Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab.pdf), страница 11

C. Adecompose it asA¼ 11A þ AT þ A AT22and show that1A þ AT is symmetric21A AT is anti-symmetricb.2a.Exercise 13Given two tensors A and B defined as0a11BA ¼ @ a21a12...01b11B.. C. A B ¼ @ b21b12...1.. C. Aa. Calculate the double inner product A : B.b. Prove that ðA þ BÞT ¼ AT þ BT and ðABÞT ¼ BT ATc. Evaluate r A þ r B.References1. Arfken G (1985) Mathematical methods for physicists, 3rd edn. Academic Press, Orlando, FL2. Aris R (1989) Vectors, tensors, and the basic equations of fluid mechanics. Dover, New York3.

Crowe MJ (1985) A history of vector analysis: the evolution of the idea of a vectorial system.Dover, New York422 Review of Vector Calculus4. Marsden JE, Tromba AJ (1996) Vector calculus. WH Freeman, New York5. Jeffreys H, Jeffreys BS (1988) methods of mathematical physics. Cambridge University Press,Cambridge, England6.

Morse PM, Feshbach H (1953) Methods of theoretical physics, Part I. McGraw-Hill,New York7. Schey HM (1973) Div, grad, curl, and all that: an informal text on vector calculus. Norton,New York8. Schwartz M, Green S, Rutledge W (1960) A vector analysis with applications to geometry andphysics. Harper Brothers, New York9. M1 Anton H (1987) Elementary linear algebra. Wiley, New York10. Bretscher O (2005) Linear algebra with applications. Prentice Hall, New Jersey11.

Bronson R (1989), Schaum’s outline of theory and problems of matrix operations.McGraw–Hill, New York12. Arnold VI, Cooke R (1992) Ordinary differential equations. Springer-Verlag, Berlin, DE; NewYork, NY13. Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge14. Brown WC (1991) Matrices and vector spaces. Marcel Dekker, New York15. Golub GH, Van Loan CF (1996) Matrix Computations.

Johns Hopkins, Baltimore16. Greub WH (1975) Linear algebra, graduate texts in mathematics. Springer-Verlag, Berlin, DE;New York, NY17. Lang S (1987) Linear algebra. Springer-Verlag, Berlin, DE; New York, NY18. Mirsky L (1990) An introduction to linear algebra. Courier Dover Publications, New York19. Nering ED (1970) Linear algebra and matrix theory. Wiley, New York20. Spiegel MR (1959) Schaum’s outline of theory and problems of vector analysis and anintroduction to tensor analysis. Schaum, New York21. Heinbockel JH (2001) Introduction to tensor calculus and continuum mechanics.

TraffordPublishing, Victoria22. Williamson R, Trotter H (2004) Multivariable mathematics. Pearson Education, Inc, New York23. Cauchy A (1846) Sur les intégrales qui s’étendent à tous les points d’une courbe fermée.Comptes rendus 23:251–25524.

Riley KF, Hobson MP, Bence SJ (2010) Mathematical methods for physics and engineering.Cambridge University Press, Cambridge25. Spiegel MR, Lipschutz S, Spellman D (2009) Vector analysis. Schaum’s Outlines, McGrawHill (USA)26. Wrede R, Spiegel MR (2010) Advanced calculus. Schaum’s Outline Series27. Katz VJ (1979) The history of stokes’s theorem. Math Mag (Math Assoc Am) 52:146–15628.

Morse PM, Feshbach H (1953) Methods of theoretical physics, Part I. McGraw-Hill, New York29. Stewart J (2008) Vector calculus, Calculus: early transcendentals. Thomson Brooks/Cole,Connecticut30. Lerner RG, Trigg GL (1994) Encyclopaedia of physics.

VHC31. Byron F, Fuller R (1992) Mathematics of classical and quantum physics. Dover Publications,New York32. Spiegel MR, Lipschutz S, Spellman D (2009) Vector analysis. Schaum’s Outlines, McGrawHill33. Flanders H (1973) Differentiation under the integral sign. Am Math Monthly 80(6):615–62734. Boros G, Moll V (2004) Irresistible integrals: symbolics, analysis and experiments in theevaluation of integrals.

Cambridge University Press, Cambridge, England35. Hijab O (1997) Introduction to calculus and classical analysis. Springer, New York36. Kaplan W (1992) Advanced calculus. Addison-Wesley, Reading, MAChapter 3Mathematical Description of PhysicalPhenomenaAbstract The chapter provides an overview of the conservation principlesgoverning fluid flow, heat and mass transfer, and other related transport phenomenaof interest in this book.

The physical laws controlling the conservation principlesare translated into mathematical relations, written in the form of partial differentialequations, representing the needed vehicle for their simulations. First the continuity,momentum, and energy equations (collectively known as the Navier-Stokesequations) expressing the principles of conservation of mass, momentum, and totalenergy, respectively, are derived. This is followed by the development of a typicalconservation equation for a general scalar, vector, or tensor quantity. The mathematical properties of the various terms in these equations are also examined.Moreover, the common practice of writing the conservation equations in anon-dimensional form using dimensionless quantities is explained and some of thedimensionless groups resulting from the application of this procedure, which arevery useful for performing parametric studies of engineering problems, arediscussed.3.1 IntroductionResearchers and practitioners of computational fluid dynamics encounter and workwith the Navier-Stokes equations [1, 2] almost on daily basis.

Many do not realizethat these equations are over one hundred seventy years old. Whereas the nameNavier-Stokes initially referred to the conservation equation of linear momentum, itis used nowadays to denote collectively the conservation equations of mass,momentum, and energy. These equations can be used to model a wide range of fluidflow configurations, whether it is the flow in a hurricane or in a turbomachine,around an airplane or a submarine, in arteries or in lungs, in pumps or in compressors, the Navier-Stokes equations can describe all these phenomena.© Springer International Publishing Switzerland 2016F.

Moukalled et al., The Finite Volume Method in Computational Fluid Dynamics,Fluid Mechanics and Its Applications 113, DOI 10.1007/978-3-319-16874-6_343443 Mathematical Description of Physical Phenomena3.2 Classification of Fluid FlowsFluids, which denote liquids and gases are substances that do not permanently changeunder a large stress (force per unit area).

Whereas a solid resists an applied shear ortangential stress by deforming, a fluid cannot and a shear stress applied to a fluid putsit to motion. Moreover, unlike solids which have well-defined shapes, fluids do nothave a definite shape. While gases are fluids that completely fill their domains, liquidsare fluids that form a free surface in the presence of a gravitational field.In analyzing fluid flow phenomena [3–6], attention is focused on what happens atthe macroscopic rather than the microscopic scale. It is also assumed that the fluid is acontinuum, so that its physical and flow properties are defined at every point in space.Within this assumption, fluid flow behavior can be categorized as either Newtonian ornon-Newtonian. Newtonian fluids are characterized by a linear relationship betweenthe shear stress and the shear rate, with the molecular viscosity l, which is a measure ofthe ability of a fluid subjected to a stress to resist deformation, representing the slope ofthe linear function.

On the other hand, for non-Newtonian fluids this relationship isnonlinear. Similarly fluid flow can be classified into various classes, such asone-dimensional or multi-dimensional, single phase or multi-phase, steady orunsteady, real (viscous) or ideal (inviscid), compressible or incompressible, turbulentor laminar, and rotational or irrotational, among others. The purpose of these classifications is to simplify the process of analysis and modeling of fluid flow phenomena.Flows are also classified mathematically according to the partial differentialequations describing them. Second order partial differential equations in twoindependent variables, for example, are categorized as hyperbolic, parabolic, orelliptic.

In these equations information travels along two characteristic lines, whichmay be real and distinct, real and coincident, or complex depending on whetherthey are of the hyperbolic, parabolic, or elliptic type, respectively. This variation inthe nature of the equations necessitates different solution methodologies that shouldalso be recognized by any numerical method used to solve them.As will be shown in this chapter, fluid flows are governed by the Navier-Stokesequations, which are highly nonlinear second order partial differential equations infour independent variables since, in general, flows are unsteady and three dimensional. Therefore, the above classification does not really apply to them.Nevertheless, the same terminology is used in their categorization as they sharemany of the properties characterizing second order equations in two independentvariables.

Transient and supersonic flows are hyperbolic, boundary layer flows areparabolic, and recirculating flows are elliptic. As flows may be subsonic in a certainpart of the domain and supersonic in other parts (e.g., flow in aconverging-diverging nozzle), or viscous dominated close to walls and essentiallyinviscid in the core region, it is hard to describe a flow as falling under one of theabove three types and in general it is of the mixed type. This categorization isnumerically translated into the following: parabolic flows that are affected byupstream locations only, elliptic flows by both upstream and downstream locations,and hyperbolic flows supporting discontinuities in the solution, e.g., shock waves.3.3 Eulerian and Lagrangian Description of Conservation Laws453.3 Eulerian and Lagrangian Description of ConservationLawsThe principle of conservation states that for an isolated system certain physicalmeasurable quantities are conserved over a local region.

This conservation principleor conservation law is an axiom that cannot be proven mathematically but can beexpressed by a mathematical relation. Laws of this type govern several physicalquantities such as mass, momentum, and energy (i.e., the Navier-Stokes equations).The conservation laws involving fluid flow and related transfer phenomena canbe mathematically formulated following either a Lagrangian (material volume, MV)or an Eulerian (control volume) approach [7].

In the Lagrangian specification of theflow field (Fig. 3.1a), the fluid is subdivided into fluid parcels and every fluid parcelis followed as it moves through space and time. These parcels are tagged using atime-independent position vector field x0 , usually selected to be the parcels’ centreof mass at some initial time t0 , and the flow is described by a function xðt; x0 Þ. Thepath line described by a fluid parcel (Fig.

3.1a) is obtained as the collection ofpositions occupied at different times.On the other hand, the Eulerian approach (Fig. 3.1b) focuses on specific locations in the flow region as time passes. Thus the flow variables are functions ofposition x and time t and the flow velocity is represented by vðt; xÞ.