, thresholds, etc.) on the ARCHE page of the module...