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Laplace equation in 3D with numerous Non-Homogeneous BC(s) [Strategy Check]

1

$begingroup$

I need to solve the three-dimensional Laplace equation ($nabla^2T = 0$) where $nabla^2=frac{partial^2}{partial x^2} +frac{partial^2}{partial y^2}+frac{partial^2}{partial z^2}$ in the domain where $xin[0,L];yin[0,l]$ and $zin[0,mu]$. The boundary conditions are
$$Tvert_{0,y,z} = T_{hi}, T{(L,y,z)} = 0 tag A$$
$$Tvert_{x,0,z} = T_{ci}, T{(x,l,z)} = 0tag B$$
$$frac{partial T}{partial z}vert_{x,y,0} = k_c(Tvert_{x,y,0}-T_{c,av}) tag C$$
$$frac{partial T}{partial z}vert_{x,y,mu} = k_h(T_{h,av} - Tvert_{x,y,mu}) tag D$$

Here, $$T_{c,av} = frac{1}{2}Bigg(T_{ci}+e^{-b_c}Bigg[T_{ci}+frac{b_c}{l}int_0^l e^{b_c s/l}T(x,s,z)mathrm{d}sBigg]Bigg) tag E$$

and, $$T_{h,av} = frac{1}{2}Bigg(T_{hi}+e^{-b_h}Bigg[T_{hi}+frac{b_h}{L}int_0^L e^{b_h s/L}T(s,y,z)mathrm{d}sBigg]Bigg) tag F$$

I have decided to sub-divide the problem into three parts and adding up the solution of each sub-problem finally. Can be better understood from the following schematic:

SP1,SP2 are identical problems. The last two figures at the end describe SP3 with each figure showing $z=0$ and $z=mu$ face respectively.

---

SP3

The other B.C.(s) for SP3 are:

$$T(0,y,z) = T(L,y,z) = T(x,0,z) = T(x,l,z) = 0 tag G$$

along with B.C.(s) $bfmathrm(C),mathrm(D)$

Hence, SP3 has two non-homogeneous Robin type BC at $z=0$ and $z=mu$ respectively.
I assume a preliminary temp. distribution as follows:

$$T(x,y,z) = sum_{n,m=1}^{infty}T_{nm}(z)sinbigg(frac{npi x}{L}bigg)sinbigg(frac{mpi y}{l}bigg) tag H$$.
where,

$$T_{nm}(z) = A_{nm}e^{gamma z} + B_{nm}e^{-gamma z} tag I$$

I plan on evaluating $A_{nm}$ and $B_{nm}$ using the two linear equations in two unknowns that would be generated on applying $(mathrm{H})$ in the boundary conditions : $bfmathrm{(C)}$ and $bfmathrm{(D)}$ .

---

My questions are:

1. Is this method of sub-dividing the problems logical or is there any conceptual flaw in this approach ?


2. The physical situation that these equations describe should not allow the temperature on $z=mu$ wall go above $T_{hi}$ and $z=0$ wall go below $T_{ci}$. My solution defies these constraints. Can anyone give this a try and help me out by verifying this ?

---

proof-verification pde boundary-value-problem heat-equation laplacian

share|cite|improve this question

edited Mar 25 at 2:37

Indrasis Mitra

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

50111

$endgroup$

add a comment | 

1

$begingroup$

I need to solve the three-dimensional Laplace equation ($nabla^2T = 0$) where $nabla^2=frac{partial^2}{partial x^2} +frac{partial^2}{partial y^2}+frac{partial^2}{partial z^2}$ in the domain where $xin[0,L];yin[0,l]$ and $zin[0,mu]$. The boundary conditions are
$$Tvert_{0,y,z} = T_{hi}, T{(L,y,z)} = 0 tag A$$
$$Tvert_{x,0,z} = T_{ci}, T{(x,l,z)} = 0tag B$$
$$frac{partial T}{partial z}vert_{x,y,0} = k_c(Tvert_{x,y,0}-T_{c,av}) tag C$$
$$frac{partial T}{partial z}vert_{x,y,mu} = k_h(T_{h,av} - Tvert_{x,y,mu}) tag D$$

Here, $$T_{c,av} = frac{1}{2}Bigg(T_{ci}+e^{-b_c}Bigg[T_{ci}+frac{b_c}{l}int_0^l e^{b_c s/l}T(x,s,z)mathrm{d}sBigg]Bigg) tag E$$

and, $$T_{h,av} = frac{1}{2}Bigg(T_{hi}+e^{-b_h}Bigg[T_{hi}+frac{b_h}{L}int_0^L e^{b_h s/L}T(s,y,z)mathrm{d}sBigg]Bigg) tag F$$

I have decided to sub-divide the problem into three parts and adding up the solution of each sub-problem finally. Can be better understood from the following schematic:

SP1,SP2 are identical problems. The last two figures at the end describe SP3 with each figure showing $z=0$ and $z=mu$ face respectively.

---

SP3

The other B.C.(s) for SP3 are:

$$T(0,y,z) = T(L,y,z) = T(x,0,z) = T(x,l,z) = 0 tag G$$

along with B.C.(s) $bfmathrm(C),mathrm(D)$

Hence, SP3 has two non-homogeneous Robin type BC at $z=0$ and $z=mu$ respectively.
I assume a preliminary temp. distribution as follows:

$$T(x,y,z) = sum_{n,m=1}^{infty}T_{nm}(z)sinbigg(frac{npi x}{L}bigg)sinbigg(frac{mpi y}{l}bigg) tag H$$.
where,

$$T_{nm}(z) = A_{nm}e^{gamma z} + B_{nm}e^{-gamma z} tag I$$

I plan on evaluating $A_{nm}$ and $B_{nm}$ using the two linear equations in two unknowns that would be generated on applying $(mathrm{H})$ in the boundary conditions : $bfmathrm{(C)}$ and $bfmathrm{(D)}$ .

---

My questions are:

1. Is this method of sub-dividing the problems logical or is there any conceptual flaw in this approach ?


2. The physical situation that these equations describe should not allow the temperature on $z=mu$ wall go above $T_{hi}$ and $z=0$ wall go below $T_{ci}$. My solution defies these constraints. Can anyone give this a try and help me out by verifying this ?

---

proof-verification pde boundary-value-problem heat-equation laplacian

share|cite|improve this question

edited Mar 25 at 2:37

Indrasis Mitra

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

50111

$endgroup$

add a comment | 

$begingroup$

I need to solve the three-dimensional Laplace equation ($nabla^2T = 0$) where $nabla^2=frac{partial^2}{partial x^2} +frac{partial^2}{partial y^2}+frac{partial^2}{partial z^2}$ in the domain where $xin[0,L];yin[0,l]$ and $zin[0,mu]$. The boundary conditions are
$$Tvert_{0,y,z} = T_{hi}, T{(L,y,z)} = 0 tag A$$
$$Tvert_{x,0,z} = T_{ci}, T{(x,l,z)} = 0tag B$$
$$frac{partial T}{partial z}vert_{x,y,0} = k_c(Tvert_{x,y,0}-T_{c,av}) tag C$$
$$frac{partial T}{partial z}vert_{x,y,mu} = k_h(T_{h,av} - Tvert_{x,y,mu}) tag D$$

Here, $$T_{c,av} = frac{1}{2}Bigg(T_{ci}+e^{-b_c}Bigg[T_{ci}+frac{b_c}{l}int_0^l e^{b_c s/l}T(x,s,z)mathrm{d}sBigg]Bigg) tag E$$

and, $$T_{h,av} = frac{1}{2}Bigg(T_{hi}+e^{-b_h}Bigg[T_{hi}+frac{b_h}{L}int_0^L e^{b_h s/L}T(s,y,z)mathrm{d}sBigg]Bigg) tag F$$

I have decided to sub-divide the problem into three parts and adding up the solution of each sub-problem finally. Can be better understood from the following schematic:

SP1,SP2 are identical problems. The last two figures at the end describe SP3 with each figure showing $z=0$ and $z=mu$ face respectively.

---

SP3

The other B.C.(s) for SP3 are:

$$T(0,y,z) = T(L,y,z) = T(x,0,z) = T(x,l,z) = 0 tag G$$

along with B.C.(s) $bfmathrm(C),mathrm(D)$

Hence, SP3 has two non-homogeneous Robin type BC at $z=0$ and $z=mu$ respectively.
I assume a preliminary temp. distribution as follows:

$$T(x,y,z) = sum_{n,m=1}^{infty}T_{nm}(z)sinbigg(frac{npi x}{L}bigg)sinbigg(frac{mpi y}{l}bigg) tag H$$.
where,

$$T_{nm}(z) = A_{nm}e^{gamma z} + B_{nm}e^{-gamma z} tag I$$

I plan on evaluating $A_{nm}$ and $B_{nm}$ using the two linear equations in two unknowns that would be generated on applying $(mathrm{H})$ in the boundary conditions : $bfmathrm{(C)}$ and $bfmathrm{(D)}$ .

---

My questions are:

1. Is this method of sub-dividing the problems logical or is there any conceptual flaw in this approach ?


2. The physical situation that these equations describe should not allow the temperature on $z=mu$ wall go above $T_{hi}$ and $z=0$ wall go below $T_{ci}$. My solution defies these constraints. Can anyone give this a try and help me out by verifying this ?

---

proof-verification pde boundary-value-problem heat-equation laplacian

share|cite|improve this question

edited Mar 25 at 2:37

Indrasis Mitra

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

50111

$endgroup$

share|cite|improve this question

edited Mar 25 at 2:37

Indrasis Mitra

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

50111

share|cite|improve this question

edited Mar 25 at 2:37

Indrasis Mitra

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

50111

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

50111

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

50111