Model rigid vessel flow with vessels of varying diameter and length in series and in parallel. Model 'Two_Resistors' uses two vessels in series and model 'Four_resistors' has two vessels in parallel and two more in series.

## Description

The model simulates fluid flow, F, through rigid vessels, in series, of resistance, R1 and R2, given a pressure drop across the length of the vessel Delta_P. The flow is related to the resistance by the fluid equivalent of Ohm's Law. F = Delta_P / (R1+R2) where Delta_P is the difference in pressure between the beginning and end of the vessel and R is determined from Poiseuille's Law as: R = 128*mu*L / pi*D^4 where mu is the fluid viscosity, L is the vessel length, and D is the vessel diameter. Model 'Four_resistors' (a part of this JSim project file) simulates flow through four resistors (four vessels of varying diameter and length), two of which are in parallel. The geometry is of one vessel emptying into two separate vessels which empty into one.

**Figure:**

(Two Resistors): F Pin R1 P1 ---> R2 Pout o-------/\/\/\/\----o-------/\/\/\/\-----o (Four Resistors): F Pin R1 P1 R2 P2 ---> R4 Pout o---/\/\/\/\-o--------/\/\/\/\-----o--------/\/\/\/\-----o | ---> | | F2 | | R3 | |--------/\/\/\/\-----| ---> F3 LEGEND: /\/\/\/\ = resistance, F = flow, P = pressure

## Equations

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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Ohm GS. Die galvanische Kette mathematisch bearbeitet, 1827 Burattini R, Di Salvia PO, Development of systemic arterial mechanical properties from infancy to adulthood interpreted by four-element windkessel models, J Appl Physiol 103:66-79, 2007

Please cite https://www.imagwiki.nibib.nih.gov/physiome** **in any publication for which this software is used and send one reprint to the address given below:

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.