Diffusion in one dimension with asymmetrical consumption is modeled using a partial differential equation.
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.
Partial Differential Equation
Left Boundary Condition
Right Boundary 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.
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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, 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.