Blood Flow in an Inclined Tapered Stenosed Porous Artery under the Influence of Magnetic Field and Heat Transfer

Blood Flow in an Inclined Tapered Stenosed Porous Artery under the Influence of Magnetic Field and Heat Transfer Adeyemi Damilare Adeoye ([email protected]) African Institute for Mathematical Sciences (AIMS) Cameroon Supervised by: Dr. Jos Usman Abubakar University of Ilorin, Ilorin, Nigeria 18 May 2018 Submitted in Partial Fulfillment of a Structured Masters Degree at AIMS-Cameroon Abstract A tapered inclined porous artery with stenosis was considered under the influence of magnetic field and heat transfer. The mathematical formulation for the momentum and energy equations of the blood flow considered to be Newtonian were obtained. The energy equation which was obtained by taking an extra factor of heat source and the nonlinear momentum equation were simplified under the assumption of mild stenosis. These equations were non-dimensionalized and solved using Differential Transform Method (DTM) to obtain expressions for velocity, temperature and volumetric flow rate. The graphs of the expressions were plotted against radius of the artery to simulate the effects of magnetic field, heat transfer and other fluid parameters on the velocity, temperature and the volumetric flow rate of the blood. It was observed that as the magnetic field parameter (M) increases, the velocity, temperature and the volumetric flow rate of the blood increase but wall shear stress decreases at the stenosis throat. It was further observed that the effects of heat transfer and magnetic field resulted into a greater variation in the volumetric flow of an inclined artery in the converging region than in the diverging region. Declaration I, the undersigned, hereby declare that the work contained in this essay is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly. Adeyemi Damilare Adeoye, 18 May 2018. i Contents Abstract i 1 General Introduction 1 1.1 Introduction ......................................... 1 1.2 Aim and Objectives ..................................... 1 1.3 Background of Study .................................... 1 2 Concepts, Assumptions and Basic Equations 3 2.1 Introduction ......................................... 3 2.2 Description of a Flow .................................... 3 2.3 Viscosity: Newtonian and Non-Newtonian Fluids ...................... 4 2.4 Variable Viscosity ...................................... 5 2.5 Governing Equations ..................................... 6 2.6 Basic Assumptions ...................................... 7 2.7 Mathematical Formulation ................................. 13 3 Differential Transform Method (DTM) 18 3.1 Introduction ......................................... 18 3.2 Principle of Differential Transform Method (DTM) .................... 18 3.3 Solution using DTM ..................................... 19 3.4 Volumetric Flow Rate .................................... 22 3.5 Wall Shear Stress ...................................... 23 4 Results, Summary and Conclusion 24 4.1 Results and Summary .................................... 24 4.2 Conclusion .......................................... 29 Acknowledgements 31 References 33 ii 1. General Introduction 1.1 Introduction Blood is a suspension of erythrocytes (red blood cells), leukocytes (white blood cells), and platelets in an aqueous electrolyte solution known as plasma. In normal blood, erythrocytes constitute 45% of the total volume of blood. This volumetric fraction of the erythrocytes defines an important variable called hematocrit. Stenosis, a constriction in blood vessels, is one of the leading causes of death in many countries of Africa and the world at large. Severe stenosis reduces the blood supply, thereby causing critical flow conditions which results in serious effects called carotid artery blockage, a major contributing factor to strokes [23]. This results from the brain not receiving enough blood as the plague builds up and hardens down the artery. When the brain lacks adequate supply of blood, the cells in it begin to die. This leads to severe disability or death of an individual. Due to these risks, it is important to watch out for the symptoms of a carotid artery blockage, like the Transient Ischemic Attack (TIA) and stroke, so that proper measures can be taken before the condition gets worse. The study of fluid dynamics has enabled many researchers to examine the mathematical and physical behaviour of blood as it circulates in the blood vessels. 1.2 Aim and Objectives The aim of this work is to study the influence of external magnetic field and heat transfer on blood flow in an inclined tapered porous artery with stenosis. The following were the objectives of the study: •to develop equations of motion for modeling blood flow in a stenotic tapered inclined porous artery under the assumptions made. •to find a numerical solution to our modeled equations using the Differential Transform Method (DTM). •to analyze and simulate, using Mathematica generated codes, the effects of the inclination angle, the magnetic field and heat source parameters on blood flow velocity, temperature and volumetric flow rate through graphs. 1.3 Background of Study In the past decades, many investigators have displayed interest in problems arising from blood flow mechanism and characteristics under different conditions and influence of external factors. The idea of electromagnetic fields in medical research was first given by Kolin [9] and the possibility that the application of magnetic field to blood would regulate its movement in human system was later 1 Section 1.3. Background of Study Page 2 studied by Korchevskii and Marochnik [10]. Abdullah et al. [1] and Bose and Banerjee [3] discussed magnetic particle capture for biomagnetic fluid flow in stenosed aortic. The above mentioned authors observed that the effect of magnetic field is to slow down the speed of blood. But, they did not analyze the magnetic effect on blood flow through an inclined artery. If a magnetic field is applied to an electrically conducting fluid in motion, electric and magnetic fields are induced which interact and a body force known as Lorentz force is produced, and has a propensity to either assist or oppose the fluid motion. Chakraborty et al. [4] analyzed the suspension of blood flow through an inclined tube with an axially non-symmetric stenosis. In the case of stenosis when the cholesterol deposits on the wall of the artery, the artery-clogging blood which clots inside the lumen of the coronary artery is considered as being equivalent to a fictitious porous medium. This case is particularly examined by some researchers. El-Shahed [7] presented a model for pulsatile blood flow through a stenosed porous artery under the effect of periodic body acceleration. Akbarzadeh [2] presented a numerical simulation of the effect of periodic body acceleration and periodic body pressure gradient on magneto-hydrodynamic (MHD) blood flow through porous artery. All of these published works, however, do not include the analysis of blood flow characteristics through an inclined artery with the applied magnetic field, which was presented by Srivastava [23]. Blood viscosity, although constant in real physiological system, it may vary in ratio of hematocrit or depend on temperature and pressure [22]. Massoudi and Christie [12], Pantokratoras [16], Nadeem and Akbar [14] analyzed the influence of heat transfer with temperature dependent viscosity. Petrofsky [18] examined the effect of the moisture content of heat source on the blood flow response of the skin through data. Prakash et al. [20] developed a model for bifurcated arteries to analytically study the effects of heat source on magneto-hydrodynamic (MHD) blood flow. In this work, we present combined effects of the external heat source and magnetic field on an inclined tapered stenosed porous artery, considering variable viscosity of the blood flow. Under well-defined boundary conditions, the resulting non-linear differential equations of the model have been solved numerically using Differential Transform Method (DTM). 2. Concepts, Assumptions and Basic Equations 2.1 Introduction In this chapter, we will describe the Lagrangian and Eulerian flow concepts and define Newtonian and Non-Newtonian fluids. In our model, blood will be considered an incompressible Newtonian fluid. We will consider temperature and pressure dependent blood viscosity. Our model equations will be derived in this chapter, from the continuity equation, the momentum equations and the energy equation. The resulting equations will be solved numerically in chapter 3using the Differential Transform Method. 2.2 Description of a Flow In order to describe a flow, we either use the ’Lagrangian’ description or the ’Eulerian’ description [5]: 2.2.1 Lagrangian Description of a Flow. Consider a fluid flow where each constituent particle carries its own flow properties such as density (ρp(t)), velocity (~vp(t)), pressure (pp(t)), etc. As the particle moves, its properties may change in time. The law of conservation of mass and Newton’s law apply to each fluid particle. This description of fluid flow in which detailed properties of each fluid particle is taken into account is known as the Lagrangian description. 2.2.2 Eulerian Description of a Flow. We can otherwise decide to keep record of how the flow properties change as time changes, at every point in space - That is, the flow properties at a specified location depend on the location and on time. For example, the velocity (~v), density (ρ), pressure (p), etc, of the flow can be described by position and time, and this may allow us to write ~v(~x, t),p(~x, t), ρ(~x, t), etc. 2.2.3 The Material (or Substantial) Derivative. The material derivative denoted by D/Dt, is a Lagrangian concept. It is defined as the rate of change of a flow property associated with a fluid particle ’p’ with respect to time. We can express the material derivative in terms of Eulerian quantities, say f(~x, t), so that the conservation laws can be applied in the Eulerian reference frame. Figure 2.1: Substantial derivative of a property frelative to the motion of a particle p 3 Section 2.3. Viscosity: Newtonian and Non-Newtonian Fluids Page 4 The rate at which fchanges with respect to time relative to a particle ’p’ moving with a velocity ~vpis the substantial derivative of fgiven by [5] Df(~xp, t) Dt = lim δt→0 f(~xp+~vpδt, t +δt)−f(~xp, t) δt .(2.2.1) By Taylor Series expansion about (~xp, t)and noting that ~vp=δ~x/δt, we have f(~xp+~vpδt, t +δt) = f(~x, t) + δt∂f(~x, t) ∂t +δ~x · ∇f(~x, t) + O(δ2).(2.2.2) From (2.2.1) and (2.2.2), neglecting O(δ2)and higher order terms in (2.2.2), we see that the material derivative of fis given by: Df Dt =∂f ∂t +~vp· ∇f. In general notation, we write [5] D Dt |{z} Lagrangian =∂ ∂t +~vp· ∇ | {z } Eulerian .(2.2.3) The flow described by (2.2.3) is said to be steady if ∂/∂t ≡0. The flow is said to be incompressible if D/Dt ≡0. A steady flow is a strictly Eulerian concept while an incompressible flow is a strictly Lagrangian concept [5]. 2.3 Viscosity: Newtonian and Non-Newtonian Fluids Viscosity is defined as the internal stickiness of a fluid [19]. It is a quantity that describes a fluid’s resistance to flow. Newton (1642 - 1727) observed that if a substance is heated, it will become less viscous and if cooled, it will become more viscous. He proposed that [11], for the straight and parallel motion of a given fluid, the shear stress between two adjoining layers tangent to the direction of the flow, is proportional to the velocity gradient in a direction perpendicular to the layers. Mathematically, τ∝∂u ∂y , or τ=µ∂u ∂y ,(2.3.1) where τis the tangential stress (or viscous force) between the two layers, ∂u/∂y is the distance rate of change of velocity (i.e., the velocity gradient as δy →0) and µis called the coefficient of viscosity. Fluids that obey Newton’s law of viscosity (2.3.1) are called Newtonian fluids. Fluids that behave differently are called non-Newtonian fluids - The viscosity of a non-Newtonian fluid changes with shear rate. Non-Newtonian fluids are classified as dilatents, pseudoplastics, and ideal plastics. Although, viscosity changes with temperature (and with pressure, with little impact), it may be regarded as a constant for a particular temperature for Newtonian fluids. As fluid comes in contact with the solid boundary of the adjoining layers, it sticks to the boundary and its velocity relative to this boundary Section 2.4. Variable Viscosity Page 5 Figure 2.2: Fluid viscosity Figure 2.3: Newtonian and non-Newtonian fluid. [19] becomes zero - any motion of the fluid would constitute an abrupt change. Hence, for a viscous liquid, a condition that there should be no-slipping at solid boundaries must be met. This condition is known as the no-slip condition. For blood, the no-slip condition is satisfied for blood particles as they come in contact with the wall of the artery. 2.4 Variable Viscosity Blood can be seen as a suspension of red blood cells in plasma. Consider an incompressible flow of the blood such that the density ρis uniform throughout. Viscosity, µ(r)however, varies as the blood flows in the radial direction. Einstein’s model of variable viscosity is: µ(r) = µ0(1 + λh(r)),(2.4.1) where µ0is the coefficient of viscosity of plasma, λis a constant which takes the value 2.5for blood (suspension of red blood cells which are considered to be of spherical shape) and h(r)is the volume fraction of the red blood cells, known as hematocrit. Section 2.5. Governing Equations Page 6 The analysis will be carried out using the following empirical formular for hematocrit [25]: h(r) = H1−r d0m,(2.4.2) and Hr=λH, (2.4.3) where Hrepresents the maximum hematocrit at the mid line of the artery, mrepresents the parameter which determines the shape of the velocity profile of blood (m≥2) and Hris the volumetric ratio of red blood cells in the blood and is known as the hematocrit parameter. 2.5 Governing Equations Any fluid flow is governed by three basic laws namely: law of conservation of mass, law of conservation of energy, and law of conservation of momentum. These laws apply to a specified mass of the fluid and are expressed mathematically using the Lagrangian flow description. Let ρbe the fluid density, V the volume (system), Σ~ Fthe resultant force acting on the system, ˙ Qthe time rate of energy change, ˙ Wthe time rate of work, ~v the velocity component of the flow and D/Dt the Lagrangian derivative following a fluid particle. Then, we state these three laws as follows: •The law of conservation of mass states that, in a closed system, mass can neither be created nor destroyed. In other words, mass conservation law states that a quantity can neither be added nor removed from a system, hence, the mass of the system remains constant. Mathematically, 0 = D Dt Zsys ρdV. (2.5.1) •Let Edenote the energy of the fluid system of mass m. We define Eas follows: E=me, where eis the heat energy density given by (ρcpT), with cpthe heat capacity and Tthe absolute temperature. The law of conservation of energy states that the difference between the rate of heat transfer and the rate of work done by a system equals the rate of change of the energy Eof the system. In essence, the energy applied to a system and the internal energy of the system are in equilibrium, so that energy remains constant. Mathematically, ˙ Q−˙ W=D Dt Zsys eρdV. (2.5.2) •The law of conservation of momentum states that the algebraic sum of the forces acting on a system equals the rate of change of momentum of the system. Mathematically, Σ~ F=D Dt Zsys ~vρdV. (2.5.3) The law of conservation of energy and the law of conservation of momentum are respectively known as the first law of thermodynamics and Newton’s second law. Section 2.7. Mathematical Formulation Page 13 Consider the function Φvcalled the viscous dissipation function - the rate at which the work done against viscous forces in the form of kinetic energy, is converted by an irreversible process, into internal energy of the blood. Φvis defined by [21] Φv=2µ"∂vr ∂r 2 +1 r ∂vθ ∂θ +vr r2 +∂vz ∂z 2#+µr∂ ∂r vθ r+1 r ∂vr ∂θ 2 +µ1 r ∂vz ∂θ +∂vθ ∂z 2 +µ∂vr ∂z +∂vz ∂r 2 −2 3µ(∇ · ~v)2. There is no motion in θ-direction and ∇ · ~v = 0, hence, we have Φv= 2µ"∂vr ∂r 2 +vr r2+∂vz ∂z 2#+µ∂vr ∂z +∂vz ∂r 2 . Also, the blood transports heat energy as it flows in and out of the control volume, hence, we introduce the term −Qp(T−T0), where Qpis a dimensional heat parameter defined as blood perfusion mass flow rate times the specific heat of blood, Tis the local temperature of the blood, and T0is the temperature of the blood at the stenotic region. Adding these extra energy terms to (2.6.12), we obtain ρcpvr ∂T ∂r +vθ r ∂T ∂θ +vz ∂T ∂z +∂T ∂t =k∂2T ∂r2+1 r ∂T ∂r +1 r2 ∂2T ∂θ2+∂2T ∂z2+ 2µ"∂vr ∂r 2 +vr r2+∂vz ∂z 2# +µ∂vr ∂z +∂vz ∂r 2 −Qp(T−T0), ρcpvr ∂T ∂r +vz ∂T ∂z =k1 r ∂ ∂r r∂T ∂r +∂2T ∂z2+ 2µ"∂vr ∂r 2 +vr r2+∂vz ∂z 2# +µ∂vr ∂z +∂vz ∂r 2 −Qp(T−T0), or ρcpvr ∂T ∂r +vz ∂T ∂z =k r ∂ ∂r r∂T ∂r +k∂2T ∂z2+ 2µ"∂vr ∂r 2 +vr r2+∂vz ∂z 2# +µ∂vr ∂z +∂vz ∂r 2 −Qp(T−T0).(2.6.13) 2.7 Mathematical Formulation Consider a tapered stenosed porous artery inclined at an angle γwith an externally applied magnetic field (M) as shown in Figure 2.6. Section 2.7. Mathematical Formulation Page 14 (a) Geometry of the axisymmetric stenosis in a porous artery. [15] (b) Geometry of the tapered stenosed porous artery for different tapering angles. [15] Figure 2.6: Schematic diagram for the geometry of the tapered porous artery with axisymmetric stenosis inclined at an angle γ. Let d(z)be the radius of the tapered artery in the region of stenosis, with [25]: d(z) = d0+ξz, then, the geometry of the stenosis in dimensionless form is defined by [13,25]: h(z) = (d(z)1−ηbn−1(z−a)−(z−a)n, a ≤z≤a+b, d(z),otherwise,(2.7.1) where d0is the radius of the non-tapered artery in the non-stenotic region, nis a parameter which determines the shape of the constriction profile (n= 2 for a symmetrically shaped stenosis and n≥2 for a non symmetric stenosis), bis the length of the stenosis and ξis the tapering parameter defined by ξ= tan φ, where φis the tapered angle which assumes value φ < 0,φ > 0and φ= 0 in the converging, diverging and non-tapered regions, respectively. Let δbe the maximum height of the stenosis at the location: [25] z=a+b nn n−1 ,(2.7.2) then, we define the parameter ηin (2.7.1) by: [25] η=δn n n−1 d0bn(n−1). Equations (2.6.9), (2.6.10), (2.6.11) and (2.6.13) are dimensional equations of motion which govern the flow of blood in the stenosed artery under the effect of magnetic field and heat transfer. These equations are not easy to solve by the number of parameters they include. This number of parameters, however, can be reduced by introducing non-dimensional groups. We therefore introduce the following non-dimensional variables into (2.6.9), (2.6.10), (2.6.11) and Section 2.7. Mathematical Formulation Page 15 (2.6.13) to get the equivalent non-dimensional equations of motion [25]:        vr=uu0δ b, r =r0d0, z =z0b, vz=w0u0, h =h0d0, ¯ P=u0bµ0P d2 0 , Re=ρbu0 µ0,Θ = T−T0 T0, Pr=µcp k, Ec=u2 0 cpT0, Z=k1 d2 0 , M2=σH2 0d2 0 µ0, Q =Qpd2 0 k, Gr=gαd3 0T0 ν2. (2.7.3) where Re,Pr,Ec,Gr,Θ,Z,M, and Qrespectively represent the Reynolds number, Prandtl number, Eckert number, Grashof number, temperature parameter, porosity parameter, magnetic field parameter and dimensionless heat source parameter. The term ν=µ0/ρ is the dynamic viscosity, αis the coefficient of thermal expansion, and k1is the permeability of the porous medium. In order to adopt simple notations, we will drop the prime 0on r,z,wand h. This basically changes nothing since we now have them in their dimensionless forms and by considering d0= 1. We assumed a symmetrically shaped mild stenosis in which case we have δ d0 1,(2.7.4) and the conditions [25,26] Reδn 1 n−1 b1,(2.7.5) d0n 1 n−1∼ O(1),(2.7.6) then, the pressure gradient in the r-direction is negligible compared to the pressure gradient in the z-direction, that is, ∂P ∂r ∂P ∂z . Also, condition (2.7.4) implies ∂vr ∂z 1. Therefore, given the conditions (2.7.4), (2.7.5), (2.7.6), the no-slip boundary condition, and that h(z), defined in (2.7.1), is the geometry of the stenosis in nondimensional form, then (2.6.9), (2.6.10), (2.6.11), and (2.6.13) in their non-dimensional forms become •Continuity Equation ∂vz ∂z = 0.(2.7.7) •Momentum Equation (r-direction) ∂P ∂r = 0.(2.7.8) •Momentum Equation (z-direction) From (2.4.2) and the non-dimensional group (2.7.3), taking µ0=µm= 1, and with d0= 1, we obtain the following:            µ=µ0(1 + Hr[1 −rm]) =⇒∂µ ∂r =−Hrmrm−1, σ1=M2µ0 H2 0d2 0 =M2µ0 H2 0 , k1=d2 0Z=Z, T0=ν2Gr gα =⇒T−T0= ΘT0= Θν2Gr gα . (2.7.9) Section 2.7. Mathematical Formulation Page 16 Putting (2.7.9) in (2.6.11), we obtain the following: ∂P ∂z =1 r ∂ ∂r µr ∂w ∂r −M2µ0 H2 0 H2 0w+ρgαΘν2Gr gα cos γ−µw Z =∂µ ∂r ∂w ∂r +µ r ∂w ∂r +µ∂ ∂r ∂w ∂r −M2w−(1 + Hr(1 −rm)) w Z+GrΘ cos γ[µ2 0/ρ taken as 1] =−Hrmrm−1∂w ∂r +1 r+Hr1 r−rm−1∂w ∂r +1 + Hr(1 −rm)∂ ∂r ∂w ∂r  −wM2+1 Z+Hr Z(1 −rm)+GrΘ cos γ =1 r+Hr1 r−rm−1−Hrmrm−1∂w ∂r +1 + Hr(1 −rm)∂ ∂r ∂w ∂r  −wM2+1 Z+Hr Z(1 −rm)+GrΘ cos γ that is, the momentum equation in dimensionless form is: ∂P ∂z =1 r+Hr1 r−(m+ 1)rm−1∂w ∂r +1 + Hr(1 −rm)∂ ∂r ∂w ∂r  −wM2+1 Z+Hr Z(1 −rm)+GrΘ cos γ. (2.7.10) •Energy Equation From the non-dimensional group (2.7.3), taking µ0=u0= 1, and with d0= 1, we obtain the following:        T=T0Θ + T0=⇒∂ ∂r r∂T ∂r =T0∂ ∂r r∂Θ ∂r  1 cp=EcT0 u2 0 =⇒µ=k cpPr=kT0EcPr, Qp=kQ =⇒Qp(T−T0) = kT0QΘ. (2.7.11) where, Br=EcPris the ratio of the heat produced by viscous dissipation to that transported by molecular conduction, i.e., the ratio between viscous heat generation and external heating. Putting (2.7.11) in (2.6.13), we get the following: 0 = k rT0 ∂ ∂r r∂Θ ∂r +kT0EcPr∂w ∂r 2 −kT0QΘ, and dividing through by kT0, we obtain the dimensionless energy equation: 1 r ∂ ∂r r∂Θ ∂r +EcPr∂w ∂r 2 −QΘ = 0.(2.7.12) To solve the momentum and energy equations (2.7.10) and (2.7.12), we use the axisymmetric boundary conditions of the flow axial velocity at the mid line of the artery: ∂w ∂r = 0,∂Θ ∂r = 0 at r = 0,(2.7.13) and the no-slip boundary conditions at the wall of the artery: w= 0,Θ = 0, at r =h(z),(2.7.14) Section 2.7. Mathematical Formulation Page 17 where h(z), defined by h(z) = (1 + ξ0z)[1 −η0((z−a0)−(z−a0)n)] when a0≤z≤a0+ 1,(2.7.15) is the geometry of the stenosis when the radius d0of the artery is of unit length. In (2.7.15), η0,a0and ξ0are defined as follows: η0=δ0nn n−1 (n−1) with δ0=δ d0 , a0=a b, ξ0=ξb d0 . 3. Differential Transform Method (DTM) 3.1 Introduction In mathematics and nature, we come up with problems which are essentially nonlinear. These problems can be solved through analytical or numerical approach. In the past years, many analytical methods have been presented for solving nonlinear problems, some of which are the [27] Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM), Adonian Decomposition Method (ADM), Weighted Residual Method (WRM), etc. The Differential Transform Method (DTM) is one of the numerical methods for solving differential and integral equations. The method was first introduced by Zhou [27] who used it to solve linear and nonlinear electrical circuits problem, and since then, many scientists have developed immense interests in the applications of the DTM to solve various scientific problems. The DTM is based on the Taylor series expansion of a function and is indeed, an alternative method for obtaining analytic Taylor series solution of differential equations. One major advantage of the DTM is that it can be used to solve nonlinear differential equations without going through the process of linearization and discretization, hence, errors due to discretization are inherently avoided by the DTM. In this chapter, we will present the principle of Differential Transform Method for solving differential equations, and use it to solve the energy and momentum equations (2.7.12) and (2.7.10). 3.2 Principle of Differential Transform Method (DTM) In order for us to understand the concept of DTM, we suppose that a function f(x)is analytic in a domain D, and that xiis any point in the domain. The Taylor series expansion of f(x)near x=xiis defined by f(x) = ∞ X k=0 (x−xi)k k!f(k)(xi),(3.2.1) where f(k)(xi)represents the k-th derivative of f(x)at x=xi. By taking xi= 0, we obtain the Maclaurin series: f(x) = ∞ X k=0 xk k!f(k)(0).(3.2.2) 3.2.1 Definition (Differential Transform of f(x)).The differential transform of the function f(x)at the point x= 0 is defined as follows: F(k) = Hk k!f(k)(0),(3.2.3) where F(k)is the transformed function and f(x)is the original function. The differential spectrum of F(k)is defined in the interval x∈[0, H], where His a constant whose value is mostly taken to be 1. 18 Section 3.3. Solution using DTM Page 19 3.2.2 Definition (Differential Inverse Transform of F(k)).The differential inverse transform of F(k) is defined as follows: f(x) = ∞ X k=0 x HkF(k).(3.2.4) In real applications, the function f(x)is expressed by a finite series, hence, we write (3.2.4) as: f(x) = n X k=0 x HkF(k).(3.2.5) Some important mathematical operations resulting from definitions (3.2.1) and (3.2.2) are presented in the following theorems [24], and will be used in the next session to solve our blood flow model equations (2.7.10) and (2.7.12): 3.2.3 Theorem. If f(x) = αu(x)±βv(x), then F(k) = αU(k)±βV (k).(3.2.6) where αand βare constants. 3.2.4 Theorem. If f(x) = ∂r ∂xru(x), then for ran integer, F(k) = (k+ 1)(k+ 2) · · · (k+r)U(k+r).(3.2.7) 3.2.5 Theorem. If f(x) = u(x)v(x), then F(k) = k X l=0 U(l)V(k−l).(3.2.8) 3.2.6 Theorem. If f(x) = xr, then for ran integer, F(k) = δ(k−r) = (1,if k=r, 0,if k6=r. (3.2.9) 3.3 Solution using DTM Let the differential transform of w(r)and Θ(r)be W(k)and G(k), respectively. We write the momentum equation (2.7.10) and the energy equation (2.7.12) respectively as follows: (1 + Hr)r−1∂w ∂r −Hr(m+ 1)rm−1∂w ∂r + (1 + Hr)∂2w ∂r2−Hrrm∂2w ∂r2 −M2+1 Z+Hr Zw+Hr Zrmw+GrΘ cos γ−∂P ∂z = 0, and r−1∂Θ ∂r +∂2Θ ∂r2+EcPr∂w ∂r 2 −QΘ=0, Section 3.3. Solution using DTM Page 20 and obtain their respective differential transforms as follows, using the theorems in section 3.2: (1 + Hr) k X l=0 δ(k−l+ 1)(l+ 1)W(l+ 1) −Hr(m+ 1) k X l=0 δ(k−l−m+ 1)(l+ 1)W(l+ 1) + (1 + Hr)(k+ 1)(k+ 2)W(k+ 2) −Hr k X l=0 δ(k−l−m)(l+ 1)(l+ 2)W(l+ 2) −M2+1 Z+Hr ZW(k) + Hr Z k X l=0 δ(k−l−m)W(l) + Grcos γG(k)−∂P ∂z δ(k)=0,(3.3.1) and k X l=0 δ(k−l+ 1)(l+ 1)G(l+ 1) + (k+ 1)(k+ 2)G(k+ 2) +EcPr k X l=0 (k−l+ 1)W(k−l+ 1)(l+ 1)W(l+ 1) −QG(k) = 0.(3.3.2) Applying equation (3.2.3) in the boundary conditions, taking H= 1, we find that W(0) = 1 0!w(h(z)) = a1, G(0) = 1 0!Θ(h(z)) = a2, W(1) = 1 1!w0(0) = 0, G(1) = 1 1!Θ0(0) = 0. where a1and a2are constants which can be determined by using the boundary conditions (2.7.14). From (3.3.1), we obtain: W(k+ 2) = Hr (1 + Hr)(k+ 1)(k+ 2)(m+ 1) k X l=0 δ(k−l−m+ 1)(l+ 1)W(l+ 1) +δ(k) (1 + Hr)(k+ 1)(k+ 2) ∂P ∂z +W(k) (1 + Hr)(k+ 1)(k+ 2) M2+1 Z+Hr Z +Hr (1 + Hr)(k+ 1)(k+ 2) k X l=0 δ(k−l−m)(l+ 1)(l+ 2)W(l+ 2) −Grcos γG(k) (1 + Hr)(k+ 1)(k+ 2) −Hr Z(1 + Hr)(k+ 1)(k+ 2) k X l=0 δ(k−l−m)W(l),(3.3.3) and from (3.3.2), we obtain G(k+ 2) = QG(k) (k+ 1)(k+ 2) −1 (k+ 1)(k+ 2) k X l=0 δ(k−l+ 1)(l+ 1)G(l+ 1) −EcPr (k+ 1)(k+ 2) k X l=0 δ(k−l+ 1)W(k−l+ 1)(l+ 1)W(l+ 1).(3.3.4) From (3.3.3) and (3.3.4), and taking m= 2, we get the following iterations: Section 3.3. Solution using DTM Page 21 For k= 0, W(2) = M2+1 Z+Hr ZW(0) −Grcos γG(0) + ∂P ∂z 2(1 + Hr) =a1 2(1 + Hr)M2+1 Z+Hr Z−Grcos γ 2(1 + Hr)a2+1 2(1 + Hr) ∂P ∂z , G(2) = QG(0) −EcPrW(1)2 2=Qa2 2. For k= 1, W(3) = Hr(m+ 1) 6(1 + Hr)δ(2 −m)W(1) + W(1) 6(1 + Hr)M2+1 Z+Hr Z−Grcos γG(1) 6(1 + Hr)= 0, G(3) = QG(1) −EcPr[2W(2)W(1) + 2W(1)W(2)] 6=QG(1) −4EcPrW(1)W(2) 6= 0. For k= 2, W(4) = Hr(m+ 1) 12(1 + Hr)δ(3 −m)W(1) + 2δ(2 −m)W(2)+Hr 6(1 + Hr)δ(2 −m)W(2) +W(2) 12(1 + Hr)M2+1 Z+Hr Z−Hr 12Z(1 + Hr)δ(2 −m)W(0) −Grcos γG(2) 12(1 + Hr) =1 24(1 + Hr)2a1M2+1 Z+Hr 1 Z−a2Grcos γ+∂P ∂z ·8Hr+M2+1 Z+Hr 1 Z −a1Hr 1 Z−1 2Qa2Grcos γ, G(4) = QG(2) −EcPr6W(3)W(1) + 4W(2)2 12 =Q2a2 24 −EcPr 12(1 + Hr)2a1M2+1 Z+Hr Z+∂P ∂z −a2Grcos γ2 . For k= 3, W(5) = Hr(m+ 1) 20(1 + Hr)δ(4 −m)W(1) + 2δ(3 −m)W(2) + 3δ(2 −m)W(3) +Hr 20(1 + Hr)[2δ(3 −m)W(2) + 6δ(2 −m)W(3)] + W(3) 20(1 + Hr)M2+1 Z+Hr Z −Hr 20Z(1 + Hr)δ(3 −m)W(0) + δ(2 −m)W(1)−Grcos γG(3) 20(1 + Hr)= 0 G(5) = QG(3) −EcPr[8W(4)W(1) + 12W(3)W(2)] 20 = 0. Section 3.4. Volumetric Flow Rate Page 22 Hence, w(r) = a1+a1 2(1 + Hr)M2+1 Z+Hr Z−Grcos γ 2(1 + Hr)a2+1 2(1 + Hr) ∂P ∂z r2 +1 24(1 + Hr)2a1M2+1 Z+Hr 1 Z−a2Grcos γ+∂P ∂z ·8Hr+M2+1 Z+Hr 1 Z −a1Hr 1 Z−1 2Qa2Grcos γr4,(3.3.5) Θ(r) = a2+Qa2 2r2+(Q2a2 24 −EcPr 12(1 + Hr)2a1M2+1 Z+Hr Z+∂P ∂z −a2Grcos γ2)r4. (3.3.6) The boundary conditions (2.7.14) imply: w(h(z)) = a1+a1 2(1 + Hr)M2+1 Z+Hr Z−Grcos γ 2(1 + Hr)a2+1 2(1 + Hr) ∂P ∂z h2 +1 24(1 + Hr)2a1M2+1 Z+Hr 1 Z−a2Grcos γ+∂P ∂z ·8Hr+M2+1 Z+Hr 1 Z −a1Hr 1 Z−1 2Qa2Grcos γh4= 0,(3.3.7) Θ(h(z)) = a2+Qa2 2h2+(Q2a2 24 −EcPr 12(1 + Hr)2a1M2+1 Z+Hr Z+∂P ∂z −a2Grcos γ2)h4= 0. (3.3.8) The Wolfram Mathematica 11.3 software is used to solve (3.3.7) and (3.3.8) simultaneously for a1and a2. The values obtained for a1and a2are then substituted into (3.3.5) and (3.3.6) to have expressions for the velocity, w(r)and temperature, Θ(r)which were then used to simulate velocity and temperature profiles. The expressions are not shown due to their lengths. 3.4 Volumetric Flow Rate The volumetric flow rate, Qvis defined as Qv= 2πZh 0 rwdr, (3.4.1) where wis the obtained axial velocity and ris the artery radius. Substituting (3.3.5) into (3.4.1), and integrating, we obtain Qv=a1πh2+a1π 4(1 + Hr)M2+1 Z+Hr Z−πGrcos γ 4(1 + Hr)a2+π 4(1 + Hr) ∂P ∂z h4 +π 72(1 + Hr)2a1M2+1 Z+Hr 1 Z−a2Grcos γ+∂P ∂z ·8Hr+M2+1 Z+Hr 1 Z −π 3Za1Hr−π 6Qa2Grcos γoh6.(3.4.2) The values obtained for a1and a2are substituted into (3.4.2) to have expressions for the volumetric flow rate. The expression is not shown due to its length. Section 4.2. Conclusion Page 29 increasing heat source parameter. Figure 4.3e shows the variation of the volumetric flow rate in the diverging, converging and non-tapered regions of the artery with the inclination angle of the artery. It shows that the volumetric flow rate decreases as the inclination angle increases from 0to π/3. 4.1.4 Variation of Wall Shear Stress (τs) with the Height of Stenosis (δ). Figures 4.4a,4.4b and 4.4c respectively indicate the variation of the wall shear stress at stenosis throat for different values of the magnetic field parameter (M), hematocrit parameter (Hr) and the porosity parameter (Z). (a) Effects of Wall Shear Stress at Stenosis Throat with the Magnetic Field. (b) Effects of Wall Shear Stress at Stenosis Throat with the Hematocrit Parameter. (c) Effects of Wall Shear Stress at Stenosis Throat with the Porosity Parameter. Figure 4.4: Variation of Wall Shear Stress at Stenosis Throat. Figure 4.4a shows that the wall shear stress decreases as the magnetic field parameter increases. We have seen that the magnetic field speeds up the blood flow and this allows the blood to flow with a reduced interaction with the walls of the artery, hence producing a minimal shear. Figure 4.4b indicates the behaviour of wall shear stress as the hematocrit parameter varies. It shows that the wall shear stress increases as the hematocrit parameter increases in value. Figure 4.4c displays how the wall shear stress varies as the porosity parameter varies. It shows that the wall shear stress decreases as the porosity parameter increases in value. 4.2 Conclusion The electrically conducting blood in motion and the effects of an external magnetic field and heat transfer on the blood flow in an inclined tapered stenosed porous artery have been studied through this Section 4.2. Conclusion Page 30 project. The mathematical formulation for the momentum and energy equations was obtained for the blood flow considered to be Newtonian fluid. The resulting equations of motion were solved numerically using the Differential Transform Method (DTM). Various fluid parameters were used to study the effect of heat transfer and magnetic field on the velocity, temperature, volumetric flow rate and the wall shear stress of the blood. The following are the findings obtained: 1. An increase in the magnetic field parameter, Mincreases the velocity and temperature profiles of the blood flow due to the presence of the Lorentz force which assists the motion of the blood. The curves representing the volumetric flow rate show that the volumetric flow rate increases as the magnetic field parameter increases and is greater in the converging region as compared to the diverging region. It is however observed that the wall shear stress decreases at the stenosis throat as the magnetic field parameter increases. 2. An increase in the voids present in the porous medium increases the velocity and temperature profiles of the blood. With an increasing porosity parameter, the volumetric flow rate decreases greatly in the converging region of the artery as compared to the diverging and non-tapered regions. The wall shear stress decreases at the stenosis throat with an increasing porosity parameter. 3. Under the influence of heat transfer and magnetic field, there is a greater variation in the volumetric flow rate of an inclined artery in the converging region than in the diverging region. 4. More red blood cells in the blood means lesser velocity and temperature of the blood as the hematocrit parameter, Hrvaries inversely as the velocity and temperature profiles. An increase in the hematocrit parameter increases the wall shear stress. The volumetric flow rate shows a reverse behaviour when the hematocrit parameter is increased and this behaviour is greater in the converging region than in the diverging region of the tapered artery. 5. By increasing the heat source parameter, we observed that the curves representing both the velocity and temperature profiles deviate rapidly from the origin. The volumetric flow rate varies inversely with the heat source parameter for converging, diverging and non-tapered regions of the artery. 6. It has been observed that, with heat transfer and applied magnetic field, the velocity and temperature profiles of blood in an inclined artery decrease as the Grashof number, Grincreases. 7. The Velocity and temperature profiles increase as the angle of inclination, γof the artery increases. With the increase of inclination angle of the artery, it was observed that the volumetric flow rate will increase and will be more in the converging region than in the non-tapered and diverging regions of the artery. 8. With the increase in the height of stenosis, we observed that the velocity and temperature profiles increase, as the curves representing them shift away from the origin. 9. The curves describing the velocity and temperature profiles show the remarkable variation with Brinkman number. It was observed that, the higher the value of the Brinkman number, the higher the temperature rise of the blood. The velocity of blood flow was also observed to increase with an increasing value of Brinkman number. Acknowledgements I thank the Almighty God for His grace and protection over my life, making me successfully complete this programme without any form of sickness or casualty throughout. I appreciate my project supervisor, Dr. Abubakar Usman Jos for keenly supervising my final project and for believing in my ability. Thanks to the leadership of African Institute for Mathematical Sciences for giving me the rare opportunity of being part of the family. I am grateful to the lecturers, all the tutors and staff of AIMS, Cameroon for their contribution to the success of the programme. I thank my parents for the solid support they have given to me up to this level. Finally, I appreciate all my friends and colleagues in class for being wonderful and great collaborators in the various group works. 31 References [1] Abdullah, I., Amin, N., and Hayat, T. (2011). 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