An Approach to the Mechanics of Fluids

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

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Section VII - A Steady Flow Situation

Summary for the Section :

• In this Section , we shall use { steady flow } as a pre-text :
  • to look at the microscopic nature of { Density } and { Pressure } of water .

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A { Steady Flow } situation :

• In a { steady flow } situation , the { density } , { pressure } and { velocity } at any given position is not { time-dependent } , i.e :
  • their values are dependant solely on the co-ordinates of the { point } in-question , but not on { time } .

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Density :

• Let us look briefly the { Density of Water } in a { Steady Flow } .

  This simply means that :

  • if the { finite element } in-question is of-constant-volume ,
    • then the average number of { water molecules } in the { finite element } will remain roughly constant , thru { time } .
      • so that the total mass of { water molecules } inside the { finite element } is unchanged .

• Thus , while { water molecules } may move-into and move-out-of the { finite element } ,
  • thru any of the available { faces } ,

    the { net-balance } of { water molecules } moving in-and-out-of the { finite element } is always [ zero ] , or very-close-to-it .

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Pressure :

• Let us now also look briefly at the { pressure of water } in a { Steady Flow } .

  Let us bring back this { Large Cylindrical Water Tank with a drain-pipe in-the-middle } , from a previous section , Section V :

  • but this time , we shall fill the entire tank with { lead pallets } instead ,
    • as per the diagram on-the-left , below .

  

  • We notice here that , even when none of the { lead pallets } are moving ,
    • we still have to apply a { Force } , as marked by the { blue-color-arrow } in the diagram on-the-left , above ,

      to prevent the { lead pallets } from falling thru .

• Let us now look at the diagram on-the-right , above , where the { water } is seemingly at rest .

  But we have already established back in Section II , that :

  • The { water molecules } themselves only occupy a small fraction of the available space .
    • so that water pressure does not and cannot arise solely from a simple { Hecules-shouldering-a-rock }-type situation ,

      but from the dynamic movements of the { water molecules } .

  And this should always be kept-in-mind , when developing and scrutinizing { pressure equations } .

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• Next , we shall move onto an interesting topic , non-stationary { finite elements } .

go to the next section : Section VIII - Time-Dependent { Finite Elements }

go to the last section : Section VI - { Flow Net } in Soil Mechanics

return to the Mechanics of Fluids HomePage

original dated 2008-6-08