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Atheoretical model of blood- flow through arteries can be explained by non-linear wave propagation by including the non-linear motion of both the blood vessels and the surrounding tissue [1]. A complete model of blood-flow includes the three components (i) the flow of blood driven by the pressure gradient under the control of vessel elasticity, (ii) the motion of vessel wall with surrounding tissues caused due to the action of the blood-flow on the wall, (iii) the boundary conditions between the blood and wall in contact. The flow of blood can be expressed by means of the incompressible, viscous flow equations but the motion of arterial wall is quite important and needs constitutive equations to describe the motion of arterial wall. The following four assumptions have been made: 1. The blood vessels are locally triclinic, anisotropic, elastic incompressible and axisymmetric bodies of revolution. The vessels walls are thin as compared with the overall cross-sectional dimension. 2. The forces acting on the blood vessels will not affect the local orthoisotropy of the vessels. 3. The surrounding tissue of the blood vessels is modelled as a thin elastic medium for the aorta or as an elastic medium with the finite thickness for the arteries connected with peripheral tissues, muscle and other substances, the stress wave propagation in such a medium is dispersed with the increase of the radial distance from the vessel wall. 4. Every day the human body is exposed to body acceleration, for example vibration when travelling or driving a vehicle or rapid movements of the body in the sport [2].

Analysis of non-linear wave propagations of blood vessels is how to determine of eliminate the unknown constrained stresses of the peripheral tissue in the equations of motion. It is assumed that the peripheral tissue around a blood vessel is an axisymmetric elastic medium and that the wave propagation in such a medium will be dispersed with the increase of radial distance. Several experimental investigations have been carried out by some researchers (cf. Kobu [3], Inoue and Kabaya [4], Nishimoto et al. [5]) to examine the effects of infrared radiation/ultrasonic radiation on blood flow. The effect of radiative heat transfer on blood flow in a stenosed artery was studied theoretically by Prakash and Makinde [6]. By using a numerical model, He et al. [7] discussed the effect of temperature on blood flow in human breast tumor under laser irradiation. While flowing through the arterial tree, blood carries a large quantity of heat to different parts of the body. On the skin surface, the transfer of heat can take place by any of the four processes: radiation, evaporation, conduction and convection. It is known that in the case of radiative heat transfer, energy is transferred through space by means of electromagnetic wave propagation. There are several determinants for the quantity of heat that blood can carry with it, namely, (1) heat transfer coefficient of blood, (2) density of blood, (3) velocity of blood flow, (4) radius of the artery and (5) temperature of the tissues that surround the artery. Out of these, since Reynolds number is related to the velocity and density of blood as well as the arterial radius, the quantity of heat carried by blood can be regarded as dependent only on Reynolds number and heat transfer coefficient of blood, and the temperature of the tissues surrounding the artery. It may further be mentioned that blood flow enhances when a man performs hard physical work and also when the body is exposed to excessive heat environment. In cases like these, blood circulation cannot remain normal. In order to take care of the increase in blood flow, the dimensions of the artery have to increase suitably. It is known that when the temperature of the surroundings exceeds 20⁰C, heat transfer takes place from the surface of the skin by the process of evaporation through sweating and when the temperature is below 20⁰C, the human body loses heat by conduction and radiation both. Blood flow with radiative heat transfer was discussed by Ogulu and Bestman [8] on the basis of a theoretical study. Shah et al. [9] designed an instrument for measuring the heat convection coefficient on the endothelial surface of arteries. Effects of pulsatile blood flow in arteries during thermal therapy were studied by Craciunescu and Clegg [10] as well as by Horng et al. [11]. Shrivastava and Roemer [12] studied heat transfer rate from one blood vessel to another and also that from a blood vessel to tissue.

Velocities and Accelerations of Arterial Walls:

Let v_z and v_r be the axial and radial displacements of a point on the arterial wall, then

where R^* is the radius of the midsurface in equilibrium or at undeformed state.

For an axisymmertrical motion, the velocity components of the wall can be defined as:

From the theory of continuum mechanics, the acceleration components of wall can be defined as:

Since the thickness of the wall is very small, the spatial variation of velocity components along the thickness of the wall is negligible i.ecan be ignored. Thus (3) reduces to

Equations of Motion of The Vessels Surrounding Tissue System:

An important thing in analysis of non-linear wave propagations of blood vessels is how to determine of eliminate the unknown constrained stresses of the peripheral tissue in the equations of motion. It is assumed that the peripheral tissue around a blood vessel is an axisymmetric elastic medium and that the wave propagation in such a medium will be dispersed with the increase of radial distance.

Letdenote the density of the tissue,  the acceleration components of material point in the tissue and be the physical normal stresses in cylindrical co-ordinates . Since the motion of the tissue is axisymmetric, in a cylindrical co-ordinate system there exists only one shear stressT_rz.

Wu and Lee [12] have obtained the equations of the vessel wall- tissue system as:

 being the incorporation mass per unit area. In equation (5) and (6) are the reduced coefficients of surrounding tissues,being the axial and radial inertia coefficients respectively,reduced coefficients of tissue stresses.

Equations of the Blood Flow:

The blood is incompressible fluid consisting of particulate phase of red cells, white platelets and erythrocytes. The flow of blood in general, can be described in terms of the momentum equations and the continuity equation of the incompressible viscous two-phase flow. For axisymmetric laminar flow, the governing equations can be expressed as follows:

a. Fluid Phase

Momentum equations:

Where p is the density of the blood, u_ru_z denote the components of velocity of blood-flow along the r and z directions respectively, T is the stress tensor of blood  represents the three normal stress components of T and represents the shear stress components of T. these stress components can be determined in terms of the constitutive relationship expressing the physical properties of blood.

b. Particulate Phase

The blood cells are assumed to be spherical in shapes and are considered to be uniformly distributed. Following assumptions have also been made to make the analysis simple.

i. Chemical reaction, mass transfer and radiation between the particles and fluid are neglected.

ii. The temperature is uniform within the particles.

iii. The interaction between particles has not been taken into account.

Momentum equations

Where N is the number of particles, w_r and w_z are the components of velocity of particles along r and z directions respectively, m the mass of the blood suspended particles and K the Stokes resistance coefficient (which is 6πμa for spherical particle).

Equations (7) and (8) are valid for both Newtonian fluid and non-Newtonian fluid. However, for the flow of blood in large arteries, the mechanical behaviour of blood is very near to that of Newtonian fluid. Thus, if the blood is assumed to be Newtonian fluid, then equations (7) and (8) reduce to the following equations:

Where p and v denote the pressure and the kinematic viscosity of fluid respectively.

Constitutive equations

Stresses in equations (5) and (6) or (7) and (8) are to be determined with the help of the constitutive equations of arterial walls and fluid.

1. The stress strain relationships of arterial walls:

The arterial tissue is in substance a kind of non-linear viscoelastic material but as per information available in the literature, for large arteries, the elasticity dominates indeed the non-linear mechanical properties of arterial tissues, whereas the viscosity is a minor effect. As a matter of fact with this in view, if arterial tissue is modelled as a triclinic, transversely isotropic, elastic, incompressible and axisymmetric, material, the non-linear stress strain relationships of arterial wall can be given by:

Where j, k are the indices of sum for mid-surface coordinates(Wu and Lee), S_n is the stress along the normal coordinate to the arc in the meridian direction of the mid surface of the blood vessel, p¹denote the hydrostatic pressure of arterial tissue; E_j(j=) denote the physical strains. For a thin wall vessel following equations are used:

Then incompressibility condition provides:

The coefficientsare the constant combined by Lame’s elastic constants [2]. However, the hydrostatic pressure p¹ is generally unknown. Therefore, adding a complemental condition or assumption is necessary for the use of equation (15).

2. The Constitutive equations of fluid:

For an axisymmetric laminar flow, the constitutive equations of fluid in cylindrical coordinates can be expressed as:

Where  is the coefficient of viscosity of fluid. If the blood is assumed to be Newtonian fluid,  can be taken as a constant or a function of the temperature being independent of the strain rate. However, if necessary, the blood is also modelled as a Cassonian fluid of non-Newtonian type. In that case,  is recognised as the apparent viscosity and it can be determined from:

 n_cdenotes the viscosity of Casson, T_c is a quantity equivalent to the yield stress, J₂is the second order invariant of the strain rate of tensor.

The wall stresses of the fluid are given by:

 

Where the subscripts w and ( )_w represent the values of variables at the inner boundary wall.

The equations of motion for the blood vessel and its surrounding tissues as a system are an advantage over earlier results. These equations have been written in terms of the stresses of vessel wall and fluid and the geometry of blood vessels. The stresses of vessel wall are determined by using the constitutive relationships of non-linear elasticity theory developed by Wu and Lee [14]. The formulation of non-linear visco-elasticity theory could be used, because the stresses of the vessel wall in the equations of motion are in fact the generalized ones before the substitution by the concrete stress strain rate relationships of arterial wall. Equations of blood-flow together with equations of motion of vessel walls are given the basic equations of non-linear wave propagation.

The basic equations can be used to solve the problems for the propagation of non-linear pulse wave in arteries together with the constitutive equations of fluid and vessel walls along with relevant initial boundary conditions.

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