The steady magnetohydrodynamic ternary hybrid nanofluid flow over a slender surface under the effects of activation energy, Hall current, chemical reactions, and a heat source has been reported. A numerical model is developed to increase the rate of energy transfer and boost the efficiency and outcome of heat energy dissemination for a diverse range of biological applications and commercial uses. The rheological properties and thermal conductivity of the base fluids are improved by framing an accurate combination of nanoparticles (NPs). The ternary hybrid nanofluid has been prepared, in the current analysis, by the dispersion of magnesium oxide, titanium dioxide (TiO2), and cobalt ferrite (CoFe2O4) NPs in the base fluid. The physical phenomena have been expressed in the form of a system of nonlinear PDEs, which are degraded to a dimensionless system of ODEs through the similarity replacement and numerically solved by employing the MATLAB software package bvp4c. The graphical and tabular results are estimated for velocity, mass, and energy curves vs distinct physical factors. It has been noticed that the variation in the magnetic effect enhances the energy profile while the increasing number of ternary nanocomposites (MgO, TiO2, and CoFe2O4) in water lowers the energy curve. Furthermore, the effect of both Lewis and Peclet numbers weakens the motile microbe’s profile.

The analysis of nanoliquid flow across a slender stretching sheet has abundant uses in several sectors of manufacturing and engineering, such as in polymer manufacturing, aerodynamics, extrusion of polymer, glass and fiber production, and metallic furnaces.1 Hayat et al.2 evaluated the non-linear dual stratified impact of the magnetohydrodynamic (MHD) flow of nanoliquid across an enclosed surface. It was found that the upshot velocity ratio factor results in the augmentation of the velocity field. Sharma et al.3 addressed the variable physical characteristics of viscoelastic fluid flow past an slender elongating surface, such as specific viscosity and thermal conductivity. Hou et al.4 calculated the flow of a hybridized fluid along a slender substrate. It was discovered that advanced values of the Hartmann number and the wall thickness factor generate a retardation impact, resulting in a decrease in fluid velocity for SWCNT and MWCNT hybrid nanofluid. Bilal et al.5 scrutinized the flow features of mixed convection, Darcy nanocomposite flow along a porous slender stretched surface. The thermal energy outline is magnified by increasing the electric current, and the mass transmission coefficient increases with the influence of potential energy. Elattar et al.6 documented the flow of a consistent electrically charged nanofluid flow over an elongating slender texture in the presence of varying magnetic effects, energy dissipation, and chemical reactions. The influence of the Hall current was observed to improve the axial streamlines. Saravana et al.7 documented the MHD Casson fluid motion across an extended sheet and evaluated the results of Sherwood and Nusselt numbers. Murtaza et al.8 observed the stream of second-grade nanoliquid across flexible substrates in a Riga plate with varying thicknesses and determined that the characteristics of the buoyancy forces exhibit an adversarial behavior on the velocity contour. Siddique et al.9 reviewed the effect of multiple slips on the MHD bioconvection of a hydromagnetic-based nanoliquid along a permeable medium due to an enlarging plate. It was discovered that increasing the porosity and magnetic factors enhances the intensity of the skin friction. Recently, many researchers have studied the fluid flow across a slender extending surface.10–13 

Hybrid nanofluid and ternary hybrid nanofluid are advanced types of nanocomposite-based fluid, which perform well in energy transmission. Solar energy, heat converters, heat pumps, automobile engineering, electrical coolers, generators, emitting radiation systems, transmitters, biotechnology, and ship manufacturing are a few sectors where hybrid nanofluids are applicable.14–16 Sundar et al.17 assessed the heat entropy, energy efficiency, and frictional entropy of a graphene oxide-based hybrid nanoliquid flow through a capillary tube and laminar flow. Xuan et al.18 experimentally examined the thermal efficiency of the TiO2–Al2O3–Cu/water trihybrid nanoliquid. Economic findings indicated that the most effective work liquids in the turbulent and laminar flow are TiO2–Cu–Al2O3/water trihybrid nanoliquids with 1 and 0.7 vol. %. Bilal et al.19 revealed the characteristics of electroviscous, trihybrid nanoliquid containing TiO2, Al2O3, and SiO2 nanoparticles (NPs) flowing through simultaneous infinite sheets. The energy and velocity transit rates are observed to increase with the addition of ternary NPs to conventional fluids. Animasaun et al.20 defined the effects of magnetic flux on the water-based ternary hybrid nanofluid transmitting silver, aluminum, and aluminum oxide nanomaterials of different patterns on a flat plane reflect. Fattahi et al.21 used a trihybrid nanoliquid (Fe3O4-sAl2O3–ZnO) to quantitatively assess a solar collector with a surface covering. It was discovered that such a coating could reduce the drag coefficient by about 45% whereas only a minor decrease in the Nusselt number (4.5%) is observed. Zahan et al.22 investigated the performance of ternary-nano-particulates [zinc (Zn), silver (Ag), and cobalt (Co)] through a converging–diverging (CD) injector. It was discovered that the Ag–Co–Zn nanoliquid has the maximum heat transition rate. Sarada et al.23 analyzed the thermal convection boundary layer circulation of a trihybrid nanoliquid (CNT–silver/water) flow across a contoured shrinking sheet with chemical potential. Xiu et al.24 studied the energy maintenance and energy distribution regulation using a water-based ternary nanoliquid (Al, MgO, and TiO2) on an elongating wedge substrate. Goud et al.25 investigated the thermodynamic variability in a joinery fin under slip conditions with trihybrid nanoliquid, taking the humidity ratio and temperature differences as driving pressures in mass and heat mechanisms. Adnan and Ashraf26 observed trihybrid nanofluids (CuO–Cu–Al2O3/H2O) for thermal energy storage with the innovative impact of convective heating as well as consistent magnetic field and deduced that all these fluids are indeed very valuable for commercial operations, particularly detoxification, or where mega heat transfer is necessary to achieve the workflow. Revathi et al.27 used the bvp4c package to quantitatively inspect the characteristics of TiO2 nanoparticles and the applicability of contaminant reactive filtered water due to bioconvection in industry sectors. Gupta et al.28 evaluated the 3D magneto unsteady flow of trihybrid nanoliquid induced by a spirally enlarged infinite disk under numerous slip conditions. Recently, many researchers worked on the ternary nanofluid flow.29–32 

Micro-organisms are mono-celled microbes that are found in all living things. Because they are much heavier than base fluids, microbes become a source of bioconvection. Oxytocic microbes that swim depict the bioconvection characteristic. The bioconvection framework is concerned with swimming cells that are connected to micro-organisms. The physical significance of bioconvection is effectively distributed in ethanol, biofuel, and a wide range of commercial and environmental systems. When gyrotactic microbes are involved, nanomaterial suspension stability is frequently found to be substantially enhanced. In recent decades, many authors have written about these captivating concepts.33–35 Atif et al.36 numerically analyzed the tangent hyperbolic nanocomposite fluid flow passing through a wedge-shaped exterior filled with gyrotactic microbes. Shukla et al.37 estimated the bioconvection nanoliquid flow using the homotopy analysis method. The calculations show that increasing the magnetic force and bioconvection Rayleigh number slows the flow while increasing the radiation parameter factor accelerates it. Ali et al.38 defined the impact of heat absorption and generation on the Casson nanoliquid flow comprising gyrotactic microbes, with energy and mass transmission happening through parallel plates.

We have already discussed that the ternary nanofluid flow consists of ternary nanocomposites (MgO, TiO2, and CoFe2O4) and motile microbes have several practical applications in different sectors of engineering and biomedical sciences. However, for literature review, we have concluded that very less effort has been made so far on such types of physical phenomena under the significance of activation energy, the Hall effect, and chemical reactions. Therefore, we have numerically estimated the steady MHD ternary hybrid nanoliquid flow across a slender extending sheet under the effect of variable magnetic fields, activation energy, Hall current, chemical reactions, and a heat source. A numerical model is developed to increase the rate of energy transfer and boost the efficiency and outcome of heat energy dissemination for a diverse range of biological applications and commercial uses. The physical phenomena have been expressed in the form of a system of nonlinear partial differential equations, which are numerically solved by employing the MATLAB software package bvp4c.

We examined steady and incompressible 2D MHD nanofluid flow over a slender stretching sheet. The surface of the sheet is variable. In the axial (x‐axis) direction, the sheet stretches with velocity Uwx=U0x+bn, while the y‐axis is normal to the sheet surface, as depicted in Fig. 1. Here, n is the power index. The non-uniformity of the surface is assumed as y=Ax+b1n2, in which A is the stretching coefficient and is assumed to be negligible. The Hall current is functional for trihybrid nanoliquid circulation. Furthermore, the variable magnetic effect is applied in the y-direction. The basic governing equations are specified as39 

ux+vy=0,
(1)
ρtnfuux+vuy=μtnf2uy2σtnf1+m2B2xu+mw,
(2)
ρtnfuwx+vwy=μtnf2wy2σtnf1+m2B2xmuw,
(3)
uTx+vTy=ktnfρCptnf2Ty2+τtnfDBTyCy+DTTTy2+Q0TTρCptnf,
(4)
uCx+vCy=DB2Cy2+DTT2Ty2kr2CC0TTnexpEaκT,
(5)
uNx+vNy=DN2Ny2+dWcCwCNCy.
(6)
FIG. 1.

Ternary hybrid nanofluid flow in a slender stretching sheet.

FIG. 1.

Ternary hybrid nanofluid flow in a slender stretching sheet.

Close modal

Here, m = τewe is the Hall current, u,v,w are the velocity components, Q0 is the heat source factor, Ea is the activation energy, and kr2 is the second-order chemical reaction.

The initial and boundary conditions are

u=U0x+bn=Uwx,v=0,w=0,DBCy+DTTTy=0,T=Tw,N=Nwaty=Ax+b1n2,u0,CC,w0,TT,Nasy.
(7)

The transformation variables are

η=yn+12U0νfx+bn1,ψ=2n+1νfU0x+bn+1fη,(η)=NNNwN,θη=TTTwT,w=U0x+bnhη,φη=CCCwC.
(8)

By including Eq. (8) in Eqs. (1)(7), we get

f+ϑ1ϑ2ff2nn+1f2ϑ3ϑ12Mn+11+m2f+mg=0,
(9)
g+ϑ1ϑ2fg2nn+1gfϑ3ϑ12Mn+11+m2mfg=0,
(10)
θ+Prϑ4ϑ5fθ+Nbφθ+Ntθ2+Q1θ=0,
(11)
φ+NtNbθ+LefφScσ1+εϑnφexpE1+εϑ=0.
(12)
+PrLbfPeϕ+Ωϕ+ϕ=0.
(13)

Here,

ϑ1=ρtnfρf,ϑ2=μtnfμf,ϑ3=σtnfσf,ϑ4=ρCptnfρCpf,ϑ5=ktnfkf.
(14)

The transformation conditions are

fη=η1n1+n,Nbφη+Ntθη=0,fη=1,gη=0,η=1,θη=1,f0,g0,θ0,φ0,(η)0asη.
(15)

Here, M=B02σfρfT is the magnetic field, Kr=KcDB is the chemical reaction, Pr=μfρCpfρfkf is the Prandtl number, Nt=τDTTwTνfT is the thermophoresis effect, Nb=τDBCνf is the Brownian motion, Q1=xQ0ρCp is the heat source, Le=νfDB is the Lewis number, δ=An+12U0νf is the wall thickness, Pe=dWcDN is the Peclet number and Lb=kDN is the Lewis number.

The skin friction, Nusselt number, and Sherwood number are

Cfx=2τw1Uw2ρf,Cfz=τw2Uw2ρf,Nu=qwx+bTwTkf,Sh=jwx+bCwCDB,
(16)

where

τw1=μtnfuyy=Ax+b1n2,τw2=μtnfvyy=Ax+b1n2,qw=ktnfTyy=Ax+b1n2,jw=DBCzy=Ax+b1n2.
(17)

The dimensionless form of Eq. (14) is

Cfrx=RexCfx=1ϕ12.51ϕ22.51ϕ32.52n+1f0,Cfrz=RexCfz=1ϕ12.51ϕ22.51ϕ32.52n+1g0,Nur=NuRex=ktnfkfn+12θ(0),Shr=ShRex=n+12φ(0).
(18)

The obtained set of ODEs Eqs. (9)(13) and Eq. (15) is numerically computed through the MATLAB package bvp4c as40,41

1=f(η),3=f(η),5=g(η),7=θ(η),9=φ(η),11=(η),2=f(η),4=g(η),6(η)=θ(η),8=φ(η),10=(η).
(19)

By putting (19) in Eqs. (9)(13) and Eq. (15), we get

ℏ̸3+ϑ1ϑ2ℏ̸1ℏ̸32nn+1ℏ̸22ϑ3ϑ12Mn+11+m2ℏ̸2+mℏ̸4=0,
(20)
ℏ̸5+ϑ1ϑ2ℏ̸1ℏ̸52nn+1ℏ̸4ℏ̸2ϑ3ϑ12Mn+11+m2mℏ̸2ℏ̸4=0,
(21)
ℏ̸7+Prϑ4ϑ5ℏ̸1ℏ̸7+Nbℏ̸9ℏ̸7+Ntℏ̸72+Q1ℏ̸6=0,
(22)
ℏ̸9+NtNbℏ̸7+Leℏ̸1ℏ̸9Scσ1+εϑnℏ̸8expE1+εϑ=0,
(23)
11+PrLbf10Pe119+Ω9+910=0.
(24)

The transformation conditions are

ℏ̸1η=η1n1+n,Nbℏ̸9η+Ntℏ̸7η=0,ℏ̸2η=1,ℏ̸4η=0,ℏ̸6=1,ℏ̸20,ℏ̸40,ℏ̸60,ℏ̸80asη.
(25)

The physical trend and reason behind each graphical result have been described in this segment.

Velocity profile fη:

Figures 25 illustrate the arrangement of the velocity curve fη vs parameters m, n,δ, and ϕ, respectively. Figures 2 and 3 revealed that the velocity curve fη augments with the upshot of hall current m and the power index of velocity n. Hall current (discovered by Edwin Hall in 1879) is the generation of the difference in voltage through a conductor that is placed transversely to the electric current and perpendicular to the magnetic field. That is why the implication of the Hall effect magnifies the velocity curve, as shown in Fig. 2.

FIG. 2.

Arrangement of the velocity curve fηversus the Hall current m.

FIG. 2.

Arrangement of the velocity curve fηversus the Hall current m.

Close modal
FIG. 3.

Arrangement of the velocity curve fηversus the velocity power index n.

FIG. 3.

Arrangement of the velocity curve fηversus the velocity power index n.

Close modal
FIG. 4.

Arrangement of the velocity curve fηversus the wall thickness δ.

FIG. 4.

Arrangement of the velocity curve fηversus the wall thickness δ.

Close modal
FIG. 5.

Arrangement of the velocity curve fηversus the volume fraction of nanoparticles ϕ.

FIG. 5.

Arrangement of the velocity curve fηversus the volume fraction of nanoparticles ϕ.

Close modal

Figures 4 and 5 demonstrate that the upshot of δ and the nanoparticle coefficient ϕ decrease the velocity outline of trihybrid nanoliquid. The increasing number of ternary nanocomposites (MgO, TiO2, and CoFe2O4) in the base fluid makes them denser as well as improves their viscosity, which results in the decrease in the velocity curve.

Energy profile:

Figures 613 show the nature of the energy curve θη vs parameters m, n, Q, δ, Nt, Nb, M, and ϕ, respectively. Figures 6 and 7 show that the energy framework θη augments with the outcome of m and n, respectively. Physically, the voltage difference occurs due to the upshot of the Hall effect, which generates some heat and causes the elevation of the temperature curve, as shown in Fig. 6. Figures 7 and 8 show that the energy field is boosted with the significance of the heat source, which declines with the variation in δ. The influence of the heat generation term also provides additional heat to the trihybrid nanoliquid, which causes the enhancement of energy outline, as shown in Fig. 7. Physically, the variation in wall thickness generates retardation, which causes friction and enhances the energy profile (Fig. 9).

FIG. 6.

Arrangement of the energy curve θη vs the Hall current m.

FIG. 6.

Arrangement of the energy curve θη vs the Hall current m.

Close modal
FIG. 7.

Arrangement of the energy curve θη vs the power index n.

FIG. 7.

Arrangement of the energy curve θη vs the power index n.

Close modal
FIG. 8.

Arrangement of the energy curve θη vs the heat source Q.

FIG. 8.

Arrangement of the energy curve θη vs the heat source Q.

Close modal
FIG. 9.

Arrangement of the energy curve θη vs the wall thickness δ.

FIG. 9.

Arrangement of the energy curve θη vs the wall thickness δ.

Close modal
FIG. 10.

Arrangement of the energy curve θη vs the thermodiffusion Nt.

FIG. 10.

Arrangement of the energy curve θη vs the thermodiffusion Nt.

Close modal
FIG. 11.

Arrangement of the energy curve θη vs the Brownian motion Nb.

FIG. 11.

Arrangement of the energy curve θη vs the Brownian motion Nb.

Close modal
FIG. 12.

Arrangement of the energy curve θη vs the magnetic field M.

FIG. 12.

Arrangement of the energy curve θη vs the magnetic field M.

Close modal
FIG. 13.

Arrangement of the energy curve θη vs the volume fraction of nanoparticles ϕ.

FIG. 13.

Arrangement of the energy curve θη vs the volume fraction of nanoparticles ϕ.

Close modal

Figures 10 and 11 show the exhibition of the energy curve vs the effect of Nb and thermodiffusion. The influence of both parameters drops the energy curve. Consistently, the Brownian moment of particles evenly distributes the heat inside the fluid, which decreases the average temperature of the fluid [Fig. 10]. Thermophoresis is a mechanism noted in the mixing of fluid particles, where distinct nano-size particles respond differently to the applied temperature. Hence, thermodiffusion tends to interchange the heavy particles to the light molecule region and vice versa. Thus, the average temperature of trihybrid nanoliquid decreases as a result of the thermophoresis effect, as shown in Fig. 11. Figures 12 and 13 emphasize the appearance of the energy curve θη vs M and increasing number of nanoparticles ϕ. The opposing force generated due to the magnetic impact retards the fluid motion, which sources the enhancement of the energy curve, as shown in Fig. 12. As discussed earlier, the increasing number of ternary nanocomposites (MgO, TiO2, and CoFe2O4) in the base fluid makes them denser as well as improves its viscosity, which consequently declines the energy curve.

Concentration and micro-organism profiles:

Figures 1416 show the production of mass outline φη vs parameters Kr, Sc, and E, respectively. Figures 14 and 15 show that the mass curve declines with the variation in both Sc and Kr. The diffusion rate of fluid molecules slows down with the effect of Kr and Sc, while the viscosity of trihybrid nanoliquid is augmented, which results in a decrease in the concentration outline. Figure 16 illustrates that the upshot of activation energy boosts the rate of mass outline. The activation energy is identified as the small energy used to invigorate or stimulate fluid atoms and trihybrid nanocomposites to contribute to a chemical change or renovation. That is why the influence of activation energy enhances the mass transmission rate, as shown in Fig. 16.

FIG. 14.

Arrangement of mass φη outline vs the chemical reaction term Kr.

FIG. 14.

Arrangement of mass φη outline vs the chemical reaction term Kr.

Close modal
FIG. 15.

Arrangement of mass φη outline vs the Schmidt number Sc.

FIG. 15.

Arrangement of mass φη outline vs the Schmidt number Sc.

Close modal
FIG. 16.

Arrangement of mass φη outline vs the activation energy E.

FIG. 16.

Arrangement of mass φη outline vs the activation energy E.

Close modal

Figures 17 and 18 show the motile contour φη vs parameters Lb and Pe, respectively. The upshot of both Lb and Pe factors diminishes the motile microbe’s curve. Physically, the density of motile microbes decreases with the influence of Lb and Pe, which results in the lessening of the motile microbe’s boundary layer.

FIG. 17.

Arrangement of motile microbes η outline vs the Lb.

FIG. 17.

Arrangement of motile microbes η outline vs the Lb.

Close modal
FIG. 18.

Arrangement of motile microbes η outline vs the Peclet number Pe.

FIG. 18.

Arrangement of motile microbes η outline vs the Peclet number Pe.

Close modal

Tables I and II show the numerical values that are experimentally derived for ternary nanocomposites (MgO, TiO2, and CoFe2O4) and the mathematical model used for the estimation of the results. Table III described the relative valuation of the analytic technique (HAM) and the bvp4c package. Table IV displays the arithmetical calculations of MgO, TiO2, and CoFe2O4 ternary hybrid nanoliquid for Nur, Shr, and Cfx,Cfz. It is detected that the consequence of the Hall effect enhances the energy passage rate and drag force while the wall thickness factor demonstrates the opposite setup.

Table I.

The experimental values of TiO2, CoFe2O4, MgO, and water.6 

Base fluid and Nanoparticlesρ (kg/m3)k (W/mK)Cp (j/kg K)σ (S/m)β×105K1
Pure water, H2997.1 0.613 4179 0.05 21 
Magnesium oxide, MgO 3560 45 955 1.42 × 10−3 1.26 
Titanium dioxide, TiO2 4250 8.9538 686.2 2.38 × 106 0.9 
Cobalt ferrite, CoFe2O4 4907 3.7 700 5.51 × 109  
Base fluid and Nanoparticlesρ (kg/m3)k (W/mK)Cp (j/kg K)σ (S/m)β×105K1
Pure water, H2997.1 0.613 4179 0.05 21 
Magnesium oxide, MgO 3560 45 955 1.42 × 10−3 1.26 
Titanium dioxide, TiO2 4250 8.9538 686.2 2.38 × 106 0.9 
Cobalt ferrite, CoFe2O4 4907 3.7 700 5.51 × 109  
TABLE II.

The physical model for trihybrid nanoliquid.6 

Viscosityμtnfμf=1(1ϕMgO)2.5(1ϕTiO2)2.5(1ϕCoFe2O4)2.5
Density ρtnfρf=1ϕTiO21ϕTiO21ϕCoFe2O4+ϕCoFe2O4ρCoFe2O4ρf+ϕTiO2ρTiO2ρf+ϕMgOρMgOρf 
Specific heat (ρcp)tnfρcpf=ϕMgOρcpMgOρcpf+1ϕMgO1ϕTiO21ϕCoFe2O4+ϕCoFe2O4ρcpCoFe2O4ρcpf+ϕTiO2ρcpTiO2ρcpf 
Thermal conduction ktnfkhnf=kCoFe2O4+2khnf2ϕCoFe2O4khnfkCoFe2O4kCoFe2O4+2khnf+ϕCoFe2O4khnfkCoFe2O4,khnfknf=kTiO2+2knf2ϕTiO2knfkTiO2kTiO2+2knf+ϕTiO2knfkTiO2,knfkf=kMgO+2kf2ϕMgOkfkMgOkMgO+2kf+ϕMgOkfkMgO, 
Electrical conductivity σtnfσhnf=1+3σCoFe2O4σhnf1ϕCoFe2O4σCoFe2O4σhnf+2σCoFe2O4σhnf1ϕCoFe2O4,σhnfσnf=1+3σTiO2σnf1ϕTiO2σTiO2σnf+2σTiO2σnf1ϕTiO2,σnfσf=1+3σMgOσf1ϕMgOσMgOσf+2σMgOσf1ϕMgO 
Viscosityμtnfμf=1(1ϕMgO)2.5(1ϕTiO2)2.5(1ϕCoFe2O4)2.5
Density ρtnfρf=1ϕTiO21ϕTiO21ϕCoFe2O4+ϕCoFe2O4ρCoFe2O4ρf+ϕTiO2ρTiO2ρf+ϕMgOρMgOρf 
Specific heat (ρcp)tnfρcpf=ϕMgOρcpMgOρcpf+1ϕMgO1ϕTiO21ϕCoFe2O4+ϕCoFe2O4ρcpCoFe2O4ρcpf+ϕTiO2ρcpTiO2ρcpf 
Thermal conduction ktnfkhnf=kCoFe2O4+2khnf2ϕCoFe2O4khnfkCoFe2O4kCoFe2O4+2khnf+ϕCoFe2O4khnfkCoFe2O4,khnfknf=kTiO2+2knf2ϕTiO2knfkTiO2kTiO2+2knf+ϕTiO2knfkTiO2,knfkf=kMgO+2kf2ϕMgOkfkMgOkMgO+2kf+ϕMgOkfkMgO, 
Electrical conductivity σtnfσhnf=1+3σCoFe2O4σhnf1ϕCoFe2O4σCoFe2O4σhnf+2σCoFe2O4σhnf1ϕCoFe2O4,σhnfσnf=1+3σTiO2σnf1ϕTiO2σTiO2σnf+2σTiO2σnf1ϕTiO2,σnfσf=1+3σMgOσf1ϕMgOσMgOσf+2σMgOσf1ϕMgO 
TABLE III.

Quantitative results for the skin friction.

HAMbvp4c
n Fang et al.42f″(0) Khader et al.43f″(0) Present work −f″(0) Present work −f″(0) 
0.5 0.0000 0.000 000 0.000 000 0.000 000 
1.0 0.0224 0.022 410 0.022 511 0.022 513 
1.5 0.0359 0.035 871 0.035 970 0.035 987 
2.0 0.0486 0.048 615 0.048 514 0.048 532 
2.5 0.0550 0.055 049 0.055 248 0.055 242 
3.0 0.0589 0.058 920 0.058 941 0.058 941 
4.0 0.0603 0.060 329 0.060428 0.060453 
HAMbvp4c
n Fang et al.42f″(0) Khader et al.43f″(0) Present work −f″(0) Present work −f″(0) 
0.5 0.0000 0.000 000 0.000 000 0.000 000 
1.0 0.0224 0.022 410 0.022 511 0.022 513 
1.5 0.0359 0.035 871 0.035 970 0.035 987 
2.0 0.0486 0.048 615 0.048 514 0.048 532 
2.5 0.0550 0.055 049 0.055 248 0.055 242 
3.0 0.0589 0.058 920 0.058 941 0.058 941 
4.0 0.0603 0.060 329 0.060428 0.060453 
TABLE IV.

Quantitative results for the Nusselt number Nur, the Sherwood number Shr, and Cfx,Cfz.

mδNCfxCfzNurShr
0.1   −1.689 774 0.518 366 0.425 168 2.371 833 
0.3   −1.286 626 0.785 571 0.459 010 2.234 450 
0.5   −0.965 863 0.826 486 0.489 551 2.104 660 
0.7   −0.746 347 0.865 091 0.512 454 2.006 295 
0.9   −0.596 955 0.891 046 0.529 021 1.935 581 
 0.1  −1.509 797 0.520 455 0.282 621 0.923 908 
 0.2  −1.689 774 0.518 366 0.625 168 2.371 833 
 0.3  −1.877 941 0.514 771 0.801 780 6.058 712 
 0.4  −2.074 212 0.509 746 1.403 376 10.311 253 
 0.5  −2.278 456 0.503 392 1.823 850 15.479 145 
  0.2 −1.689 774 0.518 366 0.425 168 4.271 832 
  0.4 −2.214 652 0.597 394 0.353 301 2.281 254 
  0.6 −2.686 004 0.667 703 0.365 828 1.374 133 
  0.8 −3.116 159 0.731 883 0.322 737 0.877 460 
  1.0 −3.513 877 0.791 370 0.306 128 0.372 586 
mδNCfxCfzNurShr
0.1   −1.689 774 0.518 366 0.425 168 2.371 833 
0.3   −1.286 626 0.785 571 0.459 010 2.234 450 
0.5   −0.965 863 0.826 486 0.489 551 2.104 660 
0.7   −0.746 347 0.865 091 0.512 454 2.006 295 
0.9   −0.596 955 0.891 046 0.529 021 1.935 581 
 0.1  −1.509 797 0.520 455 0.282 621 0.923 908 
 0.2  −1.689 774 0.518 366 0.625 168 2.371 833 
 0.3  −1.877 941 0.514 771 0.801 780 6.058 712 
 0.4  −2.074 212 0.509 746 1.403 376 10.311 253 
 0.5  −2.278 456 0.503 392 1.823 850 15.479 145 
  0.2 −1.689 774 0.518 366 0.425 168 4.271 832 
  0.4 −2.214 652 0.597 394 0.353 301 2.281 254 
  0.6 −2.686 004 0.667 703 0.365 828 1.374 133 
  0.8 −3.116 159 0.731 883 0.322 737 0.877 460 
  1.0 −3.513 877 0.791 370 0.306 128 0.372 586 

This study described the steady MHD ternary hybrid nanoliquid flow across a slender surface under the consequences of activation energy, Hall current, chemical reactions, and a heat source. The ternary HNF is prepared by the dispersion of MgO, TiO2, and CoFe2O4 (nps) in water. The physical phenomena have been expressed in the form of a nonlinear system of PDEs. These are degraded to the dimensionless system of ODEs through the similarity replacement and numerically solved by employing the MATLAB software package bvp4c. The graphical and tabular results are estimated for velocity, mass, and energy curves vs distinct physical factors. The key findings are as follows:

  • The velocity curve fη augments with the upshot of the hall current and power index of velocity n, while the influence of the wall thickness factor δ and the nanoparticle coefficient ϕ decrease the velocity outline of trihybrid nanoliquid.

  • The energy outline θη augments with the effect of Hall current, velocity index n, and heat source while it declines with the variation in the wall thickness factor δ.

  • The temperature outline declines with the effect of Brownian motion and thermodiffusion.

  • The variation in the magnetic effect enhances the energy profile, but the increasing number of ternary nanocomposites (MgO, TiO2, and CoFe2O4) worsens it.

  • The mass profile φη decays with the variation in both the Schmidt number and chemical reaction and boosts with the action of activation energy.

  • The upshot of both Lb and Pe factors diminishes the motile microbe’s curve.

The author(s) received no specific funding for this study.

The authors have no conflicts to disclose.

Fayza Abdel Aziz Elsebaee: Software (equal). Muhammad Bilal: Writing – original draft (equal). Samy Refahy Mahmuod: Validation (equal). Mohammed Balubaid: Conceptualization (equal). Muhammad Shuaib: Methodology (equal). Joshua K. K. Asamoah: Funding acquisition (equal); Writing – review & editing (equal). Aatif Ali: Writing – review & editing (lead).

The data that support the findings of this study are available within the article.

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