Obtaining boundary conditions
I'm trying to numerically solve the three coupled PDEs;
$frac{partialtheta}{partial t} = w + nabla^2 theta, (1)$
$frac{partial Q}{partial t} = -RaPrnabla^2_Htheta + nabla^2 Q, (2)$
$nabla^2 w = Q. (3)$
Which represent a linear stability analysis for temperature ($theta$) and z-component velocity ($w$) perturbations for a fluid initially at rest between two infinite horizontal plates. I'm looking for a particular form of periodic solution such that;
$nabla^2 = -a^2 + frac{partial^2}{partial z^2}, (4)$
$nabla^2_H = -a^2. (5)$
$Ra$, $Pr$, and $a$ are parameters that do not depend on space or time. The domain is $zin[0,1]$ and $tin[0,infty)$. I have the following boundary conditions: $theta = w = partial{w}/partial{z} = 0 $. I am trying to convert these to obtain a boundary condition on $Q$. I have tried substituting $(4)$ into $(3)$ and taking $partial/partial z$ but this does not tell me anything about $Q$ and I know nothing about $partial ^3
w/partial z^3$. I am unsure on what I can try. Any help/advice would be greatly appreciated.
differential-equations derivatives pde boundary-value-problem
asked Nov 30 at 13:08
Patrick Lewis
112
|
show 5 more comments
I'm trying to numerically solve the three coupled PDEs;
$frac{partialtheta}{partial t} = w + nabla^2 theta, (1)$
$frac{partial Q}{partial t} = -RaPrnabla^2_Htheta + nabla^2 Q, (2)$
$nabla^2 w = Q. (3)$
Which represent a linear stability analysis for temperature ($theta$) and z-component velocity ($w$) perturbations for a fluid initially at rest between two infinite horizontal plates. I'm looking for a particular form of periodic solution such that;
$nabla^2 = -a^2 + frac{partial^2}{partial z^2}, (4)$
$nabla^2_H = -a^2. (5)$
$Ra$, $Pr$, and $a$ are parameters that do not depend on space or time. The domain is $zin[0,1]$ and $tin[0,infty)$. I have the following boundary conditions: $theta = w = partial{w}/partial{z} = 0 $. I am trying to convert these to obtain a boundary condition on $Q$. I have tried substituting $(4)$ into $(3)$ and taking $partial/partial z$ but this does not tell me anything about $Q$ and I know nothing about $partial ^3
w/partial z^3$. I am unsure on what I can try. Any help/advice would be greatly appreciated.
differential-equations derivatives pde boundary-value-problem
asked Nov 30 at 13:08
Patrick Lewis
112
What is the domain of this problem? And what is $Q^{(3)}$? Since $theta$, $Q$ and $w$ are independent variables, I think that you can't obtain B.C. for $Q$ from the other B.C. Could you provide the physical context of this problem? It can be helpful to determine the B.C. for $Q$.
– rafa11111
Nov 30 at 13:43
Sorry, $Q^{(3)} = partial^3Q/partial z^3$ - I will edit that part. The domain is $zin[0,1]$ and $tin[0,infty)$. I am doing a linear stability analysis of perturbations to temperature ($theta$) and the z-component of the velocity ($w$) for fluid initially at rest between two infinite horizontal plates. These boundary conditions represent fixed temperature, no slip conditions. I hope that helps.
– Patrick Lewis
Nov 30 at 13:49
And the B.C. you provided holds at $z=0$ and $z=1$?
– rafa11111
Nov 30 at 13:50
Yes, I'll put this information in the question - my apologies. I will be solving this numerically, so I should add that I am demanding it holds. However, I'm unsure how to obtain a BC on Q from these to implement into my scheme.
– Patrick Lewis
Nov 30 at 13:53
Edit - $partial^3Q/partial z^3$ should be $partial^3w/partial z^3$, which I still know nothing about.
– Patrick Lewis
Nov 30 at 14:01
|
show 5 more comments
I'm trying to numerically solve the three coupled PDEs;
$frac{partialtheta}{partial t} = w + nabla^2 theta, (1)$
$frac{partial Q}{partial t} = -RaPrnabla^2_Htheta + nabla^2 Q, (2)$
$nabla^2 w = Q. (3)$
Which represent a linear stability analysis for temperature ($theta$) and z-component velocity ($w$) perturbations for a fluid initially at rest between two infinite horizontal plates. I'm looking for a particular form of periodic solution such that;
$nabla^2 = -a^2 + frac{partial^2}{partial z^2}, (4)$
$nabla^2_H = -a^2. (5)$
$Ra$, $Pr$, and $a$ are parameters that do not depend on space or time. The domain is $zin[0,1]$ and $tin[0,infty)$. I have the following boundary conditions: $theta = w = partial{w}/partial{z} = 0 $. I am trying to convert these to obtain a boundary condition on $Q$. I have tried substituting $(4)$ into $(3)$ and taking $partial/partial z$ but this does not tell me anything about $Q$ and I know nothing about $partial ^3
w/partial z^3$. I am unsure on what I can try. Any help/advice would be greatly appreciated.
differential-equations derivatives pde boundary-value-problem
asked Nov 30 at 13:08
Patrick Lewis
112
I'm trying to numerically solve the three coupled PDEs;
$frac{partialtheta}{partial t} = w + nabla^2 theta, (1)$
$frac{partial Q}{partial t} = -RaPrnabla^2_Htheta + nabla^2 Q, (2)$
$nabla^2 w = Q. (3)$
Which represent a linear stability analysis for temperature ($theta$) and z-component velocity ($w$) perturbations for a fluid initially at rest between two infinite horizontal plates. I'm looking for a particular form of periodic solution such that;
$nabla^2 = -a^2 + frac{partial^2}{partial z^2}, (4)$
$nabla^2_H = -a^2. (5)$
$Ra$, $Pr$, and $a$ are parameters that do not depend on space or time. The domain is $zin[0,1]$ and $tin[0,infty)$. I have the following boundary conditions: $theta = w = partial{w}/partial{z} = 0 $. I am trying to convert these to obtain a boundary condition on $Q$. I have tried substituting $(4)$ into $(3)$ and taking $partial/partial z$ but this does not tell me anything about $Q$ and I know nothing about $partial ^3
w/partial z^3$. I am unsure on what I can try. Any help/advice would be greatly appreciated.
differential-equations derivatives pde boundary-value-problem
differential-equations derivatives pde boundary-value-problem
asked Nov 30 at 13:08
Patrick Lewis
112
asked Nov 30 at 13:08
Patrick Lewis
112
edited Nov 30 at 13:56
asked Nov 30 at 13:08
Patrick Lewis
112
asked Nov 30 at 13:08
Patrick Lewis
112
asked Nov 30 at 13:08
Patrick Lewis
112
112
What is the domain of this problem? And what is $Q^{(3)}$? Since $theta$, $Q$ and $w$ are independent variables, I think that you can't obtain B.C. for $Q$ from the other B.C. Could you provide the physical context of this problem? It can be helpful to determine the B.C. for $Q$.
– rafa11111
Nov 30 at 13:43
Sorry, $Q^{(3)} = partial^3Q/partial z^3$ - I will edit that part. The domain is $zin[0,1]$ and $tin[0,infty)$. I am doing a linear stability analysis of perturbations to temperature ($theta$) and the z-component of the velocity ($w$) for fluid initially at rest between two infinite horizontal plates. These boundary conditions represent fixed temperature, no slip conditions. I hope that helps.
– Patrick Lewis
Nov 30 at 13:49
And the B.C. you provided holds at $z=0$ and $z=1$?
– rafa11111
Nov 30 at 13:50
Yes, I'll put this information in the question - my apologies. I will be solving this numerically, so I should add that I am demanding it holds. However, I'm unsure how to obtain a BC on Q from these to implement into my scheme.
– Patrick Lewis
Nov 30 at 13:53
Edit - $partial^3Q/partial z^3$ should be $partial^3w/partial z^3$, which I still know nothing about.
– Patrick Lewis
Nov 30 at 14:01
|
show 5 more comments
What is the domain of this problem? And what is $Q^{(3)}$? Since $theta$, $Q$ and $w$ are independent variables, I think that you can't obtain B.C. for $Q$ from the other B.C. Could you provide the physical context of this problem? It can be helpful to determine the B.C. for $Q$.
– rafa11111
Nov 30 at 13:43
Sorry, $Q^{(3)} = partial^3Q/partial z^3$ - I will edit that part. The domain is $zin[0,1]$ and $tin[0,infty)$. I am doing a linear stability analysis of perturbations to temperature ($theta$) and the z-component of the velocity ($w$) for fluid initially at rest between two infinite horizontal plates. These boundary conditions represent fixed temperature, no slip conditions. I hope that helps.
– Patrick Lewis
Nov 30 at 13:49
And the B.C. you provided holds at $z=0$ and $z=1$?
– rafa11111
Nov 30 at 13:50
Yes, I'll put this information in the question - my apologies. I will be solving this numerically, so I should add that I am demanding it holds. However, I'm unsure how to obtain a BC on Q from these to implement into my scheme.
– Patrick Lewis
Nov 30 at 13:53
Edit - $partial^3Q/partial z^3$ should be $partial^3w/partial z^3$, which I still know nothing about.
– Patrick Lewis
Nov 30 at 14:01
What is the domain of this problem? And what is $Q^{(3)}$? Since $theta$, $Q$ and $w$ are independent variables, I think that you can't obtain B.C. for $Q$ from the other B.C. Could you provide the physical context of this problem? It can be helpful to determine the B.C. for $Q$.
– rafa11111
Nov 30 at 13:43
What is the domain of this problem? And what is $Q^{(3)}$? Since $theta$, $Q$ and $w$ are independent variables, I think that you can't obtain B.C. for $Q$ from the other B.C. Could you provide the physical context of this problem? It can be helpful to determine the B.C. for $Q$.
– rafa11111
Nov 30 at 13:43
Sorry, $Q^{(3)} = partial^3Q/partial z^3$ - I will edit that part. The domain is $zin[0,1]$ and $tin[0,infty)$. I am doing a linear stability analysis of perturbations to temperature ($theta$) and the z-component of the velocity ($w$) for fluid initially at rest between two infinite horizontal plates. These boundary conditions represent fixed temperature, no slip conditions. I hope that helps.
– Patrick Lewis
Nov 30 at 13:49
Sorry, $Q^{(3)} = partial^3Q/partial z^3$ - I will edit that part. The domain is $zin[0,1]$ and $tin[0,infty)$. I am doing a linear stability analysis of perturbations to temperature ($theta$) and the z-component of the velocity ($w$) for fluid initially at rest between two infinite horizontal plates. These boundary conditions represent fixed temperature, no slip conditions. I hope that helps.
– Patrick Lewis
Nov 30 at 13:49
And the B.C. you provided holds at $z=0$ and $z=1$?
– rafa11111
Nov 30 at 13:50
And the B.C. you provided holds at $z=0$ and $z=1$?
– rafa11111
Nov 30 at 13:50
Yes, I'll put this information in the question - my apologies. I will be solving this numerically, so I should add that I am demanding it holds. However, I'm unsure how to obtain a BC on Q from these to implement into my scheme.
– Patrick Lewis
Nov 30 at 13:53
Yes, I'll put this information in the question - my apologies. I will be solving this numerically, so I should add that I am demanding it holds. However, I'm unsure how to obtain a BC on Q from these to implement into my scheme.
– Patrick Lewis
Nov 30 at 13:53
Edit - $partial^3Q/partial z^3$ should be $partial^3w/partial z^3$, which I still know nothing about.
– Patrick Lewis
Nov 30 at 14:01
Edit - $partial^3Q/partial z^3$ should be $partial^3w/partial z^3$, which I still know nothing about.
– Patrick Lewis
Nov 30 at 14:01
|
show 5 more comments
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What is the domain of this problem? And what is $Q^{(3)}$? Since $theta$, $Q$ and $w$ are independent variables, I think that you can't obtain B.C. for $Q$ from the other B.C. Could you provide the physical context of this problem? It can be helpful to determine the B.C. for $Q$.
– rafa11111
Nov 30 at 13:43
Sorry, $Q^{(3)} = partial^3Q/partial z^3$ - I will edit that part. The domain is $zin[0,1]$ and $tin[0,infty)$. I am doing a linear stability analysis of perturbations to temperature ($theta$) and the z-component of the velocity ($w$) for fluid initially at rest between two infinite horizontal plates. These boundary conditions represent fixed temperature, no slip conditions. I hope that helps.
– Patrick Lewis
Nov 30 at 13:49
And the B.C. you provided holds at $z=0$ and $z=1$?
– rafa11111
Nov 30 at 13:50
Yes, I'll put this information in the question - my apologies. I will be solving this numerically, so I should add that I am demanding it holds. However, I'm unsure how to obtain a BC on Q from these to implement into my scheme.
– Patrick Lewis
Nov 30 at 13:53
Edit - $partial^3Q/partial z^3$ should be $partial^3w/partial z^3$, which I still know nothing about.
– Patrick Lewis
Nov 30 at 14:01