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Velocity of water across depth

  1. Feb 15, 2017 #1
    1. The problem statement, all variables and given/known data
    Here's the velocity profile of the water across the depth . I dont understand why The max velocity occur at the top surface



    2. Relevant equations


    3. The attempt at a solution
    Is it because the friction on the top surface of water only comes from the below , whereas for the water below the top surafce , it has higher friction from the moving on the top , and moving water beneath it .... Is my concept correct ?
     

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  3. Feb 15, 2017 #2

    BvU

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    Hi.
    What's holding back the water ? The viscous friction with something that doesn't move, namely the bottom. So all transport of horizontal momentum takes place in a vertical direction - as you can see from the variation in velocity going from 0 at the bottom to max at the surface.
     
  4. Feb 15, 2017 #3
    The friction at the top surface (i.e., the shear stress) is zero. There is friction at the bottom surface because of the non-slip boundary condition.
     
  5. Feb 15, 2017 #4
    Do you mean as the depth increases , the shear stress increases, causing the velocity to decreases parabolically from top surface to the bottom >
     
  6. Feb 15, 2017 #5
    Yes
     
  7. Feb 15, 2017 #6
    can you explain further why the shear stress will increase with depth ?
     
  8. Feb 15, 2017 #7
    You do understand that the shear stress is zero at the upper surface, correct? And you do understand that the fluid does not slip at the wall, correct.
     
  9. Feb 15, 2017 #8
    y
    yes , i understand both .... Why the shear stress will incresae down the depth ?
     
  10. Feb 15, 2017 #9
    Since the fluid at the wall is stationary (the velocity at the wall is zero), and you have a volumetric flow rate, the velocity has to be higher away from the wall. This means that the velocity gradient at the wall must be positive. From Newton's law of viscosity, that means that there is a shear stress at the wall. So you have shear stress at the wall and no shear stress at the interface with the air. So the shear stress must be increasing with distance from the wall.
     
  11. Feb 15, 2017 #10
    why ? ?
     
  12. Feb 15, 2017 #11
    If the velocity were zero everywhere, there would be no flow.
     
  13. Feb 15, 2017 #12
    why it has to increase parabolically ? Why the shape of graph cant be others ? I mean the graph has still positive gradient ...
     
  14. Feb 15, 2017 #13
    To obtain that result, you need to actually solve the flow problem involving the differential force balance and Newton's law of viscosity.
     
  15. Feb 15, 2017 #14
    I still doesnt get it . Can you show the problem involving the differential force balance and Newton's law of viscosity???
     
  16. Feb 15, 2017 #15
    It's too extensive to show here. Are you familiar with the derivation of the Hagen-Poiseuille law for laminar viscous flow in a pipe.
     
  17. Feb 15, 2017 #16
    Can you provide some link and explain briefly here ?
     
  18. Feb 15, 2017 #17
    Google is your friend.
     
  19. Feb 15, 2017 #18

    BvU

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    Bird Stewart and Lightfoot are even better friends. Whole derivation in 2.2 in my edition.
     
  20. Feb 15, 2017 #19
    What do you mean ?
     
  21. Feb 15, 2017 #20
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