**1. Introduction**

Nanofluid has served in a number of engineering applications, for example, porous materials [1,2], fuel-cell industry [3], etc. due to its significant increase in the heat-transfer rate compared to conventional engineered fluid [4]. Nanofluids are another class of fluids made by scattering at the nanometer scale materials (nanoparticles, nanofibers, nanotubes, nanowires, nanorods, nanosheets, or nanobeads) in base fluids. As it were, nanofluids are nanoscale colloidal suspensions containing dense nanomaterial. They are two-stage frameworks with one stage (solid stage) into another (fluid stage). It was discovered that nanofluids have improved thermophysical properties, for example, thermal conductivity, heat-diffusivity, thickness, and convective warmth move coefficients contrasted with those of base fluids, like oil or water. It has indicated incredible potential applications in numerous zones. Some investigations on nanofluid can be cited in [5–9]. Most of the human vessels are flexible in nature and the peristaltic flows exhibit such kind of geometries. The flows of such types are very useful in industry, engineering and medical. These flows have also immense applications in curing cancer cells. Abd Elnaby and Haroun [10] have studied the influence of conformal wall properties on peristaltic movement in a two-dimensional channel and produced the conclusion that the reverse pumping rate increases by rising the wall damping and reduces under the increasing magnitude of the wall elasticity as well as tension, which differs from the model used by Mittra and Prasad [11] and Srivastava and Srivastava [12]. Muthu et al. [13] analyzed the peristaltic movement of a micropolar fluid in circular cylindrical tubes with elastic wall properties. They suggested from the obtained measurement that viscous damping is affecting the mean flow reversal over the elastic surface. Nadeem et al. [14] obtained an analytical solution for pumping transport of Williamson nanofluid through a curved channel with compliant walls and offered the readings under the variation of curvature of the enclosure and heat transfer coefficient. Although a large number of studies on the peristaltic flow of conventional fluids are available, only a few articles have been reported on the peristaltic flow of nanofluids [15–18]. In this regard, Akbar et al. [19] investigated the copper nanoparticles impinging on a curved channel with compliant walls and peristalsis. They acquired analytical solutions for temperature distribution and nanoparticle concentration. Due to the importance of the effects Soret (thermal diffusion) and Dufour (diffusion-Thermo), many investigators have been studied which can be found in [20–22].

Collective forced, free convection (mixed convection stream) is occurred in large number of engineering and industrial processes, like solar central receivers attached to the wind potentials, cooling of electronic equipment through fans and nuclear reactors during emergency shutdown and heat transfers kept in lower-velocity surroundings. Heat and mass transfers accompanying effect on each other also produce a cross-diffusion influence. The temperature difference generates mass transfer which is known as Soret effect, on the other hand, the Dufour effect comes from the heat transfer produced by the concentration gradient. Due to wide range of aplications, peristaltic transport of Jeffrey fluid with double diffusion convection for nanofluids has been analyzed by Akram et al. [23] in the presence of a tilted magnetic field. Exact solutions are obtained for the breaking field of nanoparticles, the concentration field, the temperature field, the flow functions, the pressure gradient and the pressure increase with respect to the axial and transverse coordinates on the length restrictions of longwave and low Reynolds number. Akbar and Habib [24] have discussed the peristaltic flow induced by natural double-diffusive convection to achieve a nanofluid magnetic field analysis in an asymmetric porous channel and obtained solutions in a series of five coupled equations.

The feature of compliant wall in peristaltic flows is a key tool for governing muscle tension. This physical phenomenon has been revealed mathematically by a system of equations which are linked to compliant walls displacement [25,26]. Srinivasvas and Kothandapani [27] have investigated the transfer of heat and mass effects on wavy flow through a porous region experiencing compliant walls. Batti et al. [28] have introduced the wavy phenomenon of Jeffrey fluid in a non-uniform rectangular enclosure with the effects of variable magnetic field. They proposed the attributes of non-uniformity of channel on the flow with the incorporation of lubrication theory and obtained the exact solutions. Bhatt et al. [29] have published the hall current factor on peristaltic analysis of heated particle–fluid combined flow with compliant wall properties through numerical treatment. It is to be mentioned here that the analysis of double diffusion mixed convection for a wavy mechanism of viscoelastic nanofluid in a curved structured geometry has not been yet investigated.

Keeping in mind the importance of above-discussed literature and wide range of applications of mixed convection phenomenon with nanoparticles in peristaltic flows, the authors converted their attention to exploring the theoretical effects of double diffusion over peristaltic flow of nanofluid having third-grade fluid as a base fluid through a curved channel along with wall properties. Most probably, this study will be the best direction to efficiently use the achieved data in experimental

side. The equations of continuity, momentum, energy, and nanoparticle concentration have been modeled through some suitable physical conditions like low values of wavenumber and the Reynolds number. The observing equations are then solved analytically by using a perturbation method. The results are manipulated graphically and discussed in detail. The parameters affecting the phenomenon have been described individually.
