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 properties. ---------------------------------------------------------- REFERENCE EQUATIONS: 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
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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.