Is the author integrating constants?

[Peter Schles]

Dear Sirs,

I am currently calculating a velocity profile of an annular flow. Unfortunatelly I am not understanding the following step:

http://[url=https://postimg.org/image/vl256ffhj/][ATTACH=full]200119[/ATTACH]

That seems the author had integrated the R constant. And remains the question: why had R been realocated into the ln´s parenthesys?

Thanks,
Peter

 

[marcusl]

The first term R*r/R is obvious so I assume that your question regards the second term. Make a change of variables u = r/R, so that dr = R du. This R is pulled out in front making R^2, while int{du/u} becomes ln(u) = ln(r/R). There's no integration of constants.

 

[vanhees71]

No, he integrates rightly over $r$. How do you come to a different conclusion. Note that
$$\frac{\mathrm{d}}{\mathrm{d} r} \ln(r/R)=\frac{1}{r}$$.
Rightly he avoids a dimensionful logarithm by introducing an arbitrary constant. You need initial/boundary conditions anyway to fix the integration constant $C_2$. So that's the correct general solution of the ODE (2.4-5).

 

[Ssnow]

Hi, before the integration you can multiply and divide by $R$, one of this remain outside the parentesis so the $R^2$, after observe that $\frac{1}{R}\left(\frac{R}{r}\right)$ is $\frac{d}{dr}\ln{\frac{r}{R}}$. Yes, here $R$ is trated as constant and $C_{2}$ is the constant of integration that remains multiplied by the constant term in front of the bracket.