I have the two-dimensional temperature Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another fluid equation (which is one-dimensional). The Laplacian is defined over $x\in[0,L], y\in[0,l]$. On manipulating the second equation (which I have described in the Origins section of my question) I have managed to reduce the problem to a boundary value problem on the Laplacian subjected to the following boundary conditions

$$\frac{\partial T(0,y)}{\partial x}=\frac{\partial T(L,y)}{\partial x}=0 \tag 1$$

$$\frac{\partial T(x,0)}{\partial y}=\gamma \tag 2$$

$$\frac{\partial T(x,l)}{\partial y}=\zeta \Bigg[T(x,l)-\Bigg\{\alpha e^{-\alpha x}\Bigg(\int_0^x e^{\alpha s }T(s,y)\mathrm{d}s+\frac{t_{i}}{\alpha}\Bigg)\Bigg\}\Bigg] \tag 3$$

$\gamma, \alpha, \zeta, t_i$ are all constants $>0$. Can anyone suggest a way to solve this problem ?


The 3rd boundary condition is actually of the following form:

$$\frac{\partial T(x,l)}{\partial y}=\zeta \Bigg[T(x,l)-t\Bigg] \tag 4$$ The $t$ in $(4)$ is governed by the following equation (this is the other equation I mentioned earlier):

$$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 5$$

where it is known that $t(x=0)=t_i$. To derive $(3)$, I solved $(5)$ using the method of integrating factor and substituted in $(4)$.

My original problem is the Laplacian coupled with $(5)$.

Physical meaning

The problem describes the flow of a fluid (with temperature $t$ and described by $(5)$) over a rectangular plate (at $y=l$) heated from the bottom (at $y=0$). The fluid is thermally coupled to the plate temperature $T$ through boundary condition $(3)$ which is the convection or Robin type condition.


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