Solving Navier Stokes & energy equations with different coordinates

[shevir1]

Hi all I am conducting a fluid analysis on water flowing through a subsea pipe.

Having used navier stokes equation, i derived the equation for velocity in the r-direction (using cylindrical coordinates.

But when initially solving the energy equation to determine temperature distribution I have used the cartesian coordinates, x & y

From the picture I have attached am I correct in proceeding to solve the energy equation, if I were to just differentiate my velocity equation result and substitute back into the energy equation?
From my knowledge the Y direction in cartesian corresponds with the R direction in the cyclindrical hence my reasoning for proceeding this way.

 

[MexChemE]

From my knowledge the Y direction in cartesian corresponds with the R direction in the cyclindrical hence my reasoning for proceeding this way.

These are the actual relationships between cylindrical and cartesian coordinates.
$$x = r \cos \theta$$
$$y = r \sin \theta$$
$$r^2 = x^2 + y^2$$
Why don't you just use the energy equation in cylindrical coordinates? That's the easiest way to proceed.
$$\frac{1}{r} \frac{d}{dr} \left( r \frac{dT}{dr} \right) = - \frac{\mu}{k} \left( \frac{dv_z}{dr} \right)^2$$
However, this model is only valid if the temperature is a function of radius only. If it also depends on z, your energy equation becomes a PDE.

 

[shevir1]

yes this is what i assumed.
of course it is better to work in just one coordinate system.

thanks