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Engineering LibreTexts

10.2.3.1: Existences of Stream Functions

  • Page ID
    1289
  • [ "article:topic" ]

    The potential function in order to exist has to have demised vorticity. For two dimensional flow the vorticity, mathematically, is demised when

    \[
        \label{if:eq:zeroVortisity}
        \dfrac{\partial U_x}{\partial y} -   
        \dfrac{\partial U_x}{\partial x} = 0 \tag{80}
    \]
    The stream function can satisfy this condition when

    Stream Function Requirements

    \[
        \label{if:eq:streamRequirement}
        \dfrac{\partial}{\partial y} \left( \dfrac{\partial \psi}{\partial y} \right) +
        \dfrac{\partial}{\partial x} \left( \dfrac{\partial \psi}{\partial x} \right) = 0  
        \Longrightarrow  
        \dfrac{\partial^2\psi}{\partial y^2} +
        \dfrac{\partial^2\psi}{\partial x^2} = 0 \tag{81}
    \]

    Example 10.4

    Is there a potential based on the following stream function  
    \[
        \label{canItBePotential:streamFun}
        \psi = 3\,x^5 - 2\,y   \tag{82}
    \] 

    Solution 10.4

    Equation (81) dictates what are the requirements on the stream function. According to this equation the following must be zero
    \[
        \label{canItBePotential:check}
        \dfrac{\partial^2\psi}{\partial y^2} +
       \dfrac{\partial^2\psi}{\partial x^2} \overset{?}{=} 0 \tag{83}
    \]
    In this case it is  
    \[
        \label{canItBePotential:theCheck}
        0 \overset{?}{=}  0 + 60\,x^3   \tag{84}
    \]
    Since \(x^3\) is only zero at \(x=0\) the requirement is fulfilled and therefore this function cannot be appropriate stream function.

    Contributors

    • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.