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

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:

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:

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

Introducing the non-dimensional values

into the proceeding governed equations, we get

where

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

Basic definitions:

The one-parametric Mittage-Leffler function is

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

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

with the basic properties

Prabhakar kernel

The function

is the Prabhakar kernel [37,38].

Prabhakar integral

It is defined as follows [37,38]:

with its LT

Some convenient fractional constraint cases may be summarized as

1. 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).

2. If σ = 1 , δ = 0

   L − 1 { L { e ∅ , 1 0 ( ∅ ; t ) } } = L − 1 1 q = 1 .

3. If σ > 0 , δ = 0

   L − 1 { L { e ∅ , σ 0 ( φ ; t ) } } = L − 1 1 q σ = t σ − 1 Γ ( σ ) .

4. when σ > 0 , φ = 0 , then, property 3 will repeat as

Regularized Prabhakar derivative

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

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

3 Solution of the problem

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.

Figure 2

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

Figure 3

Temperature profile of Prandtl number Pr .

Figure 4

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

Figure 5

Velocity profile of volume fraction φ .

Figure 6

Velocity profile of Grashof number Gr .

Figure 7

Velocity profile of magnetic parameter M .

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].

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:

1. The temperature and momentum profiles show dual behavior for the fractional parameters ( α , β , γ ) at different times.

2. When Pr increases, the thermal profile decreases.

3. Velocity declines with the increase in φ .

4. Velocity increases as Gr increases and declines for greater values of M and Pr .

5. The development in temperature and momentum profile, due to NF of engine oil–GO is more progressive than engine oil–MoS₂ NF.

6. The substantial comparison of the temperature and momentum profile with the existing physical literature [42] enhances our study’s novelty.

7. 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.