I Solving Navier Stokes & energy equations with different coordinates

shevir1
Messages
18
Reaction score
0
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.
 

Attachments

  • IMG_5878.JPG
    IMG_5878.JPG
    33.3 KB · Views: 540
Physics news on Phys.org
shevir1 said:
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.
 
yes this is what i assumed.
of course it is better to work in just one coordinate system.

thanks
 
There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...
Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
Back
Top