# Math Help - PDE: Filling in missing steps.

1. ## PDE: Filling in missing steps.

Hi everyone,

I'm a new poster! I'm studying biomedical engineering and we're currently looking at fluid mechanics. In solving a particular problem I'm having trouble following the worked solutions, which I have uploaded as an image file below.

Your patient help would be most appreciated!

Thanks.

2. Originally Posted by NZJaguar
Hi everyone,

I'm a new poster! I'm studying biomedical engineering and we're currently looking at fluid mechanics. In solving a particular problem I'm having trouble following the worked solutions, which I have uploaded as an image file below.

Your patient help would be most appreciated!

Thanks.

Which part? The first step or the second?

3. Both of them. I suspect there's something sneaky going on when expanding those brackets. As for turning it into an ODE I'm at a loss. Thanks.

4. Anyone? Danny don't leave me now!

Or even if any of you think that it doesn't make sense that would be helpful! =)

5. You have $\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial v_\theta}{\partial r}\right)- \frac{V_\theta}{r^2}= 0$

Now, by the product rule, $\frac{\partial}{\partial r}\left(r\frac{\partial v}{\partial r}\right)= \frac{\partial r}{\partial r}\frac{\partial v_\theta}{\partial r}+ r \frac{\partial^2 v_\theta}{\partial r^2}$ $= r\frac{\partial^2 v_\theta}{\partial r^2}+ \frac{\partial v_\theta}{\partial r}$.

$\frac{1}{r}$ times that gives
$\frac{\partial^2 v_\theta}{\partial r^2}+ \frac{1}{r}\frac{\partial v_\theta}{\partial r}$

Subtracting $\frac{v_\theta}{r^2}$ from that gives the second line: $\frac{\partial^2 v_\theta}{\partial r^2}+ \frac{1}{r}\frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r^2}$

To see that this is the same as the third line, use the quotient rule:
$\frac{\partial}{\partial r}\frac{v_\theta}{r}= \frac{r\frac{\partial v_\theta}{\partial r}- v_\theta\frac{\partial r}{\partial r}}{r^2}$ $= \frac{1}{r}\frac{\partial v_\theta}{\partial r}- \frac{v_\theta}{r^2}$
and, of course, if we know that $v_\theta$ is a function of r only, we can change the partial derivatives symbols to common derivatives: $\partial$ to d.