Nonhomogeneous Neumann Boundary Conditions for the 3D Heat Equation

Question

I have recently came accross an interesting boundary problem regarding the temperature content inside a cake in an oven. The solution assumes the boundary conditions to be fixed at the oven temperature and imposes homogeneous Dirichlet conditions by substracting the oven temperature from all temperatures in the cake. However, I don't think fixed boundary conditions would be physically accurate. Is there a way to define the boundary conditions so that it shows the outside layer of the cake changes temperature over time as the cake can't just be at the oven temperature. This seems to result in a nonhomogeneous Neumann boundary condition problem. Since nonhomogeneous conditions are a bit harder to solve, especially in 3D, is there a way of using a similar method as in the image I linked? Would substracting the oven temperature from the boundary conditions as well not imply insulated cake surfaces which would mean that the outside of the cake does not undergo any change whatsoever? I would really appreciate it if you helped me understand how I should approach the issue.

Answer 1

The heat-flow boundary conditions that you have suggested $$\partial_xu(t,0)=\partial_xu(t,L)=T_{\text{oven}}$$ Are wrong, for two reasons:

First, the dimensions are wrong. The LHS has dimensions of temperature/length, but the RHS has dimensions of temperature.

Second, these boundary conditions mean that the cake is heated from one side and cooled from the other side at an equal rate, meaning that it won't cook at all (as shown by Goncalo). This is because you forgot to take into account the orientation of the normal vectors.

The rate at which heat leaves the cake is given by $\boldsymbol n\cdot \nabla u$, where $\boldsymbol n$ is the unit outward normal to the cake's surface. Conversely, the rate at which heat enters the cake is given by $-\boldsymbol n\cdot \nabla u$. In one dimension, this unit outward normal is simply the number $+1$ on the right side of the cake and $-1$ on the left side. So, given our heat flow rate $\gamma$ , your BCs should be

$$-(+1)\cdot\partial_xu(t,L)=\gamma~~~(\text{right side}) \\ -(-1)\cdot \partial_x u(t,0)=\gamma~~~(\text{left side})$$

I.e $$\partial_x u(t,0)=-\gamma \\ \partial_xu(t,L)=\gamma$$

Answer 2

Too long for a comment:

A more realistic boundary condition for this problem is $$ \frac{\partial u}{\partial n}=-\alpha(u-T_0)\,\,\text{on}\,\,\partial\Omega, \tag{1} $$ where $\frac{\partial u}{\partial n}$ is the outward normal derivative of $u$ at the boundary $\partial\Omega$ of the region $\Omega$, and $\alpha$ is a positive constant. Repeating with this BC the calculation I did in my previous comment, it's possible to show that the average temperature of the body decreases over time if its surface temperature is higher than the ambient temperature $T_0$, and increases in the opposite case. Also, it's possible to transform $(1)$ into a homogeneous BC by subtracting $T_0$ from $u$.

(A remark: $(1)$ is known as Robin boundary condition.)