SharedNSE_2D_cylindrical.ipynbOpen in CoCalc
Jupyter notebook NSE_2D_cylindrical.ipynb
from sympy import *
#init_printing(use_latex='mathjax')
init_printing()

## variables
r, phi, z, t = symbols("r phi z t", real=True)
## functions
p, ur, uphi, uz, Psi, Gamma = symbols("p u_r u_phi u_z Psi Gamma", cls=Function)
p     = Function('p')(r,z,t)
Psi   = Function('Psi')(r,z,t)
Gamma = Function('Gamma')(r,z,t)
ur    = Function('u_r')(r,z,t)
uphi  = Function('u_phi')(r,z,t)
uz    = Function('u_z')(r,z,t)
## constants
rho, mu, gz  = symbols('rho mu g_z', integer=True)


show that the continuity equation is fulfilled automatically:¶

conti = Function('conti')
conti = 1/r*ur + diff(ur,r) + diff(uz,z)
conti

$\frac{\partial}{\partial r} \operatorname{u_{r}}{\left (r,z,t \right )} + \frac{\partial}{\partial z} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{1}{r} \operatorname{u_{r}}{\left (r,z,t \right )}$
conti = conti.replace(ur,-1/r*diff(Psi,z)).replace(uphi,1/r*Gamma).replace(uz,1/r*diff(Psi,r))
conti

$\frac{\partial}{\partial z}\left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) + \frac{\partial}{\partial r}\left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) - \frac{1}{r^{2}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}$
conti.doit() == 0

True

the navier-stokes-equations:¶

combine first and third equation by cross-differentiation and subtraction.¶

eqr = rho*(diff(ur,t)+ur*diff(ur,r)+uz*diff(ur,z)-uphi**2/r) - mu*(1/r*diff((r*diff(ur,r)),r)+diff(ur,z,z)-ur/r**2) + diff(p,r)
eqr

$- \mu \left(\frac{\partial^{2}}{\partial z^{2}} \operatorname{u_{r}}{\left (r,z,t \right )} + \frac{1}{r} \left(r \frac{\partial^{2}}{\partial r^{2}} \operatorname{u_{r}}{\left (r,z,t \right )} + \frac{\partial}{\partial r} \operatorname{u_{r}}{\left (r,z,t \right )}\right) - \frac{1}{r^{2}} \operatorname{u_{r}}{\left (r,z,t \right )}\right) + \rho \left(\operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial}{\partial r} \operatorname{u_{r}}{\left (r,z,t \right )} + \operatorname{u_{z}}{\left (r,z,t \right )} \frac{\partial}{\partial z} \operatorname{u_{r}}{\left (r,z,t \right )} + \frac{\partial}{\partial t} \operatorname{u_{r}}{\left (r,z,t \right )} - \frac{1}{r} \operatorname{u_{\phi}}^{2}{\left (r,z,t \right )}\right) + \frac{\partial}{\partial r} p{\left (r,z,t \right )}$
eqz = rho*(diff(uz,t)+ur*diff(uz,r)+uz*diff(uz,z)) - mu*(1/r*diff(r*diff(uz,r),r)+diff(uz,z,z))  + diff(p,z) - rho*gz
eqz = eqz.collect(rho)
eqz

$- \mu \left(\frac{\partial^{2}}{\partial z^{2}} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{1}{r} \left(r \frac{\partial^{2}}{\partial r^{2}} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{\partial}{\partial r} \operatorname{u_{z}}{\left (r,z,t \right )}\right)\right) + \rho \left(- g_{z} + \operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial}{\partial r} \operatorname{u_{z}}{\left (r,z,t \right )} + \operatorname{u_{z}}{\left (r,z,t \right )} \frac{\partial}{\partial z} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{\partial}{\partial t} \operatorname{u_{z}}{\left (r,z,t \right )}\right) + \frac{\partial}{\partial z} p{\left (r,z,t \right )}$
eqrz = diff(eqr,z)-diff(eqz,r)
eqrz = eqrz.collect(mu).collect(rho)
eqrz

$\mu \left(- \frac{\partial^{3}}{\partial z^{3}} \operatorname{u_{r}}{\left (r,z,t \right )} + \frac{\partial^{3}}{\partial r\partial z^{2}} \operatorname{u_{z}}{\left (r,z,t \right )} - \frac{1}{r} \left(r \frac{\partial^{3}}{\partial r^{2}\partial z} \operatorname{u_{r}}{\left (r,z,t \right )} + \frac{\partial^{2}}{\partial r\partial z} \operatorname{u_{r}}{\left (r,z,t \right )}\right) + \frac{1}{r} \left(r \frac{\partial^{3}}{\partial r^{3}} \operatorname{u_{z}}{\left (r,z,t \right )} + 2 \frac{\partial^{2}}{\partial r^{2}} \operatorname{u_{z}}{\left (r,z,t \right )}\right) - \frac{1}{r^{2}} \left(r \frac{\partial^{2}}{\partial r^{2}} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{\partial}{\partial r} \operatorname{u_{z}}{\left (r,z,t \right )}\right) + \frac{1}{r^{2}} \frac{\partial}{\partial z} \operatorname{u_{r}}{\left (r,z,t \right )}\right) + \rho \left(\operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r\partial z} \operatorname{u_{r}}{\left (r,z,t \right )} - \operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r^{2}} \operatorname{u_{z}}{\left (r,z,t \right )} + \operatorname{u_{z}}{\left (r,z,t \right )} \frac{\partial^{2}}{\partial z^{2}} \operatorname{u_{r}}{\left (r,z,t \right )} - \operatorname{u_{z}}{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r\partial z} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{\partial}{\partial r} \operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial}{\partial z} \operatorname{u_{r}}{\left (r,z,t \right )} - \frac{\partial}{\partial r} \operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial}{\partial r} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{\partial}{\partial z} \operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial}{\partial z} \operatorname{u_{z}}{\left (r,z,t \right )} - \frac{\partial}{\partial r} \operatorname{u_{z}}{\left (r,z,t \right )} \frac{\partial}{\partial z} \operatorname{u_{z}}{\left (r,z,t \right )} + \frac{\partial^{2}}{\partial t\partial z} \operatorname{u_{r}}{\left (r,z,t \right )} - \frac{\partial^{2}}{\partial r\partial t} \operatorname{u_{z}}{\left (r,z,t \right )} - \frac{2}{r} \operatorname{u_{\phi}}{\left (r,z,t \right )} \frac{\partial}{\partial z} \operatorname{u_{\phi}}{\left (r,z,t \right )}\right)$
eqrz = eqrz.replace(ur,-1/r*diff(Psi,z)).replace(uphi,1/r*Gamma).replace(uz,1/r*diff(Psi,r))
eqrz

$\mu \left(\frac{\partial^{3}}{\partial r\partial z^{2}} \left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) - \frac{\partial^{3}}{\partial z^{3}} \left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) + \frac{1}{r} \left(r \frac{\partial^{3}}{\partial r^{3}} \left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) + 2 \frac{\partial^{2}}{\partial r^{2}} \left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right)\right) - \frac{1}{r} \left(r \frac{\partial^{3}}{\partial r^{2}\partial z} \left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) + \frac{\partial^{2}}{\partial r\partial z} \left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right)\right) - \frac{1}{r^{2}} \left(r \frac{\partial^{2}}{\partial r^{2}} \left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) + \frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right)\right) + \frac{1}{r^{2}} \frac{\partial}{\partial z}\left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right)\right) + \rho \left(- \frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) \frac{\partial}{\partial z}\left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) - \frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) \frac{\partial}{\partial r}\left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) + \frac{\partial}{\partial z}\left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) \frac{\partial}{\partial z}\left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) + \frac{\partial}{\partial r}\left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) \frac{\partial}{\partial z}\left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) - \frac{\partial^{2}}{\partial r\partial t} \left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) + \frac{\partial^{2}}{\partial t\partial z} \left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) - \frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r\partial z} \left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) + \frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial z^{2}} \left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) + \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r^{2}} \left(\frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) - \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r\partial z} \left(- \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right) - \frac{2}{r^{2}} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z}\left(\frac{1}{r} \Gamma{\left (r,z,t \right )}\right)\right)$
eqrz = eqrz.doit().expand()
eqrz

$\frac{\mu}{r} \frac{\partial^{4}}{\partial r^{4}} \Psi{\left (r,z,t \right )} + \frac{2 \mu}{r} \frac{\partial^{4}}{\partial r^{2}\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{\mu}{r} \frac{\partial^{4}}{\partial z^{4}} \Psi{\left (r,z,t \right )} - \frac{2 \mu}{r^{2}} \frac{\partial^{3}}{\partial r^{3}} \Psi{\left (r,z,t \right )} - \frac{2 \mu}{r^{2}} \frac{\partial^{3}}{\partial r\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{3 \mu}{r^{3}} \frac{\partial^{2}}{\partial r^{2}} \Psi{\left (r,z,t \right )} - \frac{3 \mu}{r^{4}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} - \frac{\rho}{r} \frac{\partial^{3}}{\partial r^{2}\partial t} \Psi{\left (r,z,t \right )} - \frac{\rho}{r} \frac{\partial^{3}}{\partial t\partial z^{2}} \Psi{\left (r,z,t \right )} - \frac{\rho}{r^{2}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r^{2}\partial z} \Psi{\left (r,z,t \right )} - \frac{\rho}{r^{2}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial z^{3}} \Psi{\left (r,z,t \right )} + \frac{\rho}{r^{2}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r^{3}} \Psi{\left (r,z,t \right )} + \frac{\rho}{r^{2}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{\rho}{r^{2}} \frac{\partial^{2}}{\partial r\partial t} \Psi{\left (r,z,t \right )} - \frac{2 \rho}{r^{3}} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} + \frac{\rho}{r^{3}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r\partial z} \Psi{\left (r,z,t \right )} - \frac{3 \rho}{r^{3}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r^{2}} \Psi{\left (r,z,t \right )} - \frac{2 \rho}{r^{3}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{3 \rho}{r^{4}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}$
print(eqrz)

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
eqrz2 = mu*(Derivative(Psi, r, r, r, r)/r + 2*Derivative(Psi, r, r, z, z)/r + Derivative(Psi, z, z, z, z)/r - 2*Derivative(Psi, r, r, r)/r**2 - 2*Derivative(Psi, r, z, z)/r**2 + 3*Derivative(Psi, r, r)/r**3 - 3*Derivative(Psi, r)/r**4) + rho*(-Derivative(Psi, r, r, t)/r - Derivative(Psi, t, z, z)/r - Derivative(Psi, r)*Derivative(Psi, r, r, z)/r**2 - Derivative(Psi, r)*Derivative(Psi, z, z, z)/r**2 + Derivative(Psi, z)*Derivative(Psi, r, r, r)/r**2 + Derivative(Psi, z)*Derivative(Psi, r, z, z)/r**2 + Derivative(Psi, r, t)/r**2 - 2*Gamma*Derivative(Gamma, z)/r**3 + Derivative(Psi, r)*Derivative(Psi, r, z)/r**3 - 3*Derivative(Psi, z)*Derivative(Psi, r, r)/r**3 - 2*Derivative(Psi, z)*Derivative(Psi, z, z)/r**3 + 3*Derivative(Psi, r)*Derivative(Psi, z)/r**4)
eqrz2

$\mu \left(\frac{1}{r} \frac{\partial^{4}}{\partial r^{4}} \Psi{\left (r,z,t \right )} + \frac{2}{r} \frac{\partial^{4}}{\partial r^{2}\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{1}{r} \frac{\partial^{4}}{\partial z^{4}} \Psi{\left (r,z,t \right )} - \frac{2}{r^{2}} \frac{\partial^{3}}{\partial r^{3}} \Psi{\left (r,z,t \right )} - \frac{2}{r^{2}} \frac{\partial^{3}}{\partial r\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{3}{r^{3}} \frac{\partial^{2}}{\partial r^{2}} \Psi{\left (r,z,t \right )} - \frac{3}{r^{4}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right) + \rho \left(- \frac{1}{r} \frac{\partial^{3}}{\partial r^{2}\partial t} \Psi{\left (r,z,t \right )} - \frac{1}{r} \frac{\partial^{3}}{\partial t\partial z^{2}} \Psi{\left (r,z,t \right )} - \frac{1}{r^{2}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r^{2}\partial z} \Psi{\left (r,z,t \right )} - \frac{1}{r^{2}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial z^{3}} \Psi{\left (r,z,t \right )} + \frac{1}{r^{2}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r^{3}} \Psi{\left (r,z,t \right )} + \frac{1}{r^{2}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial r\partial t} \Psi{\left (r,z,t \right )} - \frac{2}{r^{3}} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} + \frac{1}{r^{3}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r\partial z} \Psi{\left (r,z,t \right )} - \frac{3}{r^{3}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r^{2}} \Psi{\left (r,z,t \right )} - \frac{2}{r^{3}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{3}{r^{4}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right)$
# show that the manual collect() of mu and rho is correct:
(eqrz - eqrz2).doit().expand() == 0

True

Equation (A.2) from Lopez1998:

How it is published: $D(\eta/r) = \frac{1}{\operatorname{Re}} \left[ \nabla^2(\eta/r) + \frac{2}{r}(\eta/r)_r \right] + (\Gamma^2/r^4)_z$

corrected: $D(\eta/r) = \frac{1}{\operatorname{Re}} \left[ \nabla^2(\eta/r) + \frac{{\color{red}4}}{r}(\eta/r)_r \right] + (\Gamma^2/r^4)_z$

eta = Function('eta')(r,z,t)
eqrz3 = diff(eta/r,t)-1/r*diff(Psi,z)*diff(eta/r,r)+1/r*diff(Psi,r)*diff(eta/r,z)-diff(Gamma**2/r**4,z)-mu/rho*(diff(eta/r,r,r)-1/r*diff(eta/r,r)+diff(eta/r,z,z)+4/r*diff(eta/r,r))
(eqrz3*r).expand().collect(mu/rho)

$\frac{\mu}{\rho} \left(- \frac{\partial^{2}}{\partial r^{2}} \eta{\left (r,z,t \right )} - \frac{\partial^{2}}{\partial z^{2}} \eta{\left (r,z,t \right )} - \frac{1}{r} \frac{\partial}{\partial r} \eta{\left (r,z,t \right )} + \frac{1}{r^{2}} \eta{\left (r,z,t \right )}\right) + \frac{\partial}{\partial t} \eta{\left (r,z,t \right )} + \frac{1}{r} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial}{\partial z} \eta{\left (r,z,t \right )} - \frac{1}{r} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial}{\partial r} \eta{\left (r,z,t \right )} + \frac{1}{r^{2}} \eta{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} - \frac{2}{r^{3}} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )}$

die korrigierte "4" schlägt sich in den Vorzeichen von $-\frac{1}{r}\frac{\partial \eta}{\partial r} + \frac{1}{r^2}\eta$ nieder

eqrz3 = eqrz3.replace(eta,-1/r*(diff(Psi,r,r)-1/r*diff(Psi,r)+diff(Psi,z,z))).doit().expand()
eqrz3

$\frac{\mu}{r^{2} \rho} \frac{\partial^{4}}{\partial r^{4}} \Psi{\left (r,z,t \right )} + \frac{2 \mu}{r^{2} \rho} \frac{\partial^{4}}{\partial r^{2}\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{\mu}{r^{2} \rho} \frac{\partial^{4}}{\partial z^{4}} \Psi{\left (r,z,t \right )} - \frac{2 \mu}{r^{3} \rho} \frac{\partial^{3}}{\partial r^{3}} \Psi{\left (r,z,t \right )} - \frac{2 \mu}{r^{3} \rho} \frac{\partial^{3}}{\partial r\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{3 \mu}{r^{4} \rho} \frac{\partial^{2}}{\partial r^{2}} \Psi{\left (r,z,t \right )} - \frac{3 \mu}{r^{5} \rho} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} - \frac{1}{r^{2}} \frac{\partial^{3}}{\partial r^{2}\partial t} \Psi{\left (r,z,t \right )} - \frac{1}{r^{2}} \frac{\partial^{3}}{\partial t\partial z^{2}} \Psi{\left (r,z,t \right )} - \frac{1}{r^{3}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r^{2}\partial z} \Psi{\left (r,z,t \right )} - \frac{1}{r^{3}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial z^{3}} \Psi{\left (r,z,t \right )} + \frac{1}{r^{3}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r^{3}} \Psi{\left (r,z,t \right )} + \frac{1}{r^{3}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{3}}{\partial r\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{1}{r^{3}} \frac{\partial^{2}}{\partial r\partial t} \Psi{\left (r,z,t \right )} - \frac{2}{r^{4}} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} + \frac{1}{r^{4}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r\partial z} \Psi{\left (r,z,t \right )} - \frac{3}{r^{4}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial r^{2}} \Psi{\left (r,z,t \right )} - \frac{2}{r^{4}} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} \frac{\partial^{2}}{\partial z^{2}} \Psi{\left (r,z,t \right )} + \frac{3}{r^{5}} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}$

show that Lopez' equation is correct:

(eqrz-eqrz3*rho*r).expand().doit() == 0

True

second equation:¶

eqphi = rho*(diff(uphi,t)+ur*diff(uphi,r)+uz*diff(uphi,z)+ur*uphi/r) - mu*(1/r * diff(r*diff(uphi,r),r) + diff(uphi,z,z) - uphi/r**2)
eqphi

$- \mu \left(\frac{\partial^{2}}{\partial z^{2}} \operatorname{u_{\phi}}{\left (r,z,t \right )} + \frac{1}{r} \left(r \frac{\partial^{2}}{\partial r^{2}} \operatorname{u_{\phi}}{\left (r,z,t \right )} + \frac{\partial}{\partial r} \operatorname{u_{\phi}}{\left (r,z,t \right )}\right) - \frac{1}{r^{2}} \operatorname{u_{\phi}}{\left (r,z,t \right )}\right) + \rho \left(\operatorname{u_{r}}{\left (r,z,t \right )} \frac{\partial}{\partial r} \operatorname{u_{\phi}}{\left (r,z,t \right )} + \operatorname{u_{z}}{\left (r,z,t \right )} \frac{\partial}{\partial z} \operatorname{u_{\phi}}{\left (r,z,t \right )} + \frac{\partial}{\partial t} \operatorname{u_{\phi}}{\left (r,z,t \right )} + \frac{1}{r} \operatorname{u_{\phi}}{\left (r,z,t \right )} \operatorname{u_{r}}{\left (r,z,t \right )}\right)$
eqphi = eqphi.replace(ur,-1/r*diff(Psi,z)).replace(uphi,1/r*Gamma).replace(uz,1/r*diff(Psi,r))
eqphi.doit()

$- \mu \left(\frac{1}{r} \left(\frac{\partial^{2}}{\partial r^{2}} \Gamma{\left (r,z,t \right )} - \frac{1}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )} + \frac{1}{r^{2}} \Gamma{\left (r,z,t \right )}\right) + \frac{1}{r} \frac{\partial^{2}}{\partial z^{2}} \Gamma{\left (r,z,t \right )} - \frac{1}{r^{3}} \Gamma{\left (r,z,t \right )}\right) + \rho \left(- \frac{1}{r} \left(\frac{1}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )} - \frac{1}{r^{2}} \Gamma{\left (r,z,t \right )}\right) \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} + \frac{1}{r} \frac{\partial}{\partial t} \Gamma{\left (r,z,t \right )} + \frac{1}{r^{2}} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} - \frac{1}{r^{3}} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )}\right)$
eqphi2 = (eqphi*r).doit().expand()
eqphi2

$- \mu \frac{\partial^{2}}{\partial r^{2}} \Gamma{\left (r,z,t \right )} - \mu \frac{\partial^{2}}{\partial z^{2}} \Gamma{\left (r,z,t \right )} + \frac{\mu}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )} + \rho \frac{\partial}{\partial t} \Gamma{\left (r,z,t \right )} - \frac{\rho}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} + \frac{\rho}{r} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}$
eqphi3 = eqphi2.expand().collect(mu).collect(rho)
eqphi3

$\mu \left(- \frac{\partial^{2}}{\partial r^{2}} \Gamma{\left (r,z,t \right )} - \frac{\partial^{2}}{\partial z^{2}} \Gamma{\left (r,z,t \right )} + \frac{1}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )}\right) + \rho \left(\frac{\partial}{\partial t} \Gamma{\left (r,z,t \right )} - \frac{1}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} + \frac{1}{r} \frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )}\right)$
print(latex(eqphi3))

\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):

$\frac{\partial}{\partial t} \Gamma{\left (r,z,t \right )} - \frac{1}{r} \frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial z} \Psi{\left (r,z,t \right )} + \frac{1}{r}\frac{\partial}{\partial z} \Gamma{\left (r,z,t \right )} \frac{\partial}{\partial r} \Psi{\left (r,z,t \right )} = \frac{\mu}{\rho} \left(\frac{\partial^{2}}{\partial r^{2}} \Gamma{\left (r,z,t \right )} + \frac{\partial^{2}}{\partial z^{2}} \Gamma{\left (r,z,t \right )} - \frac{1}{r}\frac{\partial}{\partial r} \Gamma{\left (r,z,t \right )}\right)$

solution from [Lopez1998] eq. A.1:

$\left( \partial_t - \frac{1}{r}\Psi_z\partial_r + \frac{1}{r}\Psi_r\partial_z \right) \Gamma = \frac{1}{\operatorname{Re}} \left( \partial_r^2 - \frac{1}{r} \partial_r + \partial_z^2 \right) \Gamma$

mit $\frac{1}{\operatorname{Re}} = \frac{\mu}{\rho}$

$q.e.d.$

third equation (subsitution of second whirl entry) $\omega_\phi = \eta$¶