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Cardiac Physiome Society workshop: November 6-9, 2017 , Toronto

Diffusion1DpdeConsumption

Diffusion in one dimension with asymmetrical consumption is modeled using a partial differential equation.

Model number: 0364

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Description

This model illustrates using function generators to
generator an initial condition and a parameter that
are spatial functions in x. It also generates a
comparison function, Ctest, which shows that the
solution in the presence of consumption evolves into
a profile which is approximately Gaussian although the
mean has been shifted downstream.

The diffusion of a substance in one dimension over a finite
length is modeled. The solution is plotted as 
(1) Contours in the x-t plane,
(2) As functions of distance at specific times, and
(3) As functions of time and specific locations.

The initial values are given as
C(x) = 5,  0.049<=x<=0.051, 
C(x) = 0,  x<0.049 or x>0.051.

The boundary conditions are the zero-flux condition
(boundaries are reflective).

The consumption in the model is controlled by the area of a Gaussian curve
centered at 1/4 the length from the entrance of the capillary. Set the
area to 1e-8 to remove the effect.

Equations

 

     Partial Differential Equation


    Left Boundary Condition

.

    Right Boundary Condition


     Initial Condition

.

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

Download JSim model project file

References



Related Models

Key Terms

1d, 1-d, 1D, 1-D, one dimension, diffusion, PDE, consumption, function generator, optimize, no flux, boundary condition, tutorial

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Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send an email with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
Or send a copy to:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

[This page was last modified 02Nov16, 2:31 pm.]

Model development and archiving support at physiome.org provided by the following grants: NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.