Heat Flow Out of a Cylindrical Surface

The rate of heat flow across a surface  
 drawn in a material is  
\[\int_S \mathbf{\nabla} \cdot \mathbf{n} dS\]
. Suppose we have a temperature distribution  
\[T(x,y,z)=x^2 +y^2\]
  throughout a cylinder  
\[x^2 +y^2 =1, -1 \leq z \leq 1\]

The total heat flow is the sum of the heat flows through the top, curved surface and bottom.
\[\begin{equation} \begin{aligned} \int_S (\mathbf{\nabla} T) \cdot \mathbf{n} dS &= \int_{S_{TOP}} (\mathbf{\nabla} T) \cdot \mathbf{n} dS_{TOP} \\ &+ \int_{S_{CURVED SURFACE}} (\mathbf{\nabla} T) \cdot \mathbf{n} dS_{CURVED SURFACE} + \int_{S_{BOTTOM}} (\mathbf{\nabla} T) \cdot \mathbf{n} dS_{BOTTOM} \end{aligned} \end{equation}\]

\[T(x,y,z)=x^2 +y^2 \rightarrow (\mathbf{\nabla} T) = 2x \mathbf{i} +2y \mathbf{j} \]

Fro the top and bottom surfaces  
\[\mathbf{n} = pm \mathbf{k}\]
  respectively, then  
\[(\mathbf{\nabla} T) \cdot \mathbf{n} = (2x \mathbf{i} _ 2y \mathbf{j} ) \cdot pm \mathbf{k}=0 \]
  and the only contribution to the integral is from the curved surface. For the curved surface  
\[\mathbf{n} = x \mathbf{i} + y \mathbf{j}\]
\[|\mathbf{n}| =x^2 +y^2=1\]

\[\begin{equation} \begin{aligned} \int_S (\mathbf{\nabla} T) \cdot \mathbf{n} dS &= \int_{S_{CURVED SURFACE}} (\mathbf{\nabla} T) \cdot \mathbf{n} dS_{CURVED SURFACE} \\ &= \int_{S_{CURVED SURFACE}} (2x \mathbf{i} + 2y \mathbf{j}) \cdot (x \mathbf{i} + y \mathbf{j}) dS_{CURVED SURFACE} \\ &= \int_{S_{CURVED SURFACE}} 2x^2 + 2y^2 dS_{CURVED SURFACE} \end{aligned} \end{equation}\]

Now change to cylindrical polar coordinates. The integral becomes
\[\begin{equation} \begin{aligned} \int^1_{-1} \int^{2 \pi}_0 2r^2 r d \theta dz &= 4 \pi 1^3 \int^1_{-1} dz \\ &= 2 \pi r^3 \times 2 \\ &= 8 \pi\end{aligned} \end{equation}\]

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