# AXIOMATIC FORMULATIONS

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24-Feb-2016Category

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AXIOMATIC FORMULATIONSGraciela Herrera Zamarrn1SCIENTIFIC PARADIGMS2Generality Clarity Simplicity

AXIOMATIC FORMULATION OF MODELS

33MACROSCOPIC PHYSICSThere are two major branches of Physics:MicroscopicMacroscopic

The approach presented belongs to the field of Macroscopic Physics44GENERALITYThe axiomatic method is the key to developing effective procedures to model many different systemsIn the second half of the twentieth century the axiomatic method was developed for macroscopic physicsThe axiomatic formulation is presented in the books:Allen, Herrera and Pinder "Numerical modeling in science and engineering", John Wiley, 1988Herrera and Pinder "Fundamentals of Mathematical and computational modeling", John Wiley, in press556BALANCES ARE THE BASIS OF THE AXIOMATIC FORMULATION OF MODELS6EXTENSIVE AND INTENSIVE PROPERTIES

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Estensive property: Any that can be expressed as a volume integralIntensive proporty: Any extensive per unit volumen; this is, 7FUNDAMENTAL AXIOMA BALANCE CONDITION8An extensive property can change in time, exclusively, because it enters into the body through its boundary or it is produced in its interior. 8BALANCE CONDITIONS IN TERMS OF THE EXTENSIVE PROPERTY 9

910BALANCE CONDITIONSIN TERMS OF THE INTENSIVE PROPERTY

Balance differential equation10THE GENERAL MODEL OF MACROSCOPIC MULTIPHASE SYSTEMSAny continuous system is characterized by a family of extensive properties and a family of phasesEach extensive property is associated with one and only one phaseThe basic mathematical model is obtained by applying to each of the intensive properties the corresponding balance conditionsEach phase moves with its own velocity1111THE GENERAL MODEL OF MACROSCOPIC SYSTEMS12

Balance differential equationsIntensive properties

12SIMPLICITY PROTOCOL OF THE AXIOMATIC METHOD FOR MAKING MODELS OF MACROSCOPIC PHYSICS:Identificate the family of extensive propertiesGet a basic model for the system Express the balance condition of each extensive property in terms of the intensive propertiesIt consists of the system of partial differential equations obtainedThe properties associated with the same phase move with the same velocityIncorporate the physical knowledge of the system through the Constitutive Relations1313CONSTITUTIVE EQUATIONS14Are the relationships that incorporate the scientific and technological knowledge available about the system in question14THE BLACK OIL MODEL1515GENERAL CHARACTERISTICS OF THE BLACK-OIL MODELIt has three phases: water, oil and gasIn the oil phase there are two components: non-volatile oil and dissolved gasIn each of the other two phases there is only one componentThere is exchange between the oil and gas phases: the dissolved gas may become oil and vice versaDiffusion is neglected1616FAMILY OF EXTENSIVE PROPERTIES OF THE BLACK-OIL MODELWater mass (in the water phase) Non-volatile oil mass (in the oil phase)Dissolved gas mass (in the oil phase)Gas mass (in the gas phase) 1717MATHEMATICAL EXPRESSION OF THE FAMILY OF EXTENSIVE PROPERTIES

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1819BASIC MATHEMATICAL MODEL

19FAMILY OF INTENSIVE PROPERTIES

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2021BASIC MATHEMATICAL MODEL

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AXIOMATIC FORMULATION OF DOMAIN DECOMPOSITION METHOD2222PARALELIZATION METHODSDomain decomposition methods are the most effective way to parallelize boundary value problems Split the problem into smaller boundary value problems on subdomains 23DOMAIN DECOMPOSITION METHODS