numerical models
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MATHEMATICAL MODEL
General Purpose
Introduction
A mathematical model is defined as a description from the point of view of mathematics of a fact or phenomenon of the real world, the size of the population to physical phenomena such as speed, acceleration or density.
Process to develop a mathematical model
Mental model: first perception of the problem atmental. It is the first play of ideas that intuitively are generated on the issue at hand.
Graphic model: the set of images and graphics support that allow you to place the functional relations that prevail in the system to be studied.
Verbal Model: it is the first attempt to formalize languagecharacteristics of the problem. In this model performsfirst formal approach to the problem.
Model Types
Mathematical Model: seeks to formalize in mathematicallanguagerelations and variations you want to represent and analyze.Normally expressed as differential equations andthis reason may also be known as differential model.
Analytical Model: arises when the differential model can be solved. This is not always possible, especially when the models are posed on differential differentialPartial.
Physical Model: consists of an assembly that can function as a real testbed. They are usually simplified laboratory scale allow more detailed observations. Not always easy to build.
Number Model: Arises when using numerical techniquesto solve differential models. They are very common when the model has no analytical solution differential.
Computational Model: Refers to a computer program that allows analytical or numerical models, can be solved more quickly. They are very useful for implementing numerical models because these are based oniterative process can be long and tedious.
Model Types
1. Linear Models2. Polynomials3. Power Functions4. Rational functions5. Trigonometric functions6. Exponential Functions7. Logarithm functions8. Transcendent functions
Fig. 1. http://webcache.googleusercontent.com/search?q=cache:trQqOqKXXucJ:www.monografias.com/trabajos12/moma/moma.shtml+CUAL+ES+EL+OBJETIVO+DE+UN+MODEL
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Example. linear function
Components of a Mathematical model
Dependent variables
Independent Variables
Parameter
Functions of force
Operators
Gradient
Fig.2 Elkin slides santafé.2009.Metodos Numerical Engineering
function given its gradient vector
Gradient
Scalar field
Divergence
Rotacinal
Laplacian
Scalar field
Vector field
Finite difference approximation
GEOMETRIC DEFINITION
Fig.3 Elkin slides santafé.2009.Metodos Numerical Engineering
progressive
regressive
central
Types of differential equations
Elliptical
Satellite
Hyperbolic
Laplace equation(Steady statetwo-dimensionalspace)
Equationheat conduction(Variable time anddimensionspatial)
Wave equation(Variable time anddimensionspatial)
Fundamental Flow Equation
GEOMETRIC FACTOR
FLUID DENSITY
FLOW RATE
POROSITY
SOURCES AND / ORSINKS
Flow in porous media: Darcy's Law
c
c
In 1856, in the French city of Dijon, the engineer Henry Darcy was responsible for study of the supply network to the city. It seems that it also had to design filters sand to purify water, so I was interested in the factors influencing the flow of water through the sandy, and presented the results of his work as a appendix to his report of the distribution network. That little appendix has been the basis of all subsequent physico-mathematical studies on groundwater flow.In today's laboratories have equipment similar to that used Darcy, and are called constant head permeameter
Fig.4 constant head permeameter
Q = CaudalΔh = Potential Difference entre A y BΔl = Distance between A y B
Hydraulic gradient =
section
Basically a permeameter is a recipient of constant section which makesFlush connecting one end of a high constant level tank. In theother end is regulated by an outflow valve in each experimentalso maintains the flow constant. Finally, measure the height of the water columnat various points
Section
If the environment changes but the relationship is fulfilled K constant changes.This was called permeability.
Conservation of Momentum
caudal section x velocity
Velocidad Darcy : Caudal / Saccion total
This is false because the water does not circulatethroughout the cross section
Limitations of Darcy's law
The proportionality constant K is not proper or other featurethe porous medium.
PERMEABILITYINTRINSIC
SPECIFIC WEIGHT FLUID
VISCOSITYDYNAMICSFLUID
In some circumstances the relationship between Q and the gradient Hydraulic non-linear. This can happen when the value of K is very low or very high speeds.
Darcy is met
Darcy is not met
can be fulfilled or not
Incompressible fluid
Fluid slightlycompressible
Compressible fluid
constant
Equations of State
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Elkin slides santafé.2009.Metodos Numerical Engineering
F. Javier Sánchez San Román‐‐Dpto. Geología‐‐Univ. Salamanca (España) http://web.usal.es/javisan/hidro
Bibliography