An analytical study of effect of heat source on MHD blood flow through bifurcated arteries has been done. The blood flowing through arteries is treated to be unsteady Newtonian flow. The coupled linear partial differential equations are solved by converting into ordinary linear differential equations by choosing the axial velocity, normal velocity and temperature field as a functions of y and t along with corresponding boundary conditions. The expressions are obtained for axial velocity, normal velocity and temperature field. The effects of various parameters like Prandtl Number (Pr), Heat Source Parameter (S) and Magnetic Field (M) on axial velocity, normal velocity and temperature field are investigated. It was found that heat source and magnetic field modify the flow patterns and increase the temperature of the blood.

Topics
Thermodynamic states and processes, Mathematical modeling, Partial differential equations, Newtonian flow, Fluid flows, Diseases and conditions, Haemodynamics, Organs, Blood pressure, Cardiovascular system

The literature of biomathematics has provided huge number of applications in medicine and biology. MHD application reduces rate of flow of blood in human arterial system, which is useful in treatment of certain cardiovascular disorders (Korchevskii and Marchounik9) and in the problems which increase the rate of circulation of blood like hemorrhage and hypertension etc. Extensive research work has been done on the fluid dynamics of biological fluid in the presence of magnetic field. Huge number of magnetic devices has been devoloped for cell separation, drug carriers, cancer tumor treatment etc. Heat transfer in blood has so many applications in muscle and skin tissues and thermal therapy etc.

The idea of electromagnetic fields in medical research was firstly given by Kolin8 and later Korchevskii et al.9 discussed the possibility of regulating the movement of blood in human system by applying magnetic field. Vardhanyan16 studied the effect of magnetic field on blood flow. Halder7 analyzed the effect of magnetic field on blood flow through an indented tube in presence of erythrocytes. A mathematical model for biomagnetic fluid dynamics, suitable for the description of the Newtonian blood flow under the action of an applied magnetic field has been proposed by Tzirtzilakis.15 Singh and Rathee13 studied two-dimensional of blood flow with variable viscosity through stenotic artery in the presence of transverse magnetic field in the porous medium. A numerical study on effect of magnetic field on the blood flow in artery having multiple stenoses has been done by Varshney et al..17 

In the past, there have been a number of studies to examine heat transfer in blood vessels. Barcroft and Edholm2 studied the effect of temperature on blood flow and deep temperature in the human forearm. They gave the variation in blood flow as a result of changes in temperature of the surrounding water.

The effect of local temperature on blood flow in the human foot discussed by Allwood and Burry.1 Charm et al.5 experimentally investigated heat transfer in small tubes of diameter 0.6 mm in a water bath, while Victor and Shah18 computed heat transfer for uniform heat flux and uniform wall temperature cases for fully developed flow and in the entrance region. The correlation equations for estimating the heat transfer under different configurations and diameters of blood vessels were developed by Chato.6 Lagendijk10 analyzed temperature distributions in the entrance region around the vessels during hyperthemia. Barozzi and Dumas3 calculated heat transfer in the entrance region considering the rheological properties of the blood stream and a cell free peripheral plasma layer at the vessel wall.

Ogulu and Abbey12 have analyzed the simulation of heat transfer on an oscillatory blood flow in an indented porous artery. A mathematical model describing the dynamic response of heat and mass transfer in blood flow through bifurcated arteries under stenotic conditions has been proposed by Chakravarty and Sen.4 Obdulia and Taehong Kim11 presented calculations of the temperature distributions in an atherosclerotic plaque experiencing an inflammatory process; it analyzes the presence of hot spots in the plaque region and their relationship to blood flow, arterial geometry, and inflammatory cell distribution. A mathematical model for heat transfer to blood flow in a small tube proposed byWang.19 

The present contribution is the combined effect of heat transfer and magntic field on blood flow through bifurcated arteries. Our work is an extensive study of Suri and Suri14 with heat transfer under the condition defined in our model. Suri and Suri14 analyzed a bifuration model to study the magnetic field on the nature of local disturbances and high shear forces which causes certain cardiovascular lesions like atherosclerotic plaques, aneurysms and intimal cushions etc. Zamir and Roach20 proposed a mathematical model to predict the nature of many cardiovascular lesions such as aneurysms, intimal cushions, and atherosclerotic plaques. They considered the flow in a two-dimensional bifurcation with a symmetrical flow divider perfused with steady flow at variable Reynolds numbers.

The real blood circulation system consists of three-dimensional elastic tubes of varying cross-section and angle of bifurcation. The angle of bifurcation in the branched arteries varies too much. Arterial junction geometries are not fixed because elastic walls yield to the unsteady pressure pulses of the flowing blood. But for the sake of mathematical convenience, a two dimensional bifurcation model of similar geometry as taken by Zamir and Roach20 is considered. In the model it was selected to provide analysis for the effect of heat in unsteady blood flow in branch dynamics superseding their analysis for steady flow.

In the present attempt, blood is considered to be Newtonian, incompressible, homogeneous and viscous fluid flowing in a non-conducting parallel plate channel in such a way that the flow is from trunk to the branches. The rate of mass flow at any cross-section perpendicular to the direction of flow is m = 2bρV, where V is the mean velocity of flow, b is branch diameter and ρ is blood density. To simplify, the angle of bifurcation is taken to be zero so that a parallel stream is divided into two streams.

In addition, thickness of the bifurcating wall is considered to be negligibly small in order to have the rate of mass flow at any cross-section of branched channel as m/2. Since the breadth of the channel under consideration (which is parallel to large artery) is much larger than 1mm, the viscosity of blood can be treated to as constant throughout this analysis- that is by ignoring the Fahreus-Lindquist effect.

With the above assumptions, the unsteady flow of blood in the presence of heat source and magnetic field is governed by two dimensional boundary layer equations where u* and v* are the velocity components in the direction of x* and y* respectively at time t in the flow field. ρ and η is the density and viscosity of the blood while p* stands for pressure. KT is thermal conductivity, CP is the specific heat at constant pressure. Qis the quantity of heat, T is the temperature and β is the volumetric expansion parameter while θ is the temperature distribution.

\begin{equation}\frac{{\partial u^* }}{{\partial t^* }} + \frac{1}{\rho }\frac{{dp^* }}{{dx^* }} = \frac{\eta }{\rho }\frac{{\partial ^2 u^* }}{{\partial y^{*2} }} + g\beta \left( {T - T_\infty } \right)^* - \frac{{\sigma B_0^2 }}{\rho }u\end{equation}
∂u*∂t*+1ρdp*dx*=ηρ∂2u*∂y*2+gβT−T∞*−σB02ρu
(1)
\begin{equation}\frac{{\partial u^* }}{{\partial x^* }} + \frac{{\partial v^* }}{{\partial y^* }} = 0\end{equation}
∂u*∂x*+∂v*∂y*=0
(2)
\begin{equation}\frac{{\partial T^* }}{{\partial t^* }} = \frac{{K_T }}{{\rho \,C_P }}\frac{{\partial ^2 T^* }}{{\partial y^{*2} }} + \frac{Q}{{\rho C_P }}\left( {T - T_\infty } \right)^*\end{equation}
∂T*∂t*=KTρCP∂2T*∂y*2+QρCPT−T∞*
(3)

The Boundary conditions are:

\begin{equation}\begin{array}{l@\quad l@\quad l}\theta = e^{ - \lambda ^2 t} & v = e^{ - \lambda ^2 t} & at \, y = - 1 \\[3pt]\theta \to 0 & u \to 0 & at \, y = 1 \\\end{array}\end{equation}
θ=e−λ2tv=e−λ2taty=−1θ→0u→0aty=1
(4)

Let us introduce the non-dimensional variables,

\begin{equation*}x = \frac{{x^* }}{b}\quad u = \frac{{u^* }}{{(m/2b\rho)}}\quad h(x,t) = \frac{{(dp^* /dx^*)}}{{(\eta m/2b^3 \rho)}}\quad \tau = \frac{\eta }{\rho }\end{equation*}
x=x*bu=u*(m/2bρ)h(x,t)=(dp*/dx*)(ηm/2b3ρ)τ=ηρ
\begin{equation*}y = \frac{{y^* }}{b}\quad v = \frac{{v^* }}{{(m/2b\rho)}}\quad t = \frac{{t^* }}{{(b^2 \rho /\eta)}}\quad \theta = \frac{{\theta ^* \left( {2b^3 \rho ^2 } \right)}}{{m\eta }}\end{equation*}
y=y*bv=v*(m/2bρ)t=t*(b2ρ/η)θ=θ*2b3ρ2mη

Then omitting the stars, the dimensionless forms of the equations (1)–(3) are:

\begin{equation}\frac{{\partial u}}{{\partial t}} + h = \frac{{\partial ^2 u}}{{\partial y^2 }} + g\beta \theta - M^2 u\end{equation}
∂u∂t+h=∂2u∂y2+gβθ−M2u
(5)
\begin{equation}\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\end{equation}
∂u∂x+∂v∂y=0
(6)
\begin{equation}\frac{{\partial \theta }}{{\partial t}} = \frac{1}{{\tau P_r }}\frac{{\partial ^2 \theta }}{{\partial y^2 }} + \frac{S}{{\tau P_r }}\theta\end{equation}
∂θ∂t=1τPr∂2θ∂y2+SτPrθ
(7)

where, S (heat source parameter) = |$\displaystyle \frac{{Qb^2 }}{{K_T }}$|Qb2KT⁠, Pr (Prandtl number) = |$\displaystyle \frac{{\rho C_P }}{{K_T }}$|ρCPKT

Let us choose the solutions of (5)–(7) respectively as,

\begin{equation}u = F\left( y \right)e^{ - \lambda ^2 t}\end{equation}
u=Fye−λ2t
(8)
\begin{equation}v = G(y)e^{ - \lambda ^2 t}\end{equation}
v=G(y)e−λ2t
(9)
\begin{equation}\theta = H(y)e^{ - \lambda ^2 t}\end{equation}
θ=H(y)e−λ2t
(10)

The boundary conditions (4) are transformed to

\begin{equation}\begin{array}{l@\quad l@\quad l}H = 1 & F = 1 & at\,y = - 1 \\[3pt]H \to 0 & F \to 0 & at\,y = 1 \\\end{array}\end{equation}
H=1F=1aty=−1H→0F→0aty=1
(11)

By virtue of (8)–(10), the equations (5)–(7) respectively transform to,

\begin{equation}\frac{{d^2 F}}{{dy^2 }} + z^2 F = h_1 - g\beta H\end{equation}
d2Fdy2+z2F=h1−gβH
(12)
\begin{equation}G = A\,{\rm (a \;constant)}\end{equation}
G=A(a constant )
(13)
\begin{equation}\frac{{d^2 H}}{{dy^2 }} + (S + \frac{{P_r \lambda ^2 }}{\tau })H = 0\end{equation}
d2Hdy2+(S+Prλ2τ)H=0
(14)

From equation (14) we get

\begin{equation}H = \frac{1}{{2\cos S_1 }}\cos S_1 y - \frac{1}{{2\sin S_1 }}\sin S_1 y\end{equation}
H=12cosS1cosS1y−12sinS1sinS1y
(15)

where |$S_1 = \sqrt {S + \frac{{P_r \lambda ^2 }}{\tau }}$|S1=S+Prλ2τ⁠, |$z = \sqrt {\lambda ^2 - M^2 }$|z=λ2−M2

Using the boundary conditions (11), the solution of equation(12) is obtained as,

\begin{equation}F = \alpha \; + \;\alpha _1 \cos S_1 y + \alpha _2 \sin S_1 y + \alpha _3 \cos zy + \alpha _4 \sin zy\end{equation}
F=α+α1cosS1y+α2sinS1y+α3coszy+α4sinzy
(16)

From equation (8) and (16), the velocity of the flow of blood parallel to the direction of bifurcated artery is obtained as,

\begin{equation}u = (\alpha \; + \;\alpha _1 \cos S_1 y + \alpha _2 \sin S_1 y + \alpha _3 \cos zy + \alpha _4 \sin zy)e^{ - \lambda ^2 t}\end{equation}
u=(α+α1cosS1y+α2sinS1y+α3coszy+α4sinzy)e−λ2t
(17)

where

\begin{equation*}h_1 = \frac{h}{{e^{ - \lambda ^2 t} }}\;,\quad \alpha = \frac{{h_1 }}{{z^2 }}\;,\quad \alpha _1 = - \frac{{g\beta }}{{2\left( {z^2 - S^2 } \right)\cos S_1 }}\;,\quad \alpha _2 = \frac{{g\beta }}{{2\left( {z^2 - S^2 } \right)\sin S_1 }}\end{equation*}
h1=he−λ2t,α=h1z2,α1=−gβ2z2−S2cosS1,α2=gβ2z2−S2sinS1
\begin{equation*}\alpha _3 = \frac{{1 - 2\alpha - 2\alpha _1 \cos S_1 }}{{2\cos z}}\;,\quad \alpha _4 = - \frac{{1 + 2\alpha _2 \sin S_1 }}{{2\sin z}}\end{equation*}
α3=1−2α−2α1cosS12cosz,α4=−1+2α2sinS12sinz

From equation (9)and (13), the velocity of the flow of blood perpendicular to the direction of bifurcated artery is obtained as

\begin{equation}v = Ae^{ - \lambda ^2 t}\end{equation}
v=Ae−λ2t
(18)

And from equation (10) and (15), temperature distribution is given by

\begin{equation}\theta = \left( {\frac{1}{{2\cos S_1 }}\cos S_1 y - \frac{1}{{2\sin S_1 }}\sin S_1 y} \right)e^{ - \lambda ^2 t}\end{equation}
θ=12cosS1cosS1y−12sinS1sinS1ye−λ2t
(19)

Theoretical results such as axial velocity, normal velocity and temperature field for the blood are obtained in this analysis. The numerical solutions of axial velocity, normal velocity and temperature are shown graphically for different values of magnetic field parameter (M), Prandtl number (Pr) and heat source parameter (S) against y for better understanding of the problem.

Figure 1 illustrates the behavior of temperature field at |$t = 1,\,\lambda = 0.5,\tau = 0.5,\,\Pr = 0.71$|t=1,λ=0.5,τ=0.5,Pr=0.71 and for different values for heat source parameter (S=1, 1.5, 2, 2.5). It is observed that the temperature decreases with increasing the values of S upto y ⩽ 1.2 and it increases from y ⩾ 1 (for different values of S). Figure 2 describes the effect of different values of Prandtl number (Pr=0.71, 3, 7) at t = 1, λ = 0.5, τ = 0.5, S = 1. The effect of Prandtl number on temperature is same as heat source parameter. It is decreasing with increasing the values of Parndtl number upto y ⩽ 1.4 its reverse effect is observed from y ⩾ 1.2 (for different values Pr).

FIG. 1.
FIG. 1. Temperature field for different values of Heat source parameter (S).
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Temperature field for different values of Heat source parameter (S).

FIG. 1.
FIG. 1. Temperature field for different values of Heat source parameter (S).
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Temperature field for different values of Heat source parameter (S).

Close modal
FIG. 2.
FIG. 2. Temperature field for different values of Prandtl number (Pr).
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Temperature field for different values of Prandtl number (Pr).

FIG. 2.
FIG. 2. Temperature field for different values of Prandtl number (Pr).
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Temperature field for different values of Prandtl number (Pr).

Close modal

We have set these parameters (t = 1,τ= 0.5, h = 0.5,β= 0.5, g = 9.8) fix to understand the effect of heat source parameter, Prandtl number, magnetic field parameter on the axial velocity of blood. From figure 3, we observed that axial velocity of blood is increasing for increasing values of magnetic field parameter (M), while heat source parameter(S) is kept constant at(S=6). For M=1.4, the axial velocity is increasing very fast against y while for M=1 velocity is increasing slowly against y. Figure 4 indicates the effect of heat source parameter (S) on axial velocity of blood against y. From figure it is clear that the velocity increases with increasing values of heat source parameter(S) for y ⩽ 0.8 and represents reverse effect for 0.8 ⩽ y ⩽ 2. Axial velocity for different values of Prandtl number (Pr) is shown in Fig. 5. From Fig. 5, we conclude that the velocity is increasing for increasing values of Pr for y ⩽ 0.8 and decreasing for the same values of Pr for y ⩾ 0.8.

FIG. 3.
FIG. 3. Axial velocity for different values of Magnetic field parameter M.
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Axial velocity for different values of Magnetic field parameter M.

FIG. 3.
FIG. 3. Axial velocity for different values of Magnetic field parameter M.
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Axial velocity for different values of Magnetic field parameter M.

Close modal
FIG. 4.
FIG. 4. Axial velocity for different values of heat source parameter (S).
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Axial velocity for different values of heat source parameter (S).

FIG. 4.
FIG. 4. Axial velocity for different values of heat source parameter (S).
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Axial velocity for different values of heat source parameter (S).

Close modal
FIG. 5.
FIG. 5. Axial velocity for different values of Prandtl number.
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Axial velocity for different values of Prandtl number.

FIG. 5.
FIG. 5. Axial velocity for different values of Prandtl number.
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Axial velocity for different values of Prandtl number.

Close modal

It is clear from the fig.6 that the normal velocity is decreasing with increasing values of λ and also for increasing values of t. The velocity is decreasing slowly for λ= 1 while it is tending to zero very fast for λ= 2.5. It is due to the velocity is exponential function.

FIG. 6.
FIG. 6. Normal velocity for different values of decay parameter(λ).
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Normal velocity for different values of decay parameter(λ).

FIG. 6.
FIG. 6. Normal velocity for different values of decay parameter(λ).
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Normal velocity for different values of decay parameter(λ).

Close modal

The present work is effect of heat source on MHD blood flow through bifurcated arteries, which is of great interest for the pupose of medical sciences. Magnetic field applied is affecting the flow of blood which is useful for the problem like blood pressure hypertension etc. Applications of heat transfer is showing the variations in temperature of the object which is helpful for the pupose of thermal therapy in the treatment of tumor, glands etc.

Authors are extremely thankful to referee for providing valuable suggestions to improve the quality of the manuscript.

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