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Flow of blood in micro-channels: recent results based on mixture theory

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Abstract

Blood, composed of red blood cells (RBCs), white blood cells, platelets and plasma, is a non-linear fluid exhibiting complex behavior, such as plasma-skimming and the Fahraeus effect, which are observed especially in microscale applications. In this paper, we use Mixture Theory to model blood as a two-fluid (two-component) system. Plasma is treated as a viscous fluid and the RBCs are modeled as a nonlinear fluid with a shear dependent viscosity, with the effect of the hematocrit included. The drag force and the shear lift force between the RBCs and plasma are also accounted for. We present an overview of our recent studies which show very good agreement between our numerical results and available experimental results.

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Notes

  1. A complete constitutive relation for the stress tensor of the (whole) blood should not only describe the rheological characteristics of its different components, but also the biochemistry and the chemical reactions occurring in blood. To date no such comprehensive and universal constitutive relation exists. As mentioned by Anand et al. [53], “the numerous biochemical reactions that take place leading to the formation and lysis of clots, and the exact influence of hemodynamic factors in these reactions are incompletely understood.” Anand and Rajagopal [36] developed a model for blood that is capable of incorporating platelet activation. More recently, (Anand et al. [30, 54, 55]) have provided a framework whereby some of the biochemical aspects of blood, along with certain rheological (viscoelastic) properties of blood, are included in the formulation.

  2. The function, where \( 0 \le \phi < \phi_{ \hbox{max} } < 1 \), is a continuous function of position and time; in reality, \( \phi \) is either one or zero at any position and time, depending upon whether one is pointing to a particle or to the void space at that position. That is, the real volume distribution has been averaged, in some sense, over the neighborhood of any given position. It should be mentioned that in practice \( \phi \) is never equal to one; its maximum value, generally designated as the maximum packing fraction, depends on the shape, size, method of packing, etc.

  3. Finally, for a complete study of a thermo-mechanical problem, not only in the Mixture Theory, but in continuum mechanics in general, the Second Law of Thermodynamics (the Clausius–Duhem inequality) has to be considered (see Liu [56], Batra [57]). However, since in Mixture theory there is no general agreement on the form of the second law and since the Helmholtz free energy is not known, a complete thermodynamical treatment of the model used in our studies is lacking. In recent years, Rajagopal et al. (see for example, Rajagopal and Srinivasa [58, 59]) have devised a thermodynamic framework, called the Multiple Natural Configuration Theory where by maximizing the rate of entropy production they obtain a class of constitutive relations for many different types of materials.

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Acknowledgments

Funding was provided by National Institutes of Health (NIH grant 1 R01 HL089456).

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Authors and Affiliations

  1. Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Wei-Tao Wu & James F. Antaki

  2. Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA, 02115, USA

    Nadine Aubry

  3. U. S. Department of Energy, National Energy Technology Laboratory (NETL), 626 Cochrans Mill Road, P. O. Box 10940, Pittsburgh, PA, 15236, USA

    Mehrdad Massoudi

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  1. Wei-Tao Wu
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  4. Mehrdad Massoudi
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Correspondence to Mehrdad Massoudi.

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Wu, WT., Aubry, N., Antaki, J.F. et al. Flow of blood in micro-channels: recent results based on mixture theory. Int J Adv Eng Sci Appl Math 9, 40–50 (2017). https://doi.org/10.1007/s12572-016-0173-2

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