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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 ri and i indicates one of the boundaries. Type 1. Prescribed temperature (Dirichlet condition):

T(ri, t ) = fi(ri, t )    

Type 2. Prescribed heat flux (Neumann condition):

$\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right.$k$\displaystyle {\frac{\partial T}{\partial n_{i}}}$ $\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{}$ = fi(ri, t )    

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

$\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right.$k$\displaystyle {\frac{\partial T}{\partial n_{i}}}$ $\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{}$ + hiT(ri, t ) = fi(ri, t )    

Here hi is the heat transfer coefficient and specified function fi is usually equal to hiT$\scriptstyle \infty$ where T$\scriptstyle \infty$ is a fluid temperature.
Type 4. Thin, high-conductivity film at the body surface:

$\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right.$k$\displaystyle {\frac{\partial T}{\partial n_{i}}}$ $\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{}$ = fi(ri, t ) - ($\displaystyle \rho$cb)i$\displaystyle \left.\vphantom{ \frac{\partial T}{\partial t}}\right.$$\displaystyle {\frac{\partial T}{\partial t}}$ $\displaystyle \left.\vphantom{ \frac{\partial T}{\partial t}}\right\vert _{\mathbf{r}_{i}}^{}$    

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:

$\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right.$k$\displaystyle {\frac{\partial T}{\partial n_{i}}}$ $\displaystyle \left.\vphantom{ k\frac{\partial T}{\partial n_{i}}}\right\vert _{\mathbf{r}_{i}}^{}$ + hiT(ri, t ) = fi(ri, t ) - ($\displaystyle \rho$cb)i$\displaystyle \left.\vphantom{ \frac{\partial T}{\partial t}}\right.$$\displaystyle {\frac{\partial T}{\partial t}}$ $\displaystyle \left.\vphantom{ \frac{\partial T}{\partial t}}\right\vert _{\mathbf{r}_{i}}^{}$    

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.
next up previous
Next: GF Numbering System. Up: Organization of the GF Previous: Organization of the GF
Kevin D. Cole
2003-07-21