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Britta Hagmeyer (britta)
Member
Username: britta

Post Number: 5
Registered: 11-2006
Posted on Thursday, January 25, 2007 - 10:22 am:   

Hello everyone!

I have a question concerning the AREA_INTEGRAL in 2D Cylindrical geometry (ycylinder). I have constructed a system with three regions to simulate diffusion of a substance from region1 ('drop') through region2 ('pores') into region3 ('fluid'), where 'pores' is much smaller than 'drop' or 'fluid'. From the manual and from other postings in this forum I learned that there is no flux across the outer boundaries of a system, meaning that no substance can leave the system through the outer boundaries. The simulation starts with a discontinuous initial condition for the concentration c of the substance: With a ustep-function, c is set to 100 in 'drop' and to 0 everywhere else. As I see it, there is no way in which the substance can diffuse out of the system, which means that the amount of substance in the system should remain constant over time.

I now calculated the AREA_INTEGRAL of the substance concentration over the cross-section of the complete system. Here I observed something strange: While the volume integral (INTEGRAL) of the substance concentration over the whole cylinder built up by all three regions (-> the amount of subtance in the system) stays constant or even slowly increases over time, the volume integral of the cross-section of the system (AREA_INTEGRAL) slowly decreases over time. How can this be? Should it not also remain constant over time? Have I missed anything? Might there be some rounding error responsible? Or is there another reason?

Many thanks,
Britta
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Robert G. Nelson (rgnelson)
Moderator
Username: rgnelson

Post Number: 745
Registered: 06-2003
Posted on Thursday, January 25, 2007 - 03:01 pm:   

AREA_INTEGRAL in cylindrical 2D geometry is specifically NOT the volume integral, so conservation cannot be expected in the AREA_INTEGRAL. To see this, consider a fixed amount of substance moving uniformly outward in a sphere. The concentration will drop as r^2, the volume integral will remain constant, but the AREA_INTEGRAL on a cut plane will decrease because the amounts are not multiplied by the associated volume.

The FlexPDE documentation says:

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The synonymous prototype forms of volume integral functions in 2D are:

INTEGRAL ( integrand, region )
VOL_INTEGRAL ( integrand, region )

Here region can be specified by number or name, or it can be omitted, in which case the entire domain is implied.

In two-dimensional Cartesian problems, the volume element is formed by extending the two-dimensional cell a single unit in the Z-direction, so that the volume integral is the same as the area integral in the coordinate plane.

In two-dimensional cylindrical problems, the volume element is formed as 2*pi*r*dr*dz, so that the volume integral is NOT the same as the area integral in the coordinate plane. For the special case of 2D cylindrical geometry, the additional operator

AREA_INTEGRAL ( integrand, region )

computes the area integral of the integrand over the indicated region (or the entire domain) without the 2*pi*r weighting.

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