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Jupyter notebook NSE_2D_cylindrical.ipynb
Project: NSE
Path: NSE_2D_cylindrical.ipynb
Views: 39Kernel: Python 2 (SageMath)
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show that the continuity equation is fulfilled automatically:
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True
the navier-stokes-equations:
combine first and third equation by cross-differentiation and subtraction.
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mu*Derivative(Psi(r, z, t), r, r, r, r)/r + 2*mu*Derivative(Psi(r, z, t), r, r, z, z)/r + mu*Derivative(Psi(r, z, t), z, z, z, z)/r - 2*mu*Derivative(Psi(r, z, t), r, r, r)/r**2 - 2*mu*Derivative(Psi(r, z, t), r, z, z)/r**2 + 3*mu*Derivative(Psi(r, z, t), r, r)/r**3 - 3*mu*Derivative(Psi(r, z, t), r)/r**4 - rho*Derivative(Psi(r, z, t), r, r, t)/r - rho*Derivative(Psi(r, z, t), t, z, z)/r - rho*Derivative(Psi(r, z, t), r)*Derivative(Psi(r, z, t), r, r, z)/r**2 - rho*Derivative(Psi(r, z, t), r)*Derivative(Psi(r, z, t), z, z, z)/r**2 + rho*Derivative(Psi(r, z, t), z)*Derivative(Psi(r, z, t), r, r, r)/r**2 + rho*Derivative(Psi(r, z, t), z)*Derivative(Psi(r, z, t), r, z, z)/r**2 + rho*Derivative(Psi(r, z, t), r, t)/r**2 - 2*rho*Gamma(r, z, t)*Derivative(Gamma(r, z, t), z)/r**3 + rho*Derivative(Psi(r, z, t), r)*Derivative(Psi(r, z, t), r, z)/r**3 - 3*rho*Derivative(Psi(r, z, t), z)*Derivative(Psi(r, z, t), r, r)/r**3 - 2*rho*Derivative(Psi(r, z, t), z)*Derivative(Psi(r, z, t), z, z)/r**3 + 3*rho*Derivative(Psi(r, z, t), r)*Derivative(Psi(r, z, t), z)/r**4
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True
Equation (A.2) from Lopez1998:
How it is published:
corrected:
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die korrigierte "4" schlägt sich in den Vorzeichen von nieder
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show that Lopez' equation is correct:
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True
second equation:
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\mu \left(- \frac{\partial^{2}}{\partial r^{2}} \Gamma{\left (r,z,t \right )} - \frac{\partial^{2}}{\partial z^{2}} \Gamma{\left (r,z,t \right )} + \frac{1}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )}\right) + \rho \left(\frac{\partial}{\partial t} \Gamma{\left (r,z,t \right )} - \frac{1}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} + \frac{1}{r} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right)
solution from sympy (copied from cell above):
solution from [Lopez1998] eq. A.1:
mit
third equation (subsitution of second whirl entry)
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