EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6198
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Helical Estimation of Atangana-Baleanu-Caputo
Fractionalized Magnetohydrodynamic Second Grade

Fluid for Generalized Boundary Conditions under
Porous Environment

Muhammad Zafarullah1, Ilyas Khan2,3,4,∗, Kanwal Abid5, Muhammad Jamil1,6,
A. B. Albidah2,∗, Thoraya N. Alharthi7, A. F. Aljohani8, Wei Sin Koh9

1 Department of Mathematics, University of Karachi, Karachi-75270, Pakistan
2 Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952,
Saudi Arabia
3 Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan
4 Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu,
India
5 Department of Mathematics, Jinnah University for Women, Nazimabad, Karachi, 74600, Pakistan
6 Department of Mathematics, NED University of Engineering & Technology, Karachi-75270, Pakistan
7 Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922,
Saudi Arabia
8 Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia
9 INTI International University, Persiaran Perdana BBN Putra Nilai, 71800 Nilai, Negeri Sembilan,
Malaysia

Abstract. In the current work, we have estimated the flow of helices of the unsteady fractionalized
second grade fluid with MHD effect between uniaxial annular cylinders. The analytical outcomes
are evaluated for the rotational and longitudinal velocities and the shear stresses because of fluid
circulation and translating between two infinite coaxial circular cylinders, those are turning their
axis. Hardly anyone has done this before or has applied the most modern non-integer Atangana
Baleanu Caputo fractional time derivatives on the governing equation of second grade fluid with
some natural effects. The outcomes are determined by using integral transforms such as Laplace
and finite Hankel transforms. The acquired outcomes exhibited in integral and series forms with
newly defined special Ma,b

c (κ, t) function. The outcome fulfills both the governing equation and all
designated account conditions. Additionally, the respective outcome for Newtonian fluid for the
same movement is acquired in limited cases. The impact of material parameter α and kinematic
viscosity of the α is also discussed. Last, we examined the behavior of distinct parameters on fluid
movement along with graphically analogizing second grade and Newtonian fluids.
2020 Mathematics Subject Classifications: 76A05, 76A10, 76U05, 76S05, 76W05, 26A33,
35R11, 35A22
Key Words and Phrases: Second grade fluid, helical flows, MHD, porous, ABC fractional
derivatives, smart grid, transform methods

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6198

Email addresses: i.said@mu.edu.sa (I. Khan), a.albedah@mu.edu.sa (A. B. Albidah)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 2 of 41

1. Introduction

From the last decade of the seventeen century, when Leibniz[1] introduced it as a
paradox, also mentioned that it will have applicable outcomes in future. Mathematical
analysis branch[2] - fractional calculus has paid a significant contribution in many fields of
real life[3]. Ross, B.[4] cited exhaustively the basic ideas and earlier definitions of fractional
integro-differentiation of Euler, Laplace, Lacroix, Fourier and of other mathematicians who
worked on the topic till 1900. Podlubny et al.[5] titled Abel as the father of fractional cal-
culus. Abel introduced the complete conceptions of Riemann Liouville fractional integral
and Caputo fractional derivative, which were his extremely interesting discoveries. Due
to extraordinary influence in the research, Bertram Ross arranged the first conference on
fractional calculus and its applications[6] at the University of New Haven during mid of
1974, and in the same year, Oldham and Spanier were the first who edited the proceedings
and published the first monograph on the fractional calculus. Machado et al.[2] also pre-
sented the detailed analysis of journals, exclusively issues in peer-reviewed journals, books
(authored or edited), conferences, separate symposium in conferences, courses, tutorials
and plenaries, algorithms / packages, other websites and packages, some papers on com-
putational / numerical procedures for special functions, math keywords and entries in the
new MSC2010 and patents on the advancement of fractional calculus since 1974. Caputo
and Fabrizio[7] worked on time variables that are applicable for the Laplace transform and
on spatial variables that are more convenient with the Fourier transform. Atangana and
Baleanu[8] also proposed the latest fractional derivatives, applicable on the fractionalized
heat transfer framework. They recommenced the first one on Caputo aspects and the
second on Riemann-Liouville perspective a non-local and non-singular kernel[9]. The first
one is also famous as Atangana Baleanu Caputo ABC fractional derivative.

Before Atangana Baleanu Caputo ABC fractional differential operator, the previ-
ous models were an ordinary model of second grade fluid or fractionalized in the sense
of Caputo and Riemann-Liouville with respect to the power-law kernel and Caputo and
Fabrizio with respect to the exponential kernel but both have kernel singularity compli-
cation because of the constant function does not lead to zero value during differentiation.
Subsequently, some scholars investigated and applied them to different models and iden-
tified the locality issues of the related kernel, and the derivative could not explain the
memory effects. Atangana and Baleanu addressed these issues and succeeded in dealing
with the locality issues, they incorporated a non-local and non-singular kernel. Their oper-
ator is known as Atangana Baleanu Caputo differential operator, its kernel has stochastic
and deterministic properties and is generalized in terms of Mittag-Leffler law[9]. Sania
and Atangana[10] worked on the mathematical modeling of the infectious disease dengue,
caused by mosquitoes and found that Caputo, Caputo-Fabrizio, and Atangana-Baleanu-
Caputo are more effective differential operators than other classical derivatives. Hong-
Guang et al.[3] collected and reviewed in detail the significance and practice of fractional
calculus in the disciplines of physics, dynamical systems, computer science, life science,



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 3 of 41

NOMENCLATURE [in SI units]

Π1 interior cylinder radius [m] µ specific gravity
Π2 exterior cylinder radius [m] ϕ parameter of porosity
v translational velocity field [m/s] κ porous medium permeability
ω rotational velocity field [m/s] Gp porosity constant= µϕ

k
Sρz(ρ, t) tors. shear stress (time dept.) [N/s2] ρ longitudinal component
Sρθ(ρ, t) longit. shear stress (time dept.) [N/s2] α material / fluid parameter
σ electrical conductivity of fluid [S/m] β fractional parameter
Π0 magnitude of applied magnetic field [T ] Dβ

t ABC fractional operator
ϱ density of fluid [kg/m3] ω Laplace transform of ω
Gm magnetic constant= σΠ2

o
ϱ v Laplace transform of v

η dynamic viscosity [Pa s] wH Hankel transform of ω
ν = η/ϱ kinematic viscosity [m2/s] vH Hankel transform of v
Ma,b

c special function M(ω, τ)

environmental science, macroeconomic modelings, interdisciplinary materials science, and
multidisciplinary engineering. Faraz et al.[11] redesigned some models, those are regularly
adopted modern day researchers, employ fractional operators to examine an extensive
study of preventive steps for COVID-19. Anwar et al.[12] provided a more detailed expla-
nation of the technical work introduced by Caputo, Fabrizio, and Atangana on time and
spatial variables with real life cases. The most recent notable work of fractional differen-
tial operators on fluid dynamics are; Anwar et al.[13] analyzed multiparametric fractional
operator on the mixture of aluminum and titanium while Asifa et al.[14] investigated
Prabhakar fractional operator on heat transfer of hybrid nanofluid. Many research schol-
ars are also attracted towards the practical applications of fraction differential operators
on magnetohydrodynamics(MHD) and other phenomena[15–18].

Special generalized functions in fractional calculus are growing as one of the essen-
tial tools to represent the outcomes. Kiryakova[19] conducted the survey and discussed
the special functions in depth like Meijer G-function, generalized hypergeometric functions
pFq,  Wright generalized hypergeometric functions qΨp, Fox H-functions, Mittag-Leffler
type functions in fractional calculus. Mathai et al.[20] in his publication, dedicated an
exclusive section on fractional calculus in which he connected fraction differential and frac-
tional integral operators to H-function and examined thoroughly all famous fractional op-
erators. Qureshi[21] investigated the Mass-Spring-Damper mechanical engineering model
through the Caputo fractional operator, earned the exact solution and connected it with
the Fox H-function. Ali et al.[22] contemplated Casson fluid in generalized form with heat
transfer, applied Caputo fractional derivative for mathematical modeling, and established
the result in the Wright function Φ(a, α, τ). Sharma and Jain[23] explained the gener-



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 4 of 41

alized M-series α
pMβq , its relations with Wright generalized hypergeometric function, Fox

H-function, and generalized Mittag-Leffler function, and link to fractional integrals and
derivatives. Jamil et al.[24] worked on fractionalized MHD Maxwell fluid and show their
outcomes in newly defined M-function M q

p and its relationship with Wright function qΨp

is as Mp
q (z) = tbq−1

qΨp(z). Their newly prescribed M-function is the simpler form of the
Wright function.

Figure 1: Physical structure of the discussion: Helical motion of fluid in the coaxial circular cylinders

—————————————–

The helical flow of non-Newtonian fluid in the cylindrical zone is utilized regularly
in experimental and theoretical research[25, 26]. Bayat et al.[27] worked on the human
eye and examined the partial vitreous liquefaction flow. In the article, he also explained
exclusively the behavior of Newtonian and non-Newtonian fluids by the Weissenberg ef-
fect and showed the simulation and its numerical result, such flow produced helices but
in opposite directions. Alharbi et al.[28] investigated the properties of helically pressure-
induced flow in coaxial cylinders, also known as Poiseuille flow, of the Bingham fluids.
Bingham fluid is sometimes referred to as concrete in civil engineering and mud during
drilling in the ground. Worku et al.[29] developed the multi-linear regression modeling of
pharmaceutical power having ingredients in bulk quantity which is used to manufacture
the solid dosage in the form of tablets and capsules in a cylinder with the rotating he-
lical blade for mixing. Jamil et al.[30] and Kamran et al.[31] separately considered the
helical flow of Maxwell fluid in two coaxial cylindrical region, after employing the integral
transforms, they presented their result of shear stress in the more generalization of My

α,β ,
and Ga,b,c(., .) functions respectively. More recently, Javaid et al.[32] studied the flow of
fractionalized Burgers’ fluid, after working through some integral transform and modified



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 5 of 41

Bessel equations, they earned their results of velocity field and shear stress and for valida-
tion and comparison of results, they adopted the Gaver-Stehfest�s algorithm and Tzou�s
algorithm.

The influence of two more phenomena are dominant in the literature of fluid dynam-
ics, those are magnetic and cellular (porous) effects. Jamil and Zafarullah[33] wrote some
applications of magnetohydrodynamics (MHD) motion of second grade non-Newtonian
fluid and employed some integral transforms, demonstrating the result in convolution
product form of inverse Laplace transformation and endorsed by graphical behavior. Jamil
et al.[24] extended their earlier study by taking fractionalized magnetohydrodynamics of
Maxwell fluid in the cellular cylinder and validating the result by visual presentation.
In a cellular or porous environment, there are many micro cells / pores that can absorb
something on the surface of an object. Many examples of porosity are available in mod-
ern literature. Liu and Chen[34] characterized porous materials either naturally such as
sandstone, human lung and bones, eggshell and limbs of birds, plant leaves and wood,
and some marine invertebrates, or synthetically such as porous ceramics, polymer foams,
tissue, and absorbent papers, fabric, filtration devices. Cai et al.[35] reviewed the vision
2020 of InterPore. Mochalin et al.[36] worked numerically on fluid flow through filtrating
round hollows passing in spinning porous (cellular) cylinders by applying computational
fluid dynamics techniques.

In this article, we estimate the spiral and linear flow rate of a second grade un-
steady fractionalized fluid with MHD phenomenon between two uniaxial cellular cylinders.
The analytical results are evaluated for the rotational and longitudinal velocities and the
shear stress because of fluid rotation and transformation between two infinite coaxial round
porous / cellular cylinders, turning their axes. The outcomes are determined employing the
suitable integral transformations, that are effective in circular cylinders, such as Laplace
transformation and finite Hankel transformation, by one of the most recent and practica-
ble fractional or non-integer Atangana Baleanu Caputo (ABC) time fractional derivatives.
ABC differential operator is popularized and attracted in the last few years by renowned
researchers. The acquired outcomes are exhibited in integral and series forms with the
convolution product of Laplace inverse transformation and newly prescribed special gen-
eralized Ma,b

c (κ, t) function which is more simplified pattern of the Wright function. The
result meets basic equality and all accounting requirements such as initial and generalized
boundary conditions. Only a few researchers hardly ever used such generalized bound-
ary conditions that is why they rarely appeared in research. Furthermore, Additionally,
the respective outcome for Newtonian fluid for the same movement is acquired in limited
cases. The impact of alpha and kinematic viscosity of the material parameter is also dis-
cussed. Last, we examined the behavior of distinct parameters on fluid movement along
with graphically analogizing second grade and Newtonian fluids.



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 6 of 41

2. Advancement in mathematical modeling and its governing equations

The governing equations of second grade fluid flows between two cylinders together
with heat and mass transfer effect are:

∂

∂t
w(ρ, t) =

(
ν + α

∂

∂t

)(
∂2

∂ρ2
+

1

ρ

∂

∂ρ
− 1

ρ2

)
w(ρ, t)−Gmw(ρ, t)−Gp

(
ν + α

∂

∂t

)
w(ρ, t);

ρ ∈ (Π1,Π2), t > 0, (1)

∂

∂t
v(ρ, t) =

(
ν + α

∂

∂t

)(
∂2

∂ρ2
+

1

ρ

∂

∂ρ

)
v(ρ, t)−Gmv(ρ, t)−Gp

(
ν + α

∂

∂t

)
v(ρ, t);

ρ ∈ (Π1,Π2), t > 0, (2)

Sρθ =

(
µ+ α1

∂

∂t

)(
∂

∂ρ
− 1

ρ

)
w(ρ, t), (3)

Sρz =

(
µ+ α1

∂

∂t

)
∂

∂ρ
v(ρ, t), (4)

Here for the fluid, constant density is ϱ, kinematic viscosity is ν = µ/ϱ and α = α1/ϱ.
Also shear stress are τ(w) = Sρθ and τ(v) = Sρz.

Now introducing the ABC fractionalized derivatives on the governing equations of an
incompressible second grade fluid,

∂w(ρ, t)

∂t
=

(
ν + αβDβ

t

)(
∂2

∂ρ2
+

1

ρ

∂

∂ρ
− 1

ρ2

)
w(ρ, t)−Gmw(ρ, t)−Gp

(
ν + αβDβ

t

)
w(ρ, t); (5)

ρ ∈ (Π1,Π2), t > 0,



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 7 of 41

∂v(ρ, t)

∂t
=

(
ν + αβDβ

t

)(
∂2

∂ρ2
+

1

ρ

∂

∂ρ

)
v(ρ, t)−Gmv(ρ, t)−Gp

(
ν + αβDβ

t

)
v(ρ, t); (6)

ρ ∈ (Π1,Π2), t > 0,

Sρθ(ρ, t) =
(
µ+ αβDβ

t

)(
∂

∂ρ
− 1

ρ

)
w(ρ, t), (7)

Sρz(ρ, t) =
(
µ+ αβDβ

t

) ∂

∂ρ
v(ρ, t), (8)

where the α is the second grade parameter and Gm and Gp are magnetic and porous effect
dimensionless number. The fractional differential operator Dβ

t called Atangana Baleanu
fractional operator, where β are fractional parameters[8].

With initial condition, the above formula can apply to a real-world problem and will
also be very useful for physical problem when employ the Laplace transform. For β → 1
the non integers second grade fluid model condenses to the ordinary second grade model.

3. Development of the problem

By consideration fractionalized MHD second grade fluid that is incompressible,
initially at rest t = 0 in annular region of two Π1 and Π2 radii circular cylinders as presented
in Fig. 1. At t = 0+ the cylinders start to rotate or translate around their axis (ρ =
0) with the rotational velocity U1H(t) g1(t), U2H(t)g2(t) and the translational velocity
V1H(t) g3(t), V2H(t) g4(t), where g1(t), g2(t), g3(t) and g4(t) are any general functions
with the property that they are differentiable and integrable and satisfy g1(0) = g2(0) =
g3(0) = g4(0) = 0. The velocity field ∆ is of the form

∆ = ∆(ρ, t) = w(ρ, t)eθ + v(ρ, t)ez (9)

where eθ, ez are unit vectors in transverse and z-directions. The appropriate governing
equations are given by Eqs. (5 - 8). While the suitable initial and boundary conditions
are its velocity being of the form

w(ρ, 0) = v(ρ, 0) = 0; ρ ∈ (Π1,Π2), (10)

Sρθ(ρ, 0) = Sρz(ρ, 0) = 0; ρ ∈ (Π1,Π2), (11)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 8 of 41

respectively,

w(Π1, t) = U1H(t)g1(t), w(Π2, t) = U2H(t)g2(t), (12)

v(Π1, t) = V1H(t)g3(t), v(Π2, t) = V2H(t)g4(t), (13)
t, ρ ≥ 0,

where H(t) is denoted as the Heaviside function and U1, U2, V1, V2 represents constants.

Usually the boundary conditions of fluid dynamics those are taken in the litera-
ture are constants t, t2, tn, eat, sin(ωt), cos(ωt) etc. Only few papers of the fraction-
alized governing equations with Caputo fractional operator those are less difficult of the
present governing equations, are available in the literature with the above boundary con-
ditions and famous generalized functions like Mittag-Leffler Eα,β(z), Eγ

α,β(z), Lorengo
Ga,b,c(d, t), Ra,b(c, t) functions[30, 36–38]. In order to make our problem more genuine and
original work, not only we include the magnetic and porosity terms in governing equa-
tions but we consider the helical flow of the problem. Furthermore, instead of solving the
problem by taking some specific functions on the boundary conditions given above and
to facilitate the engineers and applied scientists, we take the generalized functions on the
boundary of the cylinders g1(t), g2(t), g3(t) and g4(t). Employing such type of generalized
boundary conditions is not new in the literature, for this we mention here some recent
contributions[39, 40].

4. Computational Work

4.1. For velocity field
Recall the initial condition Eq. (10), employ Laplace transform formula for frac-

tional derivative to Eqs. (5) and (6) and boundary condition Eqs.(12) and (13). We find
that

sw̄(ρ, s) =

(
ν + αβa0

sβ

sβ + a1

)(
∂2

∂ρ2
+

1

ρ

∂

∂ρ
− 1

ρ2

)
w̄(ρ, s)−Gmw̄(ρ, s)

−Gp

(
ν + αβa0

sβ

sβ + a1

)
w̄(ρ, s); ρ ∈ (Π1,Π2), (14)

sv̄(ρ, s) =

(
ν + αβa0

sβ

sβ + a1

)(
∂2

∂ρ2
+

1

ρ

∂

∂ρ

)
v̄(ρ, s)−Gmv̄(ρ, s)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 9 of 41

−Gp

(
ν + αβa0

sβ

sβ + a1

)
v(ρ, s); ρ ∈ (Π1,Π2). (15)

where the Laplace transform of the ABC fractional derivative Dβ
t is

L(Dβ
t Ψ(η, t)) =

a◦s
βL(Ψ(η, t))− sβ−1Ψ(η, 0)

sβ + a1
, where a◦ =

1

1− β
, a1 =

β

1− β
.

To achieve the conditions, w̄(ρ, s) and v̄(ρ, s) are image functions of w(ρ, t) and v(ρ, t).

w̄(Π1, s) = U1G1(s), w̄(Π2, s) = U2G2(s), (16)

v̄(Π1, s) = V1G3(s), v̄(Π2, s) = V2G4(s), (17)

and we denote the Hankel transformation of w̄(ρ, s) and v̄(ρ, s) in the following:

w̄H(ρn, s) =

∫ Π2

Π1

ρw̄(ρ, s)D1(ρ, ρn)dρ, (18)

and
v̄H(ρm, s) =

∫ Π2

Π1

ρv̄(ρ, s)D0(ρ, ρm)dρ, (19)

where

D1(ρ, ρn) = J1(ρρn)Y1(Π2ρn)− J1(Π2ρn)Y1(ρρn), (20)

D0(ρ, ρm) = J0(ρρm)Y0(Π2ρm)− J0(Π2ρm)Y0(ρρm). (21)

The transcendental equation Di(Π1, ρ), i = 1, 2 has ρn and ρm positive roots. In the
above relations, Bessel functions Jp(.)and Yp(.) are the first and second kind of order p
respectively. ρD1(ρ, ρn) and ρD0(ρ, ρm) multiply both sides to Eqs. (14) and (15) respec-
tively and think about the conditions of Eq.(16), integrating with respect to ρ from Π1 to
Π2



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 10 of 41

∫ Π2

Π1

ρ

(
∂2

∂ρ2
+

1

ρ

∂

∂ρ
− 1

ρ2

)
w̄(ρ, s)D1(ρ, ρn)dρ

= −ρ2nw̄H(ρn, s) + 2
π

[
w̄(Π2)J1(Π1ρn)−w̄(Π1)J1(Π2ρn)

J1(Π1ρn)

]
, (22)

∫ Π2

Π1

ρ

(
∂2

∂ρ2
+

1

ρ

∂

∂ρ

)
v̄(ρ, s)D0(ρ, ρm)dρ

= −ρ2mw̄H(ρm, s) + 2
π

[
v̄(Π2)J0(Π1ρm)−v̄(Π1)J0(Π2ρm)

J0(Π1ρm)

]
. (23)

Now, here we applying the finite Hankel transform.

w̄H

= 2
π

[
U2G2(s)J1(Π1ρn)−U1G1(s)J1(Π2ρn)

J1(Π1ρn)

]
ν+αβa0

sβ

sβ+a1

(ν+αβa0
sβ

sβ+a1
)ρ2n+Gm+Gp(ν+αβa0

sβ

sβ+a1
)+s

, (24)

v̄H

= 2
π

[
V2G4(s)J0(Π1ρm)−V1G3(s)J0(Π2ρm)

J0(Π1ρm)

]
ν+αβa0

sβ

sβ+a1

(ν+αβa0
sβ

sβ+a1
)ρ2m+Gm+Gp(ν+αβa0

sβ

sβ+a1
)+s

. (25)

For working w̄(ρ, s) and v̄(ρ, s) employ the inverse Hankel Transform.Hower ever,
present a result in more suitable form,we firstly rewrite the Eqs. (24) and (25) in the
following equivalent form as follows.

wH =
2

π

1

ρ2n
(1− ξn)

[
U2G2(s)J1(Π1ρn)− U1G1(s)J1(Π2ρn)

J1(Π1ρn)

]
(26)

vH =
2

π

1

ρ2m
(1− ξm)

[
V2G4(s)J0(Π1ρm)− V1G3(s)J0(Π2ρm)

J0(Π1ρm)

]
, (27)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 11 of 41

where,

ξn =
(ν(sβ + a1) + αβa0s

β)Gp + (sβ + a1)(Gm + s)

(ν(sβ + a1) + αβa0sβ)(ρ2n +Gp) + (sβ + a1)(Gm + s)
,

and
ξm =

(ν(sβ + a1) + αβa0s
β)Gp + (sβ + a1)(Gm + s)

(ν(sβ + a1) + αβa0sβ)(ρ2m +Gp) + (sβ + a1)(Gm + s)
,

and apply the inverse Hankel transform formula

w(ρ, s) =
π2

2

∞∑
n=1

ρ2nJ
2
1 (Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)
wH(ρ, s), (28)

v(ρ, s) =
π2

2

∞∑
m=1

ρ2nJ
2
0 (Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)
vH(ρ, s), (29)

we get w(ρ, s) and v(ρ, s) in the following equivalent form.

w(ρ, s) =
U1G1(s)Π1(Π

2
2 − ρ2) + U2G2(s)Π2(ρ

2 −Π2
1)

(Π2
2 −Π2

1)ρ
−π

∞∑
n=1

(1−ξn)
J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
[
U2G2(s)J1(Π1ρn)− U1G1(s)J1(Π2ρn)

]
(30)

v̄(ρ, s) =
V1G3(s) ln(Π2/ρ) + V2G4(s) ln(ρ/Π1)

ln(Π2/Π1)
− π

∞∑
m=1

(1− ξm)
J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
[
V2G4(s)J0(Π1ρm)− V1G3(s)J0(Π2ρm)

]
(31)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 12 of 41

The equivalent forms of the factor of Eqs. (30) and (31) are

(1− ξn) =
∞∑
i=0

(−ρ2n)
i

[
ν(sβ + a1) + αβa0s

β

(ν(sβ + a1) + αβa0sβ)Gp + (sβ + a1)(Gm + s)

]i
,

=

∞∑
i=0

(−ρ2n)
i

(νsβGp + αβa0sβGp +Gmsβ + sβ+1)i

i∑
j=0

i!

j!(i− j)!
νi−j(αβa0s

β)j

×
∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1s

β(i−j−k)

[
1 +

νa1Gp + a1Gm + a1s

νsβGp + αβa0sβGp +Gmsβ + sβ+1

]−i

,

=

∞∑
i=0

(−ρ2n)
i

i∑
j=0

i!

j!(i− j)!
νi−j(αβa0s

β)j
∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1s

β(i−j−k)
∞∑
l=0

(−1)l

× (νa1Gp + a1Gm + a1s)
l

(νsβGp + αβa0sβGp +Gmsβ + sβ+1)l+i
,

=

∞∑
i=0

(−ρ2n)
i

i∑
j=0

i!

j!(i− j)!
νi−j(αβa0)

jsβj
∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1s

β(i−j−k)
∞∑
l=0

(−a1)
l

×
l∑

h=0

l!

l!(l − h)!
(νGp +Gm)hsl−h 1

sβ(l+i)(s+ νGp + αβa0Gp +Gm)l+i
,

=
∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

×
l∑

h=0

l!

l!(l − h)!
(νGp +Gm)h

s−β(k+l)+l−h

(s+ νGp + αβa0Gp +Gm)l+i
. (32)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 13 of 41

For the functions w(ρ, s) and v(ρ, s), we possess the inverse Laplace transformation by
using the following formula[41].

L−1

[
sc

(sa − κ)b

]
= Ma,b

c (κ, t); Re (ac− b) > 0, Re(s) > 0,

∣∣∣∣ dsa
∣∣∣∣ < 1, (33)

where the generalized Ma,j
c (., t) function is defined by[41]

Ma,b
c (κ, t) =

∞∑
j=0

κjΓ(c+ j)t(c+j)a−b−1

Γ(c)Γ(j + 1)Γ[(c+ j)a− b]
. (34)

Finally, we employ Laplace inverse transform L−1{w(ρ, s)} and L−1{v(ρ, s)} which are
w(ρ, t) and v(ρ, t) respectively then using the convolution theorem to Eqs. (30) and (31)
with

Ψ =

i∑
j=0

i!

j!(i− j)!

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νGp +Gm)h

and
Ω = M1,l+i

−β(k+l)+l−h(−(νGp + αβa0Gp +Gm), δ)

the resultant expressions of the velocity field are

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i Ψ

∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]Ω dδ, (35)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i Ψ

∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V2J0(Π2ρm)g3(t− δ)]Ω dδ. (36)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 14 of 41

4.2. For shear stresses
To employ Laplace transform to Eqs. (7) and (8)

S̄ρθ(ρ, s) =

(
µ+ αβa0

sβ

sβ + a1

)(
∂

∂ρ
− 1

ρ

)
w̄(ρ, s), (37)

S̄ρz(ρ, s) =

(
µ+ αβa0

sβ

sβ + a1

)
∂

∂ρ
v̄(ρ, s), (38)

and using the fact that

∂

∂ρ
D1(ρ, ρn) = ρnD1(ρ, ρn)−

1

ρ
D1(ρ, ρn) and ∂

∂ρ
D0(ρ, ρm) = −ρmD0(ρ, ρm)

where

D1(ρ, ρn) = J0(ρρn)Y1(Π2ρn)− J1(Π2ρn)Y0(ρρn),

D0(ρ, ρm) = J1(ρρm)Y0(Π2ρm)− J0(Π2ρm)Y1(ρρm),

and if

D̃1(ρ, ρn) =
2

ρ
D1(ρ, ρn)− ρnD1(ρ, ρn), D̃0(ρ, ρm) = −ρmD0(ρ, ρm),

then, we have

∂w̄(ρ, s)

∂ρ
− 1

ρ
w̄(ρ, s) =

2U2G2(s)Π2Π
2
1 − 2U1G1(s)Π1Π

2
2

(Π2
2 −Π2

1)ρ
2

+π
∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)
[U2G2(s)J1(Π1ρn)− U1G1(s)J2(Π2ρn)]ξn (39)

∂v̄(ρ, s)

∂ρ
=

V2g4(s)− V1g3(s)

ρ ln(Π2/Π1)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 15 of 41

+π

∞∑
m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

[
V2G4(s)J0(Π1ρm)− V1G3(s)J0(Π2ρm)

]
ξm (40)

are obtained from Eqs.(26) and (27) after applying inverse Hankel transformation.
Introducing Eqs.(39) and (40) into Eqs. (37) and (38) consequently, employing the inverse
Laplace transform then convolution theorem, we gained the shear stress Sρθ(ρ, t) and
Sρz(ρ, t) with

Υ =

i∑
j=0

i!

j!(i− j)!

(
αβa0s

β

ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
×(να1Gp+α1Gm)h

and

M =
(
µ+ αβa0

)
M1,l+i

−β(k+l)+l−h(−(νGp + αβaoGp +Gm), δ)

+µa1M1,l+i
−β(1+k+l)+l−h (−(νGp + αβaoGp +Gm), δ)

then;

Sρθ(ρ, t) =
2H(t)Π1Π2

(Π2
2 −Π2

1)ρ
2
[U2g2(t)Π1 − U1g1(t)Π2][µ+ αβao]−

αβa0
(Π2

2 −Π2
1)ρ

2

×
∫ t

0
[U2Π1g2(t− δ)− U1Π2g1(t− δ)]Mβ,1

0 (−a1, δ)dδ + πH(t)

∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i Υ

∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)] M dδ, (41)

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)](µ+ αβao)−

H(t)αβa0
ρ ln(Π2/Π1)

×
∫ t

0
[V2g4(t− δ)− V1g3(t− δ)]Mβ,1

0 (−a1, δ)dδ + πH(t)
∞∑

m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 16 of 41

×
∞∑
i=0

(−νρ2n)
i Υ

∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V1J0(Π2ρm)g3(t− δ)] M dδ. (42)

5. Important Limiting cases

5.1. Ordinary Second grade fluid (β → 1)
By assuming β → 1 in Eqs. (35), (36), (41) and (42), we obtained the outcomes

for ordinary second grade fluid.

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νGp+Gm)h

×
∫ t

0
[U2J1(Π1ρn)g2(t−δ)−U1J1(Π2ρn)g1(t−δ)]M1,l+i

−(k+l)+l−h(−(νGp+αa0Gp+Gm), δ)dδ,

(43)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i
i∑

j=0

i!

j!(i− j)!

(
αa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νGp+Gm)h

×
∫ t

0
[V2J0(Π1ρm)g4(t−δ)−V2J0(Π2ρm)g3(t−δ)]M1,l+i

−(k+l)+l−h(−(νGp+αa0Gp+Gm), δ)dδ,

(44)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 17 of 41

Sρθ(ρ, t) =
2H(t)Π1Π2

(Π2
2 −Π2

1)ρ
2
[U2g2(t)Π1−U1g1(t)Π2][µ+αao]−

H(t)αa0
(Π2

2 −Π2
1)ρ

2

∫ t

0
[U2Π1g2(t− δ)

−U1Π2g1(t− δ)]e−a1tdδ + πH(t)
∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

∞∑
i=0

(−νρ2n)
i

×
i∑

j=0

i!

j!(i− j)!

(
αa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νa1Gp + a1Gm)h

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]

[(
µ+ αa0

)

×M1,l+i
−(k+l)+l−h(−(νGp+αaoGp+Gm), δ)+µa1M1,l+i

−(1+k+l)+l−h(−(νGp+αaoGp+Gm), δ)

]
dδ,

(45)

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)−V1g3(t)](µ+αao)−

H(t)αa0
ρ ln(Π2/Π1)

∫ t

0
[V2g4(t−δ)−V1g3(t−δ)]

×e−a1tdδ + πH(t)
∞∑

m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αa0
ν

)j

×
∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νa1Gp + a1Gm)h

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V1J0(Π2ρm)g3(t− δ)]

[(
µ+ αa0

)

×M1,l+i
−1(k+l)+l−h(−(νGp+αaoGp+Gm), δ)+µa1M1,l+i

−1(1+k+l)+l−h(−(νGp+αaoGp+Gm), δ)

]
dδ.

(46)
If we take v(Π1, t) = V1sin(Ω1t), and v(Π2, t) = V2sin(Ω2t), then the solutions given by
Eqs.(44) and (46) are equivalent to those obtained by [42] in Eqs. (28) and (29).



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 18 of 41

5.2. Newtonian fluid with Porous and Magnetic effect (α → 0)
By assuming α → 0 in Eqs. (35), (36), (41) and (42), we acquire outcomes

for Newtonian fluid with porous and magnetic effect.

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

∫ t

0
[U2J1(Π1ρn)g2(t−δ)−U1J1(Π2ρn)g1(t−δ)]M1,i

0 (−(νGp+Gm), δ)dδ, (47)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρm)g1(t− δ)]M1,i

0 (−(νGp +Gm), δ)dδ,

(48)

Sρθ(ρ, t) =
2Π1Π2

(Π2
2 −Π2

1)ρ
2
H(t)[U2g2(t)Π1−U1g1(t)Π2]µ+πµH(t)

∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]M1,i

0 (−(νGp +Gm), δ)dδ,

(49)

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)]µ+ πµH(t)

∞∑
m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2n)
i

∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρm)g1(t− δ)]M1,i

0 (−(νGp +Gm), δ)dδ.

(50)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 19 of 41

5.3. Fractionalized Second grade with Porous effect(Gm → 0)
By assuming Gm → 0 in Eqs. (35), (36), (41) and (42), so we acquire the outcomes

for second grade with porous effect

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νGp)

h

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]M1,l+i

−β(k+l)+l−h(−(νGp + αβa0Gp), δ)dδ,

(51)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i
i∑

j=0

i!

j!(i− j)!

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νGp)

h

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V2J0(Π2ρm)g3(t− δ)]M1,l+i

−β(k+l)+l−h(−(νGp + αβa0Gp), δ)dδ,

(52)

Sρθ(ρ, t) =
2H(t)Π1Π2

(Π2
2 −Π2

1)ρ
2
[U2g2(t)Π1−U1g1(t)Π2][µ+αβao]−

H(t)αβa0
(Π2

2 −Π2
1)ρ

2

∫ t

0
[U2Π1g2(t−δ)

−U1Π2g1(t− δ)]Mβ,1
0 (−a1, δ)dδ + πH(t)

∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

∞∑
i=0

(−νρ2n)
i

×
i∑

j=0

i!

j!(i− j)!

(
αβa0sβ

ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νa1Gp)

h



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 20 of 41

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]

[(
µ+ αβa0

)

×M1,l+i
−β(k+l)+l−h(−(νGp + αβaoGp), δ) + µa1M1,l+i

−β(1+k+l)+l−h(−(νGp + αβaoGp), δ)

]
dδ,(53)

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)](µ+ αβao)−

H(t)αβa0
ρ ln(Π2/Π1)

∫ t

0
[V2g4(t− δ)

−V1g3(t− δ)]Mβ,1
0 (−a1, δ)dδ + πH(t)

∞∑
m=1

J0(Π1ρmD̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

∞∑
i=0

(−νρ2n)
i

×
i∑

j=0

i!

j!(i− j)!

(
αβa0sβ

ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(νa1Gp)

h

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V1J0(Π2ρm)g3(t− δ)]

[(
µ+ αβa0

)
aoGp), δ)

×M1,l+i
−β(k+l)+l−h(−(νGp + αβ + µa1M1,l+i

−β(1+k+l)+l−h(−(νGp + αβaoGp), δ)

]
dδ. (54)

5.4. Fractionalized Second grade with Magnetic effect (Gp → 0)
By assuming Gp → 0 in Eqs. (35), (36), (41) and (42), we achieved the solutions

for second grade fluid with magnetic effect

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(Gm)h



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 21 of 41

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]M1,l+i

−β(k+l)+l−h(−Gm, δ)dδ, (55)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i
i∑

j=0

i!

j!(i− j)!

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(Gm)h

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V2J0(Π2ρm)g3(t− δ)]M1,l+i

−β(k+l)+l−h(−Gm, δ)dδ, (56)

Sρθ(ρ, t) =
2H(t)Π1Π2

(Π2
2 −Π2

1)ρ
2
[U2g2(t)Π1−U1g1(t)Π2][µ+αβao]−

H(t)αβa0
(Π2

2 −Π2
1)ρ

2

∫ t

0
[U2Π1g2(t−δ)

−U1Π2g1(t− δ)]Mβ,1
0 (−a1, δ)dδ + πH(t)

∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

∞∑
i=0

(−νρ2n)
i

×
i∑

j=0

i!

j!(i− j)!

(
αβa0sβ

ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(Gm)h

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]

[(
µ+ αβa0

)
M1,l+i

−β(k+l)+l−h(−Gm, δ)

+µa1M1,l+i
−β(1+k+l)+l−h(−Gm, δ)

]
dδ, (57)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 22 of 41

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)](µ+ αβao)−

H(t)αβa0
ρ ln(Π2/Π1)

×
∫ t

0
[V2g4(t−δ)−V1g3(t−δ)]Mβ,1

0 (−a1, δ)dδ+πH(t)
∞∑

m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

∞∑
i=0

(−νρ2n)
i

×
i∑

j=0

i!

j!(i− j)!

(
αβa0sβ

ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

l∑
h=0

l!

h!(l − h)!
(a1Gm)h

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V1J0(Π2ρm)g3(t− δ)]

[(
µ+ αβa0

)
M1,l+i

−β(k+l)+l−h(−Gm, δ)

+µa1M1,l+i
−β(1+k+l)+l−h(−Gm, δ)

]
dδ. (58)

5.5. Fractionalized Second grade Fluid (Gm → 0 and Gp → 0)
By assuming Gm → 0 and Gp → 0 to Eqs. (35), (36), (41) and (42), so we achieved

the solutions for second grade fluid.

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]M1,i

−β(k+l)(0, δ)dδ, (59)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 23 of 41

×
∞∑
i=0

(−νρ2m)i
i∑

j=0

i!

j!(i− j)!
νi−j

(
αβa0
ν

)j ∞∑
k=0

(i− j)!

k!(i− j − k)!
ak1

∞∑
l=0

(−a1)
l

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V2J0(Π2ρm)g3(t− δ)]M1,i

−β(k+l)(0, δ)dδ, (60)

Sρθ(ρ, t) =
2H(t)Π1Π2

(Π2
2 −Π2

1)ρ
2
[U2g2(t)Π1 − U1g1(t)Π2][µ+ αβao]−

H(t)αβa0
(Π2

2 −Π2
1)ρ

2

×
∫ t

0
[U2Π1g2(t− δ)− U1Π2g1(t− δ)]Mβ,1

0 (−a1, δ)dδ + πH(t)
∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αβa0sβ

ν

)j ∞∑
k=0

(i− j − 1)!

k!(i− j − 1− k)!
ak1

∞∑
l=0

(−a1)
l

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]

[(
µ+ αβa0

)
M1,i

−β(k+l)(0, δ)

+µa1M1,i
−β(1+k+l)(0, δ)

]
dδ, (61)

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)](µ+ αβao)−

H(t)αβa0
ρ ln(Π2/Π1)

∫ t

0
[V2g4(t− δ)− V1g3(t− δ)]Mβ,1

0 (−a1, δ) + πH(t)
∞∑

m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2n)
i

i∑
j=0

i!

j!(i− j)!

(
αβa0sβ

ν

)j ∞∑
k=0

(i− j − 1)!

k!(i− j − 1− k)!
ak1

∞∑
l=0

(−a1)
l



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 24 of 41

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V1J0(Π2ρm)g3(t− δ)]

[(
µ+ αβa0

)
M1,i

−β(k+l)(0, δ)

+µa1M1,i
−β(1+k+l)(0, δ)

]
dδ. (62)

5.6. Newtonian with Porous effect (α,Gm → 0)
By assuming α → 0 and Gm → 0 in Eqs. (35), (36), (41) and (42), so we acquire

the solutions for Newtonian with porous effect .

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]M1,i

0 (−νGp, δ)dδ, (63)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V2J0(Π2ρm)g3(t− δ)]M1,i

0 (−νGp, δ)dδ, (64)

Sρθ(ρ, t) =
2H(t)Π1Π2

(Π2
2 −Π2

1)ρ
2

[
U2g2(t)Π1 − U1g1(t)Π2

]
µ+ πµH(t)

∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]M1,i

0 (−νGp, δ)dδ, (65)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 25 of 41

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)]µ+ πµH(t)

∞∑
m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V1J0(Π2ρm)g3(t− δ)]M1,i

0 (−νGp, δ)dδ. (66)

5.7. Newtonian with Magnetic effect (α,Gp → 0)
By assuming α → 0 and Gp → 0 into Eqs. (35), (36), (41) and (42), so we acquire

the solutions for Newtonian with magnetic effect .

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i

∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)]M1,i

0 (−Gm, δ)dδ, (67)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∞∑
i=0

(−νρ2m)i
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V2J0(Π2ρm)g3(t− δ)]M1,i

0 (−Gm, δ)dδ, (68)

Sρθ(ρ, t) =
2Π1Π2

(Π2
2 −Π2

1)ρ
2
H(t)

[
U2g2(t)Π1−U1g1(t)Π2

]
µ+πµH(t)

∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∞∑
i=0

(−νρ2n)
i(Gm)h

∫ t

0
[U2J1(Π1ρn)g2(t−δ)−U1J1(Π2ρn)g1(t−δ)]M1,i

0 (−Gm, δ)dδ, (69)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 26 of 41

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)]µ+ πµH(t)

∞∑
m=1

ρmJ0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)
H(t)

×
∞∑
i=0

(−νρ2m)i
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V1J0(Π2ρm)g3(t− δ)]M1,i

0 (−Gm, δ)dδ. (70)

5.8. Newtonian (α,Gp, Gm → 0)
By assuming α → 0, Gp → 0 and Gm → 0 into Eqs. (35), (36), (41) and (42), so

we acquire the solutions for Newtonian.

w =
H(t)

(Π2
2 −Π2

1)ρ

[
U1g1(t)Π1(Π

2
2−ρ2)+U2g2(t)Π2(ρ

2−Π2
1)

]
−πH(t)

∞∑
n=1

J1(Π1ρn)D0(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)][1− e−νρ2nδ]dδ, (71)

v =
H(t)

ln(Π2/Π1)

[
V1g3(t) ln(Π2/ρ) + V2g4(t) ln(ρ/Π1)

]
− πH(t)

∞∑
n=1

J0(Π1ρm)D1(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∫ t

0
[V2J0(Π1ρm)g4(t− δ)− V2J0(Π2ρm)g3(t− δ)][1− e−νρ2mδ]dδ, (72)

Sρθ(ρ, t) =
2H(t)Π1Π2

(Π2
2 −Π2

1)ρ
2

[
U2g2(t)Π1 − U1g1(t)Π2

]
µ+ πµH(t)

∞∑
n=1

J1(Π1ρn)D̃1(ρ, ρn)

J2
1 (Π1ρn)− J2

1 (Π2ρn)

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)][1− e−νρ2nδ]dδ, (73)



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 27 of 41

Sρz(ρ, t) =
H(t)

ρ ln(Π2/Π1)
[V2g4(t)− V1g3(t)µ+ πµH(t)

∞∑
m=1

J0(Π1ρm)D̃0(ρ, ρm)

J2
0 (Π1ρm)− J2

0 (Π2ρm)

×
∫ t

0
[U2J1(Π1ρn)g2(t− δ)− U1J1(Π2ρn)g1(t− δ)][1− e−νρ2mδ]dδ. (74)

If we take w(Π1, t) = A1t, w(Π2, t) = A2t, then the solutions given by Eqs.(71) and
(73) are agree to those obtained by [43] in Eqs. (20) and (28). Furthermore, if we
take w(Π1, t) = W1sin(ω1t), w(Π2, t) = W2sin(ω2t), then the solutions given by Eqs.(71)
and (73) are agree to those obtained by [44] in Eqs. (24) and (25). Similarly, we take
v(Π1, t) = A1t, w(Π2, t) = A2t, then the solutions given by Eqs.(72) and (74) are agree to
those obtained by [45] in Eqs. (4.3) and (4.4).

6. Numerical Results and Discussion

The present work estimates the helical flow of fraction MHD second grade fluid
for very generalized boundary conditions under the porous nature between uniaxial an-
nular cylinders. We are living in the era of fractional calculus and concept of fractional
calculus has become essential in all sciences and engineering. Therefore, we fractionalized
governing equations of the helical flow of second grade fluid with the modern Atangana
Baleanu Caputo (ABC) fractional operator. The exact analytical outcomes determine
the rotational and longitudinal velocities and shear stresses between two infinite circular
cylinders for generalized boundary conditions on the surfaced inner and outer cylinders.
The outcomes are obtained with the help of infinite Laplace and finite Hankel transforms
while using ABC time derivatives. The outcomes are handover in integral and series forms
with generalized Ma,b

c (·, t) function. Furthermore, these outcomes need the requirement of
all natural restrictions, we can impose on them. For example, these outcomes can satisfy
the governing equations and initial and boundary conditions. We mentioned here some
important worth of present research. We first time introduced the ABC fractional opera-
tor in the helical flow between coaxial cylinders. Another is that we used the generalized
Ma,b

c (·, t) function which shortened our exact analytical solutions as compared to some
important previous work [30, 36–38, 42–45]. Another aspect of the present outcomes is
that we used a very generalized function on the surface of the inner and outer cylinders.
Additionally, the outcomes of second grade fluid and Newtonian fluid with and without
MHD and porous effect can also be obtained very easily from the present generalized so-
lution.



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 28 of 41

Figure 2: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) presented
in Eqs. (35), (36), (41) and (42), for α = 2.5, Gm = 0.5, Gp = 0.4 dissimilar rates of ρ.

In the limiting cases of our outcomes, by assigning the fractional parameter β → 1
and material parameter α → 0, we earn ordinary second grade fluid in section 5.1 and
Newtonian fluid with both porous and magnetic effects in section 5.2 respectively. By con-
sidering magnetic constant Gm → 0 and porous constant Gp → 0, we acquire the outcome
as the fractionalized second grade with porous effect in section 5.3 and fractionalized sec-
ond grade with magnetic effect in section 5.4 respectively while in this case, both porous
and magnetic effects eliminate together then it becomes fractionalized second grade in
section 5.5. The fractionalized second grade with porous and fractionalized second grade
with magnetic effect are recognized as Newtonian with porous section 5.6 and Newtonian
with magnetic effect section 5.7 respectively, when we vanish the material parameter. At
last section 5.8, we attain the Newtonian fluid as the material parameter abolished in the
fractionalized second grade fluid.



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 29 of 41

Figure 3: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) presented
in Eqs. (35), (36), (41) and (42), for α = 2.5, Gm = 0.5, Gp = 0.4 dissimilar rates of t.

What physical represent the outcomes, and what are the impacts of physical pa-
rameters that appear in the outcomes? For these concern, we made graphs for velocity
fields ω(ρ, t), v(ρ, t) and shear stresses Sρz(ρ, t), Sρθ(ρ, t). We have two choices for making
these graphs either we use independent variable radial distances ρ or time t. We select
here to represent the velocity field component and shear stresses against the time t be-
cause we take boundary conditions g1(t), g2(t), g3(t) and g4(t) as e−tsin(ωt). Such type
of boundary conditions in literature are very few but it seems to be intrinsic because the
amplitude of the oscillation tends to zero when time goes to infinity, and other parameters
fixed in all graphs are ϱ = 11.12, η = 0.042, Π1 = 0.4, Π2 = 0.8, U1 = U2 = V1 = V2 = 2,



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 30 of 41

ω = 3, α = 2.5, fractional parameter β = 0.5, Gm = 0.5, Gp = 0.4, and the radial distance
off course vary from 0.4 to 0.8 and variation in t, we consider 1 to 6, unless otherwise
stated in the diagram. For plotting these graphs, we used Mathcad program and for final
makeup, we used MS Paint.

Figure 4: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) presented
in Eqs. (35), (36), (41) and (42), for Gm = 0.5, Gp = 0.4 dissimilar rates of α.

The effect of the radial parameter of ρ for small and large values of time on ve-
locity fields and shear stresses portrait is highlighted in Fig. 2. It is noted that the
amplitude of oscillation of these entities decreases with the increase of radial distance.
However for v(ρ, t) these above comments are not appropriate as shown in Fig. 2(b). The
large time effect is clearly shown in all diagrams. It is observed that the portrait of four



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entities tends to zero for large time t, due to boundary condition e−tsin(ωt). Further,
, it has been noted that ω(ρ, t) and v(ρ, t) and similarly Sρθ(ρ, t), Sρz(ρ, t) have quite
opposite behavior in their respective portraits. It is also clear from these pictures v(ρ, t)
and Sρz(ρ, t), and ω(ρ, t) and Sρθ(ρ, t) have again quite opposite behavior in their portrait.

Figure 5: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) presented
in Eqs. (35), (36), (41) and (42), for α = 2, Gm = 5, Gp = 4 dissimilar rates of ν.

The effect of time on ABC fractionalized helical motion of second grade fluid are
shown in Fig. 3. The velocity field components ω(ρ, t) on the entire domain and 60%
of v(ρ, t) in the given domain are increasing function of time t. However, shear stresses
Sρθ(ρ, t) and Sρz(ρ, t) have opposite portraits to each other with respect to time. It is also
clear from these figures that rotational components of velocity fields and shear stresses



M. Zafarullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6198 32 of 41

have oscillating effects in their portrait as compared to translation entities.

Figure 6: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) presented
in Eqs. (35), (36), (41) and (42), for α = 2, Gp = 2 dissimilar rates of Gm.

The very internal parameter of second grade fluid is material parameter α. The
increasing values of material parameter α increase the non-Newtonian behavior of second
grade fluid. The graphs of such an important and interesting study of the present research
are depicted in Fig. 4. All graphs agree that large values of material parameter α lower
the amplitude of the motion or decay the motion. Furthermore, the rate of decay in shear
stress portraits is faster than in velocity field components. It is also clear from these graphs
that the shear stress component Sρθ decays in minimum and the velocity field decays w(t)
in maximum time.



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Figure 7: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) presented
in Eqs. (35), (36), (41) and (42), for α = 2, Gm = 2 dissimilar rates of Gp.

The other significant internal parameter that resists the flow is the kinematic vis-
cosity ν. As it is a common phenomenon in Newtonian fluid the greater the viscosity, the
thicker it is that is flow is slow. Fig. 5 shows the impact of kinematic viscosity on the
present dynamical system. Although the fluid is second grade fluid, it is clear from Figs.
5 that the larger the viscosity the slower the flow and the amplitude of the motion of fluid
reduced. However, Fig. 5(d) seems a little bit opposite here but we can ignore it as differ-
ence in the four portraits is negligible. The present study is the electronically conducting
fluid and quantitative measurement of such fluid is done by the magnetic parameter Gm.
For this purpose, we made the Figs. 6, these graphs explicitly show the great impact of



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magnetic parameters on the present system. It is obvious that increasing the magnetic
parameter Gm decreases the amplitude of the motion and the four essences in Figs. 6 tend
to zero faster than Figs. 4 and 5.

Figure 8: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) presented
in Eqs. (35), (36), (41) and (42), for α = 1.5, Gm = 3, Gp = 2 dissimilar rates of β.

The porosity Gp of the material cannot be neglected. It is the effect that de-
creases the strength of the flow. Without porosity is the ideal condition, which practically
does not exist. This resistive measure quantitatively is explained in Figs. 7. It is noted
that the present flow conditions put the effect of porosity measure in such that increasing
values of porosity will reduce the oscillating motion of the fluid. However, there are dif-
ferent portraits in Fig 7(b), which is due to the oscillating motion of the fluid. Further,



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the impact of the porosity parameter Gp on the reduction of motion of the fluid is lower
than the magnetic parameter Gm. Today’s world is a fractionalized world. All sciences,
engineering, and social-economic information are nowadays transformed into a fractional-
ized system. The reason is clear that integer order explanation is not enough to explore
the world. Very recent fractional derivative that has already discussed is the Atangana
Baleanu Caputo (ABC) operator β that is used in this study. The effect of this operator
is explored in the Figs. 8. It brings to the knowledge that three entities v(t), Sρθ and
Sρz are increasing functions of the ABC fractional operator β, however, the w(t) is quite
opposite behavior in comparison to three.

Figure 9: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) for
α = 2.5, Gm = 2.5, Gp = 1.5 for dissimilar types of fluids.



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Figure 10: Portraits of the velocity fields w(ρ, t) and v(ρ, t) and shear stresses Sρθ(ρ, t) and Sρz(ρ, t) for
α = 2.5, Gm = 2.5, Gp = 1.5 for dissimilar types of fluids.

In order to make this study more interesting and draw some more extract and
conclusions, we made Figs 9 and 10. In these graphs we present the four fluids, frac-
tionalized second grade fluid (given by Eqs. (35), (36), (41), and (42)) for β = 0.2 and
β = 0.7, second grade (obtained from Eqs. (59 - 62) by putting β → 1) and Newtonian
(given by Eqs. 71- 74) fluids with respect to time t and radial parameter ρ. Quite the
opposite phenomenon is observed in Figs. 9 that in rotational quantities w(t), Sρθ the
fractionalized second grade fluid for β = 0.2 have the largest values and Newtonian fluid
has least values. In spite of that the translational quantities v(t), Sρz have quite contrary
attitude in comparison of rotational quantities. With regard to radial distance as shown
in Figs. 10, it is observed that fractionalized second grade fluid for β = 0.2 have largest



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amplitude and Newtonian fluid has smallest. On the other hand, the fractionalized second
grade fluid for β = 0.7 and second grade fluid have similar behavior but the second grade
fluid has deeper amplitude. All these pictures are depicted in Mathcad software and using
MKS units system. At the end, we used Paint software to bring graphs in good condition.

7. Conclusion

This research paper deals with the application and influence of the newly defined
fractional derivative Atangana Baleanu Caputo (ABC) on the unsteady helical motion of
second grade fluid with MHD effect between porous annular cylinders. In this work after
using a helical model of second grade fluid, we introduced the ABC fractional derivative to
the governing equations, and analytical outcomes are obtained for rotational and transla-
tion quantities in porous annular region. Dual integral transformations including Laplace
and Hankel are used to eliminate the partial derivatives and then the resulting algebraic
expression will be solved by inverse transformations in the form of series, integral, convo-
lution product, and generalized functions. These outcomes satisfy all restrictions and give
many particular solutions of second grade and Newtonian fluid with/without MHD and
porous attachment. Some of our special solutions coincide with previous solutions, which
prove the correctness of our results. The major novel findings of the present work are:

The ABC fractional derivative is used in the helical motion of second grade and
Newtonian fluids. Introducing the generalized M functions, which are natural for such
types of calculations. The fluid motion is a decreasing function of the material parameter
α and kinematic viscosity ν. The magnetic and porous parameters Gm and Gp respec-
tively, have almost contrary impacts on the fluid vibration. The fractional parameter β
has opposite influence on velocity field and shear stress portraits. With respect to time t,
the rotational quantities have large values of fractionalized second grade fluid for β = 0.2
and Newtonian has least. With respect to radial distance ρ the fractionalized second grade
fluid for β = 0.2 and Newtonian fluids have respectively maximum and minimum oscilla-
tions.

8. Future Recommendation

This research work can be extendable to the other non-Newtonian fluid such as
Maxwell fluid, Burgers material, Oldroyd-B model by considering various geometrical
configuration like helical flow or flow in channel along with some additional effects through
Atangana Baleanu Caputo (ABC) fractional operator. The outcome can be obtained in
terms of a series of trigonometric or any other generalized Wright functions.



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Acknowledgements

The author A. B. Albidah extends the appreciation to the Deanship of Postgraduate
Studies and Scientific Research at Majmaah University for funding this research work
through the project number (R-2025-1726).

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