## 1 Introduction
On the other hand, the model is not precise for specific situations in real engineering phenomena since it produces a parabolic energy equation [24]. In 1948, [PERSON] [25] modified the Fourier model known as Maxwell-Cattaneo (MC) law by including a thermal relaxation time term to present the thermal inertia. [PERSON] [26] re-examined the MC law by considering the time derivative of heat flux which resulting in a hyperbolic energy equation. Some studies were conducted to justify the uniqueness and stability of the solutions for the energy equation using the Cattaneo-Christov heat flux model (see [27; 28; 29]). In the past few years, many researchers have investigated the boundary layer problem of various fluids due to a stretching sheet using the non-Fourier model. [PERSON] [30] used this model for Maxwell fluid and noticed that there was a relation between the thermal boundary layer thickness and the thermal relaxation time. [PERSON] et al. [31] emphasized the magnetohydrodynamics (MHD) flow of Williamson fluid with variable thickness. They discovered that the temperature profile decreased using the heat flux model compared to the classical Fourier model. [PERSON] et al. [32] considered stagnation flow of Jeffrey fluid towards a nonlinear stretching surface which resulted in the fluid temperature declining for higher thermal relaxation parameters. [PERSON] et al. [33] analyzed the MHD flow of Casson fluid and also found that the temperature decreased using the Cattaneo Christov heat flux model. Very recently, a similar study conducted for nanofluids [24; 34] and diverse non-Newtonian fluid models such as OldroydB-fluid [35], Eyring Powell [36; 37], Carreau fluid [38], viscoelastic fluid [39; 40] and second grade fluid [41; 42] resulted in the non-Fourier model leading to the reduction of the temperature distribution. [PERSON] et al. [43] designed the boundary layer flow of Casson nanofluid using Cattaneo-Christov heat flux model while [PERSON] and [PERSON] [44] utilized the Casson hybrid TiO\({}_{2}\)-CuO/EG nanofluid on the stretched flow problem with entropy generation. Double stratification of fluids appears due to the temperature and concentration variations, respectively. From the engineering perspective, many applications utilize the concept of double stratification, such as heat rejection from environment, thermal energy storage systems and solar energy because the better stratification corresponds to the higher energy performance. There are a few studies focused on mathematical analysis for the convective flow of nanofluids with the presence of stratification [45; 46; 47; 48; 49; 50; 51; 52; 53]. [PERSON] and [PERSON] [45] analyzed the natural convection and stable doubly stratified flow with the presence of nanoparticles. [PERSON] et al. [46] dealt with the mixed convective flow in a thermally stratified porous medium saturated by a nanofluid using [PERSON] and Das model. The impact of double stratification and mixed convective flow of the non-Newtonian fluids with the appearance of nanoparticles have been studied by [PERSON] et al. [47] and [PERSON] et al. [48] using Maxwell and Jeffrey nanofluid models. [PERSON] et al. [49] examined the influence of both thermal and solutal stratification on MHD nanofluid due to an exponentially stretching sheet. [PERSON] et al. [50; 51] considered the effects of viscous dissipation and Joule heating on both steady and unsteady MHD mixed convection of doubly stratified nanofluid. [PERSON] et al. [52] investigated the effect of double stratification on MHD nanofluid over a porous vertical plate. The previously published results showed that the temperature and concentration distribution are decreased with the intensity of thermal and solutal stratification, respectively. The preceding literatures give an inspiration for the authors to examine the behavior of MHD nanofluid flow with the double stratification phenomenon over a permeable shrinking/stretching sheet.