Transcript Folie 1 - uni-stuttgart.de
Motivation Coupling of micro- and macro-models for complex flow and transport processes in biological tissue Katherina Baber, Bernd Flemisch, Rainer Helmig, Klaus Mosthaf
Department of Hydromechanics and Modeling of Hydrosystems
Free flow
Stokes equation
Porous media
Darcy ‘s law
Identification of dominating forces
→
dimensional analysis
Non-dimensional Stokes equation and transport equation
Goal:
coupling free flow with flow in porous media → focus on interface structure and processes
Application:
transvascular exchange of therapeutic agents 1
Transport equations Coupling conditions
2
Mechanical equilibrium:
Normal forces
Non dimensional transport equation combined with Darcy’s law
Mechanical equilibrium:
Tangential forces B
EAVERS
-J
OSEPH
-S
AFFMANN
-condition
A selection of dimensionless numbers:
Continuity of mass
Transvascular flow and transport processes (after Junqueira et al., 2002) → exchange processes strongly depend on wall structure and size and charge of the transported substance
Continuity of mass fractions
Free-flow region (ff) - Vasculature:
Assumption: steady laminar flow of a Newtonian fluid
Porous-medium regions (pm) – Capillary wall and tissue space:
Assumption: rigid porous media, continuous mobile Newtonian fluid phase
First results
Structure of the capillary wall: variety of para- and trans-cellular pathways, structure strongly dependent on anatomic location and physiological/pathological conditions → crucial to resolve the structure of the micro-vascular wall and the occurring transport processes
Current state Description of interface layer (capillary wall) as porous medium
→ average continuum description of the whole system (REV-scale) → → coupled model for free flow and flow in porous media single-phase compositional flow (blood/plasma + therapeutic agent) Distribution of pressure , x- and y-velocity and mass fractions of a dissolved substance
Drawbacks:
fails to resolve structure and processes at the interface volume averaging procedure not applicable to a thin heterogeneous structure like the capillary wall questionable if Beavers-Joseph-Saffman condition may be applied to biological problem setting
Work in progress Decoupling of interface layer and macro-model
→ new exchange and coupling conditions → different time and length scales possible → capillary wall as dominating structure for exchange processes resolved in more detail → inertial forces can be neglected in all three domains due to laminar viscous flow → advective transport processes dominate → flow directions differ strongly and velocities in transvascular pathways are not known resolve capillary wall on the micro-scale
Coupling concept for the micro-/macro-model:
→ coupling with the help of
mortar elements
to allow the transition from pore-scale to REV-scale model 3
Outlook
decide for partitioned or monolithic approach for each layer of mortar elements include electro-chemical processes and characteristics of transcvascular pathways derive 2D-interface by means of homogenization
Literature:
[1] Baber, K. (2009). Modeling the transfer of therapeutic agents from the vascular space to the tissue compartment (a continuum approach). Diploma thesis, Universität Stuttgart, Nupus preprint No. 2009/6 [2] Mosthaf K.; Baber, K.; Flemisch, B.; Helmig, R.; Leijnse, T.; Rybak, I. and Wohlmuth, B.: A new coupling concept for two-phase compositional porous media and single-phase compositional free flow. Submitted to Journal of Fluidmechanics (2010) [3] Balhoff, M. T. ; Thomas, S.G.; Wheeler, M. F.: Mortar coupling of pore-scale models. Comput Geosci. 12, 15-27 (2008)