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Math Help - Need solution of following eqn.

  1. #1
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    Need solution of following eqn.

    I need a solution of this eqn.

    ∇.(ε∇φ)=0

    if there is no ε, then φ=x^2-y^2 can easily be taken as solution but I am confused as ε is present. Here ε is electrical conductivity and φ is electrical potential.
    Help me please.......
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  2. #2
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    Is \displaystyle \varepsilon a constant or a function?
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  3. #3
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    Actually I am trying to make simulation of flow induced by plasma actuator using PHOENICS CFD software. Here is the detailed image:



    ε is a constant. In the solution domain upper part is air and bottom shaded part is dielectric material. Same equation for both domain except different dielectric constant/electrical conductivity.

    for upper domain ε1 is 1 and bottom domain ε2 is 2.7. Possibly the solution should be made considering ε a function. because I have to define upper and bottom domain with 1 and 2.7 respectively.
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  4. #4
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    A property of dot products is

    \displaystyle (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = (c_1c_2)(\mathbf{a}\cdot \mathbf{b}).


    So that means \displaystyle \nabla \cdot (\varepsilon \nabla \varphi) = \varepsilon (\nabla \cdot \nabla \varphi).
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  5. #5
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    that means: ε ∇^2φ=0?
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  6. #6
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    Quote Originally Posted by bdcrown007 View Post
    that means: ε ∇^2φ=0?
    Yes
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  7. #7
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    so, how to proceed for solution?
    ∇.(ε∇φ)=0
    ε ∇^2φ=0
    ε(d2φ/dx2+d2φ/dy2)=0

    in this case ε=0 ?

    >?
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  8. #8
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    I think it's pretty clear that \displaystyle \varepsilon \neq 0...

    How would you solve \displaystyle \frac{\partial ^2 \varphi}{\partial x^2} + \frac{\partial ^2 \varphi}{\partial y^2} = 0?
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  9. #9
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    If I use then there is no difference between two eqn for two domain (please check image of my 2nd reply).

    confused
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  10. #10
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    in this case of
    the solution becomes: \varphi=x^2-y^2
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  11. #11
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    but  \varepsilon is necessary what should I do?
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