MathOverflow Asked on November 3, 2021
I have the two-dimensional Laplacian $(nabla^2 T(x,y)=0)$ coupled with another equation which is:
$$frac{partial t}{partial x}+alpha(t-T)=0 tag 1$$
where it is known that $t(x=0)=t_i$.
The Laplacian is defined over $xin[0,L], yin[0,l]$ subjected to the following boundary conditions
$$frac{partial T(0,y)}{partial x}=frac{partial T(L,y)}{partial x}=0 tag 2$$
$$frac{partial T(x,0)}{partial y}=gamma tag 3$$
$$frac{partial T(x,l)}{partial y}=zeta Bigg[T(x,l)-tBigg] tag 4$$
$gamma, alpha, zeta, t_i$ are all constants $>0$.
Attempt
I solved $(1)$ using the method of integrating factor and substituted in $(4)$ to get the boundary condition in the following form:
$$frac{partial T(x,l)}{partial y}=zeta Bigg[T(x,l)-Bigg{alpha e^{-alpha x}Bigg(int_0^x e^{alpha s }T(s,y)mathrm{d}s+frac{t_{i}}{alpha}Bigg)Bigg}Bigg] tag 5$$
Can someone suggest a way to handle the Laplacian subjected to b.c.(s) $(2), (3), (5)$ ?
If there exists an alternate way to handle this coupled system, those suggestions are welcome too.
Physical meaning
The problem describes the flow of a fluid (with temperature $t$ and described by $(1)$) 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 $(4)$ which is the convection or Robin type condition.
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