I need to solve system of PDE-s:
\(\displaystyle \dfrac{2}{r} \dfrac{\partial (r V_0 p_0)}{\partial r} + \dfrac{\partial (u_0 p_0)}{\partial z} =0\)
\(\displaystyle u_0 \dfrac{\partial u_0}{\partial z}=\dfrac{4}{3} \beta \dfrac{\partial^2 u_0}{\partial r^2} -\dfrac{\partial p_0}{\partial z} +\dfrac{4 \beta}{r}\dfrac{\partial u_0}{\partial r}\)
\(\displaystyle \dfrac{\partial p_0}{\partial r}=0\)
\(\displaystyle u_0 |_{r=1}=0\), \(\displaystyle p_0|_{z=0}=p_{0i}\), \(\displaystyle p_0|_{z=1} = 1 \)
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 ( \(\displaystyle r(z)=r_i-z(r_1-1)\) ), 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?
\(\displaystyle \dfrac{2}{r} \dfrac{\partial (r V_0 p_0)}{\partial r} + \dfrac{\partial (u_0 p_0)}{\partial z} =0\)
\(\displaystyle u_0 \dfrac{\partial u_0}{\partial z}=\dfrac{4}{3} \beta \dfrac{\partial^2 u_0}{\partial r^2} -\dfrac{\partial p_0}{\partial z} +\dfrac{4 \beta}{r}\dfrac{\partial u_0}{\partial r}\)
\(\displaystyle \dfrac{\partial p_0}{\partial r}=0\)
\(\displaystyle u_0 |_{r=1}=0\), \(\displaystyle p_0|_{z=0}=p_{0i}\), \(\displaystyle p_0|_{z=1} = 1 \)
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 ( \(\displaystyle r(z)=r_i-z(r_1-1)\) ), 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?
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