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Shear Flow Against a Wall (Fluid Mechanics)

  1. May 10, 2013 #1
    1. The problem statement, all variables and given/known data

    Assume a shear flow against a wall, given by U= Uo (2y/ax - y^2/((ax)^2) where a is a constant. Derive the velocity component V (x; y) assuming incompressibility.



    2. Relevant equations
    Haven't been able to find any in my course notes.


    3. The attempt at a solution
    Some googling has taught me that shear flow is the flow induced by a shear stress force gradient. But I really need some sort of equation to solve this I think.
     
  2. jcsd
  3. May 12, 2013 #2
  4. May 12, 2013 #3
    Hi Isobel2. Welcome to Physics Forums.

    You need to make use of MaxManus' suggestion, and set the divergence of the velocity vector equal to zero:

    [tex]\frac{\partial V}{\partial y}=-\frac{\partial U}{\partial x} [/tex]

    But, before you start trying to do this by brute force, first define the following parameter:

    [tex]\eta=\frac{y}{ax}[/tex]

    so that [tex]U = U_0(2\eta - \eta ^2)[/tex]

    Also note that [tex]\frac{\partial U}{\partial x}=\frac{\partial U}{\partial \eta}\frac{\partial \eta}{\partial x}=-\frac{\partial U}{\partial \eta}\frac{\eta}{x}[/tex]
    [tex]\frac{\partial V}{\partial y}=\frac{\partial V}{\partial \eta}\frac{\partial \eta}{\partial y}=\frac{\partial V}{\partial \eta}\frac{1}{ax}[/tex]

    Working with the parameter η in this way will make the "arithmetic" much simpler and less prone to error.
     
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