Investigation of magnetized convection for second-grade nanofluids via Prabhakar differentiation

Qasim Ali; Samia Riaz; Imran Qasim Memon; Irfan Ali Chandio; Muhammad Amir; Ioannis E. Sarris; Kashif Ali Abro

doi:10.1515/nleng-2022-0286

Open Access Published by De Gruyter April 5, 2023

Investigation of magnetized convection for second-grade nanofluids via Prabhakar differentiation

Qasim Ali , Samia Riaz , Imran Qasim Memon , Irfan Ali Chandio , Muhammad Amir , Ioannis E. Sarris and Kashif Ali Abro

From the journal Nonlinear Engineering

https://doi.org/10.1515/nleng-2022-0286

Abstract

The application of nanoparticles in the base fluids strongly influences the presentation of cooling as well as heating techniques. The nanoparticles improve thermal conductivity by fluctuating the heat characteristics in the base fluid. The expertise of nanoparticles in increasing heat transference has captivated several investigators to more evaluate the working fluid. This study disputes the investigation of convection flow for magnetohydrodynamics second-grade nanofluid with an infinite upright heated flat plate. The fractional model is obtained through Fourier law by exploiting Prabhakar fractional approach along with graphene oxide ( GO ) and molybdenum disulfide ( Mo S 2 ) nanoparticles and engine oil is considered as the base fluid. The equations are solved analytically via the Laplace approach. The temperature and momentum profiles show the dual behavior of the fractional parameters ( α , β , γ ) at different times. The velocity increases as Grashof number increases and declines for greater values of magnetic parameter and Prandtl number. In the comparison of different numerical methods, the curves are overlapped, signifying that our attained results are authentic. The numerical investigation of governed profiles comparison shows that our obtained results in percentages of 0.2 ≤ temperature ≤ 4.36 and velocity 0.48 ≤ 7.53 are better than those of Basit et al. The development in temperature and momentum profile, due to engine oil–GO is more progressive, than engine oil–MoS₂.

Keywords: convection flow; second-grade nanofluid; magnetic field; Fourier law; Prabhakar fractional derivative; Laplace transform

Nomenclature

C P: particular heat at constant pressure ( J kg − 1 K − 1 )
erfc ( . ): Complementary Gauss error function
g: gravitational acceleration ( m s − 2 )
Gr: Grash of number
K nf: heat conductivity of nanofluid ( kg m s − 3 K − 1 )
M: magnetic parameter
Pr: Prandtl number
p: Laplace transform variable
υ: kinematic viscosity ( m 2 s − 1 )
w: velocity field ( m s − 1 )
μ: dynamic viscosity ( kg m − 1 s − 1 )
ρ nf: density of nanofluid ( kg m − 3 )
β θ: volumetric coefficient of thermal expansion ( K − 1 )
θ: fluid temperature ( K )
θ w: fluid temperature at the plate ( K )
θ ∞: temperature away from the plate ( K )
ρ: density of fluid ( kg m − 3 )
μ nf: dynamic viscosity of nanofluid (kg m⁻¹ s⁻¹)
σ nf: electrical conductivity of nanofluid
( ρ C P ) nf: thermal ability of nanofluid
φ: volume fraction of nanofluid
δ ( t ): Dirac delta distribution

Abbreviations

AB: Atangana–Baleanu time-fractional derivative
CF: Caputo–Fabrizio time-fractional derivative
CTFD: Caputo time-fractional derivative
HNF: hybrid nanofluid
LT: Laplace transforms
MHD: magnetohydrodynamics
NF: nanofluid
ODEs: ordinary differential equations
PDEs: partial differential equations

1 Introduction

It is a familiar fact that various researchers have more curiosity to investigate non-Newtonian fluids due to their exclusive real-world applications with substantial features in science and technology. The characteristics of these fluids are verified in several engineering areas as it plays a vigorous role in manufacturing for instance melting and processing of polymers, clay coatings, waste fluid, extrusion of melted plastic, nutrition industry, and numerous suspensions. For example, biological constituents, all creams, and composite mixtures are considered as non-Newtonian fluids. This class of fluids has diverse properties and it may not be demarcated in a single model, however, for Newtonian fluid, it is conceivable to define it in a single fluid model. It is much tough to understand that how to categorize these fluids, as in the literature, numerous forms of fluids happen. But, this type (non-Newtonian) of fluids are allocated into differential, rate, and Integral. Various investigators considered these three forms of non-Newtonian fluid models and every model has its own properties. Among them, the second-grade non-Newtonian fluid model fascinated special consideration, which is a simple sub-class of the differential form of these liquids. Natural convection is an instrument that happens because of the heat changes which disturb the density, i.e., the intensification of stove liquid and decline in the serene liquid. It has also an extensive role in biomedical engineering as well as fluid mechanics. Further, it is convenient for concentration, freezing, dehydration, ventilation of building projects, etc. [1,2].

Recently, the investigation on magnetohydrodynamics (MHD) flow has fascinated the consideration of various investigators because of its importance in several engineering as well as industrial fields, for instance, MHD power generators, chemical reactions, metal casting, plasma flows, nuclear reactors, metallurgical processes, and petrol production. First, the term MHD was used by Alfven [3]. The triangle Mittage-Leffler function to discourse the ambiguity in MHD, as well as ohmic heating impact with a third-grade fluid in the existence of heat creation, nanoparticle structure, and thermal radiation, is applied through Nadeem et al. [4]. The ferro-nanofluid with the MHD was discussed by Saqib et al. [5]. The impact of nanofluid (NF) on viscous fluids in the presence of MHD was discussed by Gul et al. [6].

Choi was the first to propose NFs that incorporated nanoparticles in 1995. Various consequences in heat conductivity may be attained from NFs. These are commonly imaginary in mechanical, biomedical manufacturing, fluid simulation, etc. The free convective NF along an oblique plate was discussed by Kuznetsov and Nield [7]. Acharya et al. [8,9] studied the hydro-thermal effects of MHD-free convective laminar NF flows by employing a Galerkin method-based finite element approach inside a circular inclusion. They also studied the radiative MHD mixed convective combined stress liquid, which runs over a porous stretched tube. Attia et al. [10] employed the reproduced Hilbert kernel space technique to formulate numerical solutions for different basic fractional ordinary differential equations (ODEs) under the fractal fractional operator along with Mittag–Leffler (ML) kernel. Moreover, the consistency between the estimated and analytical solutions approves the applicability and higher presentation of the technique. Bilal et al. [11] investigated the computational and theoretical thermal evaluation of MHD Sutterby fluid comprising radiation characteristics entrenched in a stratified medium which shows the innovation of work and determined that intensifying degree of radiation parameter raises the thermal field. Farman et al. [12] applied the HIV/AIDS model with Caputo–Fabrizio time-fractional derivative (CF) and fractal fractional operator fractional-order model. The progressive method is applied for HIV/AIDS fractional-order mathematical approach to obtain consistent outcomes with the Sumudu transform method. Modanli et al. [13] studied a fractional order pseudo-parabolic partial differential equation (PDE) demarcated by the Caputo operator and the Laplace decomposition technique, which is employed to acquire the analytical solution of this equation. Qureshi et al. [14] discussed the non-Newtonian fluid flow along with orthogonal permeable surfaces. A model NFs reproduction is taken which characterizes speculative structures of materials that are gratified in lubricants construction, biomechanics lubricants constructions, biomechanics, suspension, etc. They proved that the nanolayer viscosity is a substantial impact associated with the thermal transmission rate. Shah et al. [15] studied the adumbrate characteristics of NF flow with water and the distribution of single-wall carbon nanotubes. Decay in velocity against the Hall current parameter is detected. Decline and elevation in the temperature field are demonstrated against the sink as well as heat source parameters correspondingly. Xu et al. [16] discussed the examination of the second wave of the Coronavirus fractional model in India by employing the Atangana–Baleanu time-fractional derivative (AB) as well as the Caputo derivative. Iqbal et al. [17] discussed the Newell–Whitehead–Segel equation, which describes the consistency of the solution in the framework of current philosophy, and a novel method is used to show the presence of the solution and equivalent explicit prior approximations of the Schauder method is applied. In the previous studies [18,19], some other researchers discussed the MHD-driven buoyant-induced hybrid nanofluid (HNF; Ag–MgO–H₂O) flow with a circular tube and numerous heat approaches of the internal tube i.e., adiabatic, heated, and cold are supposed to perform this examination. Moreover, the hydro-thermal behavior and entropy investigation of MHD fluid flowing through the circular cylinder and octagonal cavity were also discussed. Acharya et al. [20] evaluated the MHD mixed HNF flow over a revolving sphere close to the stagnation zone. Jie et al. [21] examined the Brinkman type NF model by Laplace and Prabhakar fractional operator and showed that momentum and temperature profiles signify a decaying trend by changing fractional parameters.

Fractional calculus, i.e., the investigation of the generality of the usual philosophy of calculus to derivatives as well as integrals, has captivated much consideration in current years from diverse fields. Fractional differential equations based on fractional operators are enormously useful for modeling several everyday life physical problems as fractional calculus have memory impacts, like problems in relaxation, oscillation, diffusion, dynamical progressions, delay phenomena in complex structures, and several more engineering phenomena; Thus, classical models may not define the state of preceding phenomena. In the previous studies maximum of the investigations are intensive on flow problems in view of numerous fractional derivatives having local and non-local kernels like Caputo, CF, AB, Prabhakar, and some others, those are specified the existing state but also on the future state of a system. Fractional derivatives, for instance, Riemann–Liouville, Caputo, Caputo–Liouville, and CF are efficiently applied in complex media abnormal diffusion models [22,23,24,25,26,27,28,29,30]. Hristov introduced an informative investigation of fractional derivatives applications in transportation phenomena [31].

Recently, Elnaqeeb et al. [32] investigated the Prabhakar fractional derivative for second-grade transportation fluid model along with Newtonian heating effects and carbon nanotubes and solved this model by Laplace transform (LT) method. Our main emphasis is to extend this fluid problem comprising different nanoparticles of graphene oxide ( GO ) and molybdenum disulfide ( Mo S 2 ) nanoparticles along with the engine oil as base fluid with non-uniform velocity on the boundary. By using the Prabhakar fractional approach, time-dependent velocity, and the considered nanoparticles, no effect is existing in the available Literature. Moreover, NFs are very important in the applications of heat transmission as compared to ordinary fluid models. The present research is about the allocation of the solid nanomaterials in Newtonian fluids and limited research papers are presented about the solid nanomaterial’s dispensation in non-Newtonian fluids. The current research purpose is to study a convection MHD second-grade NF with a Prabhakar fractional operator over an infinite flat plate. The mathematical fractional model problem is obtained by using Fourier law with Prabhakar fractional approach. By using the LT scheme, the energy and velocity fields are solved analytically. Graphically, the impacts of fractional and flow parameters are expressed in Figure 1.

Figure 1

Geometry of the flow model.

2 Construction of the problem

Let us study the thermal transfer of the unsteady flow of convection second-grade NF with Prabhakar fractional derivative over an infinite flat plate. The NFs containing GO and MoS₂ nanoparticles along with the engine oil (base fluid) to enhance the heat transfer are considered. The plate is situated at y = 0 , and it is also supposed that the magnetic field is employed with strength B o . The fluid is flowing vertically upward along x -axis and y -axis is normal to the plate, assuming that the fluid is electrically conducting. At the start of simulations t = 0 , fluid and plate have no velocity with the temperature of the fluid being the ambient temperature θ ∞ conferring to the atmosphere. After some time, the plate begins moving with some variable velocity, and the temperature of the plate increases θ w . By using Boussinesq’s approximation [32], the governing equations after neglecting the pressure gradient are stated as follows:

(1) ρ nf w t ( y , t ) = μ nf 1 + γ 1 ∂ ∂ t w y y ( y , t ) + g ( β θ ρ ) nf [ θ − θ ∞ ] − σ nf B 0 2 w ( y , t ) ,

(2) ( ρ C p ) nf θ t ( y , t ) = − δ y ( y , t ) ,

(3) δ y ( y , t ) = − k nf θ y ( y , t ) ,

where δ ( y , t ) recognizes the thermal flux through Fourier’s law. The nanoparticle’s properties are defined by Ahmed et al. [33] as follows:

(4) μ f = ( 1 − φ ) 2.5 μ nf , k nf − [ k s + 2 k f + ( k s − k f ) 2 φ ] [ k s + 2 k f − ( k s − k f ) 2 φ ] k f = 0 , ( C p ρ ) nf = ( C p ρ ) f ( 1 − φ ) ( C p ρ ) s + φ ( C p ρ ) s , σ nf − σ f − 3 σ f φ σ s σ f − 1 σ s σ f + 2 − 3 φ σ s σ f − 1 − 1 .

The physical initial as well as the boundary conditions are as follows:

(5) θ ( y , 0 ) = θ ∞ , w ( y , 0 ) = 0 , y ≥ 0 ,

(6) θ ( 0 , t ) = θ ∞ + ( θ w − θ ∞ ) h ( t ) , w ( 0 , t ) = w 0 g ( t ) , t ≥ 0 ,

(7) θ ( y , t ) → θ ∞ , w ( y , t ) → 0 , as y → ∞ .

Introducing the non-dimensional values

y ∗ = w 0 y ν f , t ∗ = t u 0 2 ν f , w ∗ = w w 0 , θ ∗ = θ − θ ∞ θ w − θ ∞ , δ ⁎ = δ δ 0 , δ 0 = ( θ w − θ ∞ ) k nf w 0 ν f

into the proceeding governed equations, we get

(8) w t ( y , t ) = 1 c 0 c 1 1 + α 1 ∂ ∂ t w y y ( y , t ) + c 2 c 0 Gr θ ( y , t ) − 1 c 0 M w ( y , t ) ,

(9) c 3 Pr θ t = − δ y ,

(10) δ ( y , t ) = − c 4 θ y ,

(11) θ ( y , 0 ) = 0 , w ( y , 0 ) = 0 , y ≥ 0 ,

(12) θ ( 0 , t ) = h ( t ) , w ( 0 , t ) = g ( t ) , t ≥ 0 , y ≥ 0 ,

(13) θ ( y , t ) → 0 , w ( y , t ) → 0 , as y → ∞ ,

where

c 0 = ρ s ρ f φ − ( φ − 1 ) , c 1 = ( 1 − φ ) − 2.5 , c 2 = ( β θ ρ ) s ( β θ ρ ) f φ − ( φ − 1 ) , c 3 = ( C p ρ ) s ( C p ρ ) f φ − ( φ − 1 ) , c 4 = k nf k f , M = ν f u 0 2 B 0 2 σ nf ρ f ν f , Pr = ( C p μ ) f k f , α 1 = ν f γ 1 ν f u 0 2 , Gr = g ( β ν ) f ( θ w − θ ∞ ) u 0 3 .

Now, we accomplished an effective mathematical model for which the impacts of thermal memory are studied through the Prabhakar operator which is constructed on Fourier law, by presenting

(14) δ ( y , t ) = − D β , γ , β α C θ ( y , t ) .

Basic definitions:

The one-parametric Mittage-Leffler function is

E φ ( z ) = ∑ p = 0 ∞ z m Γ ( 1 + φ m ) ; φ , z ∈ C , Re ( φ ) > 0

and is proposed by Mittage-Leffler [34]. Then, Wiman [35] proposed the further general one-parametric function, identified as two parametric ML functions, which is

E φ , σ ( z ) = ∑ m = 0 ∞ z m Γ ( φ m + σ ) ; φ , z ∈ C , Re ( φ ) > 0 .

Further, in a previous study [36], Prabhakar proposed the three-parametric ML function, which is usually recognized as Prabhakar fractional operator

E ∅ , σ δ ( z ) = ∑ m = 0 ∞ ( σ ) m z m n ! Γ ( ∅ m + σ ) ; φ , σ , δ , z ∈ C , Re ( φ ) > 0

with the basic properties

E φ ( z ) = E φ 1 1 ( z ) , E φ , σ ( z ) = E φ , σ δ ( z ) , E 1 , 1 1 ( z ) = exp ( z ) ,

(15) L { t σ − 1 E φ , σ − δ ( φ t φ ) } = q − σ ( 1 − φ q − φ ) δ .

Prabhakar kernel

The function

(16) e φ , σ δ ( φ ; t ) = t σ − 1 E ∅ , σ δ ( φ t φ ) ; t ∈ ℛ , φ , σ , δ , z ∈ ℂ ,

is the Prabhakar kernel [37,38].

Prabhakar integral

It is defined as follows [37,38]:

E ∅ , σ , φ δ h ( t ) = e φ , σ δ ( φ ; t ) * h ( t ) = ∫ 0 t ( t − τ ) σ − 1 E φ , σ δ ( φ ( t − τ ) ∅ ) h ( τ ) d τ

with its LT

(17) L { E ∅ , σ , φ δ h ( t ) } ( q ) = L { e φ , σ δ ( φ ; t ) } L { h ( t ) } = q ∅ δ − σ ( q φ − φ ) δ L { h ( t ) } .

Some convenient fractional constraint cases may be summarized as

If σ = δ = 0
L − 1 { L { e φ , 0 0 ( e φ , σ δ ( φ ; t ) ; t ) } } = L − 1 { 1 } = δ ( t )
(classical case).
If σ = 1 , δ = 0
L − 1 { L { e ∅ , 1 0 ( ∅ ; t ) } } = L − 1 1 q = 1 .
If σ > 0 , δ = 0
L − 1 { L { e ∅ , σ 0 ( φ ; t ) } } = L − 1 1 q σ = t σ − 1 Γ ( σ ) .
when σ > 0 , φ = 0 , then, property 3 will repeat as

L { e φ , σ δ ( 0 ) } = q − σ .

Regularized Prabhakar derivative

Suppose m = [ σ ] where m ∈ Z and h ∈ A C m ( 0 , b ) in [37,38] regularized Prabhakar operator is

D φ , σ , φ δ C h ( t ) = E φ , m − σ , φ − δ g m ( t ) = e φ , m − σ − δ ( φ ; t ) * h m ( t )

(18) = ∫ 0 t ( t − τ ) m − σ − 1 E φ , m − σ − γ ( φ ( t − τ ) φ ) h m ( τ ) d ( τ ) ,

where D φ , σ , φ δ C implies the Prabhakar operator and h m signifies the mth derivative of h ( t ) . The LT of generalized Prabhakar as well as its kernel can be obtained by using Eqs. (15)–(18) as

L { D φ , σ , φ δ C h ( t ) } = L { e ∅ , m − σ − δ ( φ ; t ) * h m ( t ) } L { h m ( t ) } = q σ − m ( 1 − φ q − φ ) δ L { h m ( t ) } ,

(19) L { e ∅ , m − σ − δ ( φ ; t ) } = q σ − m ( 1 − φ q − φ ) δ .

3 Solution of the problem

3.1 Solution of the energy field

Using Laplace transform on equations (9), (14), and on corresponding initial and boundary conditions (11)₁–(13)₁, we get

(20) c 3 Pr θ ̅ ( y , p ) p = − δ ̅ y ( y , p ) ,

(21) δ ̅ ( y , p ) = c 4 p γ ( 1 − β p − β ) α θ ̅ y ( y , p ) ,

(22) θ ̅ ( 0 , p ) = h ( p ) , θ ̅ ( y , p ) → 0 as y → ∞ .

By using Eq. (21) in Eq. (20), we get

(23) ∂ 2 θ ¯ ( y , p ) ∂ y 2 − c 3 p Pr c 4 p γ ( 1 − β p − β ) α θ ¯ ( y , p ) = 0 .

Eq. (23) with conditions (22) can be solved as follows:

(24) θ ¯ ( y , p ) = h ( p ) θ ¯ 1 ( y , p ) ,

where θ ¯ 1 ( y , p ) = e − Ψ ( p ) y [ Ψ ( p ) − 1 ] and Ψ ( p ) = c 3 p Pr c 4 p γ ( 1 − β p − β ) α .

Suppose ζ ¯ ( y , p ) = e − p y [ p − 1 ] be the auxiliary function along with its inverse Laplace

ζ ( y , t ) = e − y 2 4 t π t − e − y + t erfc t − y 2 t , because θ ¯ 1 ( y , p ) = ζ ¯ ( y , ψ ( p ) ) ,

(25) θ 1 ( y , t ) = ∫ 0 ∞ f ( ξ , t ) ζ ( y , ξ ) d ξ ,

(26) f ( ξ , t ) = L − 1 [ e − ξ Ψ ( p ) ] = ∑ j = 0 ∞ ( − ξ Pr ) j j ! E β , j ( γ − 1 ) α j ( β t β ) t j ( γ − 1 ) − 1 .

The inverse LT of Eq. (26) is

(27) θ ( y , t ) = h ( t ) ∗ θ 1 ( y , t ) = ∫ 0 t θ 1 ( y , τ ) d τ .

For classical model ( γ = α = 0 )

At γ = α = 0 , Eq. (21) takes the form as follows:

L [ e α , 0 0 ( α ; t ) ] = 1 = L { δ ( t ) } .

By employing this, the generalized Fourier law will be converted into classical Fourier law. So,

(28) θ ̅ ( y , p ) = h ( p ) e − y c 3 Pr p c 4 ,

with its LT inverse, we get

(29) θ ( y , t ) = h ( t ) * erfc y c 3 Pr 2 c 4 t .

3.2 Solution of the velocity

Applying the Laplace transform on Eq. (8) , and on corresponding initial and boundary conditions (11)₂–(13)₂, we get

(30) p w ¯ ( y , p ) = 1 c 0 c 1 ( 1 + α 1 p ) w ¯ y y ( y , p ) + c 2 c 0 Gr θ ¯ ( y , p ) − 1 c 0 M w ¯ ( y , p ) ,

(31) w ¯ ( 0 , p ) = g ( p ) , w ¯ ( y , p ) → 0 , as , y → ∞ .

By substituting Eq. (24) in Eq. (30), we get

(32) ∂ 2 w ¯ ( y , p ) ∂ y 2 − p c 0 c 1 1 + α 1 p + M c 1 1 + α 1 p w ¯ ( y , p ) = − c 1 c 2 Gr ( 1 + α 1 p ) h ( p ) e − Ψ ( p ) y .

The solution of Eq. (32) with conditions as in the previous study (31) is

(33) w ¯ ( y , p ) = g ( p ) e − Ψ 1 ( p ) y + c 1 c 2 Gr h ( p ) ( 1 + α 1 p ) × e − Ψ 1 ( p ) y − e − Ψ ( p ) y Ψ 1 ( p ) − Ψ ( p ) ,

where Ψ 1 ( p ) = ( p c 0 + M ) c 1 1 + α 1 p .

Eq. (33) is complex and not easy to solve analytically; the inverse Laplace can be found by Zakian, Stehfest, and Tzou numerical algorithms [39,40,41].

4 Results with discussion

To observe the physical understanding of the problem, the consequences of flow, thermal transmission, and effects of diverse flow parameters on energy and momentum profile can be perceived with graphs given in this section. The thermophysical characteristics of base fluid (engine oil) as well as nanoparticles of GO and MoS₂ are given in Table 1 for a graphical demonstration. The behavior of temperature profiles for different parameters is shown in Figures 2–10.

Table 1

Thermophysical characteristics of engine oil as well as different nanoparticles [31]

Material	ρ ( kg / m 3 )	C p ( 1 / kg K )	κ ( W / m K )	β × 10 − 5 ( 1 / K )
Engine oil	884	1 , 910	0.144	70
GO	1 , 800	717	5 , 000	0.284
Mo S 2	5 , 060	397.21	904.4	2.8424

$Figure 2 Temperature profile of fractional parameters ( α , β , γ ) (\alpha ,\beta ,\gamma ) .$

Figure 2

Temperature profile of fractional parameters ( α , β , γ ) .

$Figure 3 Temperature profile of Prandtl number Pr {\rm{\Pr }} .$

Figure 3

Temperature profile of Prandtl number Pr .

$Figure 4 Velocity profile of fractional parameters ( α , β , γ ) (\alpha ,\beta ,\gamma ) .$

Figure 4

Velocity profile of fractional parameters ( α , β , γ ) .

$Figure 5 Velocity profile of volume fraction φ \varphi .$

Figure 5

Velocity profile of volume fraction φ .

$Figure 6 Velocity profile of Grashof number Gr {\rm{Gr}} .$

Figure 6

Velocity profile of Grashof number Gr .

Figure 7

Velocity profile of magnetic parameter M .

$Figure 8 Velocity profile of Prandtl number Pr {\rm{\Pr }} .$

Figure 8

Velocity profile of Prandtl number Pr .

Figure 9

Comparison of different numerical algorithms.

Figure 10

Comparison of our work with Basit et al. [42].

Figure 2 displays the impacts of fractional parameters ( α , β , γ ) on the temperature profile along with the comparison of different NFs that are engine oil–MoS₂ and engine oil–GO. For low values of time, the temperature rises with the increase in the fractional parameters ( α , β , γ ) due to the kernel of the Prabhakar operator, and this behavior is reversed with the increase in time. Figure 3 illustrates the behavior of temperature for Prandtl number ( Pr ⁡ ) . As Pr has an inverse relation with the thermal conductivity, when Pr increases, the thermal conductivity decreases. Hence temperature decreases with the increase in the value of Pr . Besides, the development in temperature rate, due to engine oil–GO is more progressive, than engine oil–MoS₂ because of the physical characteristics of certain nanoparticles.

Figure 4 points out that at the smaller time, the velocity is improved with the increase in ( α , β , γ ) . For a large time, a reverse trend is exposed. The fractional parameters may be convenient to control the velocity boundary layer. Figure 5 shows that velocity decays with rising φ . This is acceptable physically as the fluid became more viscous with the rise in φ , which causes a decrease in velocity. The increasing variation in Gr is the reason for the improvement in the buoyancy effect which speeds up the velocity as seen in Figure 6.

Figure 7 shows the consequence of magnetic parameter ( M ) on the velocity. Physically, it may answer to the drag force, which has an impact on the momentum that aspects the fluid motion. Figure 8 is portrays the influences of Pr on the velocity. Here the variation in Pr is considered because non-dimensional numbers have decisive implications in various engineering as well as industrial phenomena. An increase in Pr declines the velocity. Also, the development in momentum profile, due to engine oil–GO is more progressive than engine oil–MoS₂ due to the physical characteristics of certain nanoparticles. Figure 9 shows the comparison of three dissimilar numerical schemes (Zakian, Stehfest, and Tzou) for both profiles (temperature and momentum). The significances from diverse profile curves overlap each other, signifying the validity of the current work. Figure 10 shows the validity of the results of our accomplished work by comparing the results of temperature and velocity profile with that of the previous work [42]. Through overlapping both curves, it is seen from these diagrams that our achieved results match those established in the previous work [42]. Table 2 shows the numerical investigation of governed profile comparison in the study by Basit et al. [42] and our accomplished results. We see that our results are better than previous results obtained by Basit et al. [42].

Table 2

Numerical analysis of governed profile comparison of Basit et al. [42] and our attained results

y	Temperature profile			Velocity profile
y	Previous result	Our result	% increase	Previous result	Our result	% increase
0.1	0.9496	0.9515	0.2	1.0935	1.0988	0.48
0.3	0.8542	0.8596	0.63	1.1796	1.1955	1.35
0.5	0.7666	0.7748	1.07	1.1729	1.1984	2.17
0.7	0.6868	0.6973	1.53	1.1102	1.1431	2.96
0.9	0.6146	0.6268	1.99	1.0166	1.0545	3.73
1.1	0.5495	0.563	2.46	0.9089	0.9498	4.5
1.3	0.491	0.5053	2.91	0.7979	0.8399	5.26
1.5	0.4384	0.4532	3.38	0.6905	0.7321	6.02
1.7	0.3913	0.4064	3.86	0.5906	0.6307	6.79
1.9	0.349	0.3642	4.36	0.5004	0.5381	7.53

5 Conclusion

The investigation of MHD second-grade NF with an infinite upright heated flat plate along with the comparison of different NFs that are engine oil–MoS₂ and engine oil–GO is studied. The fractional model is obtained through Fourier law by exploiting Prabhakar fractional approach. The equations are solved analytically via the Laplace approach. The key findings of the present study are as follows:

The temperature and momentum profiles show dual behavior for the fractional parameters ( α , β , γ ) at different times.
When Pr increases, the thermal profile decreases.
Velocity declines with the increase in φ .
Velocity increases as Gr increases and declines for greater values of M and Pr .
The development in temperature and momentum profile, due to NF of engine oil–GO is more progressive than engine oil–MoS₂ NF.
The substantial comparison of the temperature and momentum profile with the existing physical literature [42] enhances our study’s novelty.
In the comparison of numerical methods, the curves of both approaches overlap, signifying that our attained results are authentic.

Current developments in the analysis of fractional order approaches contain the fractal derivative, the fractional Shehu transform, as well as the improved general Taylor fractional series technique. Investigators in the future can compare their conclusions to those we calculated by applying the Prabhakar fractional approach in our study.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Rehman AU, Awrejcewicz J, Riaz MB, Jarad F. Mittag-Leffler form solutions of natural convection flow of second-grade fluid with exponentially variable temperature and mass diffusion using Prabhakar fractional derivative. Case Stud Therm Eng. 2022;34:102018.10.1016/j.csite.2022.102018Search in Google Scholar

[2] Khan Z, ul Haq S, Ali F, Andualem M. Free convection flow of second grade dusty fluid between two parallel plates using Fick’s and Fourier’s laws: A fractional model. Sci Rep. 2022;12(1):3448.10.1038/s41598-022-06153-3Search in Google Scholar PubMed PubMed Central

[3] Alfvén H. Existence of electromagnetic-hydrodynamic waves. Nature. 1942;150(3805):405–6.10.1038/150405d0Search in Google Scholar

[4] Nadeem M, Siddique I, Jarad F, Jamil RN. Numerical study of MHD third-grade fluid flow through an inclined channel with ohmic heating under fuzzy environment. Math Probl Eng. 2021;2021:9137479.10.1155/2021/9137479Search in Google Scholar

[5] Saqib M, Khan I, Shafie S, Mohamad AQ. Shape effect on MHD flow of time fractional Ferro-Brinkman type nanofluid with ramped heating. Sci Rep. 2021;11(1):1–22.10.1038/s41598-020-78421-zSearch in Google Scholar PubMed PubMed Central

[6] Gul T, Bilal M, Alghamdi W, Asjad MI, Abdeljawad T. Hybrid nanofluid flow within the conical gap between the cone and the surface of a rotating disk. Sci Rep. 2021;11(1):1–19.10.1038/s41598-020-80750-ySearch in Google Scholar PubMed PubMed Central

[7] Kuznetsov AV, Nield DA. Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci. 2010;49(2):243–7.10.1016/j.ijthermalsci.2009.07.015Search in Google Scholar

[8] Acharya N. Buoyancy driven magnetohydrodynamic hybrid nanofluid flow within a circular enclosure fitted with fins. Int Commun Heat Mass Transf. 2022;133:105980.10.1016/j.icheatmasstransfer.2022.105980Search in Google Scholar

[9] Acharya N, Mondal H, Kundu PK. Spectral approach to study the entropy generation of radiative mixed convective couple stress fluid flow over a permeable stretching cylinder. Proc Inst Mech Eng Part C. 2021;235(15):2692–704.10.1177/0954406220954893Search in Google Scholar

[10] Attia N, Akgül A, Seba D, Nour A, Asad J. A novel method for fractal-fractional differential equations. Alex Eng J. 2022;61(12):9733–48.10.1016/j.aej.2022.02.004Search in Google Scholar

[11] Bilal S, Shah IA, Akgül A, Tekin MT, Botmart T, Yahia IS. A comprehensive mathematical structuring of magnetically effected Sutterby fluid flow immersed in dually stratified medium under boundary layer approximations over a linearly stretched surface. Alex Eng J. 2022;61(12):11889–98.10.1016/j.aej.2022.05.044Search in Google Scholar

[12] Farman M, Akgül A, Tekin MT, Akram MM, Ahmad A, Mahmoud EE, et al. Fractal fractional-order derivative for HIV/AIDS model with Mittag-Leffler kernel. Alex Eng J. 2022;61(12):10965–80.10.1016/j.aej.2022.04.030Search in Google Scholar

[13] Modanli M, Göktepe E, Akgül A, Alsallami SA, Khalil EM. Two approximation methods for fractional order pseudo-parabolic differential equations. Alex Eng J. 2022;61(12):10333–39.10.1016/j.aej.2022.03.061Search in Google Scholar

[14] Qureshi ZA, Bilal S, Khan U, Akgül A, Sultana M, Botmart T, et al. Mathematical analysis about influence of Lorentz force and interfacial nano layers on nanofluids flow through orthogonal porous surfaces with injection of SWCNTs. Alex Eng J. 2022;61(12):12925–41.10.1016/j.aej.2022.07.010Search in Google Scholar

[15] Shah IA, Bilal S, Akgül A, Tekin MT, Botmart T, Zahran HY, et al. On analysis of magnetized viscous fluid flow in permeable channel with single wall carbon nano tubes dispersion by executing nano-layer approach. Alex Eng J. 2022;61(12):11737–51.10.1016/j.aej.2022.05.037Search in Google Scholar

[16] Xu C, Farman M, Hasan A, Akgül A, Zakarya M, Albalawi W, et al. Lyapunov stability and wave analysis of covid-19 omicron variant of real data with fractional operator. Alex Eng J. 2022;61(12):11787–802.10.1016/j.aej.2022.05.025Search in Google Scholar

[17] Iqbal MS, Yasin MW, Ahmed N, Akgül A, Rafiq M, Raza A. Numerical simulations of nonlinear stochastic Newell-Whitehead-Segel equation and its measurable properties. J Comput Appl Math. 2023;418:114618.10.1016/j.cam.2022.114618Search in Google Scholar

[18] Acharya N. Magnetized hybrid nanofluid flow within a cube fitted with circular cylinder and its different thermal boundary conditions. J Magn Magn Mater. 2022;564:170167.10.1016/j.jmmm.2022.170167Search in Google Scholar

[19] Acharya N. On the hydrothermal behavior and entropy analysis of buoyancy driven magnetohydrodynamic hybrid nanofluid flow within an octagonal enclosure fitted with fins: Application to thermal energy storage. J Energy Storage. 2022;53:105198.10.1016/j.est.2022.105198Search in Google Scholar

[20] Acharya N, Mabood F, Badruddin IA. Thermal performance of unsteady mixed convective Ag/MgO nanohybrid flow near the stagnation point domain of a spinning sphere. Int Commun Heat Mass Transf. 2022;134:106019.10.1016/j.icheatmasstransfer.2022.106019Search in Google Scholar

[21] Jie Z, Khan MI, Al-Khaled K, El-Zahar ER, Acharya N, Raza A, et al. Thermal transport model for Brinkman type nanofluid containing carbon nanotubes with sinusoidal oscillations conditions: A fractional derivative concept. Waves Random Complex Media. 2022;1–20. https://doi.org/10.1080/17455030.2022.2049926.10.1080/17455030.2022.2049926Search in Google Scholar

[22] Abro KA, Abro IA, Almani SM, Khan I. On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative. J King Saud Univ Sci. 2019;31(4):973–9.10.1016/j.jksus.2018.07.012Search in Google Scholar

[23] Ali Q, Al-Khaled K, Omar J, Raza A, Khan SU, Khan MI, et al. Analysis for advection–diffusion problem subject to memory effects and local and nonlocal kernels: A fractional operators approach. Int J Mod Phys B. 2022;37:2350099.10.1142/S0217979223500996Search in Google Scholar

[24] Ali Q, Riaz S, Awan AU, Abro KA. Thermal investigation for electrified convection flow of Newtonian fluid subjected to damped thermal flux on a permeable medium. Phys Scr. 2020;95(11):115003.10.1088/1402-4896/abbc2eSearch in Google Scholar

[25] Ali Q, Riaz S, Awan AU, Abro KA. A mathematical model for thermography on viscous fluid based on damped thermal flux. Z für Naturforsch A. 2021;76(3):285–94.10.1515/zna-2020-0322Search in Google Scholar

[26] Abro KA, Siyal A, Atangana A. Thermal stratification of rotational second-grade fluid through fractional differential operators. J Therm Anal Calorim. 2021;143(5):3667–76.10.1007/s10973-020-09312-8Search in Google Scholar

[27] Riaz S, Sattar M, Abro KA, Ali Q. Thermo-dynamical investigation of constitutive equation for rate type fluid: A semi-analytical approach. Int J Model Simul. 2022;1–12. https://doi.org/10.1080/02286203.2022.2056427.10.1080/02286203.2022.2056427Search in Google Scholar

[28] Awan AU, Riaz S, Abro KA, Siddiqa A, Ali Q. The role of relaxation and retardation phenomenon of Oldroyd-B fluid flow through Stehfest’s and Tzou’s algorithms. Nonlinear Eng. 2022;11(1):35–46.10.1515/nleng-2022-0006Search in Google Scholar

[29] Ali Q, Yassen MF, Asiri SA, Pasha AA, Abro KA. Role of viscoelasticity on thermoelectromechanical system subjected to annular regions of cylinders in the existence of a uniform inclined magnetic field. Eur Phys J Plus. 2022;137(7):1–10.10.1140/epjp/s13360-022-02951-wSearch in Google Scholar

[30] Zhang J, Raza A, Khan U, Ali Q, Zaib A, Weera W, et al. Thermophysical study of Oldroyd-B hybrid nanofluid with sinusoidal conditions and permeability: A Prabhakar fractional approach. Fractal Fract. 2022;6(7):357.10.3390/fractalfract6070357Search in Google Scholar

[31] Hristov J. Derivatives With Non-Singular Kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models. In: Bhalekar S, editor. Frontiers in Fractional Calculus. 1st ed. Sharjah, UAE: Bentham Science Publishers; 2017. p. 1:270–342.Search in Google Scholar

[32] Elnaqeeb T, Shah NA, Rauf A. Natural convection flows of carbon nanotube Prabhakar‐like fractional second‐grade nanofluids over an infinite plate with Newtonian heating. Math Method Appl Sci. 2020;20:1–14.10.1002/mma.6795Search in Google Scholar

[33] Ahmed N, Vieru D, Fetecau C, Shah NA. Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel. Phys Fluids. 2018;30(5):052002.10.1063/1.5032165Search in Google Scholar

[34] Mittag-Leffler GM. Sur la nouvelle fonction Eα (x). CR Acad Sci Paris. 1903;137(2):554–8.Search in Google Scholar

[35] Wiman A. About the fundamental theorem in the teorie of the functions Ea (x). Acta Math. 1905;29:191–201.10.1007/BF02403202Search in Google Scholar

[36] Garra R, Garrappa R. The Prabhakar or three parameter Mittag–Leffler function: Theory and application. Commun Nonlinear Sci Numer Simul. 2018;56:314–29.10.1016/j.cnsns.2017.08.018Search in Google Scholar

[37] Giusti A, Colombaro I. Prabhakar-like fractional viscoelasticity. Commun Nonlinear Sci Numer Simul. 2018;56:138–43.10.1016/j.cnsns.2017.08.002Search in Google Scholar

[38] Polito F, Tomovski Z. Some properties of Prabhakar-type fractional calculus operators. 2015. arXiv preprint arXiv:1508.03224.10.7153/fdc-06-05Search in Google Scholar

[39] Zakian V. Optimization of numerical inversion of Laplace transforms. Electron Lett. 1970;21(6):677–9.10.1049/el:19700471Search in Google Scholar

[40] Stehfest H. Numerical inversion of Laplace transforms algorithm 368. Commun ACM. 1970;13(1):47–9.10.1145/361953.361969Search in Google Scholar

[41] Tzou DY. Macro-to microscale heat transfer: The lagging behavior. Hoboken (NJ), USA: John Wiley & Sons; 2014.10.1002/9781118818275Search in Google Scholar

[42] Basit A, Asjad MI, Akgül A. Convective flow of a fractional second grade fluid containing different nanoparticles with Prabhakar fractional derivative subject to non‐uniform velocity at the boundary. Math Models Methods Appl Sci. 2021. https://doi.org/10.1002/mma.7461.10.1002/mma.7461Search in Google Scholar

Received: 2022-11-13

Revised: 2023-02-09

Accepted: 2023-02-27

Published Online: 2023-04-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Investigation of magnetized convection for second-grade nanofluids via Prabhakar differentiation

Abstract

Nomenclature

Abbreviations

1 Introduction

2 Construction of the problem

3 Solution of the problem

3.1 Solution of the energy field

3.2 Solution of the velocity

4 Results with discussion

5 Conclusion

References

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