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-   -   Is this problem well posed? (http://www.cfd-online.com/Forums/main/1263-problem-well-posed.html)

Thomas P. Abraham September 7, 1999 23:48

Is this problem well posed?
 
Hello Everyone:

I am working on a problem and wondering whether the problem I defined is well posed. I am working on a different problem. I will try to illustrate a much simpler problem. I am working on a pure conduction case.

Heat Flux

^^^^^^^^^^^^^^^^^^^^^^^^^^^

- -

- Aluminum Block -

- -

- -

^^^^^^^^^^^^^^^^^^^^^^^^^^^

h, Tambient

At one end, a constant heat flux is applied. On the other end, a convective boundary condition is applied through heat transfer coefficient, h and the ambient temperature, Tambient. I think by applying a heat flux boundary condition, I am specifying the heat transfer rate at that boundary. Even at the other end, I am specifying the heat transfer rate through the heat transfer coefficeint, h and the ambient temperature, Tambient. There is no reference temperature specified in the computational domain and hence the problem may be ill-posed.

on the contrary, if I were to specify a temperature boundary condition instead of the convective boundary condition, the problem will be well posed.

What do you all think?

Thanks for your feedbacks,

Thomas


D.M. Lipinski September 8, 1999 06:00

Re: Is this problem well posed?
 
If you can provide an initial condition i.e. the temperature distribution (may be uniform) in the aluminium block, you can run the unsteady heat condution calculations. You will, say, cool the block from the top (given heat flux at the top), this will influence the temperature distribution in the block (heat conduction PLUS accumulation because of finite heat capacity of aluminium). At each time step, the temperature at the bottom of the block you will get as the result of calculations. So the rate of the heat transfer at the bottom (implemented implicitly in your calculations), at each time step, will be known as the bottom temperature of the block, ambient temperature and the heat transfer coefficient are available to the program. If the steady state condition of your problem exists, you'll reach it, after some time, by running unsteady calculations. regards DML

Hongjun Li September 8, 1999 09:52

Re: Is this problem well posed?
 
Your boundary conditions are well-posted. Your Tambient will serve as your reference temperature.


Aleksey Alekseev September 8, 1999 14:29

Re: Is this problem well posed?
 
Your problem is well posed. Why did you decide that it is ill-posed? Have you some instabillity? Aleksey

clifford bradford September 8, 1999 14:43

Re: Is this problem well posed?
 
yes your problem is well posed because the steady state heat equation (laplace's eqtn) requires two boundary conditions in ach coord direction and these can be specified temp, heat flux (equivalent to temp gradient), or convection BC (which is specified correctly as you have it). whenever you solve steady state heat transfer all you need is one of these BC on every portion of the domain boundary. if you use the time marching approach your choice of initial condition does not affect the steady state temp distribution

Thomas P. Abraham September 8, 1999 14:52

Re: Is this problem well posed?
 
Thanks to everyone for your responses. Yes, the problem is well posed. I thought that I saw something strange yesterday to doubt the validity of my boundary conditions. What I saw was physical and I got it confirmed today.

Thanks again,

Thomas


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