Engineering Thermodynamics and Fluid Mechanics 
        
        
       These lecture notes have been prepared as a first course in Engineering Thermodynamics and 
       Fluid  Mechanics  up  to  the  presentation  of  the  millennium  problem  listed  by  the  Clay 
       Mathematical Institute. Only a good knowledge of classical Newtonian mechanics is assumed. 
       We  start  the  course  with  an  elementary  derivation  of  the  equations  of  ideal  Engineering 
       Thermodynamics and Fluid Mechanicsand end up with a discussion of the millennium problem 
       related to real fluids. With this document, our primary goal is to debunk this beautiful problem as 
       much  as  possible,  without  assuming  any  previous  knowledge  neither  in  Engineering 
       Thermodynamics  and  Fluid  Mechanicsof  real  fluids  nor  in  the  mathematical  formalism  of 
       Sobolev’s inequalities. All these items are introduced progressively through the document with a 
       linear increase in the difficulty. Some rigorous proofs of important partial results concerning the 
       millennium  problem  are  presented.  At  the  end,  a  very  modern  aspect  of  Engineering 
       Thermodynamics and Fluid Mechanicsis covered concerning the subtle issue of its application to 
       high energetic hadronic collisions. 
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
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       §1. Introduction 
       §2. Continuum hypothesis 
        
       §3. Mathematical functions that define the fluid state 
       §4. Limits of the continuum hypothesis §5. Closed set 
       of equations for ideal fluids 
       §6. Boundary conditions for ideal fluids 
        
       §7. Introduction to nonlinear differential equations §8. 
       Euler’s equations for incompressible ideal fluids §9. 
       Potential flows for ideal fluids 
       §10. Real fluids and Navier-Stokes equations 
       §11. Boundary conditions for real fluids 
        
       §12. Reynolds number and related properties §13. 
       The millennium problem of the Clay Institute §14. 
       Bounds and partial proofs 
       §15. Engineering Thermodynamics and Fluid Mechanicsin relativistic Heavy-Ions collisions 
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
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       §1. Introduction 
        
       Engineering Thermodynamics and Fluid Mechanicsconcerns the study of the motion of fluids (in 
       general liquids and gases) and the forces acting on them. Like any mathematical model of the real 
       world, Engineering Thermodynamics and Fluid Mechanicsmakes some basic assumptions about 
       the materials being studied. These assumptions are turned into equations that must be satisfied if 
       the assumptions are to be held true. Modern fluid mechanics, in a well-posed mathematical form, 
       was first formulated in 1755 by Euler for ideal fluids. 
        
        
       Interestingly,  it  can  be  shown  that  the  laws  of  Engineering  Thermodynamics  and  Fluid 
       Mechanicscover more materials than standard liquid and gases. Indeed, the idea of exploiting the 
       laws of ideal Engineering Thermodynamics and Fluid Mechanicsto describe the expansion of the 
       strongly  interacting  nuclear  matter  that  is  formed  in  high  energetic  hadronic  collisions  was 
       proposed in 1953 by Landau. This theory has been developed extensively in the last 60 years and 
       is  still  an  active  field  of  research.  This  gives  a  very  simple  3-steps  picture  of  a  non-trivial 
       phenomenon observed in hot nuclear matter after the collision of high energetic heavy ions, 
       composed of a large collection of charged particles. 
       (i)                
                          
                         After the collision a nuclear medium, a zone of high 
                         density of charges, is formed with high pressure in 
                         the middle (center of the collision). 
                          
       (ii)               
                          
                          
                          
                          
                         According to the laws  of fluid  mechanics,  as  we 
                         shall prove them, this implies that an acceleration 
                         field  is  generated  from  high  pressures  to  low 
                         pressures. 
                          
        
        
        
        
        
        
        
        
        
        
        
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       (iii)              
                          
                          
                          
                          
                         This  implies  that  particles  will  flow  in  a  certain 
                         transverse direction, as indicated on the figure. This 
                         is  known  as  the  transverse  flow  property,  well 
                         established experimentally. 
                          
        
       We come back on these ideas and their developments in the last section of this document. It 
       requires a relativistic formulation of fluid mechanics. Up to this section, we always assume that 
       the dynamics is non-relativistic. 
       §2. Continuum hypothesis 
        
       Engineering Thermodynamics and Fluid Mechanicsis supposed to describe motion of fluids and 
       related  phenomena  at  macroscopic  scales,  which  assumes  that  a  fluid  can  be  regarded  as  a 
       continuous medium. This means that any small volume element in the fluid is always supposed 
       so large that it still contains a very great number of molecules. Accordingly, when we consider 
       infinitely small elements of volume, we mean very small compared with the volume of the body 
       under  consideration,  but  large  compared  with  the  distances  between  the  molecules.  The 
       expressions  fluid  particle  and  point  in  a  fluid  are  to  be  understood  in  this  sense.  That  is, 
       properties such as density, pressure, temperature, and velocity are taken to be well-defined at 
       infinitely small points. 
        
        
       These properties are then assumed to vary continuously and smoothly from one point to another. 
       Consequently, the fact that the fluid is made up of discrete molecules is ignored. If, for example, 
       we deal with the displacement of some fluid particle, we do mean not the displacement of an 
       individual  molecule,  but  that  of  a  volume  element  containing  many  molecules,  though  still 
       regarded as a point in space. That’s why Engineering Thermodynamics and Fluid Mechanicsis a 
       branch of continuum mechanics, which models matter from a macroscopic viewpoint without 
       using the information that it is made out of molecules (microscopic viewpoint). 
        
        
        
        
        
        
        
        
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