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Engineering LibreTexts Existences of Stream Functions

  • Page ID
  • 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

    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
    \dfrac{\partial^2\psi}{\partial y^2} +
    \dfrac{\partial^2\psi}{\partial x^2} = 0

    Example 10.4

    Is there a potential based on the following stream function

    \[ \label{canItBePotential:streamFun}
    \psi = 3\,x^5 - 2\,y

    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

    In this case it is

    \[ \label{canItBePotential:theCheck}
    0 \overset{?}{=} 0 + 60\,x^3
    \] Since \(x^3\) is only zero at \(x=0\) the requirement is fulfilled and therefore this function cannot be appropriate stream function.

    Contributors and Attributions

    • 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.