Recent advances in nanotechnology have discovered an advanced source of energy based on utilization nanoparticles. Nanofluids have been interacted for the impressive thermal properties that turn into enhancement of energy transportation. The enhancement of thermo-physical features of conventional base liquids with the addition of micro-sized metallic particles is a relatively new and interesting development in nanotechnology. Nanoparticles attain microscopic size, having a range between 1 and 100 nm. Recently, the investigations on nano-materials become a new class of intense engineering research due to inherent significances in biomedical, chemical, and mechanical industries, electronic field, nuclear reactors, power plants, cooling systems, diagnoses, diseases, etc. The primary investigation on this topic was reported by Choi [1], which was further worked out by several scientists, especially in the current century. The convective features for nano-materials based on thermophoresis and Brownian movement phenomenon were notified by Buongiorno [2]. This investigation revealed that the role of thermophoresis and Brownian motion factors was quite essential for convective slip mechanism. Khan and Pop [3] discussed the feature of nanofluid immersed in base material confined by moving configuration. Sheikholeslami et al. [4] reported the features of thermal radiation in magneto-nanoparticle flow between circular cylinders. The slip flow in nano-material due to porous surface has been reported by Shahzadi et al. [5]. Khan et al. [6] directed their investigation regarding stability prospective of nanoliquids in a curved geometry and successfully estimated a dual solution for the formulated problem. Turkyilmazoglu [7] imposed zero mass flux constraints regarding asymmetric channels filled by nanoparticles. Vaidya et al. [8] enrolled the fundamental thermal characteristics in the three-dimensional (3D) flow of Maxwell nanofluid where analytical expressions were developed by using optimal homotopic procedure. Hayat et al. [9] focused on the thermal properties and developed the 3D flow of Oldroyd-B fluid featuring mixed convection effects. Some valuable closed-form expressions for a nanofluid flow problem in porous space have been computed by Turkyilmazoglu [10]. Krishna and Chamkha [11] investigated the ion and hall slip effects in the rotating flow of nanofluid configured by a vertical porous plate. The enhancement of heat transfer by using hybrid nanofluids having variable thermal viscosity was reported by Manjunatha et al. [12]. Sardar et al. [13] used non-Fourier's expressions for Carreau nanofluid and suggested some useful multiple numerical solutions successfully. Alwatban et al. [14] performed a numerical analysis to examine the rheological consequences in Eyring–Powell fluid subjected to the second-order slip along with activation energy. The stability analysis for bioconvection flow of nanofluid was reported by Zhao et al. [15]. Alkanhal et al. [16] involved thermal radiation and external heat source for nanofluid enclosed by a wavy shaped cavity. Kumar et al. [17] discussed the thermo-physical properties of hybrid ferrofluid in thin-film flow impacted by uniform magnetic field. Bhattacharyya et al. [18] evaluated the characteristics of different carbon nanotubes for coaxial movement of disks. Mekheimer and Ramdan [19] investigated the flow of Prandtl nanofluid in the presence of gyrotactic microorganisms over a stretching/shrinking surface.

Recently, researchers specified their attention toward the complex and interesting properties of non-Newtonian meterials due to their miscellaneous application in many industries and technologies. The non-Newtonian materials due to convoluted features attracted special attention especially in the current century. The novel physical importance of such non-Newtonian liquids in various engineering and physical processes, biological sciences, physiology, and manufacturing industries is associated due to complex rheological features. Some useful applications associated with the non-Newtonian fluids include polymer solutions, certain oil, petroleum industries, blood, honey, lubricants, and many more. It is commonly observed that distinctive features of such non-Newtonian fluid cannot be pointed out via single relation. Therefore, different non-Newtonian fluid models are suggested by investigators according to their rheology. Among these, Eyring–Powell fluid is inferred from kinetic laws of gases instead of any empirical formulas. This model reduces the viscous fluid at both low and high shear rates (see Powell and Eyring [19]). Gholinia et al. [20] carried out the homogenous and heterogeneous impact in flow of Powell–Eyring liquid due to rotating. Khan et al. [21] focused on viscosity-dependent mixed convection flow of Eyring–Powell nanofluid encountered by inclined surface. Salawu and Ogunseye [22] reported the entropy generation prospective in Eyring–Powell nanofluid featuring variable thermal consequences and electric field. Another useful continuation performed by Abegunrin et al. [23] examined the change in the boundary layer for the flow of Eyring–Powell fluid subjected by the catalytic surface reaction. Rahimi et al. [24] adopted a numerical technique to compute the numerical solution of a boundary value problem modeled due to the flow of Eyring–Powell fluid. Reddy et al. [25] involved some interesting thermal features, like activation energy, chemical reaction, and non-linear thermal radiation in the 3D flow of Eyring–Powell nanofluid induced via slandering surface. Hayat and Nadeem [26] examined the flow of Eyring–Powell fluid and suggested modification in energy and concentration expressions by using generalized Fourier's law. Ghadikolaei et al. [27] reported the Joule heating and thermal radiation features inflow of Eyring–Powell non-Newtonian fluid in a stretching walls channel.

The double diffusion convection is a natural phenomenon that encountered multiple novel applications in area soil sciences, groundwater, oceanography, petroleum engineering, food processing, etc. The double-diffusive convection refers to the intermixing of components of two fluid having different diffuse rates. However, the situation becomes quite interesting when double-diffusive convection depends upon more than two components of fluids. Examples of such multiple diffusive phenomenons include seawater, molten alloy solidification, and geothermally heated lakes. The triple diffusion flow appears in diverse engineering and scientific fields like geology, astrophysics, disposals of nuclear waste, deoxyribonucleic acid (DNA), chemical engineering, etc. [28–30].

After careful observation of the previously cited work, it is claimed that no efforts have been made to report the triple diffusion flow of Eyring–Powell nanofluid induced by an oscillatory stretching surface with variable thermal features. Although some investigations on flow that is due to periodical acceleration have been available in the literature, thermodiffusion features for Eyring–Powell nanofluid are not studied yet. Therefore, our prime objective of this contribution is to report the triple diffusion aspects of Eyring–Powell nanofluid flow by using variable thermal properties. The most interesting convergent technique homotopy analysis procedure is followed to simulate the solution [31–35]. The graphs are prepared to see the impact of different flow parameters with relevant physical consequence.

To develop governing equations for unsteady flow of Eyring–Powell nanofluid, we have considered a periodically stretching surface where x-axis is assumed along with the stretched configuration, whereas y-axis is taken normally. The source of induced flow is based on the periodically moving surface where amplitude of oscillations are assumed to be small. Let velocity of the moving surface as u = u_ω = bx sin ωt, b as stretching rate, ω being angular frequency, whereas t represent time. The uniform features of the magnetic field are reported by implementing it vertically. Let T represent the temperature, C solutal concentration, whereas Φ report the nanoparticle volume fraction. Furthermore, T_∞, C_∞, and Φ_∞ denote free stream nanoparticle temperature, free stream solutal concentration, and volume fraction of nanofluid, respectively. After using such assumptions, the flow problem is modeled through the following equations [22, 33]:

where ν is viscosity, ρ_f fluid density, (β^*, C) fluid parameters, ϑ permeability of porous medium, σ_e electrical conductivity, α₁ thermal diffusivity, DK_TC Dufour diffusivity, τ_T = (ρc)_p/(ρc)_f ratio of heat capacity of nanoparticles to heat capacity of fluid, D_B Brownian diffusion coefficients, D_s solutal diffusivity, D_T thermophoretic diffusion coefficient, whereas DK_CT Soret diffusivity.

Following boundary assumptions are articulated for current flow problem

In order to suggest modification in energy equation (3), we used the following relations for variable thermal conductivity [33, 34]

where K_∞ ambient fluid conductivity and ε thermal dependence conductivity constant. Now, before perfume analytical simulations, first, we reduce the number of independent variables in the governing equations by using the following variables:

The dimensionless set of equations in view of the previously mentioned transformations is

The boundary constraints in the non-dimensional form are

where K = 1/μβ^*C and Γ=uw2b/2νC2 denote the material parameters, Ω=σB02/ρfb+νϑ/k′b is magneto-porosity constant, S = ω/b oscillating frequency-to-stretching rate ratio, N_t = (ρc)_pD_T (T_w − T_∞)/(ρc)_fT_∞ν thermophoresis parameter, Pr = ν/α_m is Prandtl number, N_b = (ρc)_pD_B (C_w − C_∞)/(ρc)_fν Brownian motion constant, Nd = D_TC (C_w − C_∞)/α_m (T_w − T_∞) modified Dufour number, Ld = D_CT (T_w − T_∞)/α_m (C_w − C_∞) Dufour Lewis number, Le = ν/D_s regular Lewis number, whereas Ln = ν/D_B nano-Lewis number.

We define the following relations associated with the definitions of wall shear stress, local Nusselt number, Sherwood number, and nano-Sherwood number:

where q_s, j_s, and g_s stand for surface heat flux, surface mass flux, and motile microorganisms flux, respectively. The dimensionless forms of the previously mentioned physical quantities are

where Rex=uwx̄/ν is mentioned for local Reynolds number.

The structured set of non-linear partial differential equations (12–16) with boundary conditions (17–18) are simulated analytically via homotopy analysis technique. Due to efficient and convincing accuracy, various physical problems in recent years have been solved by following this procedure. The initial guesses for the present flow problem are

Following auxiliary linear operators that are followed to precede the solution

satisfying

where a₁, … a₉ are arbitrary constants.

The convergence procedure in homotopic solution is regulated with auxiliary parameters h_f, h_θ, h_φ, and h_ϕ. On this end, we prepare h−curves to report the convincible range of such parameters. It is obvious from Figure 1 that the preferable range of these such parameters can be utilized from −2 ≤ h_f ≤ 0, −1.2 ≤ h_θ ≤ −0.2, −1.4 ≤ h_φ ≤ 0 and −1.2 ≤ h_ϕ ≤ −0.2.

Table 1 shows the comparison of present results with Abbas et al. [35] as a limiting case. An excellent accuracy of results is noted with these reported investigations.

This section aims to manifest the features of some interesting flow parameters that appeared in the dimensionless equations, where material parameter K, magneto-porosity constant Ω, oscillating frequency-to-rate of stretching ratio S, Brownian motion Nb, thermophoresis constant Nt, variable thermal conductivity δ, Prandtl number Pr, Dufour Lewis number Ld, modified Dufour constant Nd, regular Lewis number Le, and nano-Lewis number Ln. During variation of each flow parameter, we fixed some numerical values to remaining parameters, like K=0.2, Ω = 0.4, S = 0.2, Nt = 0.3, Pr = 0.7, Nb = 0.4, Nd,= 0.5, Ld = 0.3, Le = 0.2, and Ln = 0.3.

We have focused on periodically accelerated unsteady flow of Eyring–Powell nanofluid with utilization of thermal diffusive features. The variable impact of thermal conductivity, porous medium, and magnetic field consequences are also utilized. The important observations from current flow problem are summarized as:

The observation based on the reported results can be used to improve thermal extrusion processes, heat exchangers, solar technology, energy production, cooling processes, etc.

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

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.