# Analysis of Fractional Bioconvection with Hybrid Nanoparticles in Channel Flow

Authors: Muhammad Imran Asjad; Muhammad Danish Ikram; Noman Sarwar; Taseer Muhammad; Sivanandam Sivasankaran; Sami Ahmed Abdu Subaihi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, MHD Brinkman-type fluid flow containing titanium dioxide and silver nanoparticle hybrid nanoparticles with generalized Mittag–Leffler kernel-based fractional derivative is investigated in the presence of bioconvection. The governing equations with dimensional analysis and fractional approach are obtained by using the fractional Fourier’s law for heat flux and Fick’s law for diffusion. As a result, the bioconvection Rayleigh number, which is responsible for the declining in the fluid velocity and fractional parameters used to control the thermal and momentum boundary layers thickness of fluid properties. The obtained solutions can be beneficial for proper analysis of real data and provide a tool for testing possible approximate solutions where needed.


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## Body

## 1. Introduction

The concepts of fractional Brinkman form models with hybrid nanoparticles through an oscillating vertical plate and a magnetic field having variable direction are not detailed, however. To fill this void, a fractional Brinkman sorting fluid show is used to blend a flow of hybrid nanofluids over a swaying vertical plate. Brinkman’s sort of fluid show was created by Brinkman in his pioneering work while investigating fluid flow due to thick constraint on the surface of a thick swarm particle [1, 2]. Saqib et al. [3] discussed the shape impact on the MHD flow of time-fractional, Ferro-Brinkman-sorted nanofluid having slope warming. Asjad et al. [4] introduced non-Newtonian fractional derivatives in a convective channel containing hybrid nanoparticles by Prabhakar. Khan et al. [5] investigated the chemical response and heat era impact of nonofluids of the Brinkman-type H2O-Cu, Ag, TiO2, and Al2O3 in a porous medium with an MHD flow. Nanjundappa et al. [6] explored the impact of dust particles on Darcy–Brinkman gravity-driven ferro-thermal-convection in a ferrofluid soaked porous layer with an inside heat source: the impact of boundaries. Sarwar et al. [7] explained a comparative ponder on a non-Newtonian fractional-order Brinkman sorted fluid with two different parts. Shafie et al. [8] blended the convection flow of Brinkman sorted hybrid nanofluid based on Atangana–Baleanu fractional demonstration. Ali et al. [9] presented the Caputo–Fabrizio fractional derivative modelling of transitory MHD Brinkman nanofluid. Saqib et al. [10] examined the attractive resistive flow of a generalized Brinkman sort of nanofluid containing carbon nanotubes with inclined warming.In many circumstances, mathematical models of integer order derivatives, including nonlinear models, do not operate satisfactorily. Fractional calculus has numerous applications in the domains of electromagnetics, viscoelasticity, fluid mechanics, signal processing, and optics. It has been used to demonstrate the physical and design shapes that can be found in a model to be best described by fractional differential conditions. The fractional derivative models are utilized for exact modelling of those frameworks that require exact modelling of damping. Boutiara et al. [11] debated a lesson of Langevin conditions within the outline of Caputo function-dependent kernel fractional derivatives through antiperiodic edge situations. Ali et al. [12] introduced controlled regulation retention of the common convection flow of hybrid nanofluids with a consistent relative Caputo-fractional derivative due to weight angle. Imran et al. [13] studied the mass and heat transport of a differentially sorted fluid with a noninteger time-fractional Caputo derivative. Ahmad et al. [14] announced the numerical modelling of (Cu-Al2O3) water-based Maxwell hybrid nanofluids with Caputo–Fabrizio fractional derivative. Mirza and Vieru [15] considered fundamental arrangements for an advection-diffusion condition with a time-fractional Caputo–Fabrizio derivative. Gul et al. [16] explored subjective investigations of certain Dirichlet boundary value problems aimed at Caputo–Fabrizio fractional differential conditions. Abdo et al. [17] offered a definite Atangana–Baleanu–Caputo derivative with nonlinear pantograph fractional differential conditions. Sweilam et al. [18] mathematically demonstrated that the Atangana–Baleanu–Caputo fractional derivative is an ideal control for cancer treatment. Sarwar et al. [19] investigated the Prabhakar derivative for convection flow Casson fluid over fluctuating plate based on the generalized Fourier law. Shah et al. [20] studied the common convection flow of Prabhakar fractional Maxwell fluid by generalized heat transportation. Elnaqeeb et al. [21] analyzed the normal convection flow of carbon nanotube Prabhakar-like fractional second-grade nanofluids over an infinite plate with Newtonian heating. Garrappa and Kaslik [22] considered the steadiness of fractional-order frameworks with Prabhakar derivatives.Choi was one of the first to show nanofluids containing nanoparticles in 1995. Nanofluids can provide various benefits, including thermal conductivity. For the most part, these are intended for biomedical engineering, mechanical design, and fluid mechanics by Choi and Eastman [23]. Ali et al. [24] presented an investigation of a scientific fragmentary demonstration of hybrid viscous nanofluids and their application in heat and mass transfer. Ahmad et al. [25] verified expository arrangements for a complimentary convection flow of Casson nanofluid over an infinite vertical plate. Gul et al. [26] studied the hybrid nanofluid flow inside the cone-shaped hole between the cone and the surface of a spinning disk. Rafique et al. [27] defined the Casson nanofluid flow over a slanted permeable inclined surface with energy and mass transport. Khan et al. [28] discussed the effects of interfacial electro kinetic MHD radiative nanofluids flow on porous microchannels by thermophoresis and Brownian movement impacts. Ali Lund et al. [29] conferred hybrid nanofluid on the nonlinear contracting sheet with double branches of an MHD three-dimensional pivoting stream. Shah et al. [30] debated the recreation of entropy optimization and the warm behavior of nanofluids through the permeable media. Kumar et al. [31] deliberated a novel approach for the examination of warm exchange development through ferromagnetic hybrid nanofluid via considering sun-oriented radiation. Dadheech et al. [32] discussed natural convection and an angled magnetic field being used to compare the heat transfer of MoS2/C2H6O2 and SiO2-MoS2/C2H6O2 nanofluids. Dhif et al. [33] conducted a deliberate thermal study of a hybrid nanofluids solar collector and storage system. Khan et al. [34] examined the squeezing flow of nanofluids using mixed convection. Bu et al. [35] conducted a squeezing flow of nanofluids in a rotating channel with mixed convection and thermal radiation. By considering diverse physical flow conditions, thermal transport in aluminum alloy nanomaterials based on radiative nanofluids was investigated by Ullah Khan et al. [36]. In a triangular enclosure with zigzags and an elliptic obstacle, MHD flow of a hybrid nanofluid was introduced by Chabani et al. [37]. According to Rajashekhar et al. [38], the peristaltic flow of a Ree–Eyring liquid is affected by the different qualities of hemodynamic flow, mass, and heat transport. In a porous lid-driven hollow with a magnetic field, the entropy formation and heat transmission of Cu–water nanofluid were described by Marzougui et al. [39].Bioconvection is defined as the wonder of macroscopic convection motion of fluid caused by the thickness angle established by the directional collective swimming of microorganisms. Plat proposed the concept of bioconvection in 1961. Bioconvection applications include biological polymer manufacture, biotechnology, and bio-sensors, as well as the testing and laboratory industries, among others [40, 41]. Ge-Jile et al. [42] analyzed nanofluid with motile microorganisms through the three-dimensional radiative bioconvective stream of a Sisko. Ramzan et al. [43] present bioconvection as a component in a three-dimensional digression hyperbolic partially ionized magnetized nanofluid stream with Cattaneo–Christov heat flux and activation vitality. Alhussain et al. [44] analyzed the warm conductivity and magneto-bioconvective augmentation in a nanofluid stream holding gyrotactic microorganisms. Farooq et al. [45] adjusted exponential space-based heat sources and Cattaneo–Christov expressions with a thermally radioactive bioconvection flow of Carreau nanofluid. Yousuf et al. [46] discussed the magneto-bioconvection flow of Williamson nanofluids through an inclined plate by entropy generation and gyrotactic microorganisms. Saqib et al. [47] inspected a Brinkman-type fluid (BTF) fractional model with hybrid nanoparticles. Danish Ikram et al. [48] calculated the heat transfer of viscous fluid with clay nanoparticles over an exponentially moving upright plate. Asjad Imran et al. [49] analyzed the thermophysical properties of clay nanofluids using a hybrid fractional operator. Ikram et al. [50] discussed the fractional model of Brinkman-type fluid (BTF) holding hybrid nanoparticles in a bounded microchannel via a constant proportional Caputo fractional operator.Channel flow is used in a wide range of industrial applications, including heat exchangers in power plants and chemical reactors in the pharmaceutical industry. Although many actual processes with Newtonian behavior in both phases can be termed two-phase flows, there are a huge number of related applications where the continuous liquid phase exhibits non-Newtonian flow properties. There are several instances in the biochemical and biomedical industries, as well as in the food processing industry [51]. The impact of a vortex generator shape on liquids and the heat transition of hybrid nanofluids in a channel were examined by Zheng et al. [52]. On the scientific scale, D’Ippolito et al. [53] investigated the resistance of open channel flow due to vegetation. Haq et al. [54] demonstrated the fractional viscous liquid influence of MHD channel flow across a porous medium using the Caputo–Fabrizio time-fractional derivative.In the absence of fractional bioconvection, the analyses above were conducted with or without fractional derivatives. The main objective, on the other hand, is to merge these two interesting subjects, fractional derivatives, and bioconvection. Recently, Asjad et al. [55] presented fractional bioconvection properties for sticky fluid over an infinite perpendicular plate with Caputo fractional derivative. In the above-mentioned literature, there is not a single study on the subject of fractional bioconvection between two parallel plates with a Prabhakar fractional derivative. As a result, our motivation is to use the Laplace transform approach to solve the fluid flow and heat transfer problems of bioconvection. Graphics are used to offer a graphical explanation of flow parameters.


## 2. Mathematical Formulation

Consider an MHD convection flow happening within a microchannel of a generalized, electrically conductive,Ag−TiO2−H2O hybrid nanoparticles situated in xy-plane as shown in Figure 1.Figure 1
Geometry of the problem.The assumptions are as follows:(i)
Microchannel is measured of length infinite by widthL(ii)
Att≤0, the temperature of system is T0(iii)
The channel is alongx−axis and normal to y−axis(iv)
Fluid flow occurs in thex-direction(v)
Aftert=0+, the temperature and concentration level of microorganism raised from T0 to Tw and N∞ to Nw, respectively(vi)
Magnetics field of strengthB0 is applied normal to the plateThe stream of electrically conductiveAg−TiO2−H2O hybrid nanofluids endures electromotive drive, which yields current. The initiated attractive fields are disregarded since there is speculation about really small Reynolds numbers. The electromagnetic force activates the electric flux concentration [56]. The relation of thermophysical properties of nano and hybrid nanofluids is defined in Tables 1 and 2, respectively.Table 1
The thermophysical properties of hybrid nanofluid and nanofluid under consideration are given in [50].
NanofluidHybrid nanofluidρnf=1−ϕρf+ϕρsρhnf=1−ϕhnfρf+ϕTiO2ρTiO2+ϕAgρAgμnf=μf/1−ϕ5/2μhnf=μf/1−ϕAg+ϕTiO25/2ρCpnf=1−ϕρCpf+ϕρCpsρCphnf=1−ϕhnfρCpf+ϕAgρcpAg+ϕTiO2ρCpTiO2ρBTnf=1−ϕρBTf+ϕρBTsρBThnf=1−ϕhnfρBTf+ϕAgρBTAg+ϕTiO2ρBTTiO2σnf/σf=1+3σs/σf−1ϕ/σs/σf+2−σs/σf−1ϕσhnf/σf=1+3ϕAgσAg+ϕTiO2σTiO2/σf−ϕhnf/ϕAgσAg+ϕTiO2σTiO2/ϕhnfσf+2−ϕAgσAg+ϕTiO2σTiO2/σf−ϕhnfKnf/Kf=ks+2kf−2ϕks−kf/ks+2kf+ϕks−kfKhnf/Kf=ϕAgkAg+ϕTiO2kTiO2/ϕhnf+2kf+2ϕAgkAg+ϕTiO2kTiO2−2ϕhnfKf/ϕAgkAg+ϕTiO2kTiO2/ϕhnf+2kf+ϕAgkAg+ϕTiO2kTiO2−ϕhnfKfTable 2
Thermophysical possessions of nanoparticles and base fluid [50].
MaterialBase fluid water (H2O)Nanoparticles silver (Ag)Nanoparticles titanium dioxide (TiO2)ρ997.110500425Cp41792356862k0.6134298.9538βT×10−5211.890.9σ0.053.6×1071×10−12Pr6.2——The governing equations of momentum and enegry are as follows [47]:(1)∂uy,t∂t+βb∗uy,tρhnf=μhnf∂2uy,t∂y2−σhnfB02uy,t+gρβThnfT−T0−γρm−ρN−N0.The energy equation is as follows:(2)ρCphnf∂Ty,t∂t=−∂qy,t∂y.The generalized Fourier’s law for thermal flux is as follows:(3)qy,t=−khnfCDα,β,aγ∂Ty,t∂y.The diffusion balance equation is as follows:(4)∂Ny,t∂t=−∂Ly,t∂y.The generalized Fick’s law for diffusion equation is as follows:(5)L=−DhnfCDα,β,aγ∂Ny,t∂y,where cDα,β,aγ denotes the Prabhakar fractional derivative and is defined as [57, 58].For (1)–(5), we consider the following initial and boundary conditions:(6)uy,0=0,Ty,0=T0,Ny,0=N0,0≤y≤L,(7)uy,t=0,T0,t=T0,N0,t=N0,t>0,(8)uL,t=0,TL,t=Tw,NL,t=Nw,t>0.By introducing dimensionless variables,(9)τ=νL2t,Y=yL,θ=T−T0Tw−T0,N∗=N−N0Nw−N0,q∗=qqo,L∗=LL0,q0=khnfTw−T0u0vhnf,L0=DhnfNw−N0u0νhnf.The dimensionless fundamental equations are obtained by substituting from (9) into (1)–(8) and ignoring the star documentation.Momentum and bioconvection equations in dimensionless form are as follows:(10)Ao•∂VY,τ∂τ+βb•VY,τ=A1•∂2VY,τ∂Y2−A2•MVY,τ+GrA3•θY,τ−RaNY,τ.Dimensionless form of energy equation is as follows:(11)B0∂θY,τ∂τ=−∂qY,τ∂Y.In dimensionless form, the generalized Fourier’s law for thermal flux [57, 58] is as follows:(12)qY,τ=−Dα,β,aγC∂θy,τ∂Y.Dimensionless diffusion balance equation is as follows:(13)Lb∂NY,τ∂τ=−∂LY,τ∂Y.Dimensionless form of Fick’s law [57, 58] is as follows:(14)LY,τ=−Dα,β,aγC∂NY,τ∂y.Constraints are associated with(15)VY,0=0,θY,0=0,NY,0=0,Y≥0,(16)V0,τ=0,θ0,τ=0,N0,τ=0,τ>0,(17)V1,τ=0,θ1,τ=1,N1,τ=1,τ>0.Variables are as follows:(18)βb•=L2βb∗ρfμf,M=L2σfB02μf,Gr=L3gBTfTw−T0νf2,Pr=μCpfkf,Lb=νfDhnf,Ra=γρm−ρNw−N0ρfBTfTw−T0,λhnf=khnfkf,A0•=1−ϕhnf+ϕAgρAg+ϕTio2ρTio2ρf,A2•=σhnfσf,A1•=11−ϕAg+ϕTio22.5,A3•=1−ϕhnf+ϕAgρBTAg+ϕTio2ρBTTio2ρBTf,B0=PrA4∗λhnf,A4•=1−ϕhnf+ϕAgρCpAg+ϕTio2ρCpTio2ρCpf,B1=A0∗A1∗,B5=B3B2,B2=A2∗M+Ao∗βbA1∗,B3=A3∗GrA1∗,B4=GrRaA1∗,B6=B4B2.In the above equations,βb∗ is the Brinkman parameter, M is the dimensionless magnetic field parameter, Pr is dimensionless Prandtl number, Gr is dimensionless Grashof number, and Ra is dimensionless bioconvection Rayleigh number, respectively.


## 3. Solution of the Problem

### 3.1. Solution of Temperature Field

We deliberateβ ∈ [0, 1) along these lines, and in the above formularies, the boundary m remains equivalent near zero. By utilizing the Laplace change strategy and applying it to equations (11) and (12) through requirements (16) and (17), and using the fractional derivative of Prabhakar, we acquire changed issue for temperature field:(19)sB0θ¯Y,s=−∂q¯Y,s∂Y,(20)q¯Y,s=−sβ1−as−αγ∂θ¯Y,s∂Y.Using (19) and (20), we have(21)∂2θ¯Y,s∂Y2−B0sθ¯Y,ssβ1−as−αγ=0.Subject to the constraints,(22)θ¯1,s=1s,θ¯0,s=0,τ>0.The general solution of equation (21) with equation (22)is as follows:(23)θ¯Y,s=1s1e−B0s/sβ1−as−αγ−eB0s/sβ1−as−αγ/2e−YB0s/sβ1−as−αγ−eYB0s/sβ1−as−αγ2.Or,(24)θ¯Y,s=1ssinhYB0s/sβ1−as−αγsinhB0s/sβ1−as−αγ.It is important that equation (24) can be written in the equivalent form(25)θ¯Y,s=1s∑n=0∞e−2n+1−YB0s/sβ1−as−αγ−e−2n+1+YB0s/sβ1−as−αγ.Moreover, equation (25) can also be expressed as a series approach, allowing us to rationally determine the inverse Laplace transform.(26)θ¯Y,s=1s+∑n=0∞∑m=1∞∑k=0∞Y−2n−1makB0m/2Γγm/2+km!k!Γγm/21sαk−m/2+βm/2+1−∑p=0∞∑n=0∞∑l=0∞−2n−1−YlapB0l/2Γγl/2+pl!p!Γγl/21sαp+βl/2−l/2+1.Taking inverse Laplace transform on equation (26), we get(27)θY,τ=1+∑n=0∞∑m=1∞∑k=0∞Y−2n−1makB0m/2Γγm/2+km!k!Γγm/2ταk+βm/2−m/2Γαk+βm/2−m/2+1−∑n=0∞∑l=0∞∑p=0∞−2n−1−YlB0l/2apΓγl/2+pl!p!Γγl/2ταp−l/2+βl/2Γαp+βl/2−l/2+1.


### 3.2. Solution of Bioconvection Field

By utilizing the Laplace change approach and applying it to equations (13) and (14) with requirements (16) and (17) and utilizing the Prabhakar derivative, we acquire the changed issue for the bioconvection field.(28)LbsN¯Y,s=−∂L¯Y,s∂Y,(29)L¯Y,s=−sβ1−as−αγ∂N¯Y,s∂Y.Using equations (28) and (29), we have(30)∂2N¯Y,s∂Y2−sLbsβ1−as−αγN¯Y,s=0.Subject to the constraints,(31)N¯1,s=1s,N¯0,s=0.The general solution of equation (30) with equation (31) is as follows:(32)N¯Y,s=1s1e−sLb/sβ1−as−αγ−esLb/sβ1−as−αγ/2e−YsLb/sβ1−as−αγ−eYsLb/sβ1−as−αγ2.Or,(33)N¯Y,s=1ssinhYsLb/sβ1−as−αγsinhsLb/sβ1−as−αγ.It is important that equation (33) can be composed within the comparable form(34)N¯Y,s=1s∑d=0∞e−2d+1−YsLb/sβ1−as−αγ−e−2d+1+YsLb/sβ1−as−αγ.Moreover, equation (34) can be communicated in an arrangement shape so that we are able to discover the Laplace inverse transform logically.(35)N¯Y,s=1s+∑d=0∞∑s1=1∞∑s2=0∞−2d−1+Ys1as2Lbs1/2Γγs1/2+s2s1!s2!Γγs1/21sαs2+βs1/2−s1/2+1−∑d=0∞∑p1=0∞∑p2=0∞−2d−1−Yp1ap2Lbp1/2Γγp1/2+p2p1!p2!Γγp1/21sαp2+βp1/2−p1/2+1.Taking inverse Laplace transform on equation (35), we get(36)NY,τ=1+∑d=0∞∑s1=1∞∑s2=0∞−2d−1+Ys1as2Lbs1/2Γγs1/2+s2s1!s2!Γγs1/2ταs2+βs1/2−s1/2Γαs2+βs1/2−s1/2+1−∑d=0∞∑p1=0∞∑p2=0∞−2d−1−Yp1ap2Lbp1/2Γγp1/2+p2p1!p2!Γγp1/2ταp2+βp1/2−p1/2Γαp2+βp1/2−p1/2+1.


### 3.3. Solution of Velocity Field

The Laplace transform is used in equation (10) using expressions from (15)–(17), and we attain(37)∂2∂Y2−B1s−B2V¯Y,s=−B3θ¯Y,s+B4N¯Y,s,which satisfies the following constraints:(38)V¯1,s=0,V¯0,s=0.The solution of equation (37) subject to constraints (38), we have(39)V¯Y,s=−B5s∑n=0∞e−2nB0s/sβ1−as−αγ−e−2n+2B0s/sβ1−as−αγ1+B1s/B2−B0s/B2sβ1−as−αγ×∑m=0∞e−2m+1−YB2+B1s−∑m=0∞e−2m+1+YB2+B1s+B6s∑d=0∞e−2dsLb/sβ1−as−αγ−e−2d+2sLb/sβ1−as−αγ1+B1s/B2−sLb/B2sβ1−as−αγ×∑m=0∞e−2m+1−YB2+B1s−∑m=0∞e−2m+1+YB2+B1s+B5s∑n=0∞e−2n+1−YB0s/sβ1−as−αγ−e−2n+1+YB0s/sβ1−as−αγ1+B1s/B2−B0s/B2sβ1−as−αγ−B6s∑d=0∞e−2d+1−YsLb/sβ1−as−αγ−e−2d+1+YsLb/sβ1−as−αγ1+B1s/B2−sLb/B2sβ1−as−αγ.Equation (39) can be written in component form as follows:(40)V¯Y,s=V¯1Y,s+V¯2Y,s+V¯3Y,s+V¯4Y,s+V¯5Y,s+V¯6Y,s+V¯7Y,s+V¯8Y,s+V¯9Y,s+V¯10Y,s+V¯11Y,s+V¯12Y,s.It is challenging to find the inverse Laplace transform of equation (40), so we can rewrite it in a suitable series.(41)VY,τ=V1Y,τ+V2Y,τ+V3Y,τ+V4Y,τ+V5Y,τ+V6Y,τ+V7Y,τ+V8Y,τ+V9Y,τ+V10Y,τ+V11Y,τ+V12Y,τ.Next, taking the inverse Laplace of equation (41), component wise, we have(42)V1Y,τ=−B5∑n=0∞∑m=0∞∑δ2=0∞∑δ1=0∞∑w2=0∞∑w1=0∞∑z1=0∞∑z2=0∞∑z3=0∞Y−2m−1w1−B1z1B0δ1/2δ1!δ2!w1!w2!z2!z3!B1z2−w2B2w2+z1−w1/2×−2nδ1−B0z2aδ2+z3ταδ2+βδ1/2−δ1/2−w2+αz3+βz2−z1Γαδ2+βδ1/2−δ1/2−w2+αz3+βz2−z1+1Γγδ1/2+δ2Γw1/2+1Γz1+1Γγz2+z3Γγδ1/2Γw1/2+1−w2Γz1+1−z2Γγz2,V2Y,τ=B5∑n=0∞∑m=0∞∑δ1=0∞∑δ2=0∞∑q1=0∞∑q2=0∞∑z1=0∞∑z2=0∞∑z3=0∞−Y+2m+1q1−B1z1B0δ1/2δ1!δ2!q1!q2!z2!z3!B1z2−q2B2q2+z1−q1/2×−2nδ1−B0z2aδ2+z3ταδ2+βδ1/2−δ1/2−q2+αz3+βz2−z1Γαδ2+βδ1/2−δ1/2−q2+αz3+βz2−z1+1Γγδ1/2+δ2Γq1/2+1Γz1+1Γγz2+z3Γγδ1/2Γq1/2+1−q2Γz1+1−z2Γγz2,V3Y,τ=B5∑n=0∞∑m=0∞∑j1=0∞∑j2=0∞∑w1=0∞∑w2=0∞∑z1=0∞∑z2=0∞∑z3=0∞Y−2m−1w1−B1z1Prj1/2j1!j2!w1!w2!z2!z3!B1z2−w2B2w2+z1−w1/2×−2n−2j1−B0z2aj2+z3ταj2+βj1/2−j1/2−w2+αz3+βz2−z1Γαj2+βj1/2−j1/2−w2+αz3+βz2−z1+1Γγj1/2+j2Γw1/2+1Γz1+1Γγz2+z3Γγj1/2Γw1/2+1−w2Γz1+1−z2Γγz2,V4Y,τ=−B5∑n=0∞∑m=0∞∑j1=0∞∑j2=0∞∑q1=0∞∑q2=0∞∑z1=0∞∑z2=0∞∑z3=0∞−Y+2m+1q1−B1z1B0j1/2j1!j2!q1!q2!z2!z3!B1z2−q2B2q2+z1−q1/2×−2n−2j1−B0z2aj2+z3ταj2+βj1/2−j1/2−q2+αz3+βz2−z1Γαj2+βj1/2−j1/2−q2+αz3+βz2−z1+1Γγj1/2+j2Γq1/2+1Γz1+1Γγz2+z3Γγj1/2Γq1/2+1−q2Γz1+1−z2Γγz2,V5Y,τ=B6∑d=0∞∑m=0∞∑n1=0∞∑n2=0∞∑w1=0∞∑w2=0∞∑h1=0∞∑h2=0∞∑h3=0∞Y−2m−1w1−B1h1Lbn1/2n1!n2!w1!w2!h2!h3!B1h2−w2B2w2+h1−w1/2×−2dn1−Lbh2an2+h3ταn2+βn1/2−n1/2−w2+αh3+βh2−h1Γαn2+βn1/2−n1/2−w2+αh3+βh2−h1+1Γγn1/2+n2Γw1/2+1Γh1+1Γγh2+h3Γγn1/2Γw1/2+1−w2Γh1+1−h2Γγh2,V6Y,τ=−B6∑d=0∞∑m=0∞∑n1=0∞∑n2=0∞∑q1=0∞∑q2=0∞∑h1=0∞∑h2=0∞∑h3=0∞−Y+2m+1q1−B1h1Lbn1/2n1!n2!q1!q2!h2!h3!B1h2−q2B2q2+h1−q1/2×−2dn1−Lbh2an2+h3ταn2+βn1/2−n1/2−q2+αh3+βh2−hΓαn2+βn1/2−n1/2−q2+αh3+βh2−h1+1Γγn1/2+n2Γq1/2+1Γz1+1Γγh2+h3Γγn1/2Γq1/2+1−q2Γh1+1−h2Γγz2,V7Y,τ=−B6∑d=0∞∑m=0∞∑m1=0∞∑m2=0∞∑w1=0∞∑w2=0∞∑h1=0∞∑h2=0∞∑h3=0∞Y−2m−1w1−B1h1Lbm1/2m1!m2!w1!w2!h2!h3!B1h2−w2B2w2+h1−w1/2×−2d−2m1−B0h2am2+h3ταm2+βm1/2−m1/2−w2+αh3+βh2−h1αm2+βm1/2−m1/2−w2+αh3+βh2−h1+1Γγm1/2+m2Γw1/2+1Γh1+1Γγh2+h3Γγm1/2Γw1/2+1−w2Γh1+1−h2Γγh2,V8Y,τ=B6∑d=0∞∑m=0∞∑m1=0∞∑m2=0∞∑q1=0∞∑q2=0∞∑h1=0∞∑h2=0∞∑h3=0∞−Y+2m+1q1−B1h1Lbm1/2m1!m2!q1!q2!h2!h3!B1h2−q2B2q2+h1−q1/2×−2d−2m1−Lbh2am2+h3ταm2+βm1/2−m1/2−q2+αh3+βh2−h1Γαm2+βm1/2−m1/2−q2+αh3+βh2−h1+1Γγm1/2+m2Γq1/2+1Γh1+1Γγh2+h3Γγm1/2Γq1/2+1−q2Γh1+1−h2Γγh2,V9Y,τ=B5∑n=0∞∑t1=0∞∑t2=0∞∑z1=0∞∑z2=0∞∑z3=0∞Y−2n−1t1−B1z1B0t1/2t1!t2!z2!z3!B1z2B2z1×−B0z2at2+z3ταt2+βt1/2−t1/2+αz3+βz2−z1Γαt2+βt1/2−t1/2+αz3+βz2−z1+1Γγt1/2+t2Γz1+1Γγz2+z3Γγt1/2Γz1+1−z2Γγz2,V10Y,τ=−B5∑n=0∞∑v1=0∞∑v2=0∞∑z1=0∞∑z2=0∞∑z3=0∞−Y+2n+1v1−B1z1B0v1/2v1!v2!z2!z3!B1z2B2z1×−B0z2av2+z3ταv2+βv1/2−v1/2+αz3+βz2−z1Γαv2+βv1/2−v1/2+αz3+βz2−z1+1Γγv1/2+v2Γz1+1Γγz2+z3Γγv1/2Γz1+1−z2Γγz2,V11Y,τ=−B6∑d=0∞∑l1=0∞∑l2=0∞∑h1=0∞∑h2=0∞∑h3=0∞Y−2d−1l1−B1h1Lbl1/2l1!l2!h2!h3!B1h2B2h1×−Lbh2al2+h3ταl2+βl1/2−l1/2+αh3+βh2−h1Γαl2+βl1/2−l1/2+αh3+βh2−h1+1Γγl1/2+l2Γh1+1Γγh2+h3Γγl1/2Γh1+1−h2Γγh2,V12Y,τ=B6∑d=0∞∑k1=0∞∑k2=0∞∑h1=0∞∑h2=0∞∑h3=0∞−Y+2d+1k1−B1h1Lbk1/2k1!k2!h2!h3!B1h2B2h1×−Lbh2ak2+h3ταk2+βk1/2−k1/2+αh3+βh2−h1Γαk2+βk1/2−k1/2+αh3+βh2−h1+1Γγk1/2+k2Γh1+1Γγh2+h3Γγk1/2Γh1+1−h2Γγh2.


## 3.1. Solution of Temperature Field

We deliberateβ ∈ [0, 1) along these lines, and in the above formularies, the boundary m remains equivalent near zero. By utilizing the Laplace change strategy and applying it to equations (11) and (12) through requirements (16) and (17), and using the fractional derivative of Prabhakar, we acquire changed issue for temperature field:(19)sB0θ¯Y,s=−∂q¯Y,s∂Y,(20)q¯Y,s=−sβ1−as−αγ∂θ¯Y,s∂Y.Using (19) and (20), we have(21)∂2θ¯Y,s∂Y2−B0sθ¯Y,ssβ1−as−αγ=0.Subject to the constraints,(22)θ¯1,s=1s,θ¯0,s=0,τ>0.The general solution of equation (21) with equation (22)is as follows:(23)θ¯Y,s=1s1e−B0s/sβ1−as−αγ−eB0s/sβ1−as−αγ/2e−YB0s/sβ1−as−αγ−eYB0s/sβ1−as−αγ2.Or,(24)θ¯Y,s=1ssinhYB0s/sβ1−as−αγsinhB0s/sβ1−as−αγ.It is important that equation (24) can be written in the equivalent form(25)θ¯Y,s=1s∑n=0∞e−2n+1−YB0s/sβ1−as−αγ−e−2n+1+YB0s/sβ1−as−αγ.Moreover, equation (25) can also be expressed as a series approach, allowing us to rationally determine the inverse Laplace transform.(26)θ¯Y,s=1s+∑n=0∞∑m=1∞∑k=0∞Y−2n−1makB0m/2Γγm/2+km!k!Γγm/21sαk−m/2+βm/2+1−∑p=0∞∑n=0∞∑l=0∞−2n−1−YlapB0l/2Γγl/2+pl!p!Γγl/21sαp+βl/2−l/2+1.Taking inverse Laplace transform on equation (26), we get(27)θY,τ=1+∑n=0∞∑m=1∞∑k=0∞Y−2n−1makB0m/2Γγm/2+km!k!Γγm/2ταk+βm/2−m/2Γαk+βm/2−m/2+1−∑n=0∞∑l=0∞∑p=0∞−2n−1−YlB0l/2apΓγl/2+pl!p!Γγl/2ταp−l/2+βl/2Γαp+βl/2−l/2+1.


## 3.2. Solution of Bioconvection Field

By utilizing the Laplace change approach and applying it to equations (13) and (14) with requirements (16) and (17) and utilizing the Prabhakar derivative, we acquire the changed issue for the bioconvection field.(28)LbsN¯Y,s=−∂L¯Y,s∂Y,(29)L¯Y,s=−sβ1−as−αγ∂N¯Y,s∂Y.Using equations (28) and (29), we have(30)∂2N¯Y,s∂Y2−sLbsβ1−as−αγN¯Y,s=0.Subject to the constraints,(31)N¯1,s=1s,N¯0,s=0.The general solution of equation (30) with equation (31) is as follows:(32)N¯Y,s=1s1e−sLb/sβ1−as−αγ−esLb/sβ1−as−αγ/2e−YsLb/sβ1−as−αγ−eYsLb/sβ1−as−αγ2.Or,(33)N¯Y,s=1ssinhYsLb/sβ1−as−αγsinhsLb/sβ1−as−αγ.It is important that equation (33) can be composed within the comparable form(34)N¯Y,s=1s∑d=0∞e−2d+1−YsLb/sβ1−as−αγ−e−2d+1+YsLb/sβ1−as−αγ.Moreover, equation (34) can be communicated in an arrangement shape so that we are able to discover the Laplace inverse transform logically.(35)N¯Y,s=1s+∑d=0∞∑s1=1∞∑s2=0∞−2d−1+Ys1as2Lbs1/2Γγs1/2+s2s1!s2!Γγs1/21sαs2+βs1/2−s1/2+1−∑d=0∞∑p1=0∞∑p2=0∞−2d−1−Yp1ap2Lbp1/2Γγp1/2+p2p1!p2!Γγp1/21sαp2+βp1/2−p1/2+1.Taking inverse Laplace transform on equation (35), we get(36)NY,τ=1+∑d=0∞∑s1=1∞∑s2=0∞−2d−1+Ys1as2Lbs1/2Γγs1/2+s2s1!s2!Γγs1/2ταs2+βs1/2−s1/2Γαs2+βs1/2−s1/2+1−∑d=0∞∑p1=0∞∑p2=0∞−2d−1−Yp1ap2Lbp1/2Γγp1/2+p2p1!p2!Γγp1/2ταp2+βp1/2−p1/2Γαp2+βp1/2−p1/2+1.


## 3.3. Solution of Velocity Field

The Laplace transform is used in equation (10) using expressions from (15)–(17), and we attain(37)∂2∂Y2−B1s−B2V¯Y,s=−B3θ¯Y,s+B4N¯Y,s,which satisfies the following constraints:(38)V¯1,s=0,V¯0,s=0.The solution of equation (37) subject to constraints (38), we have(39)V¯Y,s=−B5s∑n=0∞e−2nB0s/sβ1−as−αγ−e−2n+2B0s/sβ1−as−αγ1+B1s/B2−B0s/B2sβ1−as−αγ×∑m=0∞e−2m+1−YB2+B1s−∑m=0∞e−2m+1+YB2+B1s+B6s∑d=0∞e−2dsLb/sβ1−as−αγ−e−2d+2sLb/sβ1−as−αγ1+B1s/B2−sLb/B2sβ1−as−αγ×∑m=0∞e−2m+1−YB2+B1s−∑m=0∞e−2m+1+YB2+B1s+B5s∑n=0∞e−2n+1−YB0s/sβ1−as−αγ−e−2n+1+YB0s/sβ1−as−αγ1+B1s/B2−B0s/B2sβ1−as−αγ−B6s∑d=0∞e−2d+1−YsLb/sβ1−as−αγ−e−2d+1+YsLb/sβ1−as−αγ1+B1s/B2−sLb/B2sβ1−as−αγ.Equation (39) can be written in component form as follows:(40)V¯Y,s=V¯1Y,s+V¯2Y,s+V¯3Y,s+V¯4Y,s+V¯5Y,s+V¯6Y,s+V¯7Y,s+V¯8Y,s+V¯9Y,s+V¯10Y,s+V¯11Y,s+V¯12Y,s.It is challenging to find the inverse Laplace transform of equation (40), so we can rewrite it in a suitable series.(41)VY,τ=V1Y,τ+V2Y,τ+V3Y,τ+V4Y,τ+V5Y,τ+V6Y,τ+V7Y,τ+V8Y,τ+V9Y,τ+V10Y,τ+V11Y,τ+V12Y,τ.Next, taking the inverse Laplace of equation (41), component wise, we have(42)V1Y,τ=−B5∑n=0∞∑m=0∞∑δ2=0∞∑δ1=0∞∑w2=0∞∑w1=0∞∑z1=0∞∑z2=0∞∑z3=0∞Y−2m−1w1−B1z1B0δ1/2δ1!δ2!w1!w2!z2!z3!B1z2−w2B2w2+z1−w1/2×−2nδ1−B0z2aδ2+z3ταδ2+βδ1/2−δ1/2−w2+αz3+βz2−z1Γαδ2+βδ1/2−δ1/2−w2+αz3+βz2−z1+1Γγδ1/2+δ2Γw1/2+1Γz1+1Γγz2+z3Γγδ1/2Γw1/2+1−w2Γz1+1−z2Γγz2,V2Y,τ=B5∑n=0∞∑m=0∞∑δ1=0∞∑δ2=0∞∑q1=0∞∑q2=0∞∑z1=0∞∑z2=0∞∑z3=0∞−Y+2m+1q1−B1z1B0δ1/2δ1!δ2!q1!q2!z2!z3!B1z2−q2B2q2+z1−q1/2×−2nδ1−B0z2aδ2+z3ταδ2+βδ1/2−δ1/2−q2+αz3+βz2−z1Γαδ2+βδ1/2−δ1/2−q2+αz3+βz2−z1+1Γγδ1/2+δ2Γq1/2+1Γz1+1Γγz2+z3Γγδ1/2Γq1/2+1−q2Γz1+1−z2Γγz2,V3Y,τ=B5∑n=0∞∑m=0∞∑j1=0∞∑j2=0∞∑w1=0∞∑w2=0∞∑z1=0∞∑z2=0∞∑z3=0∞Y−2m−1w1−B1z1Prj1/2j1!j2!w1!w2!z2!z3!B1z2−w2B2w2+z1−w1/2×−2n−2j1−B0z2aj2+z3ταj2+βj1/2−j1/2−w2+αz3+βz2−z1Γαj2+βj1/2−j1/2−w2+αz3+βz2−z1+1Γγj1/2+j2Γw1/2+1Γz1+1Γγz2+z3Γγj1/2Γw1/2+1−w2Γz1+1−z2Γγz2,V4Y,τ=−B5∑n=0∞∑m=0∞∑j1=0∞∑j2=0∞∑q1=0∞∑q2=0∞∑z1=0∞∑z2=0∞∑z3=0∞−Y+2m+1q1−B1z1B0j1/2j1!j2!q1!q2!z2!z3!B1z2−q2B2q2+z1−q1/2×−2n−2j1−B0z2aj2+z3ταj2+βj1/2−j1/2−q2+αz3+βz2−z1Γαj2+βj1/2−j1/2−q2+αz3+βz2−z1+1Γγj1/2+j2Γq1/2+1Γz1+1Γγz2+z3Γγj1/2Γq1/2+1−q2Γz1+1−z2Γγz2,V5Y,τ=B6∑d=0∞∑m=0∞∑n1=0∞∑n2=0∞∑w1=0∞∑w2=0∞∑h1=0∞∑h2=0∞∑h3=0∞Y−2m−1w1−B1h1Lbn1/2n1!n2!w1!w2!h2!h3!B1h2−w2B2w2+h1−w1/2×−2dn1−Lbh2an2+h3ταn2+βn1/2−n1/2−w2+αh3+βh2−h1Γαn2+βn1/2−n1/2−w2+αh3+βh2−h1+1Γγn1/2+n2Γw1/2+1Γh1+1Γγh2+h3Γγn1/2Γw1/2+1−w2Γh1+1−h2Γγh2,V6Y,τ=−B6∑d=0∞∑m=0∞∑n1=0∞∑n2=0∞∑q1=0∞∑q2=0∞∑h1=0∞∑h2=0∞∑h3=0∞−Y+2m+1q1−B1h1Lbn1/2n1!n2!q1!q2!h2!h3!B1h2−q2B2q2+h1−q1/2×−2dn1−Lbh2an2+h3ταn2+βn1/2−n1/2−q2+αh3+βh2−hΓαn2+βn1/2−n1/2−q2+αh3+βh2−h1+1Γγn1/2+n2Γq1/2+1Γz1+1Γγh2+h3Γγn1/2Γq1/2+1−q2Γh1+1−h2Γγz2,V7Y,τ=−B6∑d=0∞∑m=0∞∑m1=0∞∑m2=0∞∑w1=0∞∑w2=0∞∑h1=0∞∑h2=0∞∑h3=0∞Y−2m−1w1−B1h1Lbm1/2m1!m2!w1!w2!h2!h3!B1h2−w2B2w2+h1−w1/2×−2d−2m1−B0h2am2+h3ταm2+βm1/2−m1/2−w2+αh3+βh2−h1αm2+βm1/2−m1/2−w2+αh3+βh2−h1+1Γγm1/2+m2Γw1/2+1Γh1+1Γγh2+h3Γγm1/2Γw1/2+1−w2Γh1+1−h2Γγh2,V8Y,τ=B6∑d=0∞∑m=0∞∑m1=0∞∑m2=0∞∑q1=0∞∑q2=0∞∑h1=0∞∑h2=0∞∑h3=0∞−Y+2m+1q1−B1h1Lbm1/2m1!m2!q1!q2!h2!h3!B1h2−q2B2q2+h1−q1/2×−2d−2m1−Lbh2am2+h3ταm2+βm1/2−m1/2−q2+αh3+βh2−h1Γαm2+βm1/2−m1/2−q2+αh3+βh2−h1+1Γγm1/2+m2Γq1/2+1Γh1+1Γγh2+h3Γγm1/2Γq1/2+1−q2Γh1+1−h2Γγh2,V9Y,τ=B5∑n=0∞∑t1=0∞∑t2=0∞∑z1=0∞∑z2=0∞∑z3=0∞Y−2n−1t1−B1z1B0t1/2t1!t2!z2!z3!B1z2B2z1×−B0z2at2+z3ταt2+βt1/2−t1/2+αz3+βz2−z1Γαt2+βt1/2−t1/2+αz3+βz2−z1+1Γγt1/2+t2Γz1+1Γγz2+z3Γγt1/2Γz1+1−z2Γγz2,V10Y,τ=−B5∑n=0∞∑v1=0∞∑v2=0∞∑z1=0∞∑z2=0∞∑z3=0∞−Y+2n+1v1−B1z1B0v1/2v1!v2!z2!z3!B1z2B2z1×−B0z2av2+z3ταv2+βv1/2−v1/2+αz3+βz2−z1Γαv2+βv1/2−v1/2+αz3+βz2−z1+1Γγv1/2+v2Γz1+1Γγz2+z3Γγv1/2Γz1+1−z2Γγz2,V11Y,τ=−B6∑d=0∞∑l1=0∞∑l2=0∞∑h1=0∞∑h2=0∞∑h3=0∞Y−2d−1l1−B1h1Lbl1/2l1!l2!h2!h3!B1h2B2h1×−Lbh2al2+h3ταl2+βl1/2−l1/2+αh3+βh2−h1Γαl2+βl1/2−l1/2+αh3+βh2−h1+1Γγl1/2+l2Γh1+1Γγh2+h3Γγl1/2Γh1+1−h2Γγh2,V12Y,τ=B6∑d=0∞∑k1=0∞∑k2=0∞∑h1=0∞∑h2=0∞∑h3=0∞−Y+2d+1k1−B1h1Lbk1/2k1!k2!h2!h3!B1h2B2h1×−Lbh2ak2+h3ταk2+βk1/2−k1/2+αh3+βh2−h1Γαk2+βk1/2−k1/2+αh3+βh2−h1+1Γγk1/2+k2Γh1+1Γγh2+h3Γγk1/2Γh1+1−h2Γγh2.


## 4. Graphical Results and Discussion

Bioconvection has been studied using an MHD effect and thermal transfer model with a Prabhakar fractional approach. Laplace transform procedures are used to provide exact solutions for dimensionless governing equations. Graphical illustrations have been used to explain some of the physical effects of flow parameters.Figures2–4 are projected to show the impact of the fractional parameters α, β, γ on bioconvection, temperature, and velocity fields. For a large time, bioconvection, temperature, and velocity decreased by increasing values of α, β, γ. This is due to the boundary layer becoming wider; therefore, bioconvection, temperature, and velocity decrease. Usually, we can say that in fluid dynamics, a fractional approach is better for controlling the boundary layer thickness of the fluid properties.Figure 2
The effects of fractional parameters on bioconvection field, whent=3,Lb=3,anda=0.2.Figure 3
The effects of fractional parameters on temperature field for large time, whent=3,Pr=6.2,anda=0.2.Figure 4
The effects of fractional parameters on velocity field for large time, when.t=2,Pr=6.2,Gr=12,a=0.2, Lb=30,M=0.01,β∗=0.006,ϕhnf=0.04,andRa=4.Figure5 shows the comparison between several base fluids (water, engine oil, and kerosene oil) on the temperature field. It is clearly noticed that the temperature of water is higher than all other Newtonian liquids, such as kerosene and engine oil. Meanwhile, viscosity and Prandtl numbers are very low for water compared to the other two, so water heats up faster than them physically.Figure 5
The comparison between different base fluids (water, kerosene oil, and engine oil) on temperature field for small time, whent=3,α=β=γ=0.5,anda=0.2.To see the impact of bioconvection Rayleigh numberRa on velocity, Figure 6 is plotted. It is proved that velocity near the plate decreases for greater values of Ra. Ra decreases the fluid velocity since the buoyancy influenced by the transference of microorganisms is decreased by Ra. The consequences of the Brinkman parameter on the velocity field will be shown in Figure 7. Velocity decreased as the Brinkman parameter’s value increased. This is because, by increasing the values of the Brinkman parameter, the drag forces become stronger, so velocity reduces.Figure 6
The effects of Rayleigh number on velocity field, when.Pr=6.2,t=5,α=0.5,β=0.5,γ=0.5,Lb=5,Gr=5,β∗=0.006,ϕhnf=0.04,a=0.2,andM=0.2.Figure 7
The effects of Brinkman parameter on velocity field, whenPr=6.2,t=5,α=β=γ=0.5,Lb=30, Gr=12,a=0.2,Ra=4,ϕhnf=0.08,andM=0.01.Figures8 and 9 show the effects of the ϕhnf volume fraction on hybrid nanoparticles. It is discovered that for higher values of ϕhnf and velocity indicated drops, the temperature can be increased. The nanofluid density has important significance in the velocity field. By mixing nanoparticles through base fluid, the consequent hybrid nanofluids improve considerably thicker which decreases velocity and increases temperature.Figure 8
The effects ofϕhnf on temperature field, when Pr=6.2,t=3,α=β=γ=0.5,anda=0.2.Figure 9
The effects ofϕhnf on velocity field, when. Pr=6.2,t=4,α=β=γ=0.5,Lb=30,a=0.2, Gr=15,Ra=4,andM=2.0.In the end, we have presented a comparison between our results and those of Saqib et al. [47]. It is clearly evident that the solution obtained with generalized Mittage–Leffler kernel in the presence of bioconvection shows stronger memory rather than exponential kernel that appeared in Caputo-Fabrizio fractional derivative as presented in Figure 10. It is concluded with the remark that our results can be enhanced in terms of memory.Figure 10
The comparison among our results and Saqib et al. [47] for velocity field, when Pr=6.2,t=2,β=γ=0.5,Lb=10, β∗=0.006,Ra=5,Gr=10,ϕhnf=0.04,andM=0.02.


## 5. Conclusions

The current study investigated bioconvection using a heat transfer and MHD effects model, as well as a Prabhakar fractional approach. Laplace transform techniques are used to provide exact solutions for dimensionless governing equations. Graphical illustrations have been used to explain some of the physical effects of flow parameters. The followings are the significant outcomes:(i)
Obtained solutions are predicted for different values of fractional parameters based on generalized Fourier’s law are responsible to attain better memory instead of artificial replacement(ii)
For a large time, fluid properties such as temperature, bioconvection, and velocity depict history/memory.(iii)
The temperature of water base nanoparticles is comparatively higher than kerosene and engine oil.(iv)
Bioconvection Rayleigh is responsible for the rapid decline in the momentum equation.(v)
The obtained solutions can be beneficial for proper analysis of real data and provide a tool for testing possible approximate solutions where needed.For the future direction of readers, this work can be extended to include a large class of fluids of non-Newtonian nature and different thermal and mechanical boundary conditions. Also, you can extend this work with the fuzzy boundary conditions.



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