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Wijb Sommer (wijb)
New member
Username: wijb

Post Number: 2
Registered: 12-2007
Posted on Thursday, December 13, 2007 - 09:56 am:   

I am working on a (density dependent) flow problem and am not sure how to put boundary conditions for an impermeable boundary.

1) In general: what derivative am i specifying with a natural BC in 2D for pressure if i put in:
natural(pressure)=0 (dp/dx or dp/dy)?

2) Explicit: For 2D the Darcy flow equation gives: discharge_vector=-K(grad(pressure)-density*gravity_vector).
So for a horizontal impermeable boundary (no flow in y direction) should that be:
natural(pressure)=density*gravity vector?

I hope you can help me.
Greetings,
Wijb
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Robert G. Nelson (rgnelson)
Moderator
Username: rgnelson

Post Number: 1020
Registered: 06-2003
Posted on Thursday, December 13, 2007 - 02:46 pm:   

As described in the FlexPDE documentation (Help->User Guide->Addressing more difficult problems->Natural Boundary Conditions), the Natural Boundary Condition derives from application of the Divergence Theorem to terms of second order.

The meaning of the Natural BC therefore depends on how you have written your PDE.

If you have written your equation as a divergence of some vector which is itself a derivative of first order in space, then the meaning of the Natural BC is the outward normal component of the vector argument of the divergence.

For an example of how this is addressed, see the notes in "Samples | Steady_State | Stress | Elasticity.pde".

I presume you have a PDE something like
Div(-K*grad(H))=Source
In this case, -K*grad(H) represents a flux of material.
Applying the Divergence Theorem, we get
Surface_Integral(-K*grad(H))=Volume_integral(Source).

The Natural Boundary condition in this case is the outward normal component of -K*grad(H), corresponding to the integrand of the surface integral generated by application of the Divergence Theorem.

You must therefore supply for the NATURAL(H) the specific value of Outward_Normal(-K*grad(H)) at each point of the boundary.
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Wijb Sommer (wijb)
Junior Member
Username: wijb

Post Number: 3
Registered: 12-2007
Posted on Tuesday, December 18, 2007 - 05:37 am:   

Hi; thanks mr. Nelson,
now it runs perfectly.
But i have another question: I'm working with mass-fraction's and it appears sometimes a negative mass fraction is calculated. Is it possible to state that if a negative mass fraction is calculated, it is automatically set to zero?

kind regards,
Wijb
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Robert G. Nelson (rgnelson)
Moderator
Username: rgnelson

Post Number: 1030
Registered: 06-2003
Posted on Tuesday, December 18, 2007 - 04:14 pm:   

This issue was discussed recently in a posting titled "Oxygen Concentration"
(http://www.pdesolutions.com/discus/messages/4/4348.html).

If the solution shape cannot be fit by polynomials, you will get ringing in the interpolation.

But the interpolated value maintains the correct volume integral of concentration. If you artificially truncate some of the interpolated excursions, you will be adding mass to the system.

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