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Heat Equation, General Case

Consider the following general boundary-value problem with vector coordinate $\mathbf{r}$ :

\begin{eqnarray*}
\alpha \nabla ^{2}T+\frac{\alpha }{k}g(\mathbf{r},t)-\alpha m^...
...ac{\partial T}{\partial t}\right\vert _{\mathbf{r}_{i}};\;i=1,2.
\end{eqnarray*}

The general boundary condition represents five different boundary conditions (type 1 through 5) by suitable choice of boundary parameters $k_{i}=0$ or $k$; $%
h_{i}=0$ or $h$; $(\rho cb)_{i}=0$ or nonzero. $ $Here $(\rho cb)_{i}$ represents properties of a high conductivity surface film (density, specific heat, thickness) which is thin enough that there is a negligible temperature gradient across the film and negligible heat flux parallel to the surface inside the film.

The Green's Function Solution Equation for temperature $T(\mathbf{r},t)$ is given by:

\begin{eqnarray*}
&&T(\mathbf{r},t)=\int_{R}G(\mathbf{r},t \left\vert  \mathbf...
...{r}_{i}}\right] ds_{j}^{\prime } \;
\mbox{(b.c. of type 1 only)}
\end{eqnarray*}

This equation applies to any orthogonal coordinate system if the correct form for differential area $ds_{i}$ and differential volume $dv$ are used. See the table below for $ds_{i}$ and $dv$ for several body shapes in the rectangular, cylindrical, and spherical coordinate systems. Spatial derivative $\partial /\partial n_{i}$ denotes differentiation along the outward normal on surface $S_{i}$, $i=1,2,\ldots  ,s$where $s$ represents the number of boundary conditions. The number of boundary conditions $s$ only includes conditions at ``real'' boundaries; the number of ``real'' boundaries does not include the boundary at $x\rightarrow \infty
$ for a semi-infinite body, for example.

Table 1. Differential area and volume for the GF Solution Equation.

Body shape coordinates $ds_{i}^{\prime }$ $dv^{\prime }$
plate $x$ $^{\ast }1$ $dx^{\prime }$
rectangle $x,y$ $dx^{\prime }\mbox{ or }dy^{\prime }$ $dx^{\prime
}dy^{\prime }$
parallelpiped $x,y,z$ $\begin{array}{l}
dx^{\prime }dy^{\prime }\mbox{ or }dx^{\prime }dz^{\prime } \\
\mbox{ or }dy^{\prime }dz^{\prime }
\end{array}$ $dx^{\prime }dy^{\prime }dz^{\prime }$
infinite cylinder $r$ $2\pi r^{\prime }$ $2\pi r^{\prime }dr^{\prime}$
thin shell $\phi$ $^{\ast }\delta \mbox{ (shell thickness)}$ $\delta ad\phi ^{\prime }$
finite cylinder $r,z$ $^{\ast }2\pi r_{i}dz^{\prime }\mbox{ or }2\pi
r^{\prime }dr^{\prime }$ $2\pi r^{\prime }dr^{\prime }dz^{\prime }$
wedge $r,\phi$ $dr^{\prime }\mbox{ or }r_{i}d\phi ^{\prime }$ $r^{\prime }dr^{\prime }d\phi ^{\prime }$
sphere $r$ $^{\ast }4\pi r_{i}^{2}\mbox{ }$ $4\pi (r^{\prime
})^{2}dr^{\prime }$
$\ast \mbox{no integral on }S_{i}$      


next up previous
Next: Laplace and Helmholtz Equation Up: HEAT EQUATION (TRANSIENT CONDUCTION) Previous: Heat Equation, 1-D Rectangular
2004-01-31