Model number

Fluid flow from an open, compliant vessel, driven only by the energy stored inthe compliant vesel wall.


This model simulates current flow generated from the discharging 
of a charged capacitor through a resistance element. It is 
analogous in the fluid flow context to applying a pressure across 
an open, compliant vessel that contains a volume of fluid and 
then letting the fluid drain out driven only by the energy 
stored in the compliant vessel wall. For example, a balloon is 
filled with water with its outlet held closed which generates an 
internal pressure, P. At a time, t=0, the outlet is allowed to open 
and the time course for the balloon outflow can be recorded as a 
function of time.

The simplest description of an elastic vessel under the influence 
of time-varying pressure must have a resistance and compliance 
element such as in this model. Here the simulation begins with 
a given volume of fluid in the vessel and at time t=0 the outlet 
of the vessel is opened and the compliant vessel drains. The flow, 
F is a function of the difference in the current volume and the 
volume at rest of the vessel as well as the compliance, C, of the 
vessel and the resistance, R, to flow out of the vessel. The 
change in vessel volume as a function of time is equal to the 
flow out of the vessel, -F. External
pressure is assumed to be zero.

The model uses a constant compliance to create a linear 
relationship between pressure and volume. In reality the 
pressure-volume curves of flexible tubes are non-linear and 
transmural pressure trends towards negative infinity as volume
goes to zero. However, there are linear portions of the P-V 
curve which can be approximated using a constant compliance or 
elastance value. A constant resistance that is independent of 
vessel geometry is also used in this model. For laminar flow 
through a cylindrical tube, resistance is dependent on fluid 
viscosity, tube length and tube radius (Poiseiulle's Law); 
however, resistance in this model remains independent of these 


  Eq. A)  Flow (mL/unit time) = change in volume / change in time 
		Basis: Definition of flow
  Eq. B)  Compliance = Change in volume / Change in transmural pressure 
      		Basis: Fluid analog of capacitance
  Eq. C)  Pressure drop = Resistance * Flow
		Basis: Fluid analog of Ohm's Law
  Eq. D)  (Sum of flows entering junction = sum of flows leaving junction)
      		Basis: Kirchhoff Junction rule
  Eq. E)  Pressure drop = (change in Flow/change in time)*Inertance
		Basis: Fluid analog of inductance


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

Key terms
Cardiovascular system
compliant wall
Poiseiulle's Law

Please cite 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 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.