CeBa - A low-dimensional RBC model

Testing a low-dimensional model for RBC suspensions

We are currently testing a model for RBCs with much less degrees of freedom (and therefore much less computation) than the membrane model. The model is similar to the "low-dimensional" model of Pan et al [1] in the way RBCs are represented as a ring of overlapping spheres. Elasticity modules for stretch and bending deformations allow to quantitatively realize most of the deformation a real RBC can undergo.

In contrast to the DPD-based point-particle model of Pan et al, the spheres in our model are full bodies and their rotational motion is fully resolved. This results in much more accurate fluid-cell boundary conditions and normal and tangential stresses on the surfaces of the spheres. It also provides a way to mimick tank-treading through 2 additional degrees of freedom representing the motion of the membrane around the cell volume. Other possible extensions include the variability of sphere diameters as a response to local normal stresses under the conservation of the total cell volume.

At the moment, we study apparent viscosities and diffusion in dense suspensions of these particles and validate them against experimental data.

In the small simulation shown here we exposed a suspension of such particles with a volume fraction of $\phi=0.4$ to an unbounded shear flow. Hydrodynamic stresses and collisions with other particles cause the particles to deform. A slice through the domain shows pressure (color) and flow velocity vectors after subtraction of the Newtonian shear-flow profile.

Shear rates in the right part are lower than in the left. This is due to a higher local volume fraction and the resulting increase in the apparent viscosity in the right region. The left part has a lower viscosity and is sheared much more easily.

Note on the visualization: The total domain is decomposed into two subdomains each with its own reference frame. These reference frames are moving in respect to each other but are shown here statically. In the actual simulation the transition is smooth and the interface region is no different from any other point in the domain.

Applicability

This rather simple model of RBCs does not resolve the membrane of the cell and is therefore not able to reproduce tank-treading or any effect related to it. However, it is able to accurately reproduce blood properties (rheology and transport) for flows in geometries with characteristics larger than $\approx 30\mu\mathrm{m}$.

Pan et al Scales of Blood Models
This figure from Pan et al [2] illustrates the applicability of different models. The membrane model ("MSM") resolves physics down to below $1\mu\mathrm{m}$. The model presented here (similar to "LDM") has its own scope of application.

References

[1] W. Pan, B. Caswell, G. Karniadakis, Soft Matter 6(18), p 4366, 2010.
[2] W. Pan, D. Fedosov, B. Caswell, G. Karniadakis, Microvascular Research 82(2), p 163, 2011.