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Thread: Runge Kutta shooting with partial differential equation

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    Runge Kutta shooting with partial differential equation

    I need to solve system of PDE-s:


    [tex]\dfrac{2}{r} \dfrac{\partial (r V_0 p_0)}{\partial r} + \dfrac{\partial (u_0 p_0)}{\partial z} =0[/tex]


    [tex]u_0 \dfrac{\partial u_0}{\partial z}=\dfrac{4}{3} \beta \dfrac{\partial^2 u_0}{\partial r^2} -\dfrac{\partial p_0}{\partial z} +\dfrac{4 \beta}{r}\dfrac{\partial u_0}{\partial r}[/tex]

    [tex]\dfrac{\partial p_0}{\partial r}=0[/tex]


    [tex]u_0 |_{r=1}=0[/tex], [tex]p_0|_{z=0}=p_{0i}[/tex], [tex]p_0|_{z=1} = 1 [/tex]


    where u_0, V_0, p_0 are dimensionless velocity in z axis, dimensionless radial velocity, and dimensionless pressure. z=0 is inlet and z=1 is outlet of this circular microtube with linear change of geometry ( [tex]r(z)=r_i-z(r_1-1)[/tex] ), where r is radial coordinate.


    I need to solve this system to get velocities and pressure, but I dont know how.
    With Runge Kutta shooting methods I have different differentials, and couple of them in equations, so how to calculate these values for unknown boundary conditions?


    And how to solve these eqautions later?
    Last edited by mmm4444bot; 02-15-2018 at 02:56 AM. Reason: fixed LaTex issues

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