East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

ARTICLE

Computational Study of MHD Blood Flow

through Bifurcated Artery Using

Caputo-Fabrizio Fractional Derivative,

Thermal Radiation, and Magnetic Field for

Tumor Therapies

ARTICLE INFO

Volume 7(1), 2026

https:

//dx.doi.org/10.4314/eajbcs.v7i1.2S

Isah Abdullahi¹, Dauda Gulibur Yakubu^1,∗, Muhammad

ARTICLE HISTORY

Shamsuddeen Dauda², Mahmood Abdulhameed³,Saidu Abubakar

Kadas⁴,Mohammed Abdulhameed⁵, and Garba Tahiru Adamu⁶

Received: March 10, 2026

Accepted: 23 May, 2026

Published Online: 10 June, 2026

CITATION

¹Department of Mathematical Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria

²Department of Biological Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria

³Department of Electrical Electronic Engineering, Abubakar Tafawa Balewa University, Bauchi, Nigeria

⁴Department of Obstetrics Gynaecology, ATBU, Teaching Hospital, Bauchi, Nigeria

⁵School of Science and Technology, The Federal Polytechnic Bauchi, Nigeria

Abdullahi et.al (2026). Computational

Study of MHD Blood Flow through

Bifurcated Artery Using Caputo-Fabrizio

Fractional Derivative, Thermal

Radiation, and Magnetic Field for Tumor

Therapies. East African Journal of

Biophysical and Computational

⁶Department of Mathematical Sciences, Bauchi State University, Gadau, Bauchi, Nigeria

^∗Corresponding author: dgyakubu@atbu.edu.ng

Sciences Volume 7(1), 2026. .https://dx.

doi.org/10.4314/eajbcs.v7i1.2S.18-26

Abstract

OPEN ACCESS

This study investigates the impact of heat sources, thermal radiation, and chemical reactions on the

magnetohydrodynamic blood flow through a bifurcated artery in the presence of a slanted magnetic

field. Using Laplace transform and the method of undetermined coefficients, the constitutive equations

for the mathematical model of Caputo-Fabrizio fractional derivative order were solved. Blood flow

velocity, temperature distribution, and concentration were found to have analytical expressions. The

effects of certain physical parameters on blood velocity, temperature and concentration are graphically

represented, and these representations accurately depict the flow disturbances. We discovered that

the bifurcation apex of the artery with a symmetrical divider has steady blood flow. This may lead

to significant shear stresses on either side of the bifurcation. Near the apex, when the flow is

substantially different, obstruction may result from the formation of boundary layers on the inner walls

of the bifurcation. Sluggish flow also occurred along the outer walls of the bifurcation. It has also

been discovered that the temperature distribution, concentration, and arterial blood flow velocity are

significantly influenced by the fractional order parameter, the slanted magnetic fields, the heat source,

and the chemical reaction parameter. This study offers significant benefits for medical applications in

biomechanical engineering, biomedical engineering, and medicine.

This work is licensed under the Creative

Commons open access license (CC

BY-NC 4.0).

East African Journal of Biophysical and

Computational Sciences (EAJBCS) is

already indexed on known databases

like AJOL, DOAJ, CABI ABSTRACTS and

FAO AGRIS.

Keywords: Chemical reaction; Heat source; MHD Blood low; Slanted magnetic ield; Thermal

radiation

(Shit & Majee, 2015). Understanding many facets (aspects) of the

medical sciences, such as homeostasis, treating cancerous tumors, and

administering medication using magnetic particles, depends on the

study of biomagnetic ﬂuid dynamics (Shaw & Murthy, 2010). Blood’s

hemoglobin molecules are regarded as biomagnetic ﬂuids with magnetic

properties. Blood can also be considered as a Newtonian ﬂuid if it ﬂows

through the bigger arteries at a high shear rate. These arteries are thought

to be homogenous whose ﬂow behavior can be described by a Newtonian

model (see, Caro et al., 2011; MacDonald, 1979). Numerous researchers

have looked into various possibilities for studying physiological ﬂuids

1 Introduction

Bio-magnetic ﬂuid dynamics (BFD), which is the study of bio-ﬂuid ﬂow in

the presence of magnetic ﬁeld, is a rapidly developing subject of study in

ﬂuid mechanics (Tzirtzilakis, 2005). This ﬁeld of study is of tremendous

importance to the ﬁeld of medical science and has the possibility

to be utilized in a diversity of domains, including the delivery of

medications via the utilization of magnetized particles, the management

of severe bleeding, and the assistance in dealing with malignant cancers

Abdullahi et.al (2026)

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

using porous media (see, Bhatti & Lu, 2019; Dash et al., 1996; Ramesh &

Devakar, 2015; Shit & Roy, 2015) developed blood ﬂow model via porous

medium. Based on Darcy’s law, Bhargava et al. (2007) and Ghasemi et al.

(2015) investigated the pulsatile ﬂow and mass transfer of an electrically

conducting Newtonian bio-ﬂuid via a channel comprising porous media

using blood as the porous medium ﬂuid. Bhatti et al. (2018) developed a

mathematical model to investigate heat transfer, mass transfer, and blood

ﬂow in a porous medium channel while accounting for the integrated

Darcy-Brinkman-Forchheimer model. Blood behaves non-Newtonian

even in larger arteries at low shear rates, as demonstrated by Liepsch

(1986). When blood ﬂows through arteries at a low shear rate, it can be

treated as Cassons ﬂuid (Srivastava & Srivastava, 1984). Many researches

have supported the Casson ﬂuid model for blood ﬂow via tiny arteries at

low shear rates (see,Hayat et al., 2016; Nagarani et al., 2006; Venkatesan

et al., 2013 ). Many authors (see, Abdulhameed et al., 2017; Misra & Shit,

2009; Mondal & Shit, 2017; Yakubu et al., 2020; Zeeshan et al., 2017 )

have regarded blood as a non-Newtonian ﬂuid, because of its electrical

conductivity, displays magneto hydrodynamic behavior.

Therefore, it wasn’t until the last few decades that a signiﬁcant number

of scholars started to highlight the fact that diﬀerential equations and

fractional derivatives have numerous applications in a variety of domains

(see, Abdulhameed et al., 2023; Imoro et al., 2024). These days, fractional

derivative order diﬀerential equation problems are the most eﬀective

and successful ways to model the nonlinear processes that emerge in

many domains of applied study, including biology, chemistry, ecology,

engineering, and many other application areas. Several mathematical

models have shown that they oﬀer a more realistic depiction of the

phenomenon under research. Examples of these models include those

employed in biomedical engineering, viscoelastic mechanics, boundary

layers, electromagnetic, and porous media. Bansi et al. (2018) investigated

a fractional blood ﬂow model in the oscillatory artery with magnetic ﬁeld

and heat radiation eﬀects. With the aid of fractional time derivative,

(Yakubu et al., 2021) examined the eﬀects of pressure gradient, body

acceleration, and magnetic ﬁeld on blood ﬂow through artery. The

eﬀects of blood ﬂow parameters, Caputo’s time fractional derivatives, and

the external magnetic ﬁeld on the cylindrical domain were studied by

(Shah et al., 2016). Ali et al. (2017) solved a fractional order model for

Cassons ﬂuid ﬂow using the Hankel transform and Laplace transform

techniques to determine the exact solutions. He and collaborators (2019)

used the fractional order Caputo derivative to investigate the complexity

of blood in arteries under various forces. In the ﬁeld of medicine, magneto

hydrodynamic ﬂow plays a crucial role. It is considered for the reduction

of bleeding from wounds and for the treatment of malignant tumors.

Kumar et al. (2021) employed a chemical reaction, heat source, and

inclined magnetic ﬁeld to cure malignancies.

Many authors considered the examination of the heat and mass

transfer occurrences generated from these processes to be a highly

relevant element with respect to modeling physiological processes

(Prasad et al., 2025) and industrial processes (Sademaki et al., 2026).

Electromagnetohydrodynamics is the study of ﬂuids whose motion is

constantly aﬀected by externally applied magnetic ﬁeld and electric

ﬁeld. In order to comprehend the impact of magnetohydrodynamic

(MHD) and electrohydrodynamic (EHD) forces on the ﬂow of normal

ﬂuids, including blood, several studies have mostly concentrated on

the theoretical, computational, and experimental aspects of these forces.

Cell-based therapies, medication delivery, and biological processes are

just a few of the ﬁelds where the application of (EHD) has shown notable

advancement. Additional force components, primarily the Lorentz and

Coulomb forces, are incorporated into momentum equations and have a

direct eﬀect on ﬂuid velocity. Magnetohydrodynamics or MHD has been

used in a wide variety of biomedical applications (Vardanyan, 1973).

The fractional order time derivative of MHD blood ﬂow via a bifurcated

artery in the presence of a slanted magnetic ﬁeld, as well as the

coupling impact of heat transfer and blood ﬂow concentration, are

described here using Newtonian ﬂuid. The goal is to investigate

how magneto-hydrodynamic blood ﬂow through a bifurcated artery is

aﬀected by thermal radiation and a slanted magnetic ﬁeld during tumor

treatments. The Laplace transform and the indeterminate coeﬃcients

approach were used to ﬁnd the exact solutions, which were then

simulated to produce graphical outputs and the implications of several

important parameters on the outcomes were explored. The study was

motivated by the fact that there is currently very little information

available on the ﬂow in arterial bifurcation since the phenomena is

currently not stringent to mathematical analysis or precise experimental

measurement. The present investigation shows that the vast number of

variables involved are the main challenge in both situations.

The heat transmission and magnetohydrodynamic (MHD) blood ﬂow in

a restricted artery were studied by Majee and Shit (2017). Akbar and Butt

(2017) considered ferromagnetic blood ﬂow in a restricted, smaller artery

with a porous wall. The radiant heat transfer that takes place in the blood

vessels must also be considered while treating hyperthermia. Oncology

professionals are familiar with the medical practice of using heat therapy

to cancer patients. Chinyoka and Makinde (2014) investigated the eﬀects

of magnetic ﬁelds and heat radiation on arterial blood ﬂow. Sinha and

Shit (2015) investigated the magnetic hydrodynamic blood ﬂow in the

presence of thermal radiation. Tabi et al. (2017) studied the combined

eﬀects of magnetic ﬁelds and external radiation on blood ﬂow in the major

blood arteries. Yakubu et al. (2022) examined blood ﬂow of Oldroyd-B

ﬂuids in order to investigate the erratic ﬂow, with magnetic ﬁeld applied

perpendicular to the ﬂow direction. Heat transfer processes were studied

in the peristaltic ﬂow of blood with variable viscosity particle-liquid

suspensions by Bhatti et al. (2016). Blood ﬂow is greatly aﬀected when

the human body is exposed to a vibratory environment, as occurs when

operating machines or traveling in spacecraft. When the human body

undergoes body acceleration, a number of health problems might arise,

such as an elevated heart rate and vision loss. In the study of the impact of

body acceleration, a number of researches have produced mathematical

simulations of oscillatory blood ﬂow (see, Chaturani & Palanisamy, 1990;

Ghasemi et al., 2016; Sud & Sekhon, 1984 ). Bhatti and Lu (2019)

investigated the propagation of a hydro elastic single wave in a channel

with uniform ﬂow. Blood ﬂow characteristics have been discovered

to promote blood velocity in a vibratory environment using fractional

order derivative diﬀerential equation problems. Fractional diﬀerential

equations are the most used method for modeling natural phenomena.

This is due to the fact that equations oﬀer the possibility for a system to

either retain memory or to be hereditary with the properties of its history,

similar to how dynamic systems work (Syed et al., 2026).

2 Methods

2.1 Physical Structure and Mathematical

formulation

Blood considered in this study, is Newtonian, incompressible,

homogeneous, sticky ﬂuid that ﬂows from the trunk to the branches.

A mass stream’s rate at any cross-section that is perpendicular to its

direction is equal to m = 2bv, where b is the stream’s radius and v is

its mean speed. The mass stream’s speed at any cross section of the

extended channel is equal to m/2, and the bifurcating divider (internal

apical curve) has no eﬀect on this (see Figure 1). The magnetic ﬁeld is

applied to the ﬂow at an angle (φ) since the evaluated magnetic Reynolds

number is low. Therefore, it is believed that the magnetic and electric

ﬁelds produced by blood ﬂow are insigniﬁcant, the angle of bifurcation

is set to zero (Θ = 0), that is, the blood ﬂow region is bifurcated into two

streams that ﬂow parallel to the principal artery (the trunk). Figure 2

demonstrates the smooth muscle ﬁbers of the three concentric layers that

make up the walls of a typical elastic artery. These ﬁbers are controlled

by the sympathetic nervous system to contract or relax.

Fractional order derivatives have been applied in many ﬁelds of study,

including the complicated dynamics and rheological properties of

diﬀerent kinds of ﬂuids. The behavior of ﬂuid ﬂow is well depicted by

substituting fractional-order derivatives for the ordinary time derivative

in the constitutive equations (see, Atangana and Baleanu, 2016; Caputo

and Fabrizio, 2015; Samko et al., 1993). The concept of fractional calculus

was initially proposed by L’Hôpital in 1695, more than four centuries ago.

Abdullahi et.al (2026)

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

where D is the diﬀusion coeﬃcient and G = −k₁(C − C_∞) represents

chemical reaction rate in the ﬂuid ﬂow. It is important to mention that the

eﬀect of an electric ﬁeld in the concentration equation was also ignored.

θ = `^−z˙

u = `^−z˙

v = `^−z˙

C = `^−z˙at y = −1,

(5)

and θ → 0, u → 0, v → 0, C → 0 at y = 1.

However, by using the proper normalizing factors, the governing

equations (1)–(4) can be converted to dimensionless form. We present

the non-dimensional parameters as follows:

Figure 1: Physical ﬂow diagram of the bifurcated artery with zero angle of

bifurcation

x¯

y¯

u¯

v¯

dp¯/dx¯

u_HSηm/2b³ρ

x = , y = , u =

v =

, h(x, t) =

mu_HS/2bρ

ꢂ

ꢃ

ꢂ

ꢃ

mηu

ρ/η

ꢀ

ꢁ

k= ^ꢀ

and

^ꢁ, C =

, θ =

, τ =

t =

ρ/η

−16δT₀³

∂T

∂y¯

q¯ =

(6)

3k⁰

Applying (5) and (6) to eqns. (1)- (4) and removing the bars we obtain:

Figure 2: The structure of an artery walls (Transverse section through an artery)

∂u

∂t

∂²u

∂y²

ꢄ

2.2 Fundamental flow equations and their

solutions

+ h =

+ gβθ + gβ⁰C − M²sin²φ −

(7)

(8)

ꢅ

ꢄ

ꢅ

∂θ

∂t

∂²θ

∂y²

τp_r

+ R

τp_r

Blood ﬂows through a porous media as two-limit layers when it is

subjected to magnetic ﬁelds and heat, with the assumptions made in the

numerical deﬁnition guiding its movement. In the stream ﬁeld headings

of x and y at time t , let u and v be the speed parts, η and ρ denote

blood density and thickness, respectively. Blood pressure is represented

by p , warm conductivity (K_T), C_pis the speciﬁc heat capacity at steady

strain, hotness is represented by Q , temperature is represented by T ,

the volumetric development boundary is represented by β , the angle of

the slanted (inclined) magnetic ﬁeld is represented by φ, and porosity

parameter is represented by K . With these, we have the equations

provided by (see, Ali et al., 2017; He et al., 2019; Kumar et al., 2021), with

some additional terms as follows:

∂u

∂x

∂v

∂y

= 0

(9)

∂C

∂t

1 ∂²C

S_C∂y²

− ωC

(10)

where

σB²

⁰, p_r=

16δT₀³

3k⁰τ

Qb²

k_T

k₁b²

ρC

k_T

M²

^ρ, R =

, S =

, S_C

, ω =

∂u

∂t

1 ∂p

ρ ∂x

η ∂²u

cB_αsin φ

+ gβ(T − T_∞) + gβ⁰(C − C_∞) −

v −

∂y

are the magnetic ﬁeld parameter, Prandtl number, thermal radiation

parameter, heat source parameter, Schmidt number, chemical reaction

parameter and θ is the temperature conveyance. Now, using the

Caputo-Fabrizio fractional derivative as stated in (Caputo and Fabrizio

2015), we consider the time fractional momentum equations as:

(1)

∂²T

ρC_o

∂q

∂y

k_T

∂T

∂t

(T − T_o) −

(2)

(3)

∂y²

ρC_o

ꢄ

ꢅ

∂u(y, τ)

∂τ

α(t − τ)

^CFD_t^αu(y, t) =

exp

−

dτ, 0 < α < 1

1 − α

∂u

∂x

∂v

∂y

= 0

su(y, s) − u(y, 0)

^CFD_t^αu(y, t)

(1 − α)s + α

∂

∂t

(11)

where,

is the material time derivative. On the other hand, the

dimensionless concentration equation for medication (concentration)

delivery in magneto hydrodynamic blood ﬂow through permeable

bifurcated artery is provided by,

u(y, 0) =

(12)

∂²C

∂y²

The Caputo-Fabrizio derivative corresponding to equations (7), (8) and

(10) are as follows:

∂C

∂ f

= D

+ G

(4)

Abdullahi et.al (2026)

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

(

)

A₇+ A cosh A₂y + B sinh A₂y + A₈cos A₆y + A₉sin A₆y +

F =

∂²u

∂y²

ꢄ

+ A₁₀cosh Λy + A₁₁sinh Λy

^CFD_t^αu(y, t) + h =

+ gβθ + gβ⁰C − M²sin²φ −

(13)

(14)

(15)

(30)

ꢅ

τp_r

∂²θ

∂y²

τp_r

^CFD_t^αθ(y, t) =

+ R

+ (

)θ

We now have blood velocity in the axial direction, using equation (30) and

equation (19) as

∂C

∂t

1 ∂²C

S_C∂y²

^CFD_t^αC (y, t) =

− ωC.

A₇+ A cosh A₂y¯ + B sinh A₂y¯ + A₈cos A₆y¯ +

u¯(y, s) =

s + λ²

A₉sin A₆y¯ + A₁₀cosh Λy¯ + A₁₁sinh Λy¯

Applying Laplace transform to equations (13)-to-(15), and using the

boundary condition in equation (12) we have;

(31)

Equation (26) and equation (21) together provide the usual direction of

blood velocity in the bifurcated artery, which is given by

su(y, s)

(1 − α)s + α

∂²u¯

∂y¯²

u¯

+ h =

+ gβθ + gβ C − M sin φ −

(16)

(17)

(18)

v¯(y, s) = A₁

(32)

s + λ²

ꢄ

ꢅ

su(y, s)

(1 − α)s + α

τp_r

∂ θ

τp_r

)θ

+ R

+ (

∂y¯²

According to equations (28) and (20), the temperature distribution in the

bifurcated artery is given as follows:

su(y, s)

(1 − α)s + α

1 ∂ C

− ωC.

S_C∂y¯²

ꢄꢄ

ꢅ

cos A₆y¯

2 cos A₆

sin A₆y¯

θ(y, s) =

−

(33)

2 sin A₆s + λ²

2.3 Exact solutions

Equations (22) and (29) provide the drug’s concentration in the ﬂowing

blood in the carotid artery as follows:

Here we assume the following as the arbitrary solutions of equations (9),

(16), (17) and (18),

u¯ = F(y)

ꢄ

ꢅ

(19)

(20)

(21)

(22)

s + λ²

cosh A₉y¯

2 cosh A₉

sinh A₉y¯

C(y, s) =

−

2 sinh A₉s + λ²

θ = H(y)

s + λ²

v¯ = G(y)

(34)

s + λ²

Equations (31) through (34) yield the inverse Laplace transform.

Using Mathcad software, we simulated the given solutions using

Gaver-Stehfest’s algorithm, and the results are shown graphically in the

next section.

C = I(y)

s + λ²

then the boundary conditions in eqns. (5) reduce to;

H = 1, I = 1, F = 1, at y = −1,

H → 0, I → 0, F → 0, at y = 1.

3 Results and Discussion

(23)

To get the ﬂow information, we simulated the solutions of equations

(31), (33), and (34) using Mathcad software. The inﬂuence of the

fractional-parameter (α) on velocity, temperature and blood concentration

are displayed graphically and discussed. Axial ﬂuid velocity, temperature

distribution, and concentration are explored as functions of several

dimensionless factors, including: slanted (inclined) magnetic ﬁeld

parameter (M), radiation parameter (R), fractional parameter (α), heat

source parameter (S), and Schmidt number (S_C). In all the dimensionless

parameter calculations, we vary the value of the fractional parameter (α),

but we maintain other values constant, such as, t = 1, S_C= 0.5, ω = 0.5,

S = 1, P_r= 2, K = 2, R = 0.5, h = 0.5, β = 0.5, φ = 30^◦.

As a result, the following are the simpliﬁed governing equations of

motions with arbitrary solutions:

d F

− A₂F = A₃− gβH − gβ I,

(24)

dy²

d H

+ A₆H = 0,

(25)

(26)

(27)

dy²

G = A₁(constant),

d I

dy²

− A₉I = 0.

3.1 Velocity Profile

Equation (23)’s boundary conditions are used to solve equations (24)

through (27) and the following solutions are obtained:

Consequently, the magnetic ﬁeld always has a greater inﬂuence on the

blood velocity proﬁle. The application of the magnetic ﬁeld to the system,

as shown in Figure 3, increases the Lorentz force, a resistive force that

primarily restricts the ﬂow of ﬂuid Bunonyo and Ebiwareme (2023) and

Vardanyan (1973). For fractional order (α = 0.4), as the magnetic ﬁeld

parameter’s strength increases, as seen in Figure 3(a), the blood velocity

reduces sharply, whereas it declines gradually for (α = 1) as shown in

Figure 3b.

ꢄ

ꢅ

cos A₆y¯

2 cos A₆

sin A₆y¯

2 sin A₆

H =

−

(28)

(29)

cosh A₉y¯

2 cosh A₉

sinh A₉y¯

2 sinh A₉

I =

−

Abdullahi et.al (2026)

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

However, for the fractional order parameter in Figure 6a and 6b, the ﬂow

velocity vanishes between angles of (80^◦to 85^◦).

Because several of the related parameters’ values have changed, the

graphs in Figure 6c and 6d behave very diﬀerently from one another.

Using Figure 6c as an example, y = 0.004, p = 4, and the fractional

parameter (α

0.2), whereas Figure 6d also includes the fractional

parameter (α = 0.4) and y = 0.004, p = 3.

(a)

(b)

Figure 3: Axial velocity proﬁle for diﬀerent values of magnetic parameter: (a)

α = 0.4 (b) α = 1, ω = 0.5, S = 1, P = 2, K = 2, R = 0.5, t = 1, h = 0.5,

β = 0.5, φ = 30^◦, Sc = 0.5.

For therapeutic purposes and treatment procedures related to

atherosclerosis, bone fractures, controlled tissue damage, and malignant

tumors, to mention a few, a regulated magnetic ﬁeld can therefore be a

useful tool (Imoro et al., 2024). For both the fractional parameter (α = 0.4)

and the integer order model blood ﬂow (α = 1), Figure 4 shows the

variation in blood ﬂow at diﬀerent heat source parameter (S) values. It is

clear that an increase in the heat source has an impact on blood velocity

and the fractional ﬂuid parameter (α = 0.4) (see Figure 4a). As seen in

Figure 4b, the axial velocity does, however, drop symmetrically as the

heat source parameter increases.

(a)

(b)

(c)

(d)

Figure 6: Axial velocity proﬁle for various angles of inclination of the magnetic ﬁeld:

(a) α = 0.4 (b) α = 1 (c) y = 0.004, p = 4 and α = 0.2 (d) y = 0.004, p = 3, α = 0.4,

ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦, Sc = 0.5.

(a)

(b)

Figure 4: Proﬁle of axial velocity for various heat source parameter: (a) α = 0.4 (b)

The blood ﬂow velocity proﬁle at two independent times, t

0.01

α = 1, ω = 0.5, M = 0.5, P = 2, K = 2, R = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦,

and 0.5, is shown in Figure 7 with ﬁve diﬀerent values of the fractional

parameter (α = 0.2, 0.4, 0.6, 0.8, and 1). It has been observed that the

fractional parameter (α) plays a critical role in regulating blood velocity.

The fractional derivative ﬂuid velocity initially moves faster than the

integer order ﬂuid model when the time is relatively small (t = 0.01).

However, for a longer period of time (t = 0.5), the reverse behavior

is seen, that is, ﬂuids with integer order have a faster velocity than

those with fractional order parameters. Naturally, this results from the

system’s stability, which can improve over longer timescales. For both

fractional order derivative ﬂuid models and integer order derivative ﬂuid

models, it is often observed that blood velocity increases with increasing

time t. Figure 7a shows the evolution of the primary velocity proﬁle,

showing how the ﬂow develops into fully formed Poiseuille ﬂow, which

is distinguished by the typical parabolic proﬁle. Figure 7b clearly depicts

the velocity proﬁle at the fork section for various values of the fractional

parameter. It is signiﬁcant to note that when (α = 1), a zone of sluggish

ﬂow appears along the outside wall and gets worse as time goes on, as

was previously observed by Gade et al. (2026).

Sc = 0.5.

Figure 5 displays the velocity distribution based on diﬀerent thermal

radiation parameters (R). The blood velocity increases as the radiation

parameter (R) increases, as indicated by both the fractional parameter

(α

0.4) and the classical order parameter (α

1). Remarkably,

comparable results for a related ﬂuid model were discussed in Tabi et al.

(2017). According to Yakubu et al. (2022), heat radiation possesses the

capability to modify the eﬀective viscosity of ﬂuids, hence potentially

causing an indirect inﬂuence on the velocity proﬁle.

(a)

(b)

Figure 5: Proﬁle of axial velocity for various heat source parameter: (a) α = 0.4 (b)

α = 1, ω = 0.5, M = 0.5, P = 2, K = 2, R = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦,

Sc = 0.5.

(a)

(b)

The applied magnetic ﬁeld parameter for various tilted values is

displayed in Figure 6. Blood ﬂow is reduced over the aﬀected area when

the applied magnetic ﬁeld’s angle of inclination is increased for both the

fractional order parameter and the classical order parameter (α = 1).

Figure 7: Axial velocity proﬁle for diﬀerent values of α at: (a) t = 0.01 (b) t = 0.5

ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦, Sc = 0.5

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

3.2 Temperature profile

It’s noteworthy to see in Figure 9b that the temperature at the middle line

of the channel decreases as the heat source’s values rise. The temperature

exhibits oscillating behavior for diﬀerent amounts of the heat source in

Figures 9c and 9d. As the values of the heat source rise, the temperature

is maximum at the center, decreases, and ﬁnally approaches zero at the

artery walls. The diﬀerent values of the fractional parameter really cause

a shift in the temperature distribution, as shown in Figure 10. It illustrates

how temperature increases with increasing fractional parameter . This

implies that the fractional order ﬂuid model’s temperature distribution is

more faster and higher over a longer period of time, which is what causes

the variation shown in Figures 10a and 10b, as mentioned earlier. The

temperature gradually decreases toward the artery’s axis in Figures 10c

and 10d, eventually tending to align with the axis.

Temperature proﬁles for diﬀerent radiation parameters (R), fractional

parameters , and heat source parameters (S) are shown in Figures 8–??.

The temperature change for diﬀerent values of the radiation parameter

R, as shown in Figure 8. It is evident that when the thermal radiation

increases, temperature increases for both fractional and integer order

derivatives. The temperature varies near the center line for both the

integer order and the fractional order derivative, as shown in Figure 8a

and 8b. Consequently, it is more visible in the graphs of Figure 8b.

(a)

(b)

Figure 8: Temperature proﬁle for diﬀerent values of radiation parameter: (a) α = 0.4

(b) α = 1, ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦,

(a)

(b)

Sc = 0.5

During hyperthermia, the temperature distribution is very important. It

is commonly recognized that hyperthermia results from a breakdown

in thermoregulation, which takes place when the body absorbs heat

from outside sources like radiation or a body temperature that is being

generated or absorbed. When a person has hyperthermia, the blood’s

internal temperature increases without damaging the tissues around

the blood vessel. We have not taken into account the temperature

exchange at the artery wall to account for this, meaning that the wall’s

temperature is zero. In light of this, the blood temperature in the current

model is low at the artery wall and high at the midline for classical

ﬂuid. Numerous theoretical and experimental studies for Newtonian

and non-Newtonian ﬂuids of integer order reported similar phenomena,

for example in (Ramesh & Devakar, 2015). Similar to radiation, the heat

source (S) another crucial factor, has a large impact on the bloodstream’s

temperature distribution. More mitochondria per cell increase the

thermal activity involved with the heat production process, as seen

in Figure 9, which raises the system’s temperature. The heat source

improves the temperature distribution and supplies more heat to the

blood ﬂow system even though the wall temperature must remain zero in

order to meet the boundary conditions. The temperature distribution at

the channel walls, which decreases and becomes more ﬂattened toward

the channel’s center line when the heat source is increased as shown in

Figure 9a, which is ampliﬁed to maintain a constant ﬂow rate, as seen in

Figure 9a.

(c)

(d)

Figure 10: Temperature proﬁle for diﬀerent values of α at: (a) t = 0.05, y = 0.004,

p = 1 and α = 1, t = 0.1 (b) t = 0.25, ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1,

h = 0.5, β = 0.5, φ = 30^◦, Sc = 0.5, y = 0.004, p = 3, α = 0.4. (c) y = 0.004, p = 2

and α = 1, t = 0.1 (d) y = 0.004, p = 10, α = 0.01, t = 0.1.

3.3 Concentration profile

The concentration proﬁle for diﬀerent values of fractional order

parameter (α) , Schmidt number (Cs) , and chemical process (ω) is

shown in Figures 11 to 13. There is a relationship between the blood

concentration and the quantity of blood cells ﬂoating in the plasma. Red

blood cells (RBCs) are important blood cells because of their size and

density in the bloodstream. RBCs assembled at the center of the vessel,

where there is a greater concentration of solutes, due to their revolving

nature. However, because the oﬀ-axis zone is an area predominantly

represented by cells that carry plasma, the solute concentration there

decreases to a minimum. This observation is displayed in all of the

concentration graphs in this section. The fractional model ﬂuid in Figure

11 reaches a greater concentration more quickly than the integer order

ﬂuid (Imoro et al., 2024).

(a)

(b)

(c)

(d)

(a)

(b)

Figure 9: Temperature proﬁle for diﬀerent values of heat source parameter: (a)

α = 0.4 (b) α = 1, ω = 0.5, R = 0.5, P = 2, K = 2, M = 0.5, t = 1, h = 0.5,

Figure 11: Concentration proﬁle for diﬀerent values of α at: (a) t = 0.1 (b) t = 0.5

β = 0.5, φ = 30^◦, Sc = 0.5

Sc = 0.5, S = 1, R = 0.5, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦

Abdullahi et.al (2026)

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

This is because a fractional order derivative that restricts ﬂuid ﬂow

is included in the model. The Schmidt number exhibits the opposite

pattern. The blood cells show an additional force of the temperature

gradient in the presence of the Schmidt number, as seen in Figure 12,

which further increases the concentration. Therefore, lower Schmidt

number values, for example in industrial applications, physically

represent hydrogen gas as the species diﬀusing (Sademaki et al., 2026).

4 Conclusion

Currently, a fractional-order model of the magneto hydrodynamic blood

ﬂow via a bifurcated artery under the inﬂuence of thermal radiation,

a slanted magnetic ﬁeld, and a heat source during tumor treatment is

being developed. The Laplace transform and the combined methods

of indeterminate coeﬃcients were used to solve the mathematical

models. The fractional order parameter has a major eﬀect on the

blood velocity proﬁles, concentration, and temperature distribution.

It has been noted that ﬂuids with fractional order can occasionally

move faster than those with integer order. Fractional model ﬂuid

ﬂow is slower than integer-order ﬂuid ﬂow over longer dimensionless

durations. The impact of ﬂuid velocity is demonstrated by the fact

that the rate of increase in ﬂuid velocity is slower at larger levels of

the magnetic ﬁeld parameter. As the chemical reaction parameter rises,

the blood ﬂow concentration falls. The blood concentration rises as the

Schmidt number rises. As the fractional parameter and the heat source

increase, the blood ﬂow’s dimensionless temperature rises, which also

aﬀects the radiation parameter. We noted that the outcomes will be

intriguing to comprehend and evaluate throughout cancer therapy using

hyperthermia. Additionally, it will be useful in understanding the drug

particle concentration phenomena for applications and administrations

involving drug delivery. Our research’s ﬁndings should serve as a

foundation for the study of increasingly sophisticated blood ﬂow models

and also serve as a basis for in vitro and in vivo testing, particularly

in the application areas like medicine, biomedical engineering, biology,

pathology, and other related domains.

(a)

(b)

Figure 12: Concentration proﬁle for diﬀerent values of Schmidt number: (a) α = 0.4

(b) α = 1, ω = 0.5, S = 1, R = 0.5, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5,

φ = 30^◦

Figure 12 illustrates how the species’ chemical molecular diﬀusivity

decreases dramatically with increasing Schmidt number (Sc) , making

it easier for the species to enter the ﬂow ﬁeld and raising the mass

transfer function. Higher Schmidt number compounds can enhance

mass transfer and dispersion properties in the bloodstream, especially

for pharmaceutical diﬀusion in pulse blood ﬂow. The amplitude of the

blood ﬂow concentration is larger for the integer order derivative. As

demonstrated in Figure 13a and 13b, this phenomena is clearly seen along

the ﬂow axis (0 ≤ y ≤ 0.5) and slowly declines in the region (0 ≤ y ≤ 1)

for both fractional and integer order derivatives, respectively. As can be

observed from all of the graphs in Figure 13, the blood ﬂow decreases

along the distensible tube’s length where the graphs begin to ﬂuctuate

because of the size of the chemical reaction parameter’s peak value

(pressure gradient) (see, Abdul-Wahab & Al-Saif, 2024). Furthermore,

it has been demonstrated that the variation is more pronounced in the

larger section of the artery wall, permitting the ﬂow to pass without

producing a perceptible pressure gradient. Nevertheless, the substantial

pressure gradient is usually required to maintain a consistent ﬂow rate as

it passes through the constrictions in the artery.

Acknowledgments

This work was supported by Tertiary Education Trust Fund (TETFund)

Ref. No. TETF/ DR&D/CE /UNI /BAUCHI /IBR /2025/ VOL.1.

Therefore, the authors gratefully acknowledged the ﬁnancial support of

the TETFUND. The authors would like to express their sincere gratitude

to the handily editor and the reviewers for their helpful and informative

comments, which have enhanced the manuscript.

Declaration of Generative AI

The authors declare that they do not used generative AI in the scientiﬁc

writing.

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