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Fluid mechanics: simple calculus issue

  1. Aug 12, 2012 #1


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    1. The problem statement, all variables and given/known data
    Consider the steady flow of an incompressible viscous fluid down an inclined plane under the action of gravity. The plane makes an angle [itex]\alpha[/itex] with the horizontal. The fluid is infinite in the z-direction (x is down the plane and y is normal to the plane). Look for a solution of the form:
    [itex]v_{x}=v_{x}(y), v_{y}=v_{y}(y), v_{z}=0, p=p(y)[/itex]

    By considering the incompressibility condition show that [itex]v_{y}\equiv0[/itex]

    2. Relevant equations
    Incompressibility condition:
    [itex]\bar{\nabla} . \bar{v}=0[/itex]

    3. The attempt at a solution
    [itex]\bar{\nabla} . \bar{v}= \frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{\partial v_{z}}{\partial z}=0[/itex]

    Since [itex]v_{x}[/itex] is a function of y only and [itex]v_{z} = 0[/itex]:
    [itex]\frac{\partial v_{y}}{\partial y}=0[/itex]

    Surely [itex]v_{y}[/itex] can be any constant, and is not necessarily 0???? Or does this have something to do with the use of [itex]\equiv[/itex] instead of [itex]=[/itex] in the question?
  2. jcsd
  3. Aug 12, 2012 #2
    you need to use a boundary condition to fix the value.
    what do you know about vy(y=0) ?
  4. Aug 13, 2012 #3


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    Thank you. Solved
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