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ORIGINAL RESEARCH article

Front. Mater., 24 February 2023
Sec. Colloidal Materials and Interfaces
Volume 10 - 2023 | https://doi.org/10.3389/fmats.2023.1114665

Thermal investigation into the Oldroyd-B hybrid nanofluid with the slip and Newtonian heating effect: Atangana–Baleanu fractional simulation

www.frontiersin.orgQasim Ali1 www.frontiersin.orgMuhammad Amir1 www.frontiersin.orgAli Raza1,2 www.frontiersin.orgUmair Khan3,4 www.frontiersin.orgSayed M. Eldin5* www.frontiersin.orgAbeer M. Alotaibi6 www.frontiersin.orgSamia Elattar7 www.frontiersin.orgAhmed M. Abed8,9
  • 1Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan
  • 2School of Mathematics, Minhaj University, Lahore, Pakistan
  • 3Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor, Malaysia
  • 4Department of Mathematics and Social Sciences, Sukkur IBA University, Sukkur, Sindh, Pakistan
  • 5Center of Research, Faculty of Engineering, Future University in Egypt, New Cairo, Egypt
  • 6Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia
  • 7Department of Industrial & Systems Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia
  • 8Department of Industrial Engineering, College of Engineering, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia
  • 9Industrial Engineering Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt

The significance of thermal conductivity, convection, and heat transportation of hybrid nanofluids (HNFs) based on different nanoparticles has enhanced an integral part in numerous industrial and natural processes. In this article, a fractionalized Oldroyd-B HNF along with other significant effects, such as Newtonian heating, constant concentration, and the wall slip condition on temperature close to an infinitely vertical flat plate, is examined. Aluminum oxide (Al2O3) and ferro-ferric oxide (Fe3O4) are the supposed nanoparticles, and water (H2O) and sodium alginate (C6H9NaO7) serve as the base fluids. For generalized memory effects, an innovative fractional model is developed based on the recently proposed Atangana–Baleanu time-fractional (AB) derivative through generalized Fourier and Fick’s law. This Laplace transform technique is used to solve the fractional governing equations of dimensionless temperature, velocity, and concentration profiles. The physical effects of diverse flow parameters are discussed and exhibited graphically by Mathcad software. We have considered 0.15α0.85,2Pr9,5Gr20,0.2ϕ1,ϕ20.8,3.5Gm8, 0.1 Sc 0.8, and 0.3λ1,λ21.7. Moreover, for validation of our present results, some limiting models, such as classical Maxwell and Newtonian fluid models, are recovered from the fractional Oldroyd-B fluid model. Furthermore, comparing the results between Oldroyd-B, Maxwell, and viscous fluid models for both classical and fractional cases, Stehfest and Tzou numerical methods are also employed to secure the validity of our solutions. Moreover, it is visualized that for a short time, temperature and momentum profiles are decayed for larger values of α, and this effect is reversed for a long time. Furthermore, the energy and velocity profiles are higher for water-based HNFs than those for the sodium alginate-based HNF.

1 Introduction

With the addition of nanometer-sized particles in various base fluids, thermophysical characteristics may improve in energy transfer schemes. This process signals an expansion in the thermal conductivity for base fluids, making it more reliable and ongoing. These significant fluids define nanofluids (NFs) with an extensive series of suggestions in several areas of science, as well as technology, with nuclear devices, heat exchangers, solar plates, vehicle heaters, and biotic and organic devices (Usman et al., 2018; Khan et al., 2022a; Khan et al., 2022b; Ahmed et al., 2022; Hassan et al., 2022; Khan et al., 2022c). First, Lee and Eastman presented the idea of NFs in 1995 (Lee et al., 1999). Numerous applications of NFs are discoursed by Kaufui et al. (Wong and Omar De Leon., 2010). Mahian et al. (2019) proposed important ideas and reflected novel innovations to completely explain the NFs. They were obsessed with innovative expansions in this field, comprehensive explanations of the thermophysical characteristics, and imitation of thermal transmission in NF flow. Waini et al. (2019) used a numerical scheme to discuss an unsteady thermal transmission flow past a shrinking sheet in an HNF. They presented different applications of NFs in numerous branches of science along with appreciated recommendations. NFs have achieved significant consideration from researchers due to their improved heat conversion characteristics. The rheological presentation of an NF using a revolving rheometer was proposed by Vallejo et al. (2019a). Different rheological characteristics of NFs are discussed in Vallejo et al. (2019b). Currently, NFs have been characterized as HNFs in several mechanisms (Rashad et al., 2018). HNFs are developed by mixing two dissimilar nanoparticles in the base liquid. Its main inspiration is to increase the thermal features of NFs. The variable thermal transmission of HNFs through magnetic influence was examined in Mohebbi et al. (2019). The heat transmission in the non-Newtonian HNF composed with entropy generation was discussed in Shahsavar et al., 2018). Furthermore, Farooq et al. (2018) deliberated on the entropy in the HNF flow in a stretching sheet.

Asogwa et al. (2021) discussed chemical reactions and heat sinks over a ramped temperature. The analytical solution of governing equations was found with the Laplace transform. Asogwa et al. (2022a) used the Laplace approach to discuss a water-based NF containing aluminum oxide and copper in a moving plate and proved that thermal absorption causes a decline in aluminum oxide NF’s thermal and momentum profiles with a copper NF. Shankar Goud et al. (2022) used the Keller–box scheme for the numerical solution along with thermal effects, momentum, and solutal slip on the thermal transmission with a description of the magnetohydrodynamic (MHD) flow of Casson fluid and an exponential porous surface with Dufour, chemical reaction, and Soret impacts. Khan et al. (2022d) studied a fractionalized electro-osmotic flow based on the Caputo operator of a Casson NF containing sodium alginate nanoparticles over a vertical microchannel with MHD effects. They proved that the inclination angle boosts the velocity. Asogwa et al. (2022b) and Asogwa et al. (2022c) considered the stimulation significance of the thermal transmission with the MHD flow of a NF through an extending sheet with MATLAB bvp4c. Furthermore, they investigated the radiative features of the MHD flow with collective heat transportation characteristics on a reactive stretching surface with the Casson NF numerically using MATLAB bvp4c. Goud et al. (2022) applied the bvp4c scheme to study the convection flow via an infinite porous plate on thermal transmission, as well as mass transmission. Asogwa et al. (2022d) discussed the influences of the movement of nanoparticles in NFs by an exponentially enhanced Riga plate. Reddy et al. (2022) calculated the effect of activation energy on a second-grade MHD NF flow over a convectively curved heated stretched surface by considering the Brownian motion and generation/absorption, and thermophoresis. They have shown that velocity and thermal profiles suggestively increase with the concurrent increasing estimation of the fluid parameter.

The fractional calculus (FC) has obtained substantial consideration from experts in previous decades. The important inventions have newly been presented in the application of the FC, where new derivatives, as well as integral operators, are hired (Awan et al., 2019). The new anticipated operators contain the generalized Mittag–Leffler function (MLF), and these features intensify the innovative constructions to achieve numerous attractive properties that are recognized in important outcomes. Subsequently, Atangana and Dumitru (2016) anticipated, the innovative and applicable time-fractional operator, which is expansively hired in numerous branches of science and engineering. It is exposed that the MLF is a more operative and vigorous screening apparatus than the exponential and power laws, constructing the AB-fractional operator, in terms of Caputo, an effective arithmetic procedure to simulate progressively perilous complex tasks. Due to their extensive implications, such fractional models are extensively identified for deriving fractional differential equations (FDEs) with no manufactured irregularities, as for Caputo, Riemann–Liouville (RL), and Caputo–Fabrizio (CF) derivatives, because of their characteristic non-orientation (Ali et al., 2021; Ali et al., 2022a; Raza et al., 2022; Zhang et al., 2022). We also perceived interest in these fractional derivatives on the topic of mathematical approaches, although scientifically approximating these operators' outcomes to compute different problems (Martyushev and Sheremet, 2012; Ali et al., 2022b).

Batool et al. (2022) discussed the thermal and mass transmission processes of a micropolar NF under magnetic and buoyancy effects across an inclusion. Rasool et al. (2022a) examined the significance of the MHD Maxwell NF flow and obtained the solution to this problem by employing the homotopy analysis technique for diverse physical parameters. Moreover, they studied an electro-magneto-hydrodynamic NF flow in a permeable medium with heating boundary conditions. Furthermore, they applied Buongiorno’s method for the flow of radiating thixotropic NFs over a horizontal surface by considering the retardational effects of Lorentz forces and using the influence of Brownian and thermophoresis diffusions (Rasool et al., 2022b; Rasool et al., 2023).

In this paper, a fractionalized Oldroyd-B HNF flow is examined by the recent definitions of the AB time-fractional derivative having a Mittage–Leffler kernel along with Newtonian heating, constant concentration, and the wall slip condition on temperature close to an infinite vertical flat plate. The AB fractional operator is introduced in the governing equations of temperature and diffusion by employing the generalized types of Fourier and Fick’s law. The developed non-dimensional fractional model is solved using the Laplace transform method. Graphical illustrations are used to depict the physical behavior of fractional derivatives and the consequence of diverse flow parameters on velocity, thermal, and concentration fields. Furthermore, for validation of our attained results, some limiting cases are considered to recover fractional derivatives, as well as classical models of Maxwell and Newtonian fluids. The impacts of diverse flow parameters on variable profiles are achieved and presented graphically with significant conclusions.

2 Mathematical formulation based on a hybrid nanofluid

Consider an unsteady and an incompressible Oldroyd-B HNF flow close to an infinite vertical flat plate. Initially, consider that the fluid and plate are at a relaxation position, with constant temperature T and concentration C. After some time, the plate is kept constant and the fluid begins to move with a temperature value T0,ta1T0,tξ=u0sinωt, where u0 is a constant that signifies the dimension of velocity. At that time, the plate obtains a temperature Tw and concentration Cw, which persist constantly. We supposed that velocity, temperature, and concentration profiles are the only functions of ξ and t. The configuration of the problem is shown in Figure 1.

FIGURE 1
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FIGURE 1. Physical flow.

By Boussinesq’s estimation (Ali et al., 2021), the governing equations for an Oldroyd-B HNF are discussed by Martyushev and Sheremet (2012). The equation of motion is as follows:

ρhbnf1+λ1tW1ξ,tt=μhbnf1+λ2t2W1ξ,tξ2+gρβ1hbnf1+λ1tTξ,tT+gρβ2hbnf1+λ1tCξ,tC.(1)

The energy balance equation is as follows (Awan et al., 2019):

ρCphbnfTξ,tt=qξ.(2)

The Fourier law (Zhang et al., 2022) for thermal conduction is as follows:

qξ,t=κhbnfTξ,tξ.(3)

The diffusion equation (Awan et al., 2019) for

Cξ,tt=jξ.(4)

The Fick law is as follows (Awan et al., 2019):

jξ,t=DhbnfCξ,tξ.(5)

The appropriate initial and boundary conditions are as follows:

W1ξ,0=0,Tξ,0=T,Cξ,0=C,ξ0,(6)
W10,t=0,T0,ta1Tξ,tξξ=0=u0sinωt,C0,t=Cw,(7)
W1ξ,t0,Tξ,tT,Cξ,tCas,ξ.(8)

Table 1 shows the properties of thermal and under-conversation fluids and nanoparticles.

ρhbnf=ρf1ϕ2×ρs1ρfϕ1+1ϕ1+ϕ2ρs2μhbnf=μf1ϕ12.51ϕ22.5,ρCphbnf=ρCpf1ϕ2×1ϕ1+ϕ1ρCps1ρCpf+ϕ2ρCps2,ρβThbnf=1ϕ2ρβTf×1ϕ1+ϕ1ρβTs1ρβTf+ϕ2ρβTs2,κhbnf=κs2+s1κbfs1ϕ2κbfκs2κs2+s1κbf+ϕ2κbfκs2κbf,κbf=κs1+s1κfs1ϕ1κfκs1κs1+s1κf+ϕ1κfκs1κf.(9)

TABLE 1
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TABLE 1. Thermal characteristics of base fluids and nanoparticles (Raza et al., 2022; Zhang et al., 2022).

The following are a set of non-dimensional parameters:

ψ*=u0υfξ,η*=u02υft,W*=W1u0,θ*=TTTwT,Φ*=CCCwC,q*=qq0,j*=jj0,λ1*=u02υfλ1,λ2*=u02υfλ2,q0=κfTwTu0υf,j0=DnfCwCu0υf,Gr=gυβ1fTwTu03,Gm=gυβ2fCwCu03,Pr=μCpfκf,Sc=υfDf.(10)

By utilizing the aforementioned variables in Eqs. 18 and after dropping the * notation, we obtain

Ω11+λ1ηWψ,ηη=Ω21+λ2η2Wψ,ηψ2+Ω31+λ1ηGrθψ,η+Ω41+λ1ηGmΦψ,η.(11)
Ω5Prθψ,ηη=qψ.(12)
qψ,η=Ω6θψ,ηψ.(13)
DηαABΦψ,η=1ϕ11ϕ2Scjψ,(14)
jψ,η=Φψ,ηψ.(15)
Wψ,0=0,θψ,0=0,Φψ,0=0,ψ0,(16)
W0,η=0,θ0,ηa1θψ,ηψψ=0=sinωη,Φ0,η=1,η>0,(17)
Wψ,η0,θψ,η0,Φψ,η0,as,ψ.(18)

where

Ω1=1ϕ2×1ϕ1+ϕ1ρs1ρf+ϕ2ρs2ρf,Ω2=11ϕ12.51ϕ22.5,Ω3=1ϕ2×1ϕ1+ϕ1ρβ1s1ρβ1f+ϕ2ρβ1s2ρβ1f,Ω4=1ϕ2×1ϕ1+ϕ1ρβ2s2ρβ2f+ϕ2ρβ2s2ρβ2f,Ω5=1ϕ2×1ϕ1+ϕ1ρCpsρCpf+ϕ2ρCps2ρCpf,Ω6=κs2+s1κbfs1ϕ2κbfκs2κs2+s1κbf+ϕ2κbfκsκbf,κbf=κs1+s1κfs1ϕ1κfκs1κs1+s1κf+ϕ1κfκs1.(19)

2.1 Fractional model based on a non-local kernel

Now, we develop a fractional Oldroyd-B HNF using Fourier and Fick’s law based on the AB-fractional operator (Atangana and Dumitru, 2016), which is explained as the following expression for a function fξ,t

DtγABfξ,t=11γ0tEγγtτγ1γfξ,τdτ,0<γ<1,(20)

and the kernel Mittage–Leffler function Eγτ is defined by

Eγτ=r=0τγΓrγ+1,0<γ<1,τC.(21)

The Laplace transform is

LDtγABfξ,t=sγLfξ,tsγ1fξ,0sγ1γ+γ,(22)

with

limγ1DtγABfξ,t=fξ,tt.(23)

The governing equations for the AB-fractional derivative are obtained by substituting the ordinary derivative with the AB derivative operator DηαAB in Eqs. 1115 as

Ω11+λ1ηDηαABWψ,η=Ω21+λ2η2Wψ,ηψ2+Ω31+λ1ηGrθψ,η+Ω41+λ1ηGmΦψ,η,(24)
Ω5PrABDηαθψ,η=qψ,(25)
qψ,η=Ω6θψ,ηψ.(26)
DηαABΦψ,η=1Scjψ,(27)
jψ,η=Φψ,ηψ.(28)

3 Solution of the problem

3.1 Energy profile

Using the Laplace transform on Eqs. 25, 26 and corresponding conditions (15)2-(17)2, we have

Ω6Prsα1αsα+αθ¯ψ,s=q¯ψ,(29)
q¯ψ,s=Ω6θ¯ψ,sψ,(30)
θ¯0,sa1θ¯ψ,sψψ=0=ωs2+ω2,θ¯ψ,s0,as,ψ,(31)

where θ¯ψ,s=0θψ,testdt is the Laplace transform for θψ,t, and s is the Laplace transform parameter (Ali et al., 2021).

The solution of Eq. (29) by using Eq. (30) and with conditions in Eq. (31) is

θ¯ψ,s=ωs2+ω211+aΠsα1αsα+αexpψΠsα1αsα+α.(32)

Eq. (32) can be written as

θ¯ψ,s=ωs2+ω211+aΛ1sexpψΛ1s,(33)

where Π=Ω5PrΩ6 and Λ1s=Πsα1αsα+α.

The Laplace inverse of Eq. (33) is shown numerically in Table 2.

TABLE 2
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TABLE 2. Numerical comparison of energy, concentration, and velocity profiles by different numerical methods.

3.2 Concentration field

By employing the Laplace transform on Eqs. 27, 28 with associated conditions defined in Eqs. (15)3– (17)3, we have

sα1αsα+αΦ¯ψ,s=1ϕ11ϕ2Scj¯ψ,sψ,(34)
j¯ψ,s=j¯ψ,sψ.(35)
Φ¯0,s=1s,Φ¯ψ,s0,as,ψ.(36)

The solution of Eq. (34) by using Eq. (35) and conditions in Eq. (36) is

Φ¯ψ,s=1sexpψSc1ϕ11ϕ2sα1αsα+α.(37)

Eq. (37) may be written as

Φ¯ψ,s=1sexpψΛ2s,(38)

where Λ2s=Sc1ϕ11ϕ2sα1αsα+α.

The Laplace inverse of Eq. (38) is computed numerically in Table 2 by invoking diverse numerical methods.

3.3 Momentum profile

Taking the Laplace transform on Eq. (24) with related conditions in Eqs. (15)1– (17)1, we have

Ω11+λ1sqα1αqα+αW¯ψ,s=Ω21+λ2s2W¯ψ,sψ2+Ω31+λ1sGrθ¯ψ,s+Ω41+λ1sGmΦ¯ψ,s,(39)
W¯0,s=0,W¯ψ,s0,as,ψ.(40)

By using temperature values from Eq. (37) and concentration from Eq. (38) and with conditions of Eq. (40), we obtain the solution of the velocity field for Eq. (40) as

W¯ψ,s=Λ4sGrΛ3sΛ1sωss+ω2eψΛ1s1+aΛ1seψΛ3s1+aΛ1s+Λ5sGmΛ3sΛ2seψΛ2sseψΛ3ss,(41)

where b1=1+λ1s1+λ2s,Λ3s=b1Ω1Ω2sα1αsα+α,Λ4s=b1Ω3Ω2,andΛ5s=b1Ω4Ω2.

Our achieved solutions of variable profiles are complex to find analytically. Different researchers employed varied numerical approaches; so to compute Laplace inversion, we also employed numerical techniques, i.e., Stehfest and Tzou numerical methods. These algorithms are defined as follows (Stehfest, 1970; Tzou, 2014):

Wψ,η=ln2ηm=1MwmW¯ψ,mln2η,(42)

where wm=1m+M2r=q+12minq,M2rM22r!M2r!r!r1!qr!2rq!,

and

Wψ,η=e4.7η12W¯ψ,4.7η+Rej=1M1jW¯ψ,4.7+jπiη.(43)

Case I. Classical Oldroyd-B fluidBy substituting α=1 in Eq. (41), the velocity solution takes the form as

W¯ψ,s=Ω3Ω61+λ1sGrΩ1Ω61+λ1sΩ2Ω5Pr1+λ2sωs2+ω2eψsΩ5PrΩ61+asΩ5PrΩ6eψ1+λ1s1+λ2sΩ1Ω2s1+asΩ5PrΩ6+Ω4Sc1+λ1sGmΩ1Sc1+λ1sΩ21+λ2seψsScseψ1+λ1s1+λ2sΩ1Ω2ss.(44)

Case II. Fractionalized Maxwell fluidBy substituting λ2=0 in Eq. (41), the velocity solution converts as follows:

W¯ψ,s=1+λ1s1αsα+αΩ3Ω6Gr1+λ1sΩ1Ω6sαΩ5Ω2Prsαωss+ω211+aΩ5PrΩ6sα1αsα+αeψΩ5PrΩ6sα1αsα+αeψ1+λ1sΩ1Ω2sα1αsα+α+1+λ1s1αsα+αΩ4ScGm1+λ1sΩ1ScsαΩ2sαeψ1Scsα1αsα+αseψ1+λ1sΩ1Ω2sα1αsα+αs.(45)

Case III. Ordinary Maxwell fluidBy substituting α=1 and λ2=0 in Eq. (41), the velocity solution converts

W¯ψ,s=1+λ1sΩ3Ω6Gr1+λ1sΩ1Ω6Ω5Ω2Prωs2+ω211+asΩ5PrΩ6eψsΩ5PrΩ6eψ1+λ1sΩ1Ω2s+1+λ1sΩ4ScGm1+λ1sΩ1ScΩ2eψsScseψ1+λ1sΩ1Ω2ss.(46)

Case IV. Fractionalized Newtonian fluidBy substituting λ1=0 in Eq. (45), the velocity solution converts

W¯ψ,s=1αsα+αΩ3Ω6GrΩ1Ω6sαΩ5Ω2Prsαωss+ω211+aΩ5PrΩ6sα1αsα+αeψΩ5PrΩ6sα1αsα+αeψΩ1Ω2sα1αsα+α+1αsα+αΩ4ScGmΩ1ScsαΩ2sαeψ1Scsα1αsα+αseψΩ1Ω2sα1αsα+αs.(47)

Case V. Ordinary Newtonian fluidBy substituting α=1 in Eq. (47), the velocity solution converts

W¯ψ,s=Ω3Ω6GrΩ1Ω6Ω5Ω2Prωss+ω211+aΩ5PrΩ6eψΩ5PrΩ6eψΩ1Ω2+Ω4ScGmΩ1ScΩ2eψ1ScseψΩ1Ω2s.(48)

4 Discussion of results

In this article, the natural convection flow of the Oldroyd-B HNF flowing close to an infinite vertical flat plate is examined. Aluminum oxide–magnetite–water (Al2O3–Fe3O4–H2O) and aluminum oxide–magnetite–sodium alginate (Al2O3–Fe3O4–C6H9NaO7)-based HNFs are considered with an AB-fractional approach. The solution of dimensionless fractional equations of energy, concentration, and momentum is obtained with the Laplace method. To observe from the physical perception, the impacts of fractional derivatives and different flow parameters on concentration, velocity, and temperature are measured and shown in Figures 215 graphically.

FIGURE 2
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FIGURE 2. Simulation to explain the temperature for changing α when Pr=3andϕ1=ϕ2=0.4.

FIGURE 3
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FIGURE 3. Simulation to explain the temperature for Pr when α=0.4.

FIGURE 4
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FIGURE 4. Simulation to explain the temperature for fluctuating ϕ1 when Pr=6,α=0.4,andϕ2=0.5.

FIGURE 5
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FIGURE 5. Simulation to explain the temperature for changing ϕ2 when Pr=2,α=0.8,andϕ1=0.2.

FIGURE 6
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FIGURE 6. Simulation to explain the velocity for fluctuating α when Pr=3.2,ϕ1=ϕ2=0.2,Gr=8,Gm=6.5,Sc=0.5,λ1=0.5,andλ1=0.5.

FIGURE 7
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FIGURE 7. Simulation to explain the velocity for changing Pr when α=0.5,Gr=8,Gm=6.5,Sc=0.5,λ1=0.5,andλ1=0.3.

FIGURE 8
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FIGURE 8. Simulation to explain the velocity for changing Gr when α=0.4,Pr=2.5,ϕ1=ϕ2=0.2,Gm=4.5,Sc=0.8,λ1=0.7,andλ1=0.3.

FIGURE 9
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FIGURE 9. Simulation to explain the velocity for changing Gm when α=0.4,Pr=3.2,ϕ1=ϕ2=0.3,Gr=8,Sc=0.2,λ1=1.2,andλ1=0.3.

FIGURE 10
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FIGURE 10. Simulation to explain the velocity for fluctuating ϕ1 when α=0.5,Pr=3.2,ϕ2=0.3,Gr=6,Gm=8,Sc=0.3,λ1=0.7,andλ1=1.

FIGURE 11
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FIGURE 11. Simulation to explain the velocity for fluctuating ϕ2 when α=0.5,Pr=3.2,ϕ1=0.2,Gr=5,Gm=6,Sc=0.2,λ1=1.7,andλ1=1.3.

FIGURE 12
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FIGURE 12. Simulation to explain the velocity for the different fluid models when Pr=3.2,ϕ1=ϕ2=0.2,Gr=8,Gm=4.5,Sc=0.1,λ1=0.7,andλ1=0.3.

FIGURE 13
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FIGURE 13. Simulation to clarify the velocity for the slip and no-slip condition when Pr=6.2,ϕ1=ϕ2=0.2,Gr=8,Gm=4.5,Sc=0.5,λ1=0.7,andλ1=0.3.

FIGURE 14
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FIGURE 14. Simulation to explain the concentration, temperature, and velocity for the comparison of different inversion numerical algorithms.

FIGURE 15
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FIGURE 15. Comparison of our results with the results by Chen et al. (2022) for validation.

Figure 2 shows the influence of α on the temperature field. By setting other parameters constant and fluctuating the value of α, it is seen that for a small time, the temperature profile declined for larger values α and this effect is reversed for a greater time. We see that fluid characteristics can be measured by fractional parameters. For a different value of α, the temperature close to the plate is extreme. The temperature declines away from the plate and is asymptotic in the growing ξ direction, which satisfies our boundary conditions. Figure 3 shows the thermal behavior for Pr. For large estimations of Pr, the temperature declines. Substantially, the heat conductivity increasing the estimations of Pr, manufacturing the fluid thicker, sources the least thickness of the heat boundary layer. Figures 4, 5 show the temperature behavior with ϕ1 and ϕ2. The temperature field represents an increasing function of ϕ1 and ϕ2. As expected, with greater values of ϕ1 and ϕ2, the capacity of the HNF expands to hold additional heat. Therefore, the heat conductivity of the NF increases and temperature increases at different times.

The fluid velocity declines as we increase α, as shown in Figure 6, when there is less time. For a long time, the velocity is enhanced. Physically, when α increases, the velocity and thermal boundary layer decline, and as a consequence, the velocity declines for a short time. Figure 7 shows the behavior of the velocity with Pr. The velocity field also decreases with increasing Pr. Enhancement in Pr decreases the thermal conductivity and increases the viscosity of the fluid because of which the momentum profile declines with Pr.

Figure 8 shows the influence of Gr on the momentum profile. By increasing Gr, the velocity profile is enhanced. Since Gr exhibits the buoyancy force that increases the natural convection, therefore the velocity grows. Figure 9 shows the impact of Gm on the velocity by considering the changing Gm with time. The ratio of the buoyant force and viscous force is named the mass Grashof number that sources unrestricted convection. Figure 9 shows that velocity is enhanced for enhancing Gm. Figures 10, 11 show the effect of ϕ1 and ϕ2 on velocity. The velocity decreases with increase in ϕ1 and ϕ2. This means that with the addition of nanoparticles to the base liquids, the resulting HNF becomes denser, so they become more viscous than the regular fluid. Also, the boundary layer of regular fluids is thinner than that of the HNF, and as a result, the velocity shows a declining behavior with increasing values of ϕ1 and ϕ2. Moreover, the impact of a water-based HNF has more progressive values as compared to that of the sodium alginate-based HNF on the profiles of energy and velocity.

Figure 12 shows a comparison of different fluid models. It is observed that the solutions of Maxwell nanofluids for both ordinary and fractional cases have developed curves as compared to Oldroyd-B and viscous nanofluids. Figure 13 shows the velocity for the slip and no-slip conditions. It can be seen that the slip condition shows a lesser profile for velocity than the no-slip conditions. Figure 14 shows the temperature and velocity behaviors for the comparison of diverse numerical techniques (Stehfest and Tzou’s algorithm). The overlapping of profiles shows that these algorithms are strongly validated with each other. Figure 15 shows the validation of our results with Chen et al. (2022). By overlapping both curves, it is observed from these graphs that our achieved results match those developed by Chen et al. (2022). The numerical comparison of energy, concentration, and velocity profiles by different numerical methods is shown in Table 2. Table 3 shows the numerical results of the Nusselt number, Sherwood number, and skin friction. The comparison of the momentum profile with the work of Chen et al. (2022) is shown in Table 4.

TABLE 3
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TABLE 3. Numerical results of the Nusselt number, Sherwood number, and skin friction.

TABLE 4
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TABLE 4. Numerical results of comparisons of the velocity field.

5 Conclusion

This article examines the investigations of the unsteady, convective flow of the Oldroyd-B HNF flowing over a flat plate with wall slip conditions on temperature and constant concentration. The model is developed using the AB-fractional operator and solved with the Laplace transform method. The Laplace inversion is computed with the well-known Stehfest and Tzou numerical schemes. Finally, the effect of diverse flow parameters is planned to estimate the physical clarification of the achieved results of governed equations. The main results from the previous section are summarized in the following:

❖ For a short time, the temperature and momentum profile decayed for a larger value of α, and this effect for both profiles is reversed for a longer time.

❖ By increasing Pr, the temperature and velocity show a decreasing behavior.

❖ By increasing Gm and Gr, the velocity profile is improved.

❖ The velocity decreases with increasing ϕ1 and ϕ2.

❖ The energy and velocity profiles are larger for a water-based HNF than those of the sodium alginate-based HNF.

❖ The graphs of Maxwell nanofluids for both classical and fractional models have more advanced curves than Oldroyd-B and viscous nanofluids.

❖ The slip condition shows a lower profile for velocity than the no-slip condition.

❖ The comparison of diverse numerical algorithms (Stehfest and Tzou) strongly validated our study’s solutions.

Chen et al. (2022), the overlapping of both curves validate the achieved results of our study.

6 Future recommendation

For extension of this fractional problem examined in this article, we idolized the following proposal based on investigation, approaches, extensions, and geometries, as demarcated in the following:

• The same problem can also be considered over a horizontal plate by using Prabhakar’s time-fractional approach with an MHD effect in a porous medium.

• A comparative study of this study can be solved by the natural and Laplace transform methods.

• The same problem may be discussed by the Keller–box scheme.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

Conceptualization, SME, AR, QA, MA, and UK; methodology, SME, AR, and UK; software, MA, QA, SME, AR, and UK; validation, SME, AR, UK, SE, MA, and AhA; formal analysis, AbA, SE, AR, and AhA; investigation, UK, AbA, SE, and AhA; resources, AbA; data curation, QA; writing—original draft preparation, MA, SME, QA, UK, AbA, SE, and AhA; writing—review and editing, AbA, QA, MA, and AhA; visualization, AR, AhA, and SE; supervision, UK; project administration, SE; funding acquisition, SE. All authors have read and agreed to the published version of the manuscript.

Funding

This work received support from Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R163), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. In addition, this study is also funded by Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

Acknowledgments

The authors are thankful for the support of Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R163), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Also, this work is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Abbreviations

W1, velocity [ms1]; β1, volumetric coefficient of thermal expansion [K1]; g, acceleration due to gravity LT2; T, temperature value away from the plate K; Tw, temperature on the plate K; C, concentration value away from the plate kgm3; T, temperature K; Cw, concentration at the plate ML3; Gr, thermal Grashof number [-]; μhbnf, dynamic viscosity of hybrid nanofluid [-]; κhbnf, thermal conductivity of hybrid nanofluid [-]; Pr, Prandtl number [-]; λ2, Maxwell parameter [-]; ϕ1,ϕ2, volumetric fractions [-]; Gm, mass Grashof number [-]; ρhbnf, density for hybrid nanofluid [-]; Cp, specific heat at constant pressure JM1K1; λ1, Oldroyd parameter [-]; s, Laplace transformed variable [-]; α, γ, fractional parameters [-]. Note: this [-] characterizes the dimensionless quantity.

References

Ahmed, Awais, Alhowaity, Sawsan, and Mohamed, E. (2022). Ghoneim, fehmi gamaoun, elsayed tag-eldin, mansour F. Yassen, and mahnoor sarfraz. "Material and wave relaxation phenomena effects on the rheology of Maxwell nanofluids. Front. Phys. 10, 886.

CrossRef Full Text | Google Scholar

Ali, Qasim, Al-Khaled, Kamel, Ijaz Khan, M., Khan, Sami Ullah, Ali, Raza, Oreijah, Mowffaq, et al. (2022). Diffusion phenomenon for natural convection flow of classical hartmann problem due to a cylindrical tube by generalized fourier’s theories: A fractional analysis. Int. J. Mod. Phys. B 13, 2350104. doi:10.1142/s0217979223501047

CrossRef Full Text | Google Scholar

Ali, Qasim, Al-Khaled, Kamel, Omar, Jiyan, Ali, Raza, Khan, Sami Ullah, Ijaz Khan, M., et al. (2022). Analysis for advection–diffusion problem subject to memory effects and local and nonlocal kernels: A fractional operators approach. Int. J. Mod. Phys. B 15, 2350099. doi:10.1142/s0217979223500996

CrossRef Full Text | Google Scholar

Ali, Qasim, Riaz, Samia, Aziz Ullah, Awan, and Ali Abro, Kashif (2021). A mathematical model for thermography on viscous fluid based on damped thermal flux. Z. für Naturforsch. A 76 (3), 285–294. doi:10.1515/zna-2020-0322

CrossRef Full Text | Google Scholar

Asogwa, Kanayo K., Bilal, Sardar M., Animasaun, Isaac L., and Mebarek-Oudina, Fateh M. (2021). Insight into the significance of ramped wall temperature and ramped surface concentration: The case of Casson fluid flow on an inclined Riga plate with heat absorption and chemical reaction. Nonlinear Eng. 10 (1), 213–230. doi:10.1515/nleng-2021-0016

CrossRef Full Text | Google Scholar

Asogwa, Kanayo Kenneth, Goud, B. Shankar, Shah, Nehad Ali, and -Jin Yook, Se (2022). Rheology of electromagnetohydrodynamic tangent hyperbolic nanofluid over a stretching riga surface featuring dufour effect and activation energy. Sci. Rep. 12 (1), 14602. doi:10.1038/s41598-022-18998-9

PubMed Abstract | CrossRef Full Text | Google Scholar

Asogwa, Kanayo Kenneth, Goud, B. Shankar, Yanala Dharmendar Reddy, , and Ibe, Amarachukwu A. (2022). Suction effect on the dynamics of EMHD casson nanofluid over an induced stagnation point flow of stretchable electromagnetic plate with radiation and chemical reaction. Results Eng. 15 (2022), 100518. doi:10.1016/j.rineng.2022.100518

CrossRef Full Text | Google Scholar

Asogwa, Kanayo Kenneth, Prasad, K. C. Rajendra, Kumar, Raman, Murtugudde, Gururaj, and Punith Gowda, R. J. (2022). Transient electromagnetohydrodynamic Nanofluid flow traveling through a moving Riga plate subject to radiation and heat absorption. Int. J. Mod. Phys. B 09, 2350168. doi:10.1142/s0217979223501680

CrossRef Full Text | Google Scholar

Asogwa, Kanayo K., Mebarek-Oudina, F., and Animasaun, I. L. (2022). Comparative investigation of water-based Al2O3 nanoparticles through water-based CuO nanoparticles over an exponentially accelerated radiative Riga plate surface via heat transport. Arabian J. Sci. Eng. 47 (7), 8721–8738. doi:10.1007/s13369-021-06355-3

CrossRef Full Text | Google Scholar

Atangana, Abdon, and Dumitru, Baleanu (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. arXiv Prepr. arXiv:1602.03408 21.

CrossRef Full Text | Google Scholar

Awan, Aziz Ullah, Shah, Nehad Ali, Ahmed, Najma, Ali, Qasim, and Riaz, Samia (2019). Analysis of free convection flow of viscous fluid with damped thermal and mass fluxes. Chin. J. Phys. 60, 98–106. doi:10.1016/j.cjph.2019.05.006

CrossRef Full Text | Google Scholar

Batool, Saima, Rasool, Ghulam, Alshammari, Nawa, Khan, Ilyas, Kaneez, Hajra, and Hamadneh, Nawaf (2022). Numerical analysis of heat and mass transfer in micropolar nanofluids flow through lid driven cavity: Finite volume approach. Case Stud. Therm. Eng. 37 (2022), 102233. doi:10.1016/j.csite.2022.102233

CrossRef Full Text | Google Scholar

Chen, C., Rehman, A. U., Riaz, M. B., Jarad, F., and Sun, X.-E. (2022). Impact of Newtonian heating via fourier and Fick’s laws on thermal transport of Oldroyd-B fluid by using generalized mittag-leffler kernel. Symmetry 14 (4), 766. doi:10.3390/sym14040766

CrossRef Full Text | Google Scholar

Farooq, U., Afridi, M. I., Qasim, M., and Lu, D. C. (2018). Transpiration and viscous dissipation effects on entropy generation in hybrid nanofluid flow over a nonlinear radially stretching disk. Entropy 20 (9), 668. doi:10.3390/e20090668

PubMed Abstract | CrossRef Full Text | Google Scholar

Goud, B. Shankar, Reddy, Y. Dharmendar, and Kenneth Asogwa, Kanayo (2022). Inspection of chemical reaction and viscous dissipation on MHD convection flow over an infinite vertical plate entrenched in porous medium with Soret effect. Biomass Convers. Biorefinery 29, 1–12. doi:10.1007/s13399-022-02886-3

CrossRef Full Text | Google Scholar

Hassan, Ali, Hussain, Azad, Arshad, Mubashar, Gouadria, Soumaya, Jan, Awrejcewicz, Galal, Ahmed M., et al. (2022). Insight into the significance of viscous dissipation and heat generation/absorption in magneto-hydrodynamic radiative casson fluid flow with first-order chemical reaction. Front. Phys. 10, 605. doi:10.3389/fphy.2022.920372

CrossRef Full Text | Google Scholar

Khan, Dolat, Kenneth Asogwa, Kanayo, Akkurt, Nevzat, Kumam, Poom, Watthayu, Wiboonsak, and Sitthithakerngkiet, Kanokwan (2022). Development of generalized fourier and Fick’s law of electro-osmotic MHD flow of sodium alginate based casson nanofluid through inclined microchannel: Exact solution and entropy generation. Sci. Rep. 12 (1), 18646. doi:10.1038/s41598-022-21854-5

PubMed Abstract | CrossRef Full Text | Google Scholar

Khan, Muhammad Naveed, Ahammad, N. Ameer, Ahmad, Shafiq, and Mohamed Abdelghany, Elkotb (2022). Elsayed Tag-eldin, Kamel Guedri, Khaled A. Gepreel, and Mansour F. Yassen. "Thermophysical features of Ellis hybrid nanofluid flow with surface-catalyzed reaction and irreversibility analysis subjected to porous cylindrical surface. Front. Phys. 10, 795.

CrossRef Full Text | Google Scholar

Khan, M. N., Alhowaity, S., Wang, Z., Alqahtani, A. M., Tag-eldin, E., and Yassen, M. F. (2022). Alqahtani, Elsayed Tag-eldin, and Mansour F. Yassen. "Significance of multiple solutions on the dynamics of ethylene glycol conveying gold and copper nanoparticles on a shrinking surface. Front. Phys. 10, 928.

CrossRef Full Text | Google Scholar

Khan, Umar, Ahmed, Naveed, and Khan, Ilyas (2022). Heat transfer evaluation in MgZn6Zr/C8H18 [(Magnesium-Zinc-Zirconium)/Engine oil] with non-linear solar thermal radiations and modified slip boundaries over 3-dimensional convectively heated surface. Front. Energy Res. 12, 351.

Google Scholar

Lee, S., Choi, S. U.-S, Li, S., and Eastman, J. A (1999). Measuring thermal conductivity of fluids containing oxide nanoparticles 2, 280–289.

CrossRef Full Text | Google Scholar

Mahian, Omid, Kolsi, Lioua, Amani, Mohammad, Estellé, Patrice, Ahmadi, Goodarz, Kleinstreuer, Clement., et al. (2019). Recent advances in modeling and simulation of nanofluid flows-Part I: Fundamentals and theory. Phys. Rep. 790, 1–48. doi:10.1016/j.physrep.2018.11.004

CrossRef Full Text | Google Scholar

Martyushev, Semen G., and Sheremet, Mikhail A. (2012). Characteristics of Rosseland and P-1 approximations in modeling nonstationary conditions of convection-radiation heat transfer in an enclosure with a local energy source. J. Eng. Thermophys. 21 (2), 111–118. doi:10.1134/s1810232812020026

CrossRef Full Text | Google Scholar

Mohebbi, Rasul, Mehryan, S. A. M., Izadi, Mohsen, and Mahian, Omid (2019). Natural convection of hybrid nanofluids inside a partitioned porous cavity for application in solar power plants. J. Therm. Analysis Calorim. 137, 1719–1733. doi:10.1007/s10973-019-08019-9

CrossRef Full Text | Google Scholar

Rashad, A. M., Chamkha, Ali J., Ismael, Muneer A., and Taha, Salah. (2018). Magnetohydrodynamics natural convection in a triangular cavity filled with a Cu-Al2O3/water hybrid nanofluid with localized heating from below and internal heat generation. J. Heat Transf. 140, 7. doi:10.1115/1.4039213, no.

CrossRef Full Text | Google Scholar

Rasool, Ghulam, Ahammad, N. Ameer, Ali, Mohamed R., Shah, Nehad Ali, Wang, Xinhua, Shafiq, Anum, et al. (2023). Hydrothermal and mass aspects of MHD non-Darcian convective flows of radiating thixotropic nanofluids nearby a horizontal stretchable surface: Passive control strategy. Case Stud. Therm. Eng. 42 (2023), 102654. doi:10.1016/j.csite.2022.102654

CrossRef Full Text | Google Scholar

Rasool, Ghulam, Shafiq, Anum, Hussain, Sajjad, Zaydan, Mostafa, Wakif, Abderrahim, Ali, J., et al. (2022). Significance of rosseland’s radiative process on reactive Maxwell nanofluid flows over an isothermally heated stretching sheet in the presence of Darcy–forchheimer and Lorentz forces: Towards a new perspective on buongiorno’s model. Micromachines 133, 368. doi:10.3390/mi13030368, no.

CrossRef Full Text | Google Scholar

Rasool, G., Ali Shah, N., El-Zahar, E. R., and Wakif, A. (2022). Numerical investigation of EMHD nanofluid flows over a convectively heated riga pattern positioned horizontally in a Darcy-Forchheimer porous medium: Application of passive control strategy and generalized transfer laws. Waves Random Complex Media 17, 1–20.

CrossRef Full Text | Google Scholar

Raza, A., Almusawa, M. Y., Ali, Q., Haq, A. U., Al-Khaled, K., and Sarris, I. E. (2022). Solution of water and sodium alginate-based casson type hybrid nanofluid with slip and sinusoidal heat conditions: A prabhakar fractional derivative approach. Symmetry 1412, 2658. doi:10.3390/sym14122658, no.

CrossRef Full Text | Google Scholar

Reddy, S. C., Asogwa, K. K., Yassen, M. F., Adnan, , Iqbal, Z., M-Eldin, S., et al. (2022). Dynamics of MHD second-grade nanofluid flow with activation energy across a curved stretching surface. Front. ENERGY Res. 10. doi:10.3389/fenrg.2022.1007159

CrossRef Full Text | Google Scholar

Shahsavar, Amin, Moradi, Mehdi, and Bahiraei, Mehdi (2018). Heat transfer and entropy generation optimization for flow of a non-Newtonian hybrid nanofluid containing coated CNT/Fe3O4 nanoparticles in a concentric annulus. J. Taiwan Inst. Chem. Eng. 84, 28–40. doi:10.1016/j.jtice.2017.12.029

CrossRef Full Text | Google Scholar

Shankar Goud, B., Dharmendar Reddy, Y., and Kenneth Asogwa, Kanayo (2022). Chemical reaction, Soret and Dufour impacts on magnetohydrodynamic heat transfer Casson fluid over an exponentially permeable stretching surface with slip effects. Int. J. Mod. Phys. B, 2350124. doi:10.1142/s0217979223501242

CrossRef Full Text | Google Scholar

Stehfest, Harald. (1970). Algorithm 368: Numerical inversion of Laplace transforms [D5]. Commun. ACM 13 (1), 47–49. doi:10.1145/361953.361969

CrossRef Full Text | Google Scholar

Tzou, Da Yu (2014). Macro-to microscale heat transfer: The lagging behavior.Hoboken, NJ, USA John Wiley & Sons.

Google Scholar

Usman, M., Hamid, M., Zubair, T., Ul Haq, Rizwan, and Wang, Wei. (2018). Cu-AlO/Water hybrid nanofluid through a permeable surface in the presence of nonlinear radiation and variable thermal conductivity via LSM. Int. J. Heat Mass Transf. 126, 1347–1356. doi:10.1016/j.ijheatmasstransfer.2018.06.005

CrossRef Full Text | Google Scholar

Vallejo, Javier P., Zyla, G., Fernandez-Seara, J., and Lugo, L. (2019). Influence of six carbon-based nanomaterials on the rheological properties of nanofluids. Nanomaterials 9 (2), 146. doi:10.3390/nano9020146

PubMed Abstract | CrossRef Full Text | Google Scholar

Vallejo, J. P., Álvarez-Regueiro, E., Cabaleiro, D., Fernández-Seara, J., Fernández, J., and Lugo, L. (2019). Functionalized graphene nanoplatelet nanofluids based on a commercial industrial antifreeze for the thermal performance enhancement of wind turbines. Appl. Therm. Eng. 152, 113–125. doi:10.1016/j.applthermaleng.2019.02.046

CrossRef Full Text | Google Scholar

Waini, Iskandar, Ishak, Anuar, and Pop, Ioan (2019). Unsteady flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid. Int. J. heat mass Transf. 136, 288–297. doi:10.1016/j.ijheatmasstransfer.2019.02.101

CrossRef Full Text | Google Scholar

Wong, Kaufui V., and Omar De, Leon (2010). Applications of nanofluids: Current and future. Adv. Mech. Eng. 2, 519659. doi:10.1155/2010/519659

CrossRef Full Text | Google Scholar

Zhang, Juan, Ali, Raza, Khan, Umair, Ali, Qasim, Zaib, Aurang, Weera, Wajaree, et al. (2022). Thermophysical study of Oldroyd-B hybrid nanofluid with sinusoidal conditions and permeability: A prabhakar fractional approach. Fractal Fract. 67, 357. doi:10.3390/fractalfract6070357, no.

CrossRef Full Text | Google Scholar

Keywords: fractionalized hybrid Oldroyd-B fluid, AB time-fractional derivative, Newtonian heating, Laplace transform method, hybrid nanofluid

Citation: Ali Q, Amir M, Raza A, Khan U, Eldin SM, Alotaibi AM, Elattar S and Abed AM (2023) Thermal investigation into the Oldroyd-B hybrid nanofluid with the slip and Newtonian heating effect: Atangana–Baleanu fractional simulation. Front. Mater. 10:1114665. doi: 10.3389/fmats.2023.1114665

Received: 02 December 2022; Accepted: 07 February 2023;
Published: 24 February 2023.

Edited by:

Noor Saeed Khan, University of Education Lahore, Pakistan

Reviewed by:

Ghulam Rasool, Beijing University of Technology, China
Kanayo Kenneth Asogwa, Nigeria Maritime University, Nigeria

Copyright © 2023 Ali, Amir, Raza, Khan, Eldin, Alotaibi, Elattar and Abed. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Sayed M. Eldin, sayed.eldin22@fue.edu.eg

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