Electrohydrodynamic Instability in Dielectric Oldroydian hybrid Nanofluid Layer PDF

**Electrohydrodynamic Instability in Dielectric Oldroydian hybrid Nanofluid Layer** #### Abstract The influence of rheology on the initiation of thermal convection in a hybrid nanofluid layer heated beneath with the inclusion of an AC electric field (vertical) is studied **by employing** linear st...

**Electrohydrodynamic Instability in Dielectric Oldroydian hybrid Nanofluid Layer** #### Abstract The influence of rheology on the initiation of thermal convection in a hybrid nanofluid layer heated beneath with the inclusion of an AC electric field (vertical) is studied **by employing** linear stability analysis in which bounding surfaces are taken as stress-free. The stationary convective stability of the rheological nanofluid is customarily established utilizing the Buongiorno model for nanoparticles and the Oldroyd model for the rheological behavior of regular fluid. The Buongiorno model deployed for nanofluids incorporates the impact of thermophoresis and the Brownian motion. In the continuation of nanofluids research, the researchers have also tried to use [[hybrid nanofluid]](https://www.sciencedirect.com/topics/engineering/hybrid-nanofluid) recently, which is engineered by suspending dissimilar [[nanoparticles]](https://www.sciencedirect.com/topics/materials-science/nanoparticle) either in mixture or composite form.  The eigen-value problem is solved for exact solutions analytically by using single term Galerkin technique and the expressions of the thermal Rayleigh number for the start of stationary convection for both bottom-heavy and top-heavy nanoparticles configuration are obtained and are presented graphically. **Keywords**: **Nanofluid,** Oldroyd model**, Hybrid nanofluid, Brownian motion,** electric field**.** 1. **Introduction** Interdisciplinary fields of interactions between thermal hydrodynamics and electro-hydrodynamics (EHD) can be established through experiments on a dielectric fluid placed horizontally with an applied electric field along the dielectric fluid. The aims is in controlling the augmentation of heat transfer and examining the stability of a system, thereby giving advantages like those of reduced operational costs, less power consumption and less time utilization of the procedure. Several authors have worked to establish the influence of the external electric field on natural convection. A gradient in the electrical conductivity as well as a dielectric constant are produced in a dielectric regular fluid. This convection, which arises in an electrically conducting and infinitely extending horizontal layer subjected to an electric field (AC) externally and the field due to gravity, is termed as electroconvection, analogous to Bénard convection \[1\]. The electroconvection occurs as a result of the combined action of thermal buoyancy and electric forces with the inclusion of an electric field. Motivated by this fact, Roberts \[2\] was the first to study the initiation of electroconvection, in which the electric conductivity and dielectric constant are considered to vary linearly as a function of temperature. To analyze the influence of the electric field on thermal instability, a variety of research has been conducted both analytically and experimentally \[3-4\]. Previous studies are mainly concerned with micrometer-sized or millimeter-sized particles. Researchers have shown great interest in utilizing nanofluids in the present arena due to their extraordinary characteristics in enhancing thermal conductivity with the dispersion of nanoscale particles (oxide ceramics, metal carbides, nitrides and metals or carbon nanotubes) of diameter less than 100 nm to a base fluid (ethylene or triethylene glycols, water, alcohol and polymer solutions), which was first postulated by Choi \[11\]. Buongiorno\[12\] was the first to study convective instability in nanofluids and establish equations of equilibrium models of nanofluids that accounted for the impacts of thermophoresis and Brownian motion. This model was utilized by various researchers \[13-16\] to investigate the rise in thermal conductivity of nanofluids by including various parameters, individually or simultaneously and distinct combinations of boundary conditions. A substantial amount of research has been conducted on nanofluids over the past few decades, encompassing preparation, characterization, modeling, convective and boiling heat transfer, and various applications. However, hybrid nanofluids represent a relatively recent advancement in the field. These can be created by either (i) dispersing two or more types of nanoparticles in a base fluid or (ii) suspending hybrid (composite) nanoparticles within a base fluid. A hybrid material combines both the physical and chemical properties of different substances, presenting these characteristics within a homogeneous phase. Synthetic hybrid nanomaterials possess exceptional physicochemical properties that are absent in their individual components. Extensive research has been carried out on these composites \[65\], particularly those consisting of carbon nanotubes (CNTs), which have found applications in electrochemical sensors, biosensors, nanocatalysts, and more \[66\]. Despite this, the application of hybrid nanomaterials in nanofluids remains underdeveloped. Research on hybrid nanofluids is still in its early stages, with ongoing experimental studies. The primary aim of synthesizing hybrid nanofluids is to combine the beneficial properties of their constituent materials. A single material often lacks all the desirable traits needed for specific applications, typically excelling in either thermal or rheological properties. However, in many practical scenarios, balancing various properties becomes essential, which is where hybrid nanofluids prove advantageous. Additionally, hybrid nanofluids are anticipated to offer superior thermal conductivity compared to individual nanofluids due to the synergistic effects. Carbon nanotubes exhibit a range of exceptional properties, including remarkable physical strength, chemical stability, mechanical resistance, and extremely high electrical and thermal conductivity. These attributes have garnered significant interest from researchers, leading to the development of hybrid nanomaterials that integrate carbon nanotubes with metallic, semi-conductive, or non-conductive nanoparticles. To the best of the authors\' knowledge, no comprehensive review of hybrid nanofluids has been conducted to date. Modern materials play a crucial role in enhancing our quality of life. Some of these materials are utilized in contemporary vehicles and aircraft to reduce weight, increase safety, and improve fuel efficiency compared to earlier models. Among these advanced materials are nano-composites. A nano-composite consists of a matrix or curing phase combined with nano-particles, which are in powder form, suspension, or dispersion, to enhance the properties of the base material. The incorporation of nano-particles into a polymer matrix enhances its performance. This technique is highly effective in producing high-performance composites. Various nanoparticles, such as carbon nanotubes, graphene, and molybdenum disulfide, are commonly used as reinforcing agents to create robust, eco-friendly nanocomposites for cartilage tissue engineering. Even at low concentrations, the inclusion of these nanoparticles significantly improves the compressive strength of nanocomposites. The convective heat transfer ability of nanofluids primarily depends on the thermal and physical properties of the base fluid, the shape and size of the suspended nanoparticles, their volume fraction, and the flow configuration. These characteristics make nanofluids highly valuable in various real-world applications, including residential, commercial, transportation, and industrial sectors. Tzou \[1, 2\] studied thermal stability in a nanofluid layer consistently heated from below. He employed the transport equations developed by Buongiorno \[3\], discovering that the combined effects of nanoparticle thermophoresis and Brownian motion had a destabilizing influence. Nield and Kuznetsov \[4\] revisited thermal stability in a horizontal nanofluid layer by introducing several dimensionless numbers. Dhananjay et al. \[5\] examined Rayleigh--Bénard stability in a nanofluid layer. Numerous authors have investigated the influence of various parameters on thermal instability in nanofluid layers. Yadav et al. \[6\] explored the effect of rotation on the stability of nanofluids, showing that rotation stabilizes the system for certain values of the nanofluid parameters. Nield and Kuznetsov \[7\] analyzed thermal stability in a nanofluid-saturated porous layer using Darcy's model, which was later extended by Kuznetsov and Nield \[8\] through the application of Brinkman's model. Bhadauria and Agarwal \[9, 10\] studied the effects of porous media on nanofluid convection. Nield and Kuznetsov \[11\] further examined the double-diffusive phenomenon in nanofluid layers using a Galerkin approach and derived stability boundaries. The influence of swirl on the thermal stability in a horizontal porous layer of nanoliquid was investigated by Chand and Rana \[12\], revealing that porosity exhibits a destabilizing effect during stationary convection. Sheikholeslami et al. \[13\] explored heat transfer in nanoparticles using innovative turbulators while considering entropy generation. Similarly, Sheikholeslami et al. \[14\] also examined heat transfer in nanoparticles using innovative turbulators and entropy generation. Shankar et al. \[15\] analyzed the MHD instability in pressure-driven flow of non-Newtonian fluids. Faraz et al. \[16\] considered the MHD effects on axisymmetric Casson nanofluids and analyzed heat transfer over a sheet. Sheu \[17\] focused on the linear thermal stability in a viscoelastic nanofluid layer. Kumar and Awasthi \[18\] studied the triple-diffusive phenomenon affecting the stability of a nanoliquid layer, while Awasthi et al. \[19\] examined the impact of the triple-diffusive phenomenon on a Maxwell fluid layer in the presence of a heat source. Recently, nano-composite materials have garnered significant attention due to their applications in research, industry, and other fields. Polymer nanocomposites, which consist of an organic polymer combined with inorganic nanoparticles, have drawn even more interest due to their exceptional properties resulting from the synergy of organic and inorganic hybrid materials. By integrating the functionalities of both components along with the nanostructure of the particles, these nanocomposites are expected to exhibit enhanced and novel properties. These resultant nanocomposites are widely applicable in automotive industries, optoelectronics, biomedical fields, sensors, and more. Numerous experimental studies on nanocomposite materials have been reported in the literature, but this work represents the first attempt to perform a mathematical stability analysis of nanocomposite materials. To the best of our knowledge, the thermal stability in a composite horizontal nanoliquid layer has not yet been studied. Therefore, this study seeks to investigate the onset of thermal convection in a composite nanoliquid layer heated from below. The following assumptions are made in the mathematical analysis of the problem: 1. The nanoliquid is assumed to be Newtonian, incompressible, and in laminar flow conditions. 2. Surface charge technology is employed to suspend nanoparticles in the base liquid. 3. There is no chemical reaction between the liquid and the nanoparticles during thermal convection. 4. No force exists between the two nanoparticle suspensions. 5. The density in each term of the nanofluid momentum equation is treated as constant, except in the external force term, while other thermophysical parameters such as viscosity, specific heat, and thermal conductivity are assumed constant (Boussinesq approximation). 6. The nanoparticles and the base fluid are assumed to be in thermal equilibrium. 7. In this study, the nanoparticles are considered spherical. 8. The effect of radiative heat transfer between the sides of the rigid boundary is neglected, as it is exceedingly small. Figure 2 presents a comparison between the Rayleigh numbers for ordinary nanofluid and composite nanofluid under conditions where , , , , , and. Positive values of and indicate a top-heavy distribution. It can be observed that the Rayleigh number for composite nanofluid is higher than that of ordinary nanofluid, suggesting that a mixture of two different nanoparticles enhances the stabilization of stationary convection more effectively than single nanoparticles in top-heavy distributions. In Figure 3, the Rayleigh numbers for ordinary nanofluid and composite nanofluid are compared under conditions where , , , , , and , with negative values of and representing a bottom-heavy distribution. The results show that the Rayleigh number for composite nanofluid is lower than that of ordinary nanofluid, indicating that the mixture of two different nanoparticles tends to destabilize stationary convection more than single nanoparticles in bottom-heavy distributions Figure 4 shows the neutral curves for different values of the first thermo-nanofluid Lewis number, while keeping other parameters constant. It can be observed that as the first thermo-nanofluid Lewis number increases, the stationary Rayleigh number decreases. This indicates that the first thermo-nanofluid Lewis number has a destabilizing effect on stationary convection in a top-heavy distribution but stabilizes it in a basement-heavy distribution. Similarly, Figure 5 presents the neutral curves for varying values of the second thermo-nanofluid Lewis number, with other parameters held constant. The results show that an increase in the second thermo-nanofluid Lewis number leads to a decrease in the stationary Rayleigh number. Therefore, the second thermo-nanofluid Lewis number destabilizes stationary convection in a top-heavy distribution and stabilizes it in a basement-heavy distribution. **Instability is not influenced by Brownian motion or thermophoresis. It arises solely from the interaction between buoyancy effects and the conservation of nanoparticles.** **The first and second thermo-nanofluid Lewis numbers have a destabilizing effect on stationary convection when the distribution is top-heavy, but they stabilize stationary convection when the distribution is basement-heavy.** **The Concentration Rayleigh number contributes to the destabilization of stationary convection. However, stationary convection is possible in both top-heavy and basement-heavy nanoparticle distributions.** **Oscillatory convection does not occur if both the first and second thermo-nanofluid Lewis numbers are considered.** **The critical Rayleigh number remains unchanged with increases in the modified particle density.** **The conditions required to prevent over-stability are specific values for the parameters and.** The application of mono nanofluids in heat transfer has been widely studied and reviewed by many researchers. The concept of hybrid nanofluids, however, is relatively recent and has gained increased research focus. Reviews on HyNF applications are still limited. Previous review articles have addressed the synthesis and preparation methods of selected HyNFs, but their focus has primarily been on a few applications, particularly those related to heat exchangers. Additionally, an accessible thermophysical properties database for various HyNFs is lacking in current literature. Therefore, this review presents synthesis and preparation techniques for different HyNFs along with a database of their thermophysical properties. Commonly used equations for calculating density, specific heat, viscosity, and thermal conductivity are also summarized. A comprehensive review is provided on the use of hybrid nanofluids in a variety of heat transfer applications, including heat exchangers, heat sinks, heat pipes, photovoltaic modules, natural convection enclosures, refrigeration and air-conditioning systems, boiling applications, jet impingement cooling systems, and thermal energy storage systems. Lastly, the challenges related to HyNF and future research directions are discussed Thermal conductivity variance on convective flow nanofluid (water-alumina) in an annulus has been studied by Parvin et al.\[17\] and a considerable increase in nanoparticle volume fraction and thermal Prandtl number has been reported due to the presence of nanoparticles. **Yadav et al. \[18\] explored thermal instability in an electric field-affected nanofluid layer and observed that the AC electric Rayleigh number completely determines the convection cells size. Saad et al. \[19\] analyzed the influence of an external electric field on the initiation of nanofluids convection inside flat, tiny heat pipe with an axial groove in steady-state.** In the aforesaid studies, the regular fluid has been taken to be Newtonian. #### Thermal convection of nanofluids with nanoparticle suspension and non-Newtonian base fluid has a lot of applications nowadays, including oil recovery, dispersal of environmental toxins, packed food production and chemical and material manufacturing processes and are characterized by variety of models Because of its stability and features, the Oldroyd \[20\] model describes fluid elasticity and viscosity parameters that stay constant throughout a broad range of shear rates. The model of this type has a significance in various fields like chemical, electronics, material industries and electrical. Different researchers \[21-24\] have investigated the variability in creeping and interial flows configurations of Oldroyd-B fluid. Kolodner \[25\] demonstrated experimentally that convection patterns of DNA suspensions can be described spatially as travelling and localised standing waves with short periods and long enough oscillation periods. Gupta et al. \[26\] explored the rheological behavior of nanofluids with Oldroyd-B as base fluid and observed strain retardation delays the start of oscillatory modes, whilst stress relaxation shows reverse impact. Sharma et al. \[27\] investigated electro-convection in dielectric nanofluids with Oldroydian base fluid and top-/bottom-heavy nanoparticles configurations. Khan et al. \[28\] studied the  heat transport in a rheological nanofluid in stretching sheet which generates motion. They discovered that increasing the Rayleigh number and buoyancy ratio increases the nanoparticles temperature considerably. The Oldroyd-B nanofluid flow through a stratified media, considering gyrotactic bacteria has been investigated by Elanchezhian et al. \[31\]. Yasir et al. \[32\] studied a problem on time-dependent Oldroyd-B fluid flow saturated with nanoparticles using gyrotactic microorganisms. #### Motivated by the applications of various pertinent parameters mentioned above, the onset of thermal convection in a rheological nanofluid with an external electric field and uniform vertical rotation is established for stress free bounding surfaces in the present work The rheological behavior of the base fluid is characterized by the Oldroyd model. 2. **Mathematical Formulation** An incompressible and electrically conducting (infinitely) horizontal Oldroydian layer of hybrid nanofluid of depth ***d*** separated by two parallel horizontal planes heated uniformly beneath with temperature and nanoparticles volumetric fraction are and at and , respectively, which is shown in Figure 1. A force due to gravity and rotation (uniform) pervade the system along - axis with inclusion of an external vertical AC applied uniformaly across the fluid. The bottom plane is taken under an electric circuit and at the top plane, the root mean square potential, is considered. Except for density, all of the fluids relevant thermophysical variables are held constant for analytical analysis. **Figure 1**: Geometric configurationof Oldroydian hybrid nanofluid layer under vertical AC electric field. The equations of continuity, momentum, heat energy and hybrid nanoparticles relevant to the governing physical system \[Buongiorno\[12\], Oldroyd \[20\], Chandrasekhar \[33\], Pundir et al. (2022)\] are: , (1) [\$\\begin{matrix} \\rho\_{f}\\left\\lbrack \\frac{\\partial v}{\\partial t} + \\left( v.\\nabla \\right)v \\right\\rbrack\\left( 1 + \\overset{¯}{\\lambda\_{1}}\\frac{\\partial}{\\partial t} \\right) = \\left\\lbrack - \\nabla P - f\_{e} + \\left( \\phi\_{1}\\rho\_{p\_{1}} + \\phi\_{2}\\rho\_{p\_{2}} + \\left( 1 - \\phi\_{1} - \\phi\_{2} \\right)\\rho\_{f}\\left\\{ 1 - \\beta\\left( T - T\_{0} \\right) \\right\\} \\right)g \\right\\rbrack\\left( 1 + \\overset{¯}{\\lambda\_{1}}\\frac{\\partial}{\\partial t} \\right) \\\\ + \\left( 1 + \\overset{¯}{\\lambda\_{2}}\\frac{\\partial}{\\partial t} \\right)\\mu\\nabla\^{2}v, \\\\ \\end{matrix}\$]{.math.inline} (2) [\$\\rho\\left( 1 + \\overline{\\lambda\_{1}}\\frac{\\partial}{\\partial t} \\right)\\left\\lbrack \\frac{\\partial\\mathbf{v}}{\\partial t} + \\left( \\mathbf{v}.\\nabla \\right)\\mathbf{v} \\right\\rbrack = \\left( 1 + \\overline{\\lambda\_{1}}\\frac{\\partial}{\\partial t} \\right)\\left\\lbrack - \\left( \\nabla p + \\mathbf{f}\_{e} \\right) + \\left( \\phi\\rho\_{p} + \\left( 1 - \\phi \\right)\\rho\_{0}\\left\\{ 1 - \\beta\\left( T - T\_{1} \\right) \\right\\} \\right)\\mathbf{g} \\right\\rbrack + \\left( 1 + \\overline{\\lambda\_{2}}\\frac{\\partial}{\\partial t} \\right)\\mu\\nabla\^{2}\\mathbf{v},\$]{.math.inline}here [\$v,\\mu,\\rho\_{f},p,T,\\ g,\\rho\_{p\_{1}},\\ \\rho\_{p\_{2}},\\beta,\\phi\_{1},\\ \\phi\_{1}\\overset{¯}{,\\lambda\_{1}}\$]{.math.inline}and denote the fluid velocity, coefficient of viscosity, base fluid density, pressure, temperature, acceleration due to gravity, density of hybrid nanoparticles, adverse temperature gradient, volume fractions of hybrid nanopartcles, stress relaxation and strain retardation. In Eq. (2), the electric origin force is expressed as [\$\\mathbf{f}\_{e} = \\rho\_{e}\\mathbf{E} - \\frac{1}{2}\\mathbf{E}\^{2}\\nabla\\epsilon + \\frac{1}{2}\\nabla\\left( \\rho\\frac{\\partial\\epsilon}{\\partial t}\\mathbf{E}\^{2} \\right),\$]{.math.inline} (3) where and denote, respectively charged particles density, dielectric constant and electric field, and the pressure term is modified as (4) dielectric constant and fluid density are described as , (5) where denote the dielectric constant coefficient with very small relative temperature variations and and represent the coefficient for volume expansion and fluid density at lower layer. The tendency of a charged body to move along the electric lines of force in lieu of an externally applied electric field (AC) and transfer force to the neighbouring fluid, Maxwell Eqs. take the form **, (6)** **The electric field is derived from scalar potential** , represented by and using this value in Eq. (6), one gets (7) where [*φ*]{.math.inline} denote electric potential (root mean square value). The hybrid nanoparticles conservation equations are [\$\\left\\lbrack \\frac{\\partial v}{\\partial t} + \\left( v.\\nabla \\right)v \\right\\rbrack\\phi\_{1} = D\_{B\_{1}}\\nabla\^{2}\\phi\_{1} + \\left( \\frac{D\_{T\_{1}}}{T\_{0}} \\right)\\nabla\^{2}\\text{T.}\$]{.math.inline}[\$\\left\\lbrack \\frac{\\partial\\mathbf{v}}{\\partial t} + \\left( \\mathbf{v}.\\nabla \\right)\\mathbf{v} \\right\\rbrack\\phi = D\_{B}\\nabla\^{2}\\phi + \\frac{D\_{T}}{T\_{1}}\\nabla\^{2}\\text{T.}\$]{.math.inline} (8) [\$\\left\\lbrack \\frac{\\partial v}{\\partial t} + \\left( v.\\nabla \\right)v \\right\\rbrack\\phi\_{2} = D\_{B\_{1}}\\nabla\^{2}\\phi\_{2} + \\left( \\frac{D\_{T\_{2}}}{T\_{0}} \\right)\\nabla\^{2}\\text{T.}\$]{.math.inline}[\$\\left\\lbrack \\frac{\\partial\\mathbf{v}}{\\partial t} + \\left( \\mathbf{v}.\\nabla \\right)\\mathbf{v} \\right\\rbrack\\phi = D\_{B}\\nabla\^{2}\\phi + \\frac{D\_{T}}{T\_{1}}\\nabla\^{2}\\text{T.}\$]{.math.inline} (8) Thermal energy equation of hybrid nanofluid is [\$\\left( \\text{ρc} \\right)\_{f}\\left\\lbrack \\frac{\\partial v}{\\partial t} + \\left( v.\\nabla \\right) \\right\\rbrack T = k\_{f}\\nabla\^{2}T + \\left( \\text{ρc} \\right)\_{p\_{1}}\\left\\lbrack D\_{B\_{1}}\\nabla\\phi.\\nabla T + \\frac{D\_{T\_{1}}}{T\_{0}}\\nabla T.\\nabla T \\right\\rbrack + \\left( \\text{ρc} \\right)\_{p\_{2}}\\left\\lbrack D\_{B\_{2}}\\nabla\\phi.\\nabla T + \\frac{D\_{T\_{2}}}{T\_{0}}\\nabla T.\\nabla T \\right\\rbrack,\$]{.math.inline}. (9) here the parameters , [(ρc)~*p*~1~~]{.math.inline}, [(ρc)~*p*~2~~]{.math.inline} , [*D*~*B*~1~~]{.math.inline}, [*D*~*B*~2~~]{.math.inline}, [*D*~*T*~1~~]{.math.inline} and [*D*~*T*~2~~ ]{.math.inline}denote the heat capacities of fluid and hybrid nanoparticles, thermophoretic and Brownian Diffusion coefficients of hybrid nanoparticles. The parameter represents thermal conductivity. The relevant boundary conditions are [\$w = T = \\phi\_{1} = \\phi\_{2} = \\frac{\\partial\^{2}w}{\\partial z\^{2}} = \\frac{\\partial\\varphi}{\\partial z} = 0\$]{.math.inline}when and (10) **3. Primary Flow and Disturbed Equations** The basic flow is supposed to be steady motionless, unaffected by time and quiescent (i.e., the settling of dispersed nanoscale particles is negligible). As a result, the temperature, pressure, electric field dielectric constant and electric potentials fluctuate in vertical direction only. Thus, solutions corresponding to this state using Eqs. (1)-(2) and (7)- (9) are [\$\\begin{matrix} v = v\_{b} = 0,T = T\_{b}\\left( z \\right) = T\_{0} - \\beta z,\\ \\phi\_{1} = {\\phi\_{1}}\_{b},\\phi\_{2} = {\\phi\_{2}}\_{b},p = p\_{b}\\left( z \\right),\\gamma = \\ \\ \\gamma\_{0}\\left( 1 + \\text{eβz} \\right) = \\gamma\_{b}\\left( z \\right), \\\\ \\ \\\\ \\end{matrix}\$]{.math.inline} [\$E = \\frac{E\_{0}\\widehat{k}}{1 + e\_{0}\\left( \\text{βz} \\right)} = E\_{b}\\left( z \\right)\\ ,\\varphi = - \\frac{E\_{0}\\log\\left( 1 + e\_{0}\\text{βz} \\right)}{e\_{0}\\beta} = \\varphi\_{b}\\left( z \\right)\\ ,\$]{.math.inline} (11) where, [\$\\beta = \\frac{\\left( T\_{0} - T\_{1} \\right)}{d}\$]{.math.inline}and represent the adverse thermal gradient and electric field( root mean square value) at and subscript indicates basic state Infinitesimal perturbatios are superimposed to the fundamental (basic) state flow in order to check the system stability, taken as [*ϕ*~1~ = *ϕ*~1*b*~ + *ϕ*′~1~, *ϕ*~2~ = *ϕ*′~2*b*~]{.math.inline} (12) where,[*ϕ*′~1~]{.math.inline} , [*ϕ*′~2~]{.math.inline} ,, and denote the disturbed physical quantities. Substituting perturbations from Eq. (12) and using (11), Eqs. (2), (7)-(9), ignoring the product of perturbations and eliminating pres sure term,we obtain linear non-dimensional equations as (for convenience, asterisks are removed) [\$\\begin{matrix} \\frac{1}{P\_{r}}\\left( 1 + \\lambda\_{1}\\frac{\\partial}{\\partial t} \\right)\\frac{\\partial}{\\partial t}\\left( \\nabla\^{2}w \\right) = \\left\\lbrack \\left( 1 + \\lambda\_{1}\\frac{\\partial}{\\partial t} \\right)\\left\\{ R\_{e}\\nabla\_{1}\^{2}\\left( T - \\frac{\\partial\\varphi}{\\partial z} \\right) + R\_{a}\\nabla\_{1}\^{2}T - R\_{n\_{1}}\\nabla\_{1}\^{2}\\phi\_{1} - R\_{n\_{2}}\\nabla\_{1}\^{2}\\phi\_{2} \\right\\} \\right\\rbrack \\\\ + \\left( 1 + \\lambda\_{2}\\frac{\\partial}{\\partial t} \\right)\\left( \\nabla\^{4}w \\right), \\\\ \\end{matrix}\$]{.math.inline} (13) [\$\\frac{\\partial T\^{\'}}{\\partial t} - v\^{\'}\\ = \\nabla\^{2}T\^{\'} + \\frac{N\_{B\_{1}}}{L\_{n\_{1}}}\\left( \\frac{\\partial T\^{\'}}{\\partial z} - \\frac{\\partial\\phi\^{\'}}{\\partial t} \\right) - \\frac{2N\_{A\_{1}}N\_{B\_{1}}}{L\_{n\_{1}}}\\frac{\\partial T\^{\'}}{\\partial z} + \\frac{N\_{B\_{2}}}{L\_{n\_{2}}}\\left( \\frac{\\partial T\^{\'}}{\\partial z} - \\frac{\\partial\\phi\^{\'}}{\\partial t} \\right) - \\frac{2N\_{A\_{2}}N\_{B\_{2}}}{L\_{n\_{2}}}\\frac{\\partial T\^{\'}}{\\partial z}\$]{.math.inline}, (14) [\$\\frac{\\partial T\'}{\\partial\\mathrm{t}} - w\^{\'} = \\nabla\^{2}T\' + \\frac{N\_{B}}{L\_{n}}\\left( \\frac{\\partial T\'}{\\partial z} - \\frac{\\partial\\phi\^{\'}}{\\partial\\mathrm{t}} \\right) - \\frac{2N\_{A}N\_{B}}{L\_{n}}\\frac{\\partial T\'}{\\partial z}\$]{.math.inline}[\$\\frac{\\partial{\\phi\_{1}}\^{\'}}{\\partial t} + v\' = \\frac{1}{L\_{n\_{1}}}\\nabla\^{2}{\\phi\_{1}}\^{\'} + \\frac{N\_{B\_{1}}}{L\_{n\_{1}}}\\nabla\^{2}T\^{\'},\$]{.math.inline} (15) [\$\\frac{\\partial{\\phi\_{2}}\^{\'}}{\\partial t} + v\' = \\frac{1}{L\_{n\_{2}}}\\nabla\^{2}{\\phi\_{2}}\^{\'} + \\frac{N\_{B\_{2}}}{L\_{n\_{2}}}\\nabla\^{2}T\^{\'},\$]{.math.inline} () [\$\\frac{\\partial\\phi\^{\'}}{\\partial\\mathrm{t}} + w\^{\'} = \\frac{1}{L\_{n}}\\nabla\^{2}\\phi\^{\'} + \\frac{N\_{B}}{L\_{n}}\\nabla\^{2}T\',\$]{.math.inline} (16) where [\${\\phi\_{1}}\^{\*} = \\frac{\\left( \\phi\_{1} - \\phi\_{1\_{0}} \\right)}{\\phi\_{1\_{0}}}{\\phi\_{2}}\^{\*} = \\frac{\\left( \\phi\_{2} - \\phi\_{2\_{0}} \\right)}{\\phi\_{2\_{0}}}\$]{.math.inline}, *T*[\$\\ = \\frac{\\left( T\^{\*} - T\_{1} \\right)}{T\_{0} - T\_{1}}\$]{.math.inline} where are the dimensionless variables and the Taylor number, [\$P\_{r} = \\frac{\\mu}{\\rho\_{0}\\alpha}\$]{.math.inline}, the thermal Prandtl number, , the Rayleigh number due to electric field, , the nanofluid Lewis number, [\$R\_{n} = \\frac{\\left( \\rho\_{p} - \\rho\_{f0} \\right)\\left( \\phi\_{1} - \\phi\_{0} \\right)gd\^{3}}{\\text{μα}}\$]{.math.inline}[\$R\_{m} = \\frac{\\left\\lbrack {\\phi\_{1}}\_{0}\\rho\_{P\_{1}} + \\ {\\phi\_{2}}\_{0}\\rho\_{P\_{2} +}\\rho\_{f}(1 - {\\phi\_{1}}\_{0} - {\\phi\_{2}}\_{0}) \\right\\rbrack gd\^{3}}{\\mu\\alpha\_{m}}\$]{.math.inline}, the basic density-Rayleigh number [\$N\_{B\_{1}} = \\frac{\\left( \\text{ρc} \\right)\_{P\_{1}}\\left( \\phi\_{1\_{1}} - \\phi\_{1\_{0}} \\right)}{{(\\rho c)}\_{f}}\$]{.math.inline}[\$N\_{B} = \\frac{\\rho\_{p}c\_{p}}{\\mathrm{\\text{ρc}}}\\left( \\phi\_{1} - \\phi\_{0} \\right)\$]{.math.inline}, [\$N\_{B\_{2}} = \\frac{\\left( \\text{ρc} \\right)\_{P\_{2}}\\left( \\phi\_{2\_{1}} - \\phi\_{2\_{0}} \\right)}{{(\\rho c)}\_{f}}\$]{.math.inline}, increments in the particle density of hybrid nanofluid, [\${R\_{n}}\_{1} = \\frac{\\left( \\rho\_{p\_{1}} - \\rho\_{f0} \\right)\\left( \\phi\_{1\_{1}} - \\phi\_{1\_{0}} \\right)gd\^{3}}{\\text{μα}}\$]{.math.inline}, [\${R\_{n}}\_{2} = \\frac{\\left( \\rho\_{p\_{2}} - \\rho\_{f0} \\right)\\left( \\phi\_{2\_{1}} - \\phi\_{2\_{0}} \\right)gd\^{3}}{\\text{μα}}\$]{.math.inline}, concentration Rayleigh numbers of hybrid nanofluid ,the thermal Rayleigh number [\$N\_{A\_{1}} = \\frac{D\_{T\_{1}}\\ (\\ T\_{0} - T\_{1})}{D\_{B\_{1}}\\ T\_{0}\\ \\left( \\phi\_{1\_{1}} - \\phi\_{1\_{0}} \\right)}\$]{.math.inline}, [\$N\_{A\_{2}} = \\frac{D\_{T\_{2}}\\ (\\ T\_{0} - T\_{1})}{D\_{B\_{2}}\\ T\_{0}\\ \\left( \\phi\_{2\_{1}} - \\phi\_{2\_{0}} \\right)}\$]{.math.inline}, the modified diffusivity ratios of hybrid nanofluids, , the stress-relaxation time and , strain retardation time, respectively, are the dimensionless parameters. and are the two-dimensional and three-dimensional Laplacian operators, respectively. **3.3. Normal Mode Method** The coupled differential Eqs. *(14)-(16) and (19)* are solved by analyzing the perturbed physical quantities as normal modes *and is described as* [(*w*^′^,*T*^′^,*ϕ*^′^,*φ*^′^) = \[*W*(*z*),*Θ*(*z*),*Φ*(*z*),*Ψ*(*z*)\] × exp \[*σt*+*i*(*k*~*x*~*x*+*k*~*y*~*y*)\],]{.math.inline}[(*w*^′^,*T*^′^,*ϕ*~1~^′^,*ϕ*′~2~,*φ*^′^) = (*W*,*Θ*,*Φ*~1~,*Φ*~2~,*Ψ*)(*z*) × *e*^{*i*(*k*~*x*~*x*+*k*~*y*~*y*)+σt}^,]{.math.inline} (20) where , and indicate the respective perturbation wave number along the x- and y- axes and the complex growth rate. denotes resultant wave number, Using expression (20) in Eqs. (14) --(16) and (19), we get a set of linearized ordinary differential Eqs. as [\$\\begin{matrix} \\left( D\^{2} - a\^{2} \\right)\\left\\lbrack \\left( 1 + \\lambda\_{2}\\sigma \\right)\\left( D\^{2} - a\^{2} \\right) - \\frac{\\sigma}{P\_{r}}\\left( 1 + \\lambda\_{1}\\sigma \\right) \\right\\rbrack\^{2}W - \\left\\lbrack \\left( 1 + \\lambda\_{2}\\sigma \\right)\\left( D\^{2} - a\^{2} \\right) - \\frac{\\sigma}{P\_{r}}\\left( 1 + \\lambda\_{1}\\sigma \\right) \\right\\rbrack \\\\ \\left\\lbrack a\^{2}\\left( R\_{a} + R\_{e} \\right)\\Theta - a\^{2}R\_{n\_{1}}\\Phi\_{1} - a\^{2}R\_{n\_{2}}\\Phi\_{2} + a\^{2}R\_{e}\\text{DΨ} \\right\\rbrack = 0, \\\\ \\end{matrix}\$]{.math.inline} (21) [\$W + \\left\\lbrack - \\sigma + \\left( D\^{2} - a\^{2} \\right) + \\frac{N\_{B\_{1}}}{L\_{n\_{1}}}D + \\frac{N\_{B\_{2}}}{L\_{n\_{2}}}D - \\frac{2N\_{A\_{1}}N\_{B\_{1}}}{L\_{n\_{1}}} - \\frac{2N\_{A\_{2}}N\_{B\_{2}}}{L\_{n\_{2}}} \\right\\rbrack\\Theta - \\left( \\frac{N\_{B\_{1}}}{L\_{n\_{1}}} \\right)D\\Psi - \\frac{N\_{B\_{2}}}{L\_{n\_{2}}}D\\Psi = 0,\$]{.math.inline}[\$W + \\left\\lbrack \\frac{N\_{B}}{L\_{n}}D + D\^{2} - a\^{2} - \\frac{2N\_{A}N\_{B}}{L\_{n}} - \\sigma \\right\\rbrack\\Theta - \\frac{N\_{B}}{L\_{n}}D\\Psi = 0,\$]{.math.inline} (22) [\$W - \\frac{N\_{A}}{L\_{n}}\\left( D\^{2} - a\^{2} \\right)\\Theta - \\left\\lbrack \\frac{1}{L\_{n}}\\left( D\^{2} - a\^{2} \\right) - \\sigma \\right\\rbrack\\Phi = 0,\$]{.math.inline}[\$W + \\left( - \\frac{N\_{A\_{1}}}{L\_{n\_{1}}} \\right)\\left( D\^{2} - a\^{2} \\right)\\Theta + \\left\\lbrack - \\frac{1}{L\_{n\_{1}}}\\left( D\^{2} - a\^{2} \\right) - \\sigma \\right\\rbrack\\Phi\_{1} = 0,\$]{.math.inline} (23) [\$W - \\frac{N\_{A}}{L\_{n}}\\left( D\^{2} - a\^{2} \\right)\\Theta - \\left\\lbrack \\frac{1}{L\_{n}}\\left( D\^{2} - a\^{2} \\right) - \\sigma \\right\\rbrack\\Phi = 0,\$]{.math.inline}[\$W + \\left( - \\frac{N\_{A\_{2}}}{L\_{n\_{2}}} \\right)\\left( D\^{2} - a\^{2} \\right)\\Theta + \\left\\lbrack - \\frac{1}{L\_{n\_{2}}}\\left( D\^{2} - a\^{2} \\right) - \\sigma \\right\\rbrack\\Phi\_{2} = 0,\$]{.math.inline} [(*D*^2^−*a*^2^)*Ψ* + *DΘ* = 0.]{.math.inline}. (24) After utilizing (20), the boundaries (10) yield [*W* = *D*^2^*W* = *Θ* = *Φ* = *DΨ* = 0 at *z* = 0 and *z* = 1.]{.math.inline}[*W* = *Θ* = *Φ*~1~ = *Φ*~2~ = *D*^2^*W*= *DΨ* = 0]{.math.inline} at and. (25) **The trial functions of Eqs.(21)-(24) using GWRM satisfying boundary conditions (25) are taken as** [*Φ*~1~ = *C*~1~sin (πz), *Φ*~2~ = *C*′~1~sin (πz), *Ψ* = *D*~1~cos (πz)]{.math.inline}, (26) here , [*C*′~1~]{.math.inline}and represent unknown constants. Using these solutions (26) in Eqs. (21)- (24) and integrating each equation between the limits 0 to 1 i.e., *(0 ˂ z \< 1)* individually and using boundary conditions (25), one obtains [\$\\begin{bmatrix} (K\_{2}J\^{2} + \\frac{\\sigma}{p\_{r}}JK\_{1}) & - a\^{2}(R\_{a} + R\_{e})K\_{1} & a\^{2}R\_{n\_{1}}K\_{1} & a\^{2}R\_{n\_{2}}K\_{1} & - R\_{e}a\^{2}\\pi K\_{1} \\\\ 1 & - J - \\sigma & 0 & 0 & 0 \\\\ 1 & \\frac{N\_{A\_{1}}}{L\_{n\_{1}}}J & \\frac{J}{L\_{n\_{1}}} + \\sigma & 0 & 0 \\\\ 1 & \\frac{N\_{A\_{2}}}{L\_{n\_{2}}}J & 0 & \\frac{J}{L\_{n\_{2}}} + \\sigma & 0 \\\\ 0 & - \\pi & 0 & 0 & - J \\\\ \\end{bmatrix}\\begin{bmatrix} A\_{1} \\\\ B\_{1} \\\\ C\_{1} \\\\ D\_{1} \\\\ \\end{bmatrix} = 0,\$]{.math.inline} (27) where[*J* = (*a*^2^+*π*^2^)]{.math.inline}, [*J* = (*a*^2^+*π*^2^), *K*~1~ = (1+*λ*~1~*σ*)]{.math.inline} and Equation (27) gives as [\$R\_{a} = \\frac{\\pi\^{2}R\_{e}}{J} - R\_{e} - \\frac{\\left( J + \\sigma + \\frac{a\^{2}N\_{A}}{L\_{n}} + \\frac{\\pi\^{2}N\_{A}}{L\_{n}} \\right)R\_{n}}{\\left( \\sigma + \\frac{J}{L\_{n}} \\right)} - \\frac{J\\left( - J - \\sigma \\right)\\left\\{ \\frac{\\sigma\\left( 1 + \\sigma\\lambda\_{1} \\right)}{P\_{r}} + J\\left( 1 + \\sigma\\lambda\_{2} \\right) \\right\\}}{a\^{2}\\left( 1 + \\sigma\\lambda\_{1} \\right)}\$]{.math.inline} (28) **4. Mathematical Analysis** **4.1 Oscillatory Convection** **The convection through pure oscillatory modes is characterized by taking real part of zero. Then putting** in Eq. (28) and after some mathematical simplifications, we get [*R*~*a*~ = *Δ*~1~ + *iΔ*~2~*ω*]{.math.inline}, [\$\\Delta\_{1} = - \\frac{Jw\^{2}}{a\^{2}p\_{r}} + \\frac{( - J + \\pi\^{2})R\_{e}}{J} - \\frac{(L\_{n\_{1}}(J\^{2} + w\^{2}L\_{n\_{1}}) + J\^{2}N\_{A\_{1}})R\_{n\_{1}}}{J\^{2} + w\^{2}L\_{n\_{1}}\^{2}} - \\frac{(L\_{n\_{2}}(J\^{2} + w\^{2}L\_{n\_{2}}) + J\^{2}N\_{A\_{2}})R\_{n\_{2}}}{J\^{2} + w\^{2}L\_{n\_{2}}\^{2}}\$]{.math.inline} [\$+ \\frac{J\^{2}(J + w\^{2}\\lambda\_{1})}{a\^{2}(1 + w\^{2}\\lambda\_{1}\^{2})} + \\frac{J\^{2}w\^{2}( - 1 + J\\lambda\_{1})\\lambda\_{2}}{a\^{2}(1 + w\^{2}\\lambda\_{1}\^{2})}\$]{.math.inline} (29) [\$\\Delta\_{2} = Jw(\\frac{J}{a\^{2}p\_{r}} + \\frac{L\_{n\_{1}}( - 1 + L\_{n\_{1}} + N\_{A\_{1}})R\_{n\_{1}}}{J\^{2} + w\^{2}L\_{n\_{1}}\^{2}} + \\frac{L\_{n\_{2}}( - 1 + L\_{n\_{2}} + N\_{A\_{2}})R\_{n\_{2}}}{J\^{2} + w\^{2}L\_{n\_{2}}\^{2}}) + \\frac{J\^{2}w - J\^{3}w\\lambda\_{1}}{a\^{2}(1 + w\^{2}\\lambda\_{1}\^{2})}\\frac{J\^{2}w(J + w\^{2}\\lambda\_{1})\\lambda\_{2}}{a\^{2}(1 + w\^{2}\\lambda\_{1}\^{2})}\$]{.math.inline} Comparing real and imaginary parts of Eq. (29), gives oscillatory thermal Rayleigh number [\$\\begin{matrix} R\_{a}\^{\\text{osc}} = - \\frac{Jw\^{2}}{a\^{2}p\_{r}} + \\frac{( - J + \\pi\^{2})R\_{e}}{J} - \\frac{(L\_{n\_{1}}(J\^{2} + w\^{2}L\_{n\_{1}}) + J\^{2}N\_{A\_{1}})R\_{n\_{1}}}{J\^{2} + w\^{2}L\_{n\_{1}}\^{2}} - \\frac{(L\_{n\_{2}}(J\^{2} + w\^{2}L\_{n\_{2}}) + J\^{2}N\_{A\_{2}})R\_{n\_{2}}}{J\^{2} + w\^{2}L\_{n\_{2}}\^{2}} \\\\ \\\\ \\end{matrix}\$]{.math.inline} [\$+ \\frac{J\^{2}(J + w\^{2}\\lambda\_{1})}{a\^{2}(1 + w\^{2}\\lambda\_{1}\^{2})} + \\frac{J\^{2}w\^{2}( - 1 + J\\lambda\_{1})\\lambda\_{2}}{a\^{2}(1 + w\^{2}\\lambda\_{1}\^{2})}\$]{.math.inline} (30) and [*a*~1~*ω*^4^ + *a*~2~*ω*^2^ + *a*~3~ = 0,]{.math.inline}[*a*~1~*ω*^6^ + *a*~2~*ω*^4^ + *a*~3~*ω*^2^ + *a*~4~ = 0]{.math.inline} [*a*~1~ = *J*^2^*w*^7^*L*~*n*~1~~^2^*L*~*n*~2~~^2^*λ*~1~(*λ*~1~ + *p*~*r*~*λ*~2~)]{.math.inline} [*a*~2~ = *Jw*^5^(*a*^2^*L*~*n*~1~~*L*~*n*~2~~^2^( − 1 + *N*~*A*~1~~)*p*~*r*~*R*~*n*~1~~*λ*~1~^2^ + *J*^3^*L*~*n*~2~~^2^*λ*~1~(*λ*~1~ + *p*~*r*~*λ*~2~) + *L*~*n*~1~~^2^(*a*^2^*L*~*n*~2~~( − 1 + *N*~*A*~2~~)*p*~*r*~*R*~*n*~2~~*λ*~1~^2^ + *J*^3^*λ*~1~(*λ*~1~ + *p*~*r*~*λ*~2~) + *L*~*n*~2~~^2^(*J* + *p*~*r*~(*J* − *J*^2^*λ*~1~ + *a*^2^(*R*~*n*~1~~ + *R*~*n*~2~~)*λ*~1~^2^ + *J*^2^*λ*~2~))))]{.math.inline} [*a*~3~ = *Jw*^3^(*a*^2^*L*~*n*~1~~( − 1 + *N*~*A*~1~~)*p*~*r*~*R*~*n*~1~~(*L*~*n*~2~~^2^ + *J*^2^*λ*~1~^2^) + *L*~*n*~1~~^2^(*J*^3^ + *p*~*r*~(*J*^3^ − *J*^4^*λ*~1~ + *a*^2^(*L*~*n*~2~~(*L*~*n*~2~~*R*~*n*~1~~ + ( − 1 + *L*~*n*~2~~ + *N*~*A*~2~~)*R*~*n*~2~~) + *J*^2^*R*~*n*~1~~*λ*~1~^2^) + *J*^4^*λ*~2~)) + *J*^2^(*a*^2^*L*~*n*~2~~( − 1 + *N*~*A*~2~~)*p*~*r*~*R*~*n*~2~~*λ*~1~^2^ + *J*^3^*λ*~1~(*λ*~1~ + *p*~*r*~*λ*~2~) + *L*~*n*~2~~^2^(*J* + *p*~*r*~(*J* − *J*^2^*λ*~1~ + *a*^2^*R*~*n*~2~~*λ*~1~^2^ + *J*^2^*λ*~2~))))]{.math.inline} \ [*a*~4~ = *J*^3^*w*(*J*^3^ + *p*~*r*~(*a*^2^(*L*~*n*~1~~( − 1 + *L*~*n*~1~~ + *N*~*A*~1~~)*R*~*n*~1~~ + *L*~*n*~2~~( − 1 + *L*~*n*~2~~ + *N*~*A*~2~~)*R*~*n*~2~~) + *J*^3^(1 − *Jλ*~1~ + *Jλ*~2~)))]{.math.display}\ (31) **4.2. Stationary Convection** stationary convection sets in the marginal state is characterized by substituting in Eq. (28) and the thermal Rayleigh of non-oscillatory mode, , given by [\${R\_{a}}\^{s} = \\frac{\\left( a\^{2} + \\pi\^{2} \\right)\^{3}}{a\^{2}} - \\frac{{a\^{2}R}\_{e}}{\\left( a\^{2} + \\pi\^{2} \\right)} - R\_{n}\\left( N\_{A} + L\_{n} \\right).\$]{.math.inline}[\${R\^{s}}\_{a} = \\frac{\\left( a\^{2} + \\pi\^{2} \\right)\^{3}}{a\^{2}} - (L\_{n\_{1}} + N\_{A\_{1}})R\_{n\_{1}} - \\frac{a\^{2}R\_{e}}{a\^{2} + \\pi\^{2}} - (L\_{n\_{2}} + N\_{A\_{2}})R\_{n\_{2}}.\$]{.math.inline} (32) From expression (32), it is apparent that depends on non-dimentional parameters, namely [*a*, *L*~*n*~1~~, *N*~*A*~1~~, *R*~*n*~1~~, *R*~*e*~, *L*~*n*~2~~, *N*~*A*~2~~ ]{.math.inline}and [*R*~*n*~2~~.]{.math.inline} **Special cases:** **Case I:** For nanofluid without electric field i.e.,[*R*~*e*~ = 0]{.math.inline} Eq.(32) reduces to [\${R\_{a}}\^{s} = \\frac{\\left( a\^{2} + \\pi\^{2} \\right)\^{3}}{a\^{2}} - R\_{n}\\left( N\_{A} + L\_{n} \\right)\$]{.math.inline}[\$R\_{a}\^{s} = \\frac{\\left( a\^{2} + \\pi\^{2} \\right)\^{3}}{a\^{2}} - (L\_{n\_{1}} + N\_{A\_{1}})R\_{n\_{1}} - (L\_{n\_{2}} + N\_{A\_{2}})R\_{n\_{2}}.\$]{.math.inline} (33) Which resembles with earlier result of Kumar and awasthi (18) **Case II:** For rheological regular fluid, i.e., [*L*~*n*~1~~= *N*~*A*~1~~ = *R*~*n*~1~~ = *L*~*n*~2~~ = *N*~*A*~2~~=]{.math.inline} [*R*~*n*~2~~ = 0, ]{.math.inline}Eq (33) further shrinks to [\$R\_{a} = \\frac{\\left( a\^{2} + \\pi\^{2} \\right)\^{3}}{a\^{2}}\$]{.math.inline}[\$R\_{a}\^{s} = \\frac{\\left( a\^{2} + \\pi\^{2} \\right)\^{3}}{a\^{2}},\$]{.math.inline} (34) which coincides with the prior result of Chandrashekhar \[33\]. Equation (33), yields It is depicted from equation (32) that [\$\\frac{dR\_{a}\^{s}}{dR\_{e}}\$]{.math.inline} is negative for all wave numbers, implying that [*R*~*e*~]{.math.inline}has a destabilizing influence on a system.**For the bottom-heavy/top-heavy nanoparticles (for negative/positive values of** [*R*~*n*~1~~]{.math.inline} and [*R*~*n*~2~~]{.math.inline}), **both the nanofluid Lewis number,** [*L*~*n*~1~~]{.math.inline}, [*L*~*n*~2~~]{.math.inline} **and the modified diffusivity ratio,** [*N*~*A*~1~~, *N*~*A*~2~~]{.math.inline} **stabilize/destabilize a system,** as shown by equations (33) and (34). Furthermore, for the behavior of concentration Rayleigh number, it is observed from equations (34) that [\$\\frac{dR\_{a}\^{s}}{dR\_{n\_{1}}}\$]{.math.inline} and [\$\\frac{dR\_{a}\^{s}}{dR\_{n\_{2}}}\$]{.math.inline} **always negative for** [(*N*~*A*~1~~+*L*~*n*~1~~)]{.math.inline}\>0 and [(*N*~*A*~2~~+*L*~*n*~2~~) \> 0]{.math.inline} **. Since for most of nanofluids, the value of** [*N*~*A*~1~~]{.math.inline} and [*N*~*A*~2~~]{.math.inline} **for both top/bottom-heavy arrangements is taken in the range(1≤**[*N*~*A*~1~~, *N*~*A*~2~~]{.math.inline}**≤10)/( -1≤**[*N*~*A*~1~~, *N*~*A*~2~~]{.math.inline}**≤-25) and** [*L*~*n*~1~~]{.math.inline}and [*L*~*n*~2~~]{.math.inline} **is on the order of** **to** **. This shows that** [*R*~*n*~1~~]{.math.inline} and [*R*~*n*~2~~ ]{.math.inline}always has a destabilizing effect on the onset of stationary convection. 5. **Numerical Discussion** The expressions for thermal Rayleigh number of stationary and oscillatory modes for stress free boundaries are encapsulated in Eqs. (28) and (40), respectively. These values are evaluated numerically by applying the software MATHEMATICA -12. Control in the values of [*R*~*n*~1~~]{.math.inline} and [*R*~*n*~2~~ ]{.math.inline}depend on the concentration of hybrid nanoparticles dispersed at the bounding surfaces and the gap between these surfaces. The permissible experimential values of all the involved parameters used by Roberts \[2\], Sharma et al. \[27\], Sharma et al. \[29\], Chandrashekhar \[33\], Pundir et al. (2022) are taken as [*L*~*n*~1~~ = 200, *N*~*A*~1~~ =  ± 3, *R*~*n*~1~~ =  ± 0.4 , *L*~*n*~2~=~300, *N*~*A*~2~~ =  ± 6  ]{.math.inline}and [*R*~*n*~2~~ =  ± 0.6, *R*~*e*~ = 100]{.math.inline}. To understand stability of the convective phenomena of the nanofluid layer completely, stationary and oscillatory curves are plotted in plane. Figure 3 compares stationary convection in a regular nanofluid with that in a composite nanofluid, under a top-heavy distribution for various parameter values: Ln1​=100,200, Ln2​=0,400, Rs​=5, NA1​=4, NA2​=0,6,8, Rn1​=0.5,0.2, and Rn2​=0,0.5,0.6. Here, Rn1​ and Rn2 are positive in the top-heavy configuration. Likewise, both NA1​ and NA2​ are positive for a fluid layer heated from below. It is worth noting that the thermal Rayleigh number decreases during stationary convection in the presence of both types of nanoparticles in the top-heavy distribution. This indicates that a blend of two different nanoparticles reduces stationary convection compared to a single type of nanoparticle in the same arrangement. Higher concentrations of both nanoparticle types further destabilize the system, which can be stabilized by cooling the bottom layer relative to the top. Similarly, Figure 4 presents a comparison of stationary convection between ordinary nanofluids and hybrid nanofluids for different parameter values: Ln1​=100, Ln2​=0,100,200, Rs​=10, NA1​=5, NA2​=0,6,9, Rn1​=−0.5, and Rn2​=0,−0.5,−0.6. In this case, negative values of Rn1​ and Rn2Rn\_2Rn2​ correspond to a bottom-heavy distribution. It is worth noting that the stationary thermal Rayleigh number in the hybrid nanofluid layer is higher than that in the ordinary nanofluid under a bottom-heavy distribution. This indicates that, in such a configuration, a hybrid nanofluid layer (a mixture of two nanoparticles) is more stable than a layer composed of a single type of nanoparticle. Similar findings were reported by Kumar and Awasthi.18  **Figure 2**: behavior of stationary Rayleigh number **Figure 3**: Graph of stationary Rayleigh **for ordinary and hybrid nanofluids** number **for ordinary and hybrid nanofluids**  **Figure 4**: Graph of Rayleigh number **Figure 5**: Graph of Rayleigh number versus wave number as a function of versus wave number as a function of electric Rayleigh number for bottom- concentration Rayleigh number [*R*~*n*~2~~]{.math.inline} for top-heavy case. /bottom- case. Figure 3 displays the impact of distinct values of [*R*~*e*~ = 100, 200, 300]{.math.inline} on versus. The figure indicates a decrement in with a rise in electric Rayleigh number, which postpones the onset of convection for stationary modes. Hence, destabilization of a system is depicted by the electric Rayleigh number,. This occurs because the physical system electrostatic energy increases as a result of a greater electric field, rendering it less stable for top-heavy arrangement of hybrid nanoparticles. The concentration Rayleigh number, [*R*~*n*~]{.math.inline}is influenced by top-/bottom-heavy nanoparticle arrangements in addition to the distance between both the boundaries. The stationary thermal Rayleigh numbers has been plotted versus wavenumber in Figure 5 for various concentration Rayleigh number, [*R*~*n*~2~~ =  ± 0,  ± 0.5,  ± 0.6]{.math.inline} for both top-/bottom-heavy nanopartcles distribution, respectively. As expected, [*R*~*n*~2~~]{.math.inline} has destabilizing/stabilizing effect for the top-/ bottom-heavy nanoparticles arrangement on both the stationary mode.  **Figure 6**: Graph of Rayleigh number **Figure 7**: Graph of Rayleigh number versus wave number as a function of versus wave number as a function of hybrid nanofluid Lewis number [*L*~*n*~2~~]{.math.inline} or bottom- modified diffusivity ratio [*N*~*n*~2~~]{.math.inline} for top- heavy case. heavy/bottom-heavy case. Figure 6 represents the plot of versus for distinct values of nanofluid Lewis number, [*L*~*n*~2~~ = 0, 100, 200]{.math.inline} in bottom-/top-heavy arrangements, respectively. It has been depicted from the cuves that as [*L*~*n*~2~~]{.math.inline} rises, thermal Rayleigh number of stationary modes gets enhanced. Hence, [*L*~*n*~2~~]{.math.inline} has stabilizing impact on stationary convection in top-heavy arrangements. The impact of modified diffusivity ratio of hybrid nanofluid, [*N*~*A*~2~~ = 0,  ± 6,  ± 9]{.math.inline} on the onset of stationary convection in top-/bottom-heavy arrangements is illusrated in Figure 7 and the stationary thermal Rayleigh number is found to rise/fall as the modified diffusivity ratio increases. As a result, [*N*~*A*~2~~]{.math.inline} has expanded/contracted the stability regime for bottom-/top-heavy arrangement of hybrid nanoparticles. **6. Conclusions** The effect of an AC electric field in rheological hybrid nanofluid layer is investigated by employing Oldroyd model for both kinds of nanoparticles distribution (top-/bottom-heavy). 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