Numerical models

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

http://webcache.googleusercontent.com/search?q=cache:trQqOqKXXucJ:www.monografias.com/trabajos12/moma/moma.shtml+CUAL+ES+EL+OBJETIVO+DE+UN+MODELO+MATEMATICO&cd=1&hl=es&ct=clnk&gl=co

Elkin slides santafé.2009.Metodos Numerical Engineering

F. Javier Sánchez San Román‐‐Dpto. Geología‐‐Univ. Salamanca (España) http://web.usal.es/javisan/hidro

Bibliography