Mixing
How to solve two partial differential equations that are coupled through the boundary conditions
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My problem is a heat transfer problem between two systems that are connected through an interface. The heat balance equation for each system is a second order PDE and the systems have equal temperature and heat flux at the common interface between them. Actually, I have the final solution, but I don't know what are the steps that should be followed to solve this system of PDEs as the solution of each one depends on the solution of the second system.
PDE for the first system is:
$$rho_r C_r frac{partial T}{partial t} + left(frac{Q rho_f C_f}{2πrh} right) frac{partial T}{partial r} - K_t frac{partial^{2} T}{partial z^{2}}=0$$
The initial and boundary conditions:
begin{align}
T(r,z,0) &= T_{i} \
T(0,z,t) &= T_{0}
end{align}
The PDE of the surrounding:
$$rho_m C_m frac{partial T_m}{partial t} = K_m frac{partial^{2} T_m}{partial z^{2}}$$
The initial and boundary conditions for the surrounding:
$$T_m(z,0)=T_i$$
The common boundary conditions ($z = h$):
begin{align}
T_m(z,t) &= T(z,t) \
k_m frac{partial T_m}{partial z} &= K_t frac{partial T}{partial z}
end{align}
pde laplace-transform
edited Jul 15 '18 at 15:48
Mattos
2,79221321
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
$endgroup$
add a comment |
$begingroup$
My problem is a heat transfer problem between two systems that are connected through an interface. The heat balance equation for each system is a second order PDE and the systems have equal temperature and heat flux at the common interface between them. Actually, I have the final solution, but I don't know what are the steps that should be followed to solve this system of PDEs as the solution of each one depends on the solution of the second system.
PDE for the first system is:
$$rho_r C_r frac{partial T}{partial t} + left(frac{Q rho_f C_f}{2πrh} right) frac{partial T}{partial r} - K_t frac{partial^{2} T}{partial z^{2}}=0$$
The initial and boundary conditions:
begin{align}
T(r,z,0) &= T_{i} \
T(0,z,t) &= T_{0}
end{align}
The PDE of the surrounding:
$$rho_m C_m frac{partial T_m}{partial t} = K_m frac{partial^{2} T_m}{partial z^{2}}$$
The initial and boundary conditions for the surrounding:
$$T_m(z,0)=T_i$$
The common boundary conditions ($z = h$):
begin{align}
T_m(z,t) &= T(z,t) \
k_m frac{partial T_m}{partial z} &= K_t frac{partial T}{partial z}
end{align}
pde laplace-transform
edited Jul 15 '18 at 15:48
Mattos
2,79221321
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
$endgroup$
$begingroup$
Why don't you post the problem and your solution? As it stands, what you have written means very little.
$endgroup$
– Mattos
Jul 8 '18 at 23:42
add a comment |
$begingroup$
My problem is a heat transfer problem between two systems that are connected through an interface. The heat balance equation for each system is a second order PDE and the systems have equal temperature and heat flux at the common interface between them. Actually, I have the final solution, but I don't know what are the steps that should be followed to solve this system of PDEs as the solution of each one depends on the solution of the second system.
PDE for the first system is:
$$rho_r C_r frac{partial T}{partial t} + left(frac{Q rho_f C_f}{2πrh} right) frac{partial T}{partial r} - K_t frac{partial^{2} T}{partial z^{2}}=0$$
The initial and boundary conditions:
begin{align}
T(r,z,0) &= T_{i} \
T(0,z,t) &= T_{0}
end{align}
The PDE of the surrounding:
$$rho_m C_m frac{partial T_m}{partial t} = K_m frac{partial^{2} T_m}{partial z^{2}}$$
The initial and boundary conditions for the surrounding:
$$T_m(z,0)=T_i$$
The common boundary conditions ($z = h$):
begin{align}
T_m(z,t) &= T(z,t) \
k_m frac{partial T_m}{partial z} &= K_t frac{partial T}{partial z}
end{align}
pde laplace-transform
edited Jul 15 '18 at 15:48
Mattos
2,79221321
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
$endgroup$
My problem is a heat transfer problem between two systems that are connected through an interface. The heat balance equation for each system is a second order PDE and the systems have equal temperature and heat flux at the common interface between them. Actually, I have the final solution, but I don't know what are the steps that should be followed to solve this system of PDEs as the solution of each one depends on the solution of the second system.
PDE for the first system is:
$$rho_r C_r frac{partial T}{partial t} + left(frac{Q rho_f C_f}{2πrh} right) frac{partial T}{partial r} - K_t frac{partial^{2} T}{partial z^{2}}=0$$
The initial and boundary conditions:
begin{align}
T(r,z,0) &= T_{i} \
T(0,z,t) &= T_{0}
end{align}
The PDE of the surrounding:
$$rho_m C_m frac{partial T_m}{partial t} = K_m frac{partial^{2} T_m}{partial z^{2}}$$
The initial and boundary conditions for the surrounding:
$$T_m(z,0)=T_i$$
The common boundary conditions ($z = h$):
begin{align}
T_m(z,t) &= T(z,t) \
k_m frac{partial T_m}{partial z} &= K_t frac{partial T}{partial z}
end{align}
pde laplace-transform
pde laplace-transform
edited Jul 15 '18 at 15:48
Mattos
2,79221321
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
edited Jul 15 '18 at 15:48
Mattos
2,79221321
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
edited Jul 15 '18 at 15:48
Mattos
2,79221321
edited Jul 15 '18 at 15:48
Mattos
2,79221321
edited Jul 15 '18 at 15:48
Mattos
2,79221321
2,79221321
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
asked Jul 8 '18 at 20:55
Refaat GalalRefaat Galal
75
75
$begingroup$
Why don't you post the problem and your solution? As it stands, what you have written means very little.
$endgroup$
– Mattos
Jul 8 '18 at 23:42
add a comment |
$begingroup$
Why don't you post the problem and your solution? As it stands, what you have written means very little.
$endgroup$
– Mattos
Jul 8 '18 at 23:42
$begingroup$
Why don't you post the problem and your solution? As it stands, what you have written means very little.
$endgroup$
– Mattos
Jul 8 '18 at 23:42
$begingroup$
Why don't you post the problem and your solution? As it stands, what you have written means very little.
$endgroup$
– Mattos
Jul 8 '18 at 23:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The answer seems to be given in Yu. Melnikov and M. Melnikov monograph "Green's functions: Construction and Applications". Take a look at chapter 6 "PDE Matrices of Green's Type". Multiple examples are considered in details.
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
$endgroup$
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The answer seems to be given in Yu. Melnikov and M. Melnikov monograph "Green's functions: Construction and Applications". Take a look at chapter 6 "PDE Matrices of Green's Type". Multiple examples are considered in details.
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
$endgroup$
add a comment |
$begingroup$
The answer seems to be given in Yu. Melnikov and M. Melnikov monograph "Green's functions: Construction and Applications". Take a look at chapter 6 "PDE Matrices of Green's Type". Multiple examples are considered in details.
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
$endgroup$
add a comment |
$begingroup$
The answer seems to be given in Yu. Melnikov and M. Melnikov monograph "Green's functions: Construction and Applications". Take a look at chapter 6 "PDE Matrices of Green's Type". Multiple examples are considered in details.
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
$endgroup$
The answer seems to be given in Yu. Melnikov and M. Melnikov monograph "Green's functions: Construction and Applications". Take a look at chapter 6 "PDE Matrices of Green's Type". Multiple examples are considered in details.
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
answered Jan 16 at 16:15
Asatur KhurshudyanAsatur Khurshudyan
415211
415211
add a comment |
add a comment |
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$begingroup$
Why don't you post the problem and your solution? As it stands, what you have written means very little.
$endgroup$
– Mattos
Jul 8 '18 at 23:42