^{1}

^{2}

^{1}

^{2}

Based on ideas proposed by Massoudi and Rajagopal (M-R), we develop a model for blood using the theory of interacting continua, that is, the mixture theory. We first provide a brief review of mixture theory, and then discuss certain issues in constitutive modeling of a two-component mixture. In the present formulation, we ignore the biochemistry of blood and assume that blood is composed of red blood cells (RBCs) suspended in plasma, where the plasma behaves as a linearly viscous fluid and the RBCs are modeled as an anisotropic nonlinear density-gradient-type fluid. We obtain a constitutive relation for blood, based on the simplified constitutive relations derived for plasma and RBCs. A simple shear flow is discussed, and an exact solution is obtained for a very special case; for more general cases, it is necessary to solve the nonlinear coupled equations numerically.

Amongst the multiphase (more accurately multicomponent) flows
occurring in nature, one can identify blood, mud slides, avalanches, and so
forth, and amongst the many flows found in the chemical industries, one can
name fluidization and gasification, and in agricultural and pharmaceutical
applications, one can name the transport, storage, drying of grains, and so
forth. In most applications, the phases are not of the same material and thus
it is more appropriate and accurate to refer to these cases as “multicomponent”
problems. Historically, two distinct approaches have been used in modeling
multicomponent flows (for simplicity in the remainder of the paper, we assume a
two-component flow). In the

The large number of articles published concerning two-component
flows typically employs one of the two continuum theories developed to describe
such situations: mixture theory (the theory of interacting continua) (Rajagopal
and Tao [

Mixture theory, or the Theory
of Interacting Continua, traces its origins to the work of Fick (1855) (see
Rajagopal [

It is known that in large vessels (whole) blood behaves as a Navier-Stokes
(Newtonian) fluid (see Fung [

In this paper, we first 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 in this model. We further assume that the plasma behaves as a linearly viscous fluid and the RBCs as an anisotropic nonlinear density-gradient-type fluid. We obtain a constitutive relation for blood, based on the simplified constitutive relations derived for plasma and RBCs. It is noted that we have only discussed the development a model and that specific boundary value problems need to be solved to test the efficacy of the model. A simple shear flow is studied and an exact solution is obtained for a very special case; for more general cases, it is necessary to solve the nonlinear coupled equations numerically.

In this section, we will provide a brief review of the
essential ideas and equations of a two-component mixture. At each instant of
time,

_{1} and _{2} are the bulk densities of the mixture components given by

Assuming no interconversion of mass between the two constituents, the equations for the conservation of mass for the two components are

Let

The balance of moment of momentum implies that

Mills [

In continuum mechanics, in addition to the balance
(governing) equations, possibly subjected to some constraints, one must also
specify meaningful (physical) boundary conditions, in order to have a
well-posed problem. If the equations are nonlinear, the multiple solutions and
stability of those solutions are very often of interest. The typical boundary
conditions used in analytical/numerical procedures to solve any differential
equations are (i) Dirichlet boundary conditions
where the value of the unknowns is prescribed on the boundary, (ii) Neumann
condition where the normal gradient of the unknowns is specified, and (iii)
boundary conditions where a combination of the unknown quantities and their
normal gradients is specified. The need for additional boundary conditions
arises in many areas of mechanics, whenever nonlinear or microstructural
theories are used. For example, in higher grade fluids in the mechanics
of non-Newtonian fluids (see Rajagopal and Kaloni [

One of the fundamental questions in mixture theory is concerned
with the boundary conditions and how to split the (total) traction vector,
related to the (total) stress tensor, or the (total) velocity vector (see Ramtani [

In Rajagopal and Massoudi’s approach no couple stresses are
allowed. Nevertheless, due to the higher order gradients of volume fraction,
they also find it necessary to provide additional boundary conditions for
solving practical and simple boundary value problems (see Massoudi [

For problems with bounded domains and with a certain degree
of symmetry, such as fully developed flow in a vertical pipe (see Gudhe et al.
[

For free surface flows, in general, the location of the free surface is not known and one must find this surface as part of the solution. For steady flows, one can use the kinematical constraint

Finally, for a complete study of a thermomechanical problem,
not only in mixture theory, but in continuum mechanics in general, the second law
of thermodynamics has to be considered. In other words, in addition to other
principles in continuum mechanics such as material symmetry, and frame
indifference, the second law imposes important restrictions on the type of motion and/or the
constitutive parameters (for a thorough discussion of important concepts in constitutive
equations of mechanics, we refer the reader to the books by Antman [

To obtain the
conservation of mass for the mixture, we add (

In the next section, we will discuss the approach of Massoudi and Rajagopal, as one of the possible approaches within mixture theory, in formulating constitutive equations for the stress tensors and the interaction forces.

Mathematically, the purpose of the constitutive relations is
to supply connections between kinematical, mechanical, electromagnetic, and
thermal fields that are compatible with the balance equations and that, in
conjunction with them, provide a theory that is solvable for properly posed
problems. Deriving constitutive relations for the stress tensors and the
interaction forces is among
the outstanding issues of research in multicomponent flows. In general, the
constitutive expressions for

In this part of the paper, we provide a brief description of
Massoudi and Rajagopal [M-R] approach. We then modify the M-R model to obtain a
constitutive relation for blood. The M-R model considers a mixture of an
incompressible fluid infused with solid particles, wherein the principles of mechanics
of granular materials are used to describe the behavior of particles. This
model has been used and discussed in Massoudi [

In most practical engineering problems studied by Massoudi and Rajagopal, the fluid can be assumed to behave as a linearly viscous fluid:

The stress tensor

_{2}, _{3}, and _{5} do
not necessarily go to zero when there are no particles, the above restrictions
are necessary. Furthermore, Rajagopal
and Massoudi [

Looking at (

The interaction force, in general, depends on the fluid
pressure gradient, the density gradient (the buoyancy forces), the relative
velocity (the drag force on the particles), the relative acceleration (the
virtual mass of the particles), the magnitude of the rate of the deformation
tensor of the fluid (the lift force on the particles), the spinning motion, as
well as the translation of particles (the Faxen’s force), the particles
tendency to move toward the region of higher velocity (the Magnus force), the
history of the particle motion (the Bassett force), the temperature gradient,
and so forth. For laminar flow of a mixture of an incompressible fluid infused
with particles, the mechanical interaction force can be assumed to be of the
form (Johnson et al. [

The
physical and biological processes governing the flow of blood are intimately responsible
for safety and efficacy of all blood-wetted medical devices. The quest to
design improved cardiovascular devices is
however stifled by the inadequacies of current understanding of blood trauma
and thrombosis. Contemporary design relies upon formula for blood describing
(1) rheology, (2) cell trauma (hemolysis), and (3) thrombosis, that are based
primarily on empiricism. These relations are application-specific at best, and
are more

One of the most important
rheological consequences of the multicomponent nature of blood in small vessels
is the migration of red cells towards the core of the flow and commensurate
enhancement of platelet concentration near the boundaries. Several
investigators attempting to simulate the deposition of platelets on biological
and artificial surfaces have found that single-continuum models significantly
under-predict the concentration of platelets near the boundary (see Sorenson et
al. [

The mechanism for the above
phenomenon, whereby platelets in flowing blood are propelled towards the
surface, can be understood in terms of their interaction with red blood cells.
Extensive experimental studies dating to the late 1920s (see Fahraeus and
Lindqvist [

In general, blood can be viewed as a suspension and modeled
using the techniques of non-Newtonian fluid mechanics. We, however, assume that
blood is a two-component mixture, composed of the red blood cells (RBCs) suspended
in a (platelet rich) plasma. In the following description, the plasma in the
mixture will be represented by

We assume that the mixture is saturated, that is,

Modelling the RBCs is practically more complicated since they
are anisotropic and deformable. The main reason for this difficulty is the
orientation or the alignment of these nonspherical cells. Materials possessing
microstructures, for example, with the internal couples or couple stresses,
were first studied in the early twentieth century by D. Cosserat and F.
Cosserat (Truesdell and Toupin [

To model the stress tensor for the RBCs, we modify the
constitutive relation derived by Massoudi [

To account for the shear-thinning effects observed in blood
flow, it is reasonable to assume that the viscosity coefficient

Finally, it needs to be mentioned that the RBCs themselves actually form a mixture made of a central fluid-like region bounded by a viscoelastic solid. Many researchers have measured the viscoelastic properties of RBCs. However, for the sake of simplicity, we have decided to treat the RBCs not as a mixture, but as an isotropic granular-like material.

Blood is a complex mixture composed of plasma, red blood
cells (RBCs), white blood cells (WBCs), platelets, and other proteins. A
complete constitutive relation for the stress tensor of the (whole) blood, not
only must capture and describe the rheological characteristics of its different
components, but also must include the biochemistry and the chemical reactions
occurring. To date, no such comprehensive and universal constitutive relation
exists. As mentioned by Anand et al. [

It has been reported that at low shear rates, blood seems to
have a high apparent viscosity (due to RBC aggregation) while at high shear
rates the opposite behavior is observed (due to RBC disaggregation) (see Anand
and Rajagopal [

An alternative way is to model blood as we have done here: a two-component mixture composed of plasma and RBCs. If we are interested in describing the global behavior of blood, then a stress tensor for blood can be assumed to be given by

Obviously, the
form of (

For a simple shear
flow, that is, flow between two horizontal plates a
distance “

Recalling that the
balance of linear momentum for blood as a mixture is given by (

Equations (

To obtain an analytical solution (closed form solution) to
the above (

In this paper, we have discussed, based on the classical mixture
theory and the approach taken by Massoudi and Rajagopal, a framework for modeling
the rheological behavior of blood. The proposed constitutive relation depends
on the form of the stress tensors for the plasma and the red-blood cells. In general,
the RBCs are assumed to behave as an anisotropic density-gradient-dependent viscous
fluid. As such the equations are highly nonlinear. After making many
simplifying assumptions, a relatively simple constitutive relation for the
stress tensor for blood is obtained (see (

Furthermore, it goes without saying that the model
developed here is only appropriate for a healthy human, and it does not capture
any blood disorder. It has also been shown that with regular exercise and
physical training certain characteristics of blood can change, and blood
undergoes what is known as “fluidification” (see Ernst and Matrai [

To include
the formation and growth of clots, and lysis of blood cells in blood, in
general, the reaction-convection-diffusion equations are to be solved in
conjunction with the balance laws for mass, linear and angular momentum, and
energy (for each component). Although we have ignored the biochemical effects
of blood in this paper, in principle, the theory is amenable to extension (see
Anand et al. [

Acceleration vector

Body force vector

Symmetric part of the velocity gradient

Interaction force vector

Identity tensor

Gradient of velocity vector

Fluid pressure

Stress tensor

Velocity vector

Spin tensor

Position vector

_{f}:

Second coefficient of (fluid) viscosity

First coefficient of (fluid) viscosity

Volume fraction of the solid

Density

_{o}:

Reference density

Volume fraction of fluid

Referring to the fluid component

Referring to the solid component

Referring to the mixture

Transpose

Dimensionless quantity

Divergence operator

Gradient operator

Trace of a tensor

Outer product

Dot product

Dedicated to our esteemed teacher, Professor K. R. Rajagopal.