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In the present study, a mathematical model of unsteady blood flow through parallel plate channel under the action of an applied constant transverse magnetic field is proposed. The model is subjected to heat source. Analytical expressions are obtained by choosing the axial velocity; temperature distribution and the normal velocity of the blood depend on y and t only to convert the system of partial differential equations into system of ordinary differential equations under the conditions defined in our model. The model has been analyzed to find the effects of various parameters such as, Hartmann number, heat source parameter and Prandtl number on the axial velocity, temperature distribution and the normal velocity. The numerical solutions of axial velocity, temperature distributions and normal velocity are shown graphically for better understanding of the problem. Hence, the present mathematical model gives a simple form of axial velocity, temperature distribution and normal velocity of the blood flow so that it will help not only people working in the field of Physiological fluid dynamics but also to the medical practitioners.

The study of blood flow has been carried out by several authors. During the last decades extensive research work has been done on the fluid dynamics of biological fluids in the presence of magnetic field. For multiple reasons, applications of magnetohydrodynamics in physiological flow problems are of growing interest. Many researchers have reported that the blood is an electrically conducting fluid [1-4]. The electromagnetic force (Lorentz force) acts on the blood and this force opposes the motion of blood and there by flow of blood is impeded, so that the external magnetic field can be used in the treatment of some kinds of diseases like cardiovascular diseases and in the diseases with accelerated blood circulation such as hemorrhages and hypertension.

In general, biological systems are affected by an application of external magnetic field on blood flow through human arterial system. Many mathematical models have already been investigated by several research workers to explore the nature of blood flow under the influence of an external magnetic field. Tzirtzilakis [

Arterial MHD pulsatile flow of blood under periodic body acceleration has been studied by Das and Saha [

Heat transfer in biological systems is relevant in many diagnostic and therapeutic applications that involve changes in temperature. As we know, the cardiovascular system is sensitive to changes in the environment, and flow characteristics of blood are modified to satisfy changing demands of the orgasm. In addition to transporting of oxygen, metabolites and other dissolved sub stances to and from the tissues, blood flow alters heat transfer within the body. Adhikary and Misra [

In the present investigation, a mathematical model for the unsteady blood flow through a very narrow parallel plate channel with heat source and external transverse magnetic field is presented. This work is an extensive study of Madhu et al. [

Consider flow between non-conducting two parallel plates as shown in

Here blood is supposed to be Newtonian, incompressible, homogenous and viscous fluid. Also, the viscosity of blood is considered to be constant. The effect of mag netic field is considered in this model which is applied in a direction perpendicular to the flow of blood.

Considering u and v as velocity components in the directions of x and y respectively (axial and normal respectively) at time t in the flow field, we may write the two dimensional boundary layer equations in presence of transverse magnetic field as

Introduce the following non-dimensional variables

Substituting from Equation (4) into the Equations (1)- (3) we may write these equations after dropping the stars as

From Equation (7) we can observe that the temperature distribution has 1^{st} derivative with respect to time t. From this observation and with the help of solution of partial differential equation by separation of vari ables technique we can get the following equation

It is observed that the solution of this equation will be on the form.

Similarly, the axial velocity u has the same concept, and then the solution of the problem will take the form mentioned in Section 3 and the boundary conditions are taken as:

With the help of discussion in the previous section, let us choose the solutions of the Equations (5)-(7) respectively as

Substituting from Equations (9)-(11) into Equations (5)-(8) we obtain the following equations respectively

where

The boundary conditions become:

Solution of Equation (14) is as follows

where.

Using the boundary conditions Equation (15) we obtain

Then the final form of H(y) is

From Equation (11) and (17) then the temperature distribution is given by

Substituting from Equation (17) into Equation (12) we get

Solving the last equation to obtain F using the Equation (15) as follows The Homogenous solution:

Substitute from Equation (15) to calculate the constants

The particular solution is:

The general solution of F is

From Equation (9) and Equation (19) the axial velocity of blood is given by

(20)

Also, from Equations (10) and (13) the normal velocity is given by

where C is an arbitrary constant (C = 1).

Equations (18), (20) and (21) show the temperature distribution, the axial velocity and normal velocity respectively.

The flow investigation has been carried out by studying the effect of individual factors like heat source and magnetic field. The main objective of the study is to find the role of heat source parameter, magnetic field (Hartmann number), Prandtl number and decay parameter on tem perature distribution, axial velocity and normal velocity. To observe these effects, numerical codes are developed for the numerical evaluations of the analytic results obtained.

In

The effect of magnetic field on the axial velocity for different values of Hartmann number (Ha = 1.00, 2.00, 3.00, 4.00, 6.00) is shown in

and attains maximum at y = 0 then decreases until y = 1. While at we observe that the axial velocity decreases along y.

In the present investigation, a mathematical model for the unsteady blood flow through a very narrow parallel plate channel with heat source and external transverse magnetic field is presented. This work is an extensive study of Madhu et al. [

The main conclusions of the present paper may be summarized as follows:

• The present mathematical model gives a simple form of axial velocity, temperature distribution and normal velocity of the blood flow. Analytical expressions are

obtained by choosing the axial velocity; temperature distribution and the normal velocity of blood depend on y and t only along with corresponding boundary conditions to convert the system of partial differential equations into system of ordinary differential equations.

• The temperature field increases with increasing the heat source parameter and Prandtl number while decreases with increasing the decay parameter.

• The axial velocity increases with increasing heat source parameter and Prandtl number while decreases with increasing the Hartmann number and decay parameter.

• The normal velocity decreases with increasing the decay parameter and tending to zero very fast for higher values of the decay parameter.

Hence, the present mathematical model gives a simple form of axial velocity, temperature distribution and normal velocity of the blood flow so that it will help not only people working in the field of Physiological fluid dynamics but also to the medical practitioners.

The author would like to express deeply thankful to referee for providing valuable suggestions to improve the quality of the manuscript.

: Density of blood

: Dynamic viscosity of the blood (constant)

: Pressure of blood

: Electrical conductivity of the blood

: Intensity of the magnetic field

: Gravitational acceleration

: Coefficient of volume expansion due to temperature

: Temperature of blood

: Temperature of the wall (fixed temperature)

: Coefficient of the thermal conductivity

: Specific heat at constant pressure

: Quantity of heat

: Temperature distribution

: Decay parameter

: Kinematic viscosity

: Prandtl number

: Heat source parameter

: Hartmann number