I have two metal objects of different thickness and conductivity:
1) thicker, but poorly conductive. It's properties: $k_1, H$
2) thin and much better conductive. It's properties: $k_2, h$
Addition conditions: $k_1<<k_2$ , $H>>h$
We write the heat conduction equations for both zones: $${\frac {\partial u}{\partial t}}=k_1{\frac {\partial ^{2}u}{\partial x^{2}}}+k_1{\frac {\partial ^{2}u}{\partial y^{2}}}, 0\leq x\leq L, 0\leq y\leq H,\tag{*}\label{eq01}$$ $${\frac {\partial u}{\partial t}}=k_2{\frac {\partial ^{2}u}{\partial x^{2}}}+k_2{\frac {\partial ^{2}u}{\partial y^{2}}}, 0\leq x\leq L, -h\leq y\leq 0.\tag{**}\label{eq02}$$ Initial conditions $u(x,y,t=0) = u_0$ And following conditions on different boundaries: $$\left.\frac{\partial u}{\partial x}\right|_{x=0,y,t} = A=const, -h \leq y \leq 0 $$ $$\left.\frac{\partial u}{\partial x}\right|_{x=0,y,t} = 0, 0 \leq y \leq H $$
$$\left.\frac{\partial u}{\partial x}\right|_{x=L,y,t} = 0, -h \leq y \leq H $$ $$\left.\frac{\partial u}{\partial y}\right|_{x,y=-h,t} =\left.\frac{\partial u}{\partial y}\right|_{x,y=H,t} = 0, 0 \leq x \leq L $$
$$\left.u\right|_{x,y=-0,t} = \left.u\right|_{x,y=+0,t}, 0 \leq x \leq L $$
$$\left.k_2\frac{\partial u}{\partial y}\right|_{x,y=-0,t} =\left.k_1\frac{\partial u}{\partial y}\right|_{x,y=+0,t} = 0, 0 \leq x \leq L $$
In several books I find reduction of dimensions for this task, but I don't understand assumptions and to reduce dimensions of this system (how to get 1D equations from 2D equations):
$${\frac {\partial u}{\partial t}}=k_1{\frac {\partial ^{2}u}{\partial y^{2}}}, 0\leq x\leq L, 0\leq y\leq H, \tag{1}\label{eq1}$$ $${\frac {\partial u}{\partial t}}=k_2{\frac {\partial ^{2}u}{\partial x^{2}}} 0\leq x\leq L, -h\leq y\leq 0. \tag{2}\label{eq2}$$
In several books I have come across methods for reducing the dimension of such a task, but I do not understand the assumptions and arguments that were made in the process of reduction. With some assumptions, I can’t agree at all. For example one of them:
Due to the equality of the boundary values of $u$ in both regions at $y = 0$, the derivatives $\frac{\partial u}{\partial x}$ in these regions are of the same order, and therefore the flow $k\frac{\partial u}{\partial x}$ in the $x$ direction in the thick zone is negligible. At the same time, the heat flows in the $y$ direction coincide at $y = 0$, which can only be the case if the change of $u$ in the y direction in the thick zone occurs faster than in the thin. This implies $$\frac{\partial^2 u}{\partial y^2} >>\frac{\partial^2 u}{\partial x^2} \tag{3}\label{eq3}$$
I would be glad if somebody helps me to understand in detail how to get the \eqref{eq1}, \eqref{eq2} equations from \eqref{eq01}, \eqref{eq02} equations and proof of neglection of \eqref{eq3}.