# Coupled solid-fluid heat transfer over a rectangular plate heated from the bottom (Boundary value problem)

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 ?

Origins

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.