An Approach to the Mechanics of Fluids

by Frank C. Fung - 1st published in June, 2008.

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Section II - The Diversity of Fluid Types

Summary for the Section :

• In this Section , we shall identify three (3) common types of fluids :
  • to highlight the microscopic nature of certain fluid flows .

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Identifying 3 Common Types of Fluids :

• Let us first identify three (3) common types of fluids , for a brief comparison :
  • { Water } at room-temperature ,
  • { Atmospheric Air } at Sea-Level ,
  • { Thin Air } at the Exosphere , at an altitude of 1,000 Km to 10,000 Km , much thinner air here .

• We note , first-of-all , that { water } is { Hydrogen Oxide } , i.e. [ H-2-O ] .

  And secondly , { Atmospheric Air } is made-up-of , roughly , ( by moler content / volume ) :

  • 78.08 per cent { Nitrogen } , i.e. [ N-2 ] ;
  • 20.95 per cent { Oxygen } , i.e. [ O-2 ] ;
  • 0.934 per cent { Argon } , i.e. [ Ar ] ;
  • 0.038 per cent { Carbon Dioxide } , i.e. [ C-O-2 ] ;
  • trace amount of other gases ,
  • variable amount of water vapor , averaging around 1 per cent .

  We then have this table below for { Standard Atomic Weights } ,

  • for the afore-mentioned { Elements } above :

    Element	Symbol	Atomic Number	Standard Atomic Weight [ u ]
    Hydrogen	[ H ]	1	1.008
    Carbon	[ C ]	6	12.01
    Nitrogen	[ N ]	7	14.01
    Oxygen	[ O ]	8	16.00
    Argon	[ Ar ]	18	39.95

    ( note 1 : Standard Atomic Weights take into account the abundance of isotopes ) .

    ( note 2 : the unit for Standard Atomic Weights is roughly [ u ] = 1.660539 x 10^-27 Kg ) .

• Based on known { densities } and { molecular mass's } , we then have this table below showing the { number of molecules per Cubic Meter } ,
  • for the first 2 types of fluids mentioned above :

    Fluid Type	Density [ Kg per Cubic Meter ]	Molecular Mass [ u ]	Average Number of molecules per Cubic Meter
    Water	1,000.0	18.015	3.342 x 10^28
    Atmospheric Air	1.2	28.97	2.495 x 10^25

    ( the { molecular mass's } here are basically in-line with the { Standard Atomic Weights } presented in the previous table above . )

  • And the column on-the-very-right here then tells us that :
    • There are roughly 1,300 times more molecules in a { Cubic Meter of Water } than there are in a { Cubic Meter of Atmospheric Air } .

      Quite a vast difference here .

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The proximity of { water molecules } to one-another :

• Let us now look at how closely the { water molecules } may be bunched-up together for { water at room-temperature } .

  Based on our calculations above , we have established that there are roughly [ 3.342 x 10^28 ] { water molecules } in a { Cubic Meter of Water } .

  • Consequently , { a set of 4 water molecules } would then occupy a volume of roughly equal to [ 1.197 x 10^-28 Cubic Meters ] .

• Let us now set-up a { cube } with side-lengths equal to [ 492.8 pm ] , so that :
  • the volume of the said-Cube is roughly [ 1.197 x 10^-28 Cubic Meters ] .

    ( 1 pico-meter ( pm ) is [ 1 / 1,000,000,000,000 Meter ] , or [ 1 x 10^-12 Meter ] . )

  Let us now put a { molecule } in the middle of the said-Cube ,

  • as marked-off in { red-color } in the diagram on-the-left , below .

  

  Let us now also put-in another 12 { molecules } , positioned at the [ 12 mid-points ] of the [ 12 sides ] of the said-Cube ,

  • as marked-off in { blue-color } in the diagram in-the-middle above .

    ( This placement pattern is essentially the same as the { Face-Centered-Cubic Pattern } in Crystalline Solids . )

  We notice here that , for each of the 12 { Blue-Color Spheres } , only [ one-quarter ] of the volume of the sphere is [ inside-the-cube ] ,

  • while the remaining [ three-quarters ] of the volume is [ outside-the-cube ] .

  The count of { full-volume-molecules } [ inside-the-cube ] is therefore [ 4 ] , based on :

  • a count of [ 1 ] for the { red-color sphere } , plus :
  • a count of { [ one-quarter ] x [ 12 ] } , or a count of [ 3 ] , for the 12 { blue-color spheres } .

  Thus , the { cube } of side-lengths [ 492.8 pm ] above , hosting the equivalent of { 4 full-volume-molecules } :

  • is representative of the { maximum average spacing } of { molecules } for { water at room-temperature } .

  Let us now calculate the distance between the { red-color molecule } and any of the 12 { blue-color molecules } ,

  • as per the diagram on-the-right , above .

    And this distance is simply equal to { [ 492.8 pm ] x [ one-half the square-root-of-Two ] } ,

    • or [ 348.4 pm ] .

    Since the { Face-Centered Cubic Pattern } is one of the 2 { tightest-packing patterns } for molecules ,

    • this distance of [ 348.4 pm ] then represents the { maximum average distance } , if at all possible , between 2 { water molecules } at room-temperature .

  • Let us now compare this to the { Bond Length } for the { Hydrogen-Oxygen Bond } for [ H-2-O ] at [ 95.84 p.m. ] .

    

    This [ 348.4 pm ] distance is then roughly equal to [ 3.6 times ] the said-[ bond length ] .

• Let us now look at how tightly-packed-together are the { air molecules } at Sea Level .

  We can then go thru the same procedure here as above , using instead :

  • the [ 2.495 x 10^25 molecules per Cubic Meter of air ] figure we have above .

  And we will find here that { maximum average distance } between { air molecules } to be around [ 3,841.3 pm ] .

• And we note here that :
  • the { Bond Length } for a nitrogen [ N-2 ] molecule is roughly [ 109 pm ] ,
  • the { Bond Length } for an oxygen [ O-2 ] molecule is roughly [ 121 pm ] ,

    so that this [ 3,841.3 pm ] distance here is then roughly equal to [ 34 times ] the average { bond-length } of an { air molecule } .

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Motions of Water Molecules :

• Let us also identify , at this point , the types of motions possible for a { Water Molecule } :
  • { Translation } - the linear movement of the { center-of-mass } of the molecule in the [ X ] , [ Y ] and [ Z ] directions .

  • { Rotation } - the rotation of the { molecule } itself about its { center-of-mass } .

  • { Vibrations } - other types of vabrations as identified below :

    

  • We also note , at this point , the { dipolar nature } of a { water molecule } , namely that :
    • the { Oxygen-end } of the molecule is [ negatively charged ] as opposed to :

      the { Hydrogen-ends } of the molecule being [ positivly charged ] .

      

      This then allows for the existance of { hydrogen-bonds } attraction ,

      • as depicted by the { red-color arrow } and the { blue-color arrow } in the diagram on-the-right , above .

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A Qualitative View of Fluid Flows :

• We note , in general , that { fluid molecules } only occupy a small pecentage of the space attributable to the { fluid as-a-whole } , and
  • only weak-bonds , such as { dipolar-type bonds} mentioned above , are sometimes present .

  Thus , motions of the { fluid molecules } are important from a microscopic stand-point ,

  • and any macroscopic relation developed thereupon must always seek consistencies with the { fluid basics } .

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go to the next section : Section III - The { Cube } and other { Six-Face Object }

go to the last section : Section I - Introduction

return to the Mechanics of Fluids HomePage

original dated 2008-6-08