Computer Modeling of Water in Pipelines
In May 2007 I posted that IBM Predicts Big Changes in Water Production & Distribution in 5 Years
imho in order to have a successful 21st century water policy– desalinated water from the ocean will need to be piped to inland deserts on a vast scale. In order for this to be done economically — a way needs to be devised to cheaply create in bulk very low maintenance pipes that push water uphill over long distances with little or no added energy cost. In order to cheaply invent these pipes–a computer modeling system will have to be undertaken.
There are three variables that I can think of right off that might be modeled to push water uphill passively: some variation of hydrophobic vs hydrophillic material inside the pipe. Some variation of heat & cold conduction from the outside to the inside of the pipe. Some variation of shape inside the pipe. Nor is it clear that a pipe needs to be completely hollow. A redwood tree pushes immense amounts of water straight up daily. Finally, some allowance for solar energy to be used for pumping can be made for early models as the cost of solar power falls under the cost of coal in the next few years.
What would be the algorithms to use in the computer models? Well the idea is to find algorithms that enable researchers test new materials either singly or in combination with others–and with different shapes– as they interact with water in a pipe. What algorithms? NIST is going come out with a new library of mathematical references.
NIST releases preview of much-anticipated online mathematics reference
That’s a whole library of equations on which to base algorithms.
My suggestion would be two formulas. These are not algorithms. But they could be incorporated into algorithms. One formula models the flow of water over complex shapes and variable materials. The other formula models water in a pipe. See below.
140-year-old math problem solved by researcher
Academic makes key additions to the Schwarz-Christoffel formula
A problem which has defeated mathematicians for almost 140 years has been solved by a researcher at Imperial College London.
Professor Darren Crowdy, Chair in Applied Mathematics, has made the breakthrough in an area of mathematics known as conformal mapping, a key theoretical tool used by mathematicians, engineers and scientists to translate information from a complicated shape to a simpler circular shape so that it is easier to analyse.
This theoretical tool has a long history and has uses in a large number of fields including modelling airflow patterns over intricate wing shapes in aeronautics. It is also currently being used in neuroscience to visualise the complicated structure of the grey matter in the human brain.
A formula, now known as the Schwarz-Christoffel formula, was developed by two mathematicians in the mid-19th century to enable them to carry out this kind of mapping. However, for 140 years there has been a deficiency in this formula: it only worked for shapes that did not contain any holes or irregularities.
Now Professor Crowdy has made additions to the famous Schwarz-Christoffel formula which mean it can be used for these more complicated shapes. He explains the significance of his work, saying:
“This formula is an essential piece of mathematical kit which is used the world over. Now, with my additions to it, it can be used in far more complex scenarios than before. In industry, for example, this mapping tool was previously inadequate if a piece of metal or other material was not uniform all over - for instance, if it contained parts of a different material, or had holes.”
Professor Crowdy’s work has overcome these obstacles and he says he hopes it will open up many new opportunities for this kind of conformal mapping to be used in diverse applications.
“With my extensions to this formula, you can take account of these differences and map them onto a simple disk shape for analysis in the same way as you can with less complex shapes without any of the holes,” he added.
Professor Crowdy’s improvements to the Schwarz-Christoffel formula were published in the March-June 2007 issue of Mathematical Proceedings of the Cambridge Philosophical Society.
http://nick2.wordpress.com/2006/10/06/carbon-nanotube-research/
Navier-Stokes Equation Progress?
Penny Smith, a mathematician at Lehigh University, has posted a paper on the arXiv that purports to solve one of the Clay Foundation Millenium problems, the one about the Navier-Stokes Equation. The paper is here, and Christina Sormani has set up a web-page giving some background and exposition of Smith’s work.
Wikipedia describes Navier-Stokes Equations this way:
They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are used to model weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, and flow around an airfoil (wing). They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with Maxwell’s equations they can be used to model and study magnetohydrodynamics.
When these equations pass peer review they’ll be very helpful in algorithms that model fluids flowing in a pipe.
