- #1

jackkk_gatz

- 45

- 1

- Homework Statement
- $$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial^2T}{\partial z^2}=0$$

$$\left.-k\frac{\partial T}{\partial r}\right\rvert_{r=R}=h[T(R,z)-T_{\infty}]$$

$$\left.k\frac{\partial T}{\partial z}\right\rvert_{z=H}+h[T(r,H)-T_{\infty}]=q_s$$

$$\left.-k\frac{\partial T}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial T}{\partial r}\right\rvert_{r=0}=0$$

$where \ T_0 \ ,T_{\infty} \ and \ q_s \ are \ constants$

- Relevant Equations
- -

I did a change of variable $$\theta(r,z) = T(r,z)-T_{\infty}$$ which resulted in

$$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial \theta}{\partial r})+\frac{\partial^2 \theta}{\partial z^2}=0$$

$$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=R}=h\theta$$

$$\left.k\frac{\partial \theta}{\partial z}\right\rvert_{z=H}+h\theta=q_s$$

$$\left.-k\frac{\partial \theta}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=0}=0$$

After that I proposed $$\theta(r,z)=v(r,z)+w(r,z)$$ where w(r,z) should must satisfy

$$\left.-k\frac{\partial w}{\partial r}\right\rvert_{r=R}=hw(R,z)$$

$$\left.k\frac{\partial w}{\partial z}\right\rvert_{z=H}+hw(r,H)=q_s$$

$$\left.-k\frac{\partial w}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial w}{\partial r}\right\rvert_{r=0}=0$$

I already tried interpolation, doesn't work. I don't know how w(r,z) should look like in order to satisfy the above equations. Is there an easier method?

In short, everything I have been trying has failed and I don't know what to do anymore, I have looked for books on PDEs, all the ones I have found deal with very simple cases, which are of no use to me. I have almost no experience solving this kind of equations to know what to do or to guess how w might look like

$$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial \theta}{\partial r})+\frac{\partial^2 \theta}{\partial z^2}=0$$

$$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=R}=h\theta$$

$$\left.k\frac{\partial \theta}{\partial z}\right\rvert_{z=H}+h\theta=q_s$$

$$\left.-k\frac{\partial \theta}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=0}=0$$

After that I proposed $$\theta(r,z)=v(r,z)+w(r,z)$$ where w(r,z) should must satisfy

$$\left.-k\frac{\partial w}{\partial r}\right\rvert_{r=R}=hw(R,z)$$

$$\left.k\frac{\partial w}{\partial z}\right\rvert_{z=H}+hw(r,H)=q_s$$

$$\left.-k\frac{\partial w}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial w}{\partial r}\right\rvert_{r=0}=0$$

I already tried interpolation, doesn't work. I don't know how w(r,z) should look like in order to satisfy the above equations. Is there an easier method?

In short, everything I have been trying has failed and I don't know what to do anymore, I have looked for books on PDEs, all the ones I have found deal with very simple cases, which are of no use to me. I have almost no experience solving this kind of equations to know what to do or to guess how w might look like

Last edited: