I am now in the throws of finishing my first post-PhD paper!! It will be co-authored with Denys Dutykh who helped me to fully understand the method used to derive the question in the first place. The next stage would be to have small amplitude waves in finite depth and then onto the weakly nonlinear stuff, which then would be another paper in itself. I also found a paper with the leaky dielectric model in who also worked on this stuff and derived their own weakly nonlinear equation and I lifted their evolution equation for the charge and managed to extend my results by a little.
Jean-Marc and I are writing our first paper together, hopefully we can get it published in a well respected journal, physics of fluids or journal of fluid mechanics or something like that. When the paper has been accepted for publication, I will post a copy here.
Well I have completed the first stage of the analysis, the linear and weakly nonlinear cases. I just have to make the notation a little more clear and that should be a part of a chapter for my thesis which is nice. I may add a little bit on the numerics and then I am sorted, I can go on to what could possibly be my “main” problem where I can get some serious maths done.
Speaking of numerics though, I finally figured out where my bug was and as I suspected it was something simple. I used an equation to pick out the correct variable in my array and that is where the problem was. I have since used an if elseif statement which works beautifully. I have some rather dull number crunching to get some pretty looking waves.
I have my new problem to work on, it links in to some experimental work that some woman has been doing in France when he has been trying to make a fluid stick to an upturned electrode. I am top model the simple case when the diode, fluid and air are all dielectrics. I am supposed to look for waves in the solution and attack the linear and weakly nonlinear cases using the inviscid Euler equations and then move to to the viscous case which might be quite interesting. I come out of industry to learn lots of different mathematical techniques that I could take back into industry and be useful.