Boundary Condition Types.

Five types of boundary conditions are defined at physical boundaries, and a ``zeroth'' type designates those cases with no physical boundaries. In the equations below the coordinate at the boundary is denoted r_i and i indicates one of the boundaries. Type 1. Prescribed temperature (Dirichlet condition):

Type 2. Prescribed heat flux (Neumann condition):

$$\left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right. {\frac{\partial T}{\partial n_{i}}} \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{}$$

Here n_i is the outward-facing normal vector on the body surface.
Type 3. Convective boundary condition (sometimes called the Robin condition):

$$\left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right. {\frac{\partial T}{\partial n_{i}}} \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{}$$

Here h_i is the heat transfer coefficient and specified function f_i is usually equal to h_iT_$\infty$ where T_$\infty$ is a fluid temperature.
Type 4. Thin, high-conductivity film at the body surface:

$$\left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right. {\frac{\partial T}{\partial n_{i}}} \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{} \rho \left.\vphantom{ \frac{\partial T}{\partial t}}\right. {\frac{\partial T}{\partial t}} \left.\vphantom{ \frac{\partial T}{\partial t}}\right\vert _{\mathbf{r}_{i}}^{}$$

Here product ($\rho$cb)_i are properties of the surface film (density, specific heat, and thickness), and the surface film must have a negligible temperature gradient across it (``lumped'').
Type 5. Thin, high-conductivity film at the body surface, with the addition of convection heat losses from the surface:

$$\left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right. {\frac{\partial T}{\partial n_{i}}} \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{} \rho \left.\vphantom{ \frac{\partial T}{\partial t}}\right. {\frac{\partial T}{\partial t}} \left.\vphantom{ \frac{\partial T}{\partial t}}\right\vert _{\mathbf{r}_{i}}^{}$$

Type 0. No physical boundary. The number 0 (zero) is used where there is no physical boundary, which arises in several body shapes. For example, a semi-infinite body has ``boundary condition'' of type 0 at x $\rightarrow$ $\infty$. Another ``boundary'' of type 0 occurs at the center of a solid cylinder (or sphere), for which the coordinate has a limiting value (r = 0) but there is no physical boundary.