Implicit Function Theory Applications; Part 0
I developed analytical solutions for a couple of simple transient compressible fluid flow problems that include fluid-structure interactions. I think the problems and solutions might be candidates for standard problems / benchmarks / Method of Exact Solutions.
Implicit function theory is an important aspect of the analytical solutions. Getting the required implicit function theory results was proving to take more pages and space that the analytical solutions. I decided to document those results in these separate notes. The results needed for the analytical solutions are given in the latter part of the notes, beginning with Variable Fluid Control Volume.
I have uploaded a Table that is in landscape and that I don’t now how to get into a portrait layout-document. That Table is here and should open in a separate window.
There are tons and tons of algebra associated with this work; straightforward but tedious algebra. I have checked and re-checked but maybe haven’t cleared out all the errors. If you plan to use any of this material, let me know what is of interest and I’ll work with you to ensure that the equations are correct.
The PDF file is here and should open in a separate window.
Update February 26, 2011
A corrected version of Table 3 is here. There was a bug in the first column of the third row.
Update January 14, 2011
There’s a typo in Eq. (1.8). The X_sub i in the last line ( the bottom part of the bottom ) should be Y_sub i. That’s a strange kind of bug; a typo in a nemonic device.
In the Section, The Bridgman Method, I say:
I have uploaded the table as a PDF file and provided this URL link in my post: https://edaniel.files.wordpress.com/2011/01/testbridgmantable.pdf . You’ll have to copy-n-paste the link into your browser.
That is the same Table mentioned in the Post, and you don’t have to copy-n-paste the URL; it’ll open from the PDF.
Note that Eq. (1.33) can be written in terms of the square of the sound speed.
And in a few places following Tables of derivatives, I said that the entropy derivatives had not been reduced when in fact they are shown reduced form in the tables.
Let me know if you find any problems.
Models Methods Software 
Thanks for putting this up Dan; good stuff.
I especially like the “EOS Solution Methods” section. Am I understanding it right: you use your equation of state derivatives to estimate an initial guess for the solution at the next time-step? I’ve done similar things using time (unsteady flow problems) or temperature (equilibrium chemistry problems) as a “continuation parameter”, but that was just a really basic extrapolation based on a Taylor series. I never thought to use the EOS in that way.
I recommend Maxima; it knows the chain rule and does a good job keeping track of the signs; a few lines to do your initial background example (forgive me, I’m not sure if the formatting will turn out right on this):
(%i1) depends([y1,y2], x1,x2]);
(%o1) [y1(x1, x2), y2(x1, x2)]
(%i2) depends([F,G],[y1,y2,x1,x2]);
(%o2) [F(y1, y2, x1, x2), G(y1, y2, x1, x2)]
(%i3) diff(F,x1);
dy2 dF dy1 dF dF
(%o3) — — + — — + —
dx1 dy2 dx1 dy1 dx1
Last o-line should have been:
dy2 dF dy1 dF dF
— — + — — + —
dx1 dy2 dx1 dy1 dx1
Yes, that extrapolation can be used to get the first guess for the new-time level, or new-iteriate level, for the EOS properties. It could also be used if you want to run a linear EOS version of the model. My experience has been that it provides excellent first guesses and subsequent convergence.
I have found the more useful applications are for those situations in which you need to get the EOS properties into an implicit formulation of correlations and auxiliary models ( sometimes known as parameterizations ) when the solution variables are products of things and not simply single EOS properties.
I’ll take a look at maxima. My short experiences with computer algebra systems has been that I have to tell the system so much that it almost easier to do this kind of work by hand.
Yeah; that’s basically true. You have to tell Maxima a lot to solve a problem (I think Mathematica has more development effort devoted to “smarts” for specific problems that unloads some of this burden). I’ve found the real pay-off is dealing with the curse of dimensionality where the bookkeeping can be really tedious.
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