# additional variable in diffusion-trasnport equation

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June 20, 2013, 07:48
#1
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Maria
Join Date: Jun 2013
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Hi

I am trying to model cell diffusion in a cube 1x1. I have defined concentration as additional variable and solve it as diffusion transport equation. However I want to solve a non-linear diffusion equation which has a form as shown in the attached file.
I have defined the whole cube as a subdomain apart from the upper surface in which I have assigned inlet boundary condition as (concentration) c=1. The other surfaces of the domain are set as walls with zero flux.
In the subdomain I have defined as source to be concentration^2. Since the diffusion-transport equation also includes density I have set the domain to be consisted of a material with density =1. However for an internal point far from the boundary I take even for t=0, cell concentrations only higher than 1 which is not reasonable.

Attached Files
 equation.docx (14.9 KB, 15 views)

 June 20, 2013, 18:49 #2 Super Moderator   Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 11,282 Rep Power: 88 I cannot view the file, can you use the "Go Advanced" option and use the latex editor to put the equation directly in the post? Also, please explain what you are modelling and what you know about it. Your description of what you are trying to model sounds strange.

June 20, 2013, 19:10
#3
New Member

Maria
Join Date: Jun 2013
Posts: 3
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Hi

I have attached the equation as a png file. Is it ok now?

Actually I am trying to derive solutions of the attached diffusion equation for test case i.e., a cube 1x1x1. Initial conditions: (concentration) c=1 at the upper surface,
Boundary conditions:dc/dt=0 at the remaining surfaces.

Thanks
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 equation.png (6.7 KB, 17 views)

 June 20, 2013, 19:19 #4 Super Moderator   Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 11,282 Rep Power: 88 This is pretty straight forward - a diffusive additional variable with a source term of c^2. Do not use a diffusive transport equation as you are not modelling transport. But non-linear source terms like this sound pretty numerically unstable so you will have to be careful to get it to converge. Is the density constant? And I do not understand your comment about a point being c=0 far away. Finally, why is c>1 unphysical? If c is a concentration and therefore bounded to c<=1 then I think your entire approach is wrong - The c^2 source term is obviously an unbounded equation, so you will need to rethink your equation.

 June 20, 2013, 20:25 #5 New Member   Maria Join Date: Jun 2013 Posts: 3 Rep Power: 4 Thanks for the reply. I am not modelling transport but the general form of diffusive-transport equation that is mentioned in the help looks similar to mine if i set D=1, density ρ=1 and S=c^2. Is this right? I am not saying that c is bounded to be lower than 1;however for the transient problem (duration=1 sec, 10 timesteps), I would expect that the solution at the point (0.5,0.5,0.5) at t=0 is lower than 1. What do you mean "be careful to get it converge"?What should I do? I have set the additional variable c as specific without units.Is this ok? Best

 June 21, 2013, 03:42 #6 Super Moderator   Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 11,282 Rep Power: 88 Read the documentation carefully about the differences between specific and volumetric additional variables to make sure you have chosen the correct one. I have not looked into this in detail so cannot tell you which option to choose here, you will have to do that. To get non-linear simulations like this to converge you might need small time steps. Mesh quality might help a little but as your additional variable is just a diffusion equation it will not be a major factor. Also double precision might help.

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