In laboratories worldwide, researchers are investigating a wide range of possible applications for nanofluids. These applications include mineral oils, water, solar energy, and microelectronics. It is possible that using chemotherapy in conjunction with nanoparticles will result in the death of cancer cells. The operation of the thermal extrusion mechanism is not due to the presence of low-energy reservoirs but rather to the process by which nanoparticles are produced. Thermo-physical properties of nano-liquids, such as thermal infusibility and thermal conductivity, are among the most advanced in the industry, which could benefit various industries. It is possible to use nanomaterials in multiple applications, such as cooling of engines, processing of pharmaceuticals, operation of fuel cells, and function of residential cooling systems. Large-capacity cooling systems are now feasible thanks to the energy-saving characteristics of nano-liquids. A nano liquid-based smart material has the potential to regulate the flow of heat and act as a heat valve. In addition to these applications, there are potential uses of nano-liquids in geothermal energy production and fuel production for nuclear reactors. Both the minimum quantity cooling lubrication (MQCL) and the MQCG nanofluid technologies benefit the environment due to their reduced reliance on oil and grease. They not only cool and lubricate but also eliminate the need for cutting fluids. Choi and Eastman (1995) developed the concept of nanofluids. Nanofluids are characterized by nanoparticles suspended in liquids. The behavior of a nano-liquid can be significantly altered depending on the temperature and viscosity of the fluid’s boundary layer. Khan et al. (2021a) discussed the role of gold nanoparticles in Sisko fluid under the impacts of thermal radiation and Lorentz force using slip conditions toward a curved surface. Khan et al. (2021b) derived mono and hybrid nanomaterials in motion at the stagnation point utilizing characteristics of the microstructure of moveable frames via non-isothermal condition. Khan et al. (2022a) studied various impacts of stagnation point characteristics in thermal and mass diffusion fields under the role of magnetic fields containing graphene oxide and water nanoparticles on the surface. Khan et al. (2022b) have driven multiple features of AA7075 and AA7072 nanoparticles within heat transfer on a cylinder utilizing slip conditions. Akbar et al. (2022) described the heat transfer mechanism of unsteady peristaltic liquid, including variable thermal properties, via the exact solution approach. Akram et al. (2022a) modeled the flow of peristaltic propulsion in the motion of TiO₂/10W40 nanoparticles using mechanisms of electro-osmotics in a curved microchannel. Maraj et al. (2017) estimated the consequences of Lorentz force and thermal deposition in a vertical channel filled with carbon nanotubule (CNT) nanoparticles. Akram et al. (2022b) modeled a flow problem based on peristaltic transport that involved adding a mixture of nanoparticles considering Lorentz force in aqueous media. Shafee et al. (2021) developed a non-equilibrium theory that sheds light on how the shape factor, Lorentz force, and radiation term influence heat transfer and behavior in nanomaterials. Akbar et al. (2019) analyzed features of ethylene glycol in thermal transfer when inserting suspensions of different shaped nanoparticles in a vertical channel. Habib and Akbar (2021) investigated the use of dispersed nanoparticles to combat Staphylococcus aureus in clinical isolates.

When electrokinetic Jeffrey fluid and peristalsis are combined, a phenomenon known as the Soret–Dufour cross-diffusion effect occurs. When scientists are trying to figure out how streamlines move, they frequently consider the “trapping theory” as one of the possible explanations. Peristalsis is defined as a wave-like movement of fluids through a channel, which happens naturally in the body. The production of heat is caused by the passage of a chemically reactive liquid through a porous medium with the assistance of a semi-infinite vertical permeable plate, which also contributes to the movement of the liquid. Hall currents and a liquid called Jeffrey liquid have been studied by Babu et al. (2020). Ali et al. (2020) researched how heat and thermal energy moved through the stretching cylinder in Jeffrey fluids. During an experiment conducted by Aleem and Alex, there was a consistent flow of Jeffrey fluid between two hot plates. The existence of a strong magnetic field piqued their interest in this subject matter. According to Saif et al. (2020), the mechanisms that control the flow of magnetic hydrodynamic (MHD) fluid and heat transfer are governed by the curvature of the stretching surface. Research conducted by Gireesha et al. (2020) looked at the three-dimensional flow of radiating Jeffrey liquids on a stretched surface. According to Manjunatha et al. (2020), the effects of heat and solutes on peristaltic flow of Jeffrey fluid are not uniformly distributed throughout the system. This plan defies logic in every way. Sinha et al. (2020) investigated the effects of thermal radiation and Hall current on a nanofluid. Use of a thermal tube that vibrated and emitted moving ultrasound waves was required for that method. The effect of a specific variable electrical field on the flow of Jeffrey fluid is illustrated in Haroun (2020). As part of their experiment, they observed the non-steady and hydromagnetic flow using a flat plate and a magnetic field to observe the flow of the electrical conductor but incompressible fluid. In a porous Darcy-type medium, a third-grade nano-liquid that is optically dense and electrically conductive transfers heat to a porous surface via a Lie symmetry mechanism. This surface is located in the middle of the medium. This nano-liquid has a high density when measured in light and electricity. Maraj et al. (2022) discussed the results of rotational flow in a channel filled with hybrid nanoparticles under the impact of Lorentz forces using slip conditions. Akram et al. (2021) estimated theoretical results of thermal transfer with a mixture of hybrid nanoparticles and base fluid (water) in microchannels via electroosmotic pumping.

Bioconvection makes the growth process easier for bacteria and algae suspended in water. Microorganisms make the process of bioconvection possible. Microbes capable of bioconvection rise to the surface because they are 5–10 percent denser than water. They are referred to as “bioconvective” because of this characteristic. The action of these microorganisms causes the primary fluid to become thicker as a side effect. Examples of bioconvection can be observed in a wide variety of organic applications and microsystems, pharmaceuticals, biopolymers, environmentally friendly applications, developments in the utilization of cost-effective energy sources, microbial advanced oil recovery, biosensors and biotechnology, and continuous numerical display. One industry that uses bioconvection is the continuous numerical display (CND) industry. In their analysis, they examined MHD flow of nanofluids in the vicinity of a stretching surface, taking both velocity slip and viscous dissipation into account; in their research on the ferromagnetic materials of general Newtonian fluids, Khan et al. used bioconvection species that were chemically and paraboloid reactive. For example, Shehzad et al. (2020) have demonstrated that fluid can flow through spinning discs, analogous to the way water flows through tubes. Khan et al. (2020) proposed the hypothesis that gyrotactic microorganisms move around in nano-liquids that have thixotropic viscosity. According to Veera Krishna (2020), a magnetic field is generated when a steady convective magnetohydrodynamic flow of a viscous nanofluid is combined with a permeable porous surface that expands exponentially. As a consequence of the interaction of these two variables, the magnetic field will exhibit some movement. The Carreau–Yasuda nanofluid flow was the result of combining the findings of Hassan et al. (Waqas et al., 2020) on second-order velocity slip and moving microorganisms. Akram et al. (2022c) analyzed electroosmotic flow based on peristalsis flow of silver–water nanomaterials by implementing two approaches. Research conducted by Hosseinzadeh et al. (2020) focused on second-grade convective nanofluid flows. A square cavity that was open on all sides was used to investigate how an oxytactic microbe moved throughout the space. The Boussinesq–Darcy approximation was utilized for both the flow of heat and bioconvection. Activation energy and gyrotactic microorganisms are being used to study nanofluid rheology throughout porous media. A horizontal porous expansion sheet can influence the flow of a fluid infested with gyrotactic microorganisms in a manner comparable to that of a magnetohydrodynamic model for students using the third-grade model (Madhukesh et al., 2022).

Rostami et al. (2022) studied the hydro-thermal analysis. Moreover, the studies reported by, Hosseinzadeh et al. (2021a); Hosseinzadeh et al. (2021b); Sohail et al. (2022a); Sohail et al. (2022b); Hou et al. (2022) are prepared to notice the comportment of numerous involved parameters on momentum and thermal transport. There are no studies in the existing literature that involve synovial fluid and consider the two different nanofluid models over a two-dimensional stretched surface through use of different shaped nanoparticles. In the current study, the modeled problem is solved numerically, and the results are displayed through tables and graphs.

The following assumptions were made:

➢ SF (synovial fluid) is considered over a 2D surface;

Two models of viscosity (Salmi et al., 2022) are defined as μ C , N = e b C μ 0 γ 2 N 2 + ε n   f o r   m o d e l − I . (1) μ C , N = μ 0 γ 2 N 2 + ε n c   f o r   m o d e l − I I . (2) Here, Eqs 1, 2 are known as model-I and model II, respectively. Governing equations (Salmi et al., 2022; Wang et al., 2022) are derived utilizing conservation laws. PDEs in view of SF implicating magnetic dipole are u X + v Y = 0 , (3)

Two models regarding momentum equations are obtained using Eqs (1) and (2). Momentum equation for model-I is ρ N f u u X + v u Y = − P x + M μ N f H x + 2 μ N f u Y Y + 3 n γ 2 u Y 2 u Y Y + α * u Y Y ϕ + α * u Y ϕ Y + α * n γ 2 u Y 3 ϕ Y + 3 α * n γ 2 ϕ u Y 2 u Y Y , (4)

The momentum equation for model-II is ρ N f u u X + v u Y = − P x + M μ N f H x + 2 μ N f u Y Y − α * γ 2 2 u Y 3 ϕ Y − 3 α * γ 2 2 ϕ u Y 2 u Y Y , (5) u T X + v T Y + 1 ρ C p N f u H x + v H Y μ N f T M T = K N f ρ C p N f T y y − Q 0 ρ C p N f T − T ∞ , (6) The mass diffusion equation (Madhukesh et al., 2022) is defined as u C X + v C Y = D T y y + C − C ∞ k ν f T w − T ∞ T s y − K C − C ∞ . (7)

Developing boundary conditions (Wang et al., 2022) are C = C w , v = 0 , u = U w = a X , T = T w : y = 0 , T → T ∞ = T c , u → u e , C → C ∞ : y → ∞ . (8)

The magnetic field via magnetic dipole (Wang et al., 2022) is β 1 = δ 2 π X X 2 + Y + d 2 . (9)

Components of magnetic dipole (Wang et al., 2022) are H X = − ∂ δ ∂ X = δ 2 π X 2 − Y + d 2 X 2 + Y + d 2 2 , (10) H Y = − ∂ δ ∂ Y = δ 2 π 2 X Y + d X 2 + Y + d 2 2 . (11)

The magnitude of magnetic dipole (Wang et al., 2022) is H = ∂ δ ∂ Y 2 + ∂ δ ∂ X 2 12 , (12) ∂ H ∂ Y = − δ 2 π − 2 Y + d 3 + 4 X 2 Y + d 5 , ∂ H ∂ X = − δ 2 π − 2 X Y + d 4 , (13)

Similarity variables (Gul et al., 2020; Wang et al., 2022) are defined as u = a X F ′ , v = − a ν f 1 2 F , θ = T c − T T c − T w , η = X a ρ f μ f 1 2 , ϕ = C ∞ − C C ∞ − C w . (14)

Dimensionless representations of ODEs (Gul et al., 2020; Salmi et al., 2022; Wang et al., 2022) are 2 1 + α ϕ + n R e W e 2 F ′ 2 + 3 n R e W e 2 F ′ 2 ϕ F ‴ ] + 2 n α γ 2 F ″ ϕ + 2 n γ 2 F ″ ϕ , + 2 n R e W e 2 F ′ 3 ϕ ′ − ν N f ν f F ′ F ′ + F F ″ − θ 2 β η + Λ 4 = 0 , (15) 2 1 − 3 α R e W e 2 F ′ 2 ϕ F ‴ − α R e W e 2 F ′ 3 ϕ ′ − ν N f ν f F ′ F ′ + F F ″ − θ 2 β η + Λ 4 = 0 , (16) θ ″ + ρ C p N f k f ρ C p f k N f Pr F θ ′ − 2 F ′ θ + ρ C p N f k f ρ C p f k N f λ 2 β f θ − ϵ η + Λ 3 + k f k N f Pr H t θ = 0 , (17) 1 − χ 2.5 ϕ ″ + S c F ϕ ′ + λ 2 β f ϕ − ϵ η + Λ 3 − τ S c θ ″ ϕ + θ ′ ϕ ′ + K c S c ϕ = 0 . (18)

Correlations based on nanoparticles were captured as indicated as follows. The nanoparticle properties are listed in Table 1 and Table 2 contains information about nanoparticle shapes (Naseem et al., 2021). K N f = − 2 χ − K s + K f + 2 K f + K s 2 χ − K s + K f + 2 K f + K s , μ N f = μ f 1 − χ 2.5 , ρ N f = ρ f 1 − χ + χ ρ s ρ f , (19) D N f = D f 1 − χ 2.5 , ρ C p N f = − χ + 1 + ρ C p s ρ C p f χ ρ C p f , (20) K N f K f = K s + m − 1 K f + m − 1 K s − K f χ K s + m − 1 K f + K s − K f χ . (21)

Boundary conditions (BCs) are F η = 0 , θ η = 1 , ϕ η = 1 , F ′ η = 1 : η = 0 , F ′ η → 0 , θ η → 0 , ϕ η → 0 : η → 0 . (22)

Parameters are defined as R e = a x 2 ν f , W e = α γ 2 , β = δ 2 π μ 0 k T c − T w ρ f μ f 2 , H t = Q 0 C p f a , Pr ⁡ = μ C p k , K c , S c = ν f D f , Λ = a μ f 2 ρ f k T c − T w 1 / 2 , τ = k f T w − T ∞ T s , ϵ = T ∞ T c − T ∞ , K c = K c a , α = α * C − C ∞ .

Surface forces for model-I and model-II (Salmi et al., 2022) were delivered as R e 1 / 2 C f = 1 1 − χ 2.5 1 + α ϕ 0 F ″ + R e n W e 2 1 + α ϕ 0 F ‴ 0 3 , (23) R e 1 2 C f = 1 1 − χ 2.5 F ″ 0 + α 2 W e 2 R e F ‴ 0 3 . (24)

Temperature and concentration gradients for biological fluid (Salmi et al., 2022) are R e − 1 / 2 N U = − K N f K f θ ′ 0 ,   R e − 1 2 S h = 1 1 − χ 2.5 ϕ ′ 0 . (25)

The finite element method (FEM) was utilized to simulate numerical solution of ODE-associated BCs. Steps for explanation of the FEM are listed as follows.

Step-I: In step-I, the desired domain of the problem was discretized into a number of elements. Weak form was achieved using the concept of weighted residual. Shape functions based on linear-type polynomial were derived as follows.

The variables N , F , θ , a n d   ϕ are defined as N = ∑ j = 1 2 N j ψ j , F = ∑ j = 1 2 F j ψ j , θ = ∑ j = 1 2 θ j ψ j , ϕ = ∑ j = 1 2 ϕ j ψ j . (26)

The shape function is ψ j = − 1 j − 1 η − η j − 1 η j − η j − 1 . (27)

Step-II: Stiffness elements were calculated over each element based on the breakdown of the problem domain. Moreover, a global stiffness matrix was achieved. The Picard approach was implemented to differentiate linear systems from non-linear systems.The residual view is defined as R = M F r − 1 , N r − 1 , θ r − 1 , ϕ r − 1 F r θ r ϕ r = F . (28) ∑ i = 1 N ω r − ω r − 1 2 1 2 ∑ i = 1 N ω r 2 < 10 − 8 . (29)

Step-III: In this step, the system of linear equations is M F , N , θ , ϕ F θ ϕ = F . (30)

Step-IV: Maple 18 was used to develop code regarding the FEM. The computational domain was taken as 0 , 8 and grid size study is shown in Table 3. Validation of a present problem already published (Muhammad and Nadeem, 2017) is shown in Table 4.

This study describes the development of a 2D model associated with mass diffusion and thermal energy in two viscosity models. Shape effects based on cylinder, platelet, brick, and sphere were assessed in mineral oil. Chemical reaction and heat generation/absorption were also investigated. It is important to note that magnetic dipoles were taken out in this research project. Such considerations were used to generate a complex model, and the complex model was solved by implementing a finite element approach. Detailed outcomes based on graphical outcomes associated flow, mass diffusion, and thermal energy are discussed below.

The numerical scheme, namely, finite element algorithm, has been applied successfully for the solution of heat and mass transportation in bio-fluids, implicating magnetic dipole in the effects of shape factors and nanoparticles. Important outcomes of the current study are:

➢ Velocity field increases relative to change in magnetic dipole and heat sink, but velocity field decreases with enhancement in W e ;