Modeling and numerical simulation of blood flow using the theory of interacting continua

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Abstract

In this paper we use a modified form of the mixture theory developed by Massoudi and Rajagopal to study the blood flow in a simple geometry, namely flow between two plates. The blood is assumed to behave as a two-component mixture comprised of plasma and red blood cells (RBCs). The plasma is assumed to behave as a viscous fluid whereas the RBCs are given a granular-like structure where the viscosity also depends on the shear-rate.

Highlights

► A modified form of the mixture theory is used to study the blood flow in a simple geometry, namely flow between two plates. ► Blood is assumed to behave as a two-component mixture comprised of plasma and red blood cells (RBCs). ► Plasma is tacit to behave as gluey fluid but RBCs given granular-like structure where viscosity also depends on shear-rate.

Introduction

The safety and efficiency of blood-wetted medical devices are closely tied to the physical and biological processes governing the transport of blood. This has led to research in blood rheology, trauma, and thrombosis. Despite these efforts, a fundamental understanding of device-induced blood trauma remains incomplete. Due to a lack of predictive mathematical tools, the primary means to study the blood trauma is the costly method of experimental trial-and-error. It is therefore necessary to develop more accurate models for blood trauma and hemorheology, especially for applications in the blood-wetted devices.

It is known that in large vessels (whole) blood behaves as a Navier–Stokes (Newtonian) fluid (see [28]; however, in a vessel whose characteristic dimension (diameter for example) is about the same size as the characteristic size of blood cells, blood behaves as a non-linear fluid, exhibiting shear-thinning and stress relaxation. Thurston [82], [83] pointed out the viscoelastic behavior of blood while stating that the stress relaxation is more significant for cases where the shear rate is low. The micro-scale flow and deformation of blood have been studied for many years. Early in vitro investigations in rotational viscometers or small glass tubes revealed the characteristic rheological properties such as the reduction in the blood apparent viscosity [20], Fahraeus effect [25] and Fahraeus–Lindqvist effect [26], revealing the manifested non-homogeneity of blood in microcirculation. Similarly, the microscopic phenomenon responsible for shear-thinning is found to be the flexibility and alignment of red blood cells (RBCs) at high shear, and shear-thickening due to the aggregation of cells at low shear (as shown in Fig. 21 of [29]). A further result of the multi-component character of blood is the plasma-skimming phenomenon, whereby the hematocrit in branches of blood vessels with size below 300 μm is reduced due to a phase separation and deficit of RBCs near the wall of the parent vessel [19]. Consideration of the cellular component also involves cell–cell interaction [33] and cell–surface interaction [32]. The migration toward the centerline and the rotation of the RBCs are believed to increase the platelet diffusivity and expel the platelet to the near wall region, platelet margination [2].

Microscopic models of blood flow [14] that account for cell-scale lift and drag forces, collisions, deformation, etc. have the potential to replicate some of the phenomena causing the non-homogeneous distribution of RBCs and platelets. However, for most problems of practical value, it is prohibitive to consider the three-dimensional dynamic interactions of individual blood cells—which may amount to several thousand to millions. This has motivated the pursuit of meso-scale multi-phase (or multi-component) models as a reasonable compromise between specificity and practicality. Motivated primarily by the plasma-skimming phenomenon [19], investigators have developed four classes of multi-phase models for blood: edge-core [79], averaging [42], immersed particle [37], and effective medium approach [70].

In this paper we advocate using Mixture Theory or the theory of Interacting Continua, to propose a two-component model for blood. The large numbers of articles published concerning multi-component flows typically employ one of the two continuum theories developed to describe such situations: Mixture Theory (or the theory of interacting continua) [73] or Averaging Method(s) [38]. Both approaches are based on the underlying assumption that each component may be mathematically described as a continuum. The Averaging method directly modifies the classical transport equations to account for the discontinuities or ‘jump’ conditions at moving boundaries between the components (cf. [5], [24], [31], [43], [42]). Although the two methods seem similar, the way they approach the formulation of constitutive models are very different. In fact, as shown in Massoudi [54], many of the interaction models used by researchers in the Averaging community are not frame-indifferent, thus violating basic principles in physics. Other differences between the two approaches are explained in Refs. [40], [41], [62], [63].

In order to better understand atherosclerosis, Jung et al. in 2008 [42] used the averaging approach to simulate a three-component blood flow (RBC–WBC–plasma) in the right coronary artery using a commercial CFD package software, ANSYS Fluent [1], but there are several limitations in that study. For example, the employed drag model is valid only for solid spherical particles or for fluid particles that are sufficiently small with low concentration, whereas RBCs in flows of interest are usually non-spherical and not negligible in size. Further, the 3-dimensional Eulerian–Eulerian code used is not appropriate for such dense concentration of RBCs (45% hematocrit) as studied. Also, Jung’s definition of the relative blood mixture viscosity is not very clear. Massoudi [58] has discussed this issue in more detail. Furthermore, many of constitutive models used in the averaging methods in general and specifically in Jung et al. [43] are not frame-indifferent, for example, the virtual mass term and the shear lift force violate the principle of frame-indifference (For more details see [54]).

In Section 2 of this paper, we provide a brief review of Mixture Theory, and then discuss certain issues in constitutive modeling of blood. In the present formulation we assume blood to form a mixture consisting of RBCs suspended in plasma, while ignoring the platelets, the white blood cells (WBCs) and the proteins in the sample. No biochemical effects or interconversion of mass are considered in this model. The volume fraction (or the concentration of the RBCs) is treated as a field variable. We further assume that the plasma behaves as a viscous fluid and the RBCs as an anisotropic non-linear density-gradient-type fluid (see [60]). In Section 3, we discuss the constitutive modeling of the stress tensors and the interaction forces. In Section 4, we study and solve numerically the equations of motion for the fully developed flow of such a mixture between two horizontal flat plates. Finally, in Section 5, we present numerical solutions for a few cases.

Section snippets

A brief review of Mixture Theory

Mixture Theory, or the Theory of Interacting Continua, traces its origins to the work of Fick in 1855 (see [71]) and was first presented within the framework of continuum mechanics by Truesdell [85]. It is a means of generalizing the equations and principles of the mechanics of a single continuum to include any number of superimposed continua. More detailed information, including an account of the historical development, is available in the articles by Atkin and Craine [11], [12], Bowen [17],

Constitutive equations

Deriving constitutive relations for the stress tensors and the interaction forces are among the outstanding issues of research in multicomponent flows. In general, the constitutive expressions for T1 and T2 depend on the kinematical quantities associated with both the constituents. However, it can be assumed that T1 and T2 depend only on the kinematical quantities associated with the plasma (component 1) and the RBCs (component 2), respectively (sometimes called the principle of component

Flow between two flat plates

Substituting Eqs. (10), (17) in (6)a we obtain the dimensionless forms of the two momentum equations in their expanded forms. These are, for the plasma (component 1)(1ϕ)ρf[V1τ+(gradV1)V1]=(gradϕ)P(1ϕ)gradP+Λ[(gradϕ)divV1+(1ϕ)grad(divV1)]+(1ϕ)(gradΛ)divV1+2Re[(gradϕ)D1+(1ϕ)divD1]+ρfFr(1ϕ)b1+C1gradϕ+C2F(ϕ)(V2V1)+C3ϕ(2trD12)1/4D1(V2V1)and for the RBCs (component 2)ϕρs[V2τ+(gradV2)V2]=(gradϕ)PϕgradP+[B31(ϕ+ϕ2)+B32(ϕ+ϕ2)Π]divD2+B31[grad(ϕ+ϕ2)]D2+B32[grad(ϕ+ϕ2)Π+(ϕ+ϕ2)gradΠ]D2+ρsFrϕb

Effect of Reynolds number (Re)

The effects of the (plasma) Reynolds number (Re) on the velocity profiles of both constituents are shown in Fig. 3. The corresponding concentration profiles are plotted in Fig. 4. It can be seen that increasing Re causes a reduction of centerline velocity for plasma and RBCs, exhibited by blunting of the profiles. A similar pattern is also reported in [40] for a mixture of granular particles and a fluid, however only the fluid velocity was blunter and the granular velocity remained parabolic.

Concluding remarks

The constitutive equation used in our study for the stress tensor of the RBCs is assumed to be isotropic, but in reality the RBCs are anisotropic and deformable. The difficulty is the orientation or the alignment of the non-spherical cells [20], [90]. This could be addressed by using an anisotropic representation for the RBCs, as proposed by Massoudi and Antaki [60], in which the stress tensor is a function of the symmetric part of velocity gradient, D2, and the orientation vector, n. This was

Acknowledgment

This project was supported by NIH R01 HL089456-01.

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