Let a body $\Omega_{1}$ (space shuttle or turbine engine) be thermally insulated by a thin coating $\Omega_{2}$ of thickness $\delta$. Assume the outer boundary of the coating is subject to a high exterior temperature $H$. Let $u_{1}$ be the temperature function in $\Omega_{1}$ and $u_{2}$ be that in $\Omega_{2}$ that satisfies:
$u_{1} = u_{2}$ on $\partial \Omega_{1}$
And we can get:
$-k_{1}\frac {\partial u_{1}} {\partial \mathbf{n}} = -k_{2}\frac {\partial u_{2}} {\partial \mathbf{n}} \tag{2}$
where $n$ is the unit outer normal vector field of $\partial \Omega_{1}$.
Let the thermal conductivities of the body and the coating be $k_{1}$ and $k_{2}$, respectively.
Prove that on the boundary $\partial \Omega_{1}$ of the body, we have approximately Robin boundary condition:
$k_{1}\frac {\partial u_{1}} {\partial \mathbf{n}} + \frac {k_{2}} {\delta} (u_{1} − H) = 0 \tag{3}$
where $n$ is the unit outer normal vector field of $\partial \Omega_{1}$.
There is a Hint but I don’t know how to use it:
To insulate the body well, what should be the scaling relationship of $k_{2}$ and $\delta$? Hint: start with $(2)$; fix a point $x$ on $\partial \Omega_{1}$, and define $f(τ ) = u_{2}(x + \tau \mathbf{n})$. Then perform a Taylor expansion of $f$ at $0$.
Equation $(3)$ is called the effective boundary condition; its significance is that with it we do not need to solve, analytically or numerically, the heat equation inside the coating–we just need to solve it inside the body with $(3)$ as the B.C.
Here is the image link:
