# Studying the Blood Plasma Flow past a Red Blood Cell with the Mathematical Method of Kelvin's Transformation

Maria Hadjinicolaou (School of Science and Technology, Hellenic Open University, Patras, Greece) and Eleftherios Protopapas (School of Science and Technology, Hellenic Open University, Patras, Greece)
DOI: 10.4018/ijmstr.2014010104
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## Abstract

A mathematical tool, namely the Kelvin transformation, has been employed in order to derive analytical expressions for important hydrodynamic quantities, aiming to the understanding and to the study of the blood plasma flow past a Red Blood Cell (RBC). These quantities are the fluid velocity, the drag force exerted on a cell and the drag coefficient. They are obtained by employing the stream function ? which describes the Stokes flow past a fixed cell. The RBC, being a biconcave disk, has been modelled as an inverted prolate spheroid. The stream function is given as a series expansion in terms of Gegenbauer functions, which converge fast. Therefore we employ only the first term of the series in order to derive simple and ready to use analytical expressions. These expressions are important in medicine, for studying, for example the transportation of oxygen, or the drug delivery to solid tumors.
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## 1. Introduction

The main function of the blood as it circulates through the body is to transport oxygen, nutrients and waste. The blood flow in large arteries is considered as the flow of a homogeneous viscous fluid where its particular composition is disregarded (Yamaguchi et al., 2006). However, when the blood flows through capillaries, the flow may be described as creeping or Stokes flow. This assumption is justified by considering the values of the physical characteristics of the fluid (velocity, density, viscosity) and also by taking into account the dimensions of the RBC. Some indicative values are, for the density of the blood plasma ρ=1gr/cm3, the velocity, v≤0.1 cm/sec, the viscosity η ≈ 10−2 g/(cm⋅sec) and the characteristic length S≈10−3cm. These imply that the Reynolds number, Re, becomes Re << 10−2, which confirms the creeping flow assumption, according to which the viscous forces dominate over inertial forces. Stokes flow has also been employed for describing the flow of many other bio-fluids, (Davis et al. 2003; Marhefka et al. 2009; Sugii et al. 2005; Dassios et al. 2012).

Aiming in the better understanding of the blood behaviour, many studies consider the blood as a multiphase fluid, namely the blood plasma, where three kinds of cells are suspending in it. These are the red blood cells, RBCs, the white blood cells and the platelets. Since the RBCs occupy about the 45% of the fluid, the study of the blood plasma flow around RBCs is significant.

Furthermore, the study of the blood flow at a cellular level needs also the mathematical modelling of another parameter, which is the geometrical shape of the RBC. As the blood flows into the vessels the normal RBCs, are deformable into different shapes depending on the hydrodynamic stresses that act on them. RBCs have been considered in the literature either as rigid or solid elastic spheres (Quinlan & Dooley 2007; Wang & Skalak 1969), truncated ellipsoids (Fitz-Gerald 1972), rigid pistons (Zien 1969) or more realistically, as a biconcave disc (Bagchi 2007; Noguchi & Gompper 2005; McWhirter et al. 2009; Dassios et al. 2012;), that at rest are having the major diameter of about 8μm and thickness of at least 2 μm (Sugii et al. 2005). Another category of studies consider that the normal RBCs are deformable into different shapes, as the blood flows into the vessels, depending on the hydrodynamic stresses that act on them. Different numerical approaches for solving this problem can be found in the literature. Some of them are the boundary element method (Youngren & Acrivos 1975; Pozrikidis 2001; Pozrikidis 2003), the immersed boundary methods (Bagchi 2007; Eggleton & Popel 1998), particle methods (Tsubota et al. 2006) etc. Experimental works regarding flow visualization techniques in microcirculation can be also found (Lima et al. 2009). All these methods and techniques although describe adequately the process, their utility is sometimes limited due to some mostly technical factors (spatial resolution etc.)

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