# Thread: Formulating the BV problem

1. ## Formulating the BV problem

In a rod length L with insulated lateral surface and thermal constants $\displaystyle c, \ \rho, \ \kappa, \ K$ heat is generated at a rate uniformly proportional to the temperature, that is, at a rate $\displaystyle ru(x,t)$ per unit volume per unit time, where r is a constant and $\displaystyle u(x,t)$ is the temperature function of the rod. The ends of the rod are maintained at temperature 0 and initially the rod has uniform temperature 1.

Formulate an initial-BV problem for determining $\displaystyle u(x,t)$

$\displaystyle \text{D.E.}: \ u_t=ru_{xx}$
$\displaystyle \displaystyle\text{B.C.}=\begin{cases}u(0,t)=0\\u( L,t)=0\end{cases}$
$\displaystyle \text{I.C.}: \ u(x,0)=1$

I am not sure what to do with the r so I just put it in front of u_{xx}. Is that correct? Also, I need a K, c, and rho somewhere but not sure where and why.

2. r does belong there. It is known as Thermal Diffusivity.

It relates thermal conductivity and volumetric heat capacity together. It specifically shows how conductivity changes in comparison to the thermal bulk. As to why it belongs there, I don't know. Maybe seeing the equation will help you figure it out.

$\displaystyle r = \frac{K}{\rho c_p}$

3. Originally Posted by Aryth
r does belong there. It is known as Thermal Diffusivity.

It relates thermal conductivity and volumetric heat capacity together. It specifically shows how conductivity changes in comparison to the thermal bulk. As to why it belongs there, I don't know. Maybe seeing the equation will help you figure it out.

$\displaystyle r = \frac{K}{\rho c_p}$
But by making that substitution, I will lose r. In the final solution, there is an r, rho, K, and c.

4. The DE is

$\displaystyle \displaystyle\text{D.E.}: \ u_t=ku_{xx}+\frac{r}{c\rho}u$