Electrohydrodynamic instability in dielectric rotating Oldroydian nanofluid layer

Abstract

The influence of rheology on the initiation of thermal convection in a rotating nanofluid layer heated beneath with the inclusion of a vertical alternating current (AC) electric field is studied by employing linear stability analysis in which bounding surfaces are taken as stress-free. The stationary and oscillatory convective stability of the rheological nanofluid is customarily established utilizing the Buongiorno model for nanoparticles and the Oldroyd model for the rheological behaviour of the regular fluid. The Buongiorno model deployed for nanofluids incorporates the impact of thermophoresis and the Brownian motion. 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 both stationary and oscillatory convection for both bottom-heavy and top-heavy nanoparticles configuration are obtained and are presented graphically.

KEYWORDS:

• Nanofluid
• Oldroyd model
• Taylor number
• 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 aim 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. The 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]. Chandrasekhar [2] has provided an elaborate description of the thermal instability of a Newtonian fluid, exploring different assumptions regarding hydrodynamics and hydromagnetics. 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 [3] 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. Numerous studies, both analytical and experimental, have been conducted to analyse the influence of the electric field on thermal instability [4,5].

In a rotating system, the fluid motion induced by gravity experiences Coriolis forces perpendicular to the rotational vector, which counteract the spreading effect. This balance between buoyancy and Coriolis forces, in the absence of viscous dissipation or boundaries defined by isopotential surfaces, leads to the attainment of geotropic equilibrium. The study of rotating fluids is crucial in various physical phenomena, including atmospheric flow, deep convection in chimneys, and the convective layer of celestial bodies such as the Sun and planets in the solar system [6,7]. The joint impact of uniform rotation and an external electric field under an adverse temperature gradient was first investigated by Takashima [8]. Mansoor et al. [9] explored the influence of electric field properties on the flow behaviour of magnetohydrodynamic (MHD) Carreau-water nanofluid.

Later on, the work on an electrically conducting and insulating fluid layer confined within horizontally placed conducting electrodes with the upper part warmer was done by Stiles [10]. The influence of a uniform magnetic field on the onset of Darcy–Benard convection with horizontal throughflow – a comparison between absolute and convective instabilities and a two-parameter family of basic states in porous media that induces Darcy- Bénard convection. studied by Turkyilmazoglu [11,12]. The system was found more unstable during the initiation of stationary modes as the electric field strength attained a critical value because of body force polarization. Shivkumara et al. [13] investigated the joint impact of rotation and electric field on thermal instability in porous layer and discovered that the adequate conditions for the occurrence of oscillatory convection remain unaltered due to the electric field.

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 which was first postulated by Choi [14]. Buongiorno[15] 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 [16–21] to investigate the rise in thermal conductivity of nanofluids by including various parameters, individually or simultaneously and distinct combinations of boundary conditions. Selimefendigil and Chamkha [22] investigated magnetohydrodynamics mixed convection in a triangular cavity filled with power law nanofluid and an opening, utilizing “Tiwari and Das” nanofluid model while Nagehan Alsoy-Akgün [23] examined the influence of a homogeneous magnetic field on the transient natural convection phenomenon in a nanofluid.

Thermal conductivity variance on convective flow nanofluid (water-alumina) in an annulus has been studied by Parvin et al.[24] and a considerable increase in nanoparticle volume fraction and thermal Prandtl number has been reported due to the presence of nanoparticles. Yadav et al. [25] 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. [26] analysed the influence of an external electric field on the initiation of nanofluids convection inside a flat, tiny heat pipe with an axial groove in a steady-state. Kolsi et al. [27] examined the computational fluid dynamics investigation of magnetohydrodynamic and elastic wall influences on nanofluid convection.

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 etc. and are characterized by a variety of models because of its stability and features, the Oldroyd [28] model describes fluid elasticity and viscosity parameters that stay constant throughout a broad range of shear rates. The model of this type has significance in various fields like chemical, electronics, material industries and electrical. Umavati et al. [29] have investigated the variability in creeping and internal flow configurations of Oldroyd-B fluid while Devi [30] et al. studied electroconvection in a nanofluid. Kolodner [31] demonstrated experimentally that convection patterns of DNA suspensions can be described spatially as travelling and localized standing waves with short periods and long enough oscillation periods whereas Ebrahem et al. [32] analysed nonlinear thermal radiation and chemical reaction effects on the mixed convective flow of Casson and Oldroyd-B fluids.

Gautam et al. [33] investigated electroconvection in dielectric nanofluids with Oldroydian base fluid and top-/bottom-heavy nanoparticles configurations. Khan et al. [34] studied the heat transport in a rheological nanofluid in a stretching sheet which generates motion. They discovered that increasing the Rayleigh number and buoyancy ratio increases the nanoparticles’ temperature considerably. Kavita et al. [35] have examined the electroconvection in pulsating throughflow in a rotational nanofluid porous layer for modified realistic boundaries, analytically. The pulsating throughflow, Lewis and rotation parameters were found to shrink the size of cellular stationary modes. Yasir et al. [36] studied a problem on time-dependent Oldroyd-B fluid flow saturated with nanoparticles using gyrotactic microorganisms.

Oldroydian nanofluids exhibit improved thermal conductivity, which makes them ideal for enhancing heat transfer in various systems such as heat exchangers, cooling systems, and electronic devices and has shown potential in biomedical applications. Their unique flow behaviour and rheological properties make them suitable for drug delivery systems, tissue engineering and biomedical imaging techniques. The novel aspects of Oldroydian nanofluids lie in their viscoelastic behaviour, which distinguishes them from traditional Newtonian fluids. This unique characteristic opens up possibilities for tailoring their flow properties, optimizing heat transfer, and exploring new applications in various fields of science, engineering, and technology.

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 behaviour of the base fluid is characterized by the Oldroyd model.

2. Mathematical modelling

An incompressible and electrically conducting (infinitely) horizontal Oldroydian layer of dielectric nanofluid of depth d separated by two parallel horizontal planes heated uniformly beneath with temperature and nanoparticles volumetric fraction are $T_0,{\varphi }_0$ and $T_1,{\varphi }_1$ at $z=0$ and $z=d$_, respectively, which is shown in Figure 1. A force due to gravity $\mathbf{g}(0,0,-g)$ and rotation (uniform) with an angular velocity Ω(0,0, Ω) pervades the system along $z$- axis with the inclusion of an external vertical AC applied uniformly across the fluid. The bottom plane is taken under an electric circuit and at the top plane, the root mean square potential, $\phi$ is considered. Except for density, all of the fluids’ relevant thermophysical variables are held constant for analytical analysis.

Figure 1. Geometric configuration.

The nanofluid property equations (Buongiorno [15], Nield and Kuznetsov [17] and Sheu [18]) are. (1) $\begin{array}{rl}\rho & =\varphi {\rho }_p+(\text{1}-\varphi ){\rho }_f,\end{array}$(1) (2) $\begin{array}{rl}\mathrm{\rho c}& =\varphi (\mathrm{\rho c}{)}_p+(\text{1}-\varphi )(\mathrm{\rho c}{)}_f,\end{array}$(2) where ρ is the density of the nanofluid, $\varphi$ is the volume fraction of the nanoparticles, ρ_p density of nano particles, ρ_f density of base fluid, $(\mathrm{\rho c}{)}_f$ is the heat capacities of fluid and $(\mathrm{\rho c}{)}_p$ is the heat capacity of nanoparticles

Taking the density of the nanofluid as that of the base fluid, as adopted by (Tzou [16] and Nield and Kuznetsov [17]) the specific weight, ρg becomes. (3) $\begin{array}{r}\mathrm{\rho g}\cong (\varphi {\rho }_p+(1-\varphi )\{\rho (1-\alpha (T-T_0))\})g\end{array}$(3) where α is the coefficient of thermal expansion.

The equations of mass-balance, and momentum-balance relevant to the governing physical system (Oldroyd [20], Sheu [18] and Chand and Rana [19]) are: (4) $\begin{array}{rl}& \mathrm{\nabla }.\mathit{v}=0\end{array}$(4) (5) $\begin{array}{rl}& {\rho }_f\left[\frac{\mathrm{\partial }\mathit{v}}{\mathrm{\partial }t}+(\mathit{v}\cdot \mathrm{\nabla })\mathit{v}\right]\left(1+\overline{{\lambda }_1}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)\\ & =[2{\rho }_f(\mathit{v}\times \mathit{\Omega })-\mathrm{\nabla }P-{\mathit{f}}_{\mathit{e}}\\ & +(\varphi {\rho }_p+(1-\varphi ){\rho }_f\{1-\beta (T-T_1)\})\mathit{g}]\\ & \times \left(1+\overline{{\lambda }_1}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)+\left(1+\overline{{\lambda }_2}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)\mu {\mathrm{\nabla }}^2\mathit{v},\end{array}$(5) where $\mathit{v},\mu ,{\rho }_f,p,T,\mathit{g},{\rho }_p,\beta ,\varphi ,\mathit{\Omega }\overline{,{\lambda }_1}$ and $\overline{{\lambda }_2}$ denote, respectively, the fluid velocity, coefficient of viscosity, base fluid density, pressure, temperature, acceleration due to gravity, the density of nanoparticles, adverse temperature gradient, volume fraction of nanopartcles, angular velocity, stress relaxation and strain retardation.

In Equation (5), the electric origin force ${\mathit{f}}_{\mathit{e}}$ is expressed as (6) $\begin{array}{r}{\mathit{f}}_{\mathit{e}}={\rho }_e\mathit{E}-\frac{1}{2}{\mathit{E}}^2\mathrm{\nabla }ϵ+\frac{1}{2}\mathrm{\nabla }\left(\rho \frac{\mathrm{\partial }ϵ}{\mathrm{\partial }t}{\mathit{E}}^2\right),\end{array}$(6) where ${\rho }_e,\gamma$ and $\mathit{E}$ denote, respectively charged particles density, dielectric constant and electric field, and the pressure term is modified as (7) $\begin{array}{r}P=p+\frac{\rho }{2}\left(\frac{\mathrm{\partial }ϵ}{\mathrm{\partial }t}\right){\mathit{E}}^2.\end{array}$(7) The dielectric constant are described as. (8) $\begin{array}{r}\in ={\in }_0^{\mathrm{\prime }}\{1-{e^{\mathrm{\prime }}}_0(T-T_0)\},\end{array}$(8) where $e_0^{\mathrm{\prime }}>0$ denote the dielectric constant coefficient with very small relative temperature variations and ${\alpha }^{\mathrm{\prime }}$ represents the coefficient for volume expansion.

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 [6] equations take the form. (9) $\begin{array}{r}\mathrm{\nabla }.(\in \mathit{E})=0,\mathrm{\nabla }\times \mathit{E}=\mathit{\text{0}}.\end{array}$(9) The electric field is derived from scalar potential $\phi$, represented by $\mathit{E}=-\mathrm{\nabla }\phi$ and using this value in Equation (9), one gets (10) $\begin{array}{r}{\mathrm{\nabla }}^2\phi =0.\end{array}$(10) where $\phi$. note electric potential (root mean square value).

The nanoparticles conservation equation (Nield and Kuznetsov [17] and Sheu [18]) is. (11) $\begin{array}{r}\left[\frac{\mathrm{\partial }\mathit{v}}{\mathrm{\partial }t}+(\mathit{v}.\mathrm{\nabla })\mathit{v}\right]\varphi =D_B{\mathrm{\nabla }}^2\varphi +\left(\frac{D_T}{T_1}\right){\mathrm{\nabla }}^{2}T.\end{array}$(11) Thermal energy equation of nanofluid (Nield and Kuznetsov [17] and Sheu [18]) is (12) $\begin{array}{rl}(\mathrm{\rho c})\left[\frac{\mathrm{\partial }\mathit{v}}{\mathrm{\partial }t}+(\mathit{v}.\mathrm{\nabla })\right]T& =k_f{\mathrm{\nabla }}^{2}T+(\mathrm{\rho c}{)}_p\\ & \times \left[D_B\mathrm{\nabla \varphi }.\mathrm{\nabla T}+\frac{D_T}{T_1}\mathrm{\nabla T}.\mathrm{\nabla T}\right],\end{array}$(12) Here the parameters $D_T,D_B,(\mathrm{\rho c}{)}_f$, and $(\mathrm{\rho c}{)}_p$ denote the thermophoretic and Brownian Diffusion coefficient, and heat capacities of fluid and nanoparticles. The parameter $k_f$ represents thermal conductivity.

The relevant isothermal boundary conditions (Chandrasekhar [2]) are (13) $\begin{array}{r}w=T=\varphi =\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{z}^2}=\frac{\mathrm{\partial }\phi }{\mathrm{\partial }z}=0\text{when}z=0\text{and}z=d.\end{array}$(13)

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 the vertical direction only. Thus, solutions corresponding to this state using Equations (4), (5), (10) and (12) are (14) $\begin{array}{rl}& \mathit{v}={\mathit{v}}_b=0,T=T_b(z)=T_0-\mathrm{\beta z},p=p_b(z),\\ & ϵ={ϵ}_0(1+\mathrm{e\beta z})={ϵ}_b(z),\mathit{E}=\frac{E_{0}k}{1+e_0(\mathrm{\beta z})}={\mathit{E}}_b(z),\\ & \phi =\frac{E_0\log (1+{ϵ}_0\mathrm{\beta z})}{e_0\beta }={\phi }_b(z)\end{array}$(14) where $\beta =\frac{(T_0-T_1)}{d}$ and $E_0=-\frac{{\phi }_1{e}_0\mathrm{\beta z}}{\log (1+e_0\mathrm{\beta z})}$ represent the adverse thermal gradient and electric field(root mean square value) at $z=0.$ and subscript $b$ indicates basic state

Infinitesimal perturbations are superimposed to the fundamental (basic) state flow in order to check the system stability, taken as. (15) $\begin{array}{rl}& \mathit{q}={\mathit{q}}^{\mathrm{\prime }},\varphi ={\varphi }_b+{\varphi }^{\mathrm{\prime }},T=T_b+T^{\mathrm{\prime }},p=p_b+p^{\mathrm{\prime }},\\ & \mathit{E}={\mathit{E}}_b+{\mathit{E}}^{\mathbf{\prime }},{\in }_1={{\in }_1}_b+{{\in }_1}^{\mathrm{\prime }},\phi ={\phi }_b+{\phi }^{\mathrm{\prime }}\end{array}$(15) where ${\mathit{q}}^{\mathbf{\prime }}\mathbf{,}{\varphi }^{\mathrm{\prime }},T^{\mathrm{\prime }},p^{\mathrm{\prime }},{\mathit{E}}^{\mathbf{\prime }}\mathbf{,}{{\in }_1}^{\mathrm{\prime }}$, and ${\phi }^{\mathrm{\prime }}$ denote the disturbed physical quantities.

Substituting perturbations from Equation (15) and using (14), Equations (5), (10)–(12), ignoring the product of perturbations, we obtain linear non-dimensional equations as (for convenience, asterisks are removed) (16) $\begin{array}{rl}& \left[\left(1+{\lambda }_2\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right){\mathrm{\nabla }}^2-\frac{1}{P_r}\left(1+{\lambda }_1\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right]{\mathit{v}}^{\mathbf{\prime }}\\ & +\left(1+{\lambda }_1\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)(-\mathrm{\nabla }{p}^{\mathrm{\prime }}+R_a{T}^{\mathrm{\prime }}+R_e{T}^{\mathrm{\prime }}-R_n{\varphi }^{\mathrm{\prime }}\\ & +R_e\frac{\mathrm{\partial }{\phi }^{\mathrm{\prime }}}{\mathrm{\partial }z})+{\sqrt{T}}_a(v^{\mathrm{\prime }}\hat{i}-u^{\mathrm{\prime }}\hat{j})\left(1+{\lambda }_1\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)=0,\end{array}$(16) (17) $\begin{array}{rl}& \frac{\mathrm{\partial }{T}^{\mathrm{\prime }}}{\mathrm{\partial }t}-{\mathit{v}}^{\mathrm{\prime }}={\mathrm{\nabla }}^2{T}^{\mathrm{\prime }}+\frac{N_B}{L_n}\left(\frac{\mathrm{\partial }{T}^{\mathrm{\prime }}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{\varphi }^{\mathrm{\prime }}}{\mathrm{\partial }t}\right)-\frac{2N_A{N}_B}{L_n}\frac{\mathrm{\partial }{T}^{\mathrm{\prime }}}{\mathrm{\partial }z},\end{array}$(17) (18) $\begin{array}{rl}& \frac{\mathrm{\partial }{\varphi }^{\mathrm{\prime }}}{\mathrm{\partial }t}+{\mathit{v}}^{\mathbf{\prime }}=\frac{1}{L_n}{\mathrm{\nabla }}^2{\varphi }^{\mathrm{\prime }}+\frac{N_B}{L_n}{\mathrm{\nabla }}^2{T}^{\mathrm{\prime }},\end{array}$(18) (19) $\begin{array}{rl}& {\mathrm{\nabla }}^2{\phi }^{\mathrm{\prime }}+e{E}_0\frac{\mathrm{\partial }{T}^{\mathrm{\prime }}}{\mathrm{\partial }z}=0,\end{array}$(19) where $\begin{array}{rl}& (u,v,w){\alpha }_m=(u^{\ast },v^{\ast },w^{\ast }),(x,y,z)d=(x^{\ast },y^{\ast },z^{\ast }),\\ & \frac{t{d}^2}{{\alpha }_m}=t^{\ast },p\frac{(\mu {\alpha }_m)}{d^2}=p^{\ast },\frac{\phi }{e{E}_0\mathrm{\beta d}}={\varphi }^{\ast },\\ & \varphi =\frac{{\varphi }^{\ast }-{\varphi }_0}{{\varphi }_1-{\varphi }_0},T=\frac{T^{\ast }-T_1}{T_0-T_1},\end{array}$ where ${\alpha }_m=\frac{k_f}{{(\mathrm{\rho c})}_f}$ are the dimensionless variables and $T_a=\frac{4{\Omega }^2{{\rho }_f}^2{d}^4}{{\mu }^2},$ the Taylor number, $\mathrm{Pr}=\frac{\mu }{{{\rho }_f}_{{\alpha }_m}}$, the thermal Prandtl number, $R_e=\frac{{\gamma }_0{e_0}^2{E}_0^2{\beta }^2{d}^2}{\mu {\alpha }_m}$, the Rayleigh number due to the electric field, $L_n=\frac{{\alpha }_m}{D_B}$, the nanofluid Lewis number,$R_m=\frac{[{\varphi }_0{\rho }_P+{{\rho }_f}_{(}1-{\varphi }_0)]g{d}^3}{\mu {\alpha }_m}$, the concentration Rayleigh number $N_B=\frac{{(\mathrm{\rho c})}_P({\varphi }_1-{\varphi }_0)}{{(\mathrm{\rho c})}_f}$, increment in the particle density, $R_a=\frac{d^3\alpha {\rho }_f\mathrm{g\Delta T}}{\mu {\alpha }_m}$, the thermal Rayleigh number $N_A=\frac{D_T(T_0-T_1)}{D_B{T}_1({\varphi }_1-{\varphi }_0)}$, the modified diffusivity ratio, ${\lambda }_1=\frac{\overline{{\lambda }_1}\alpha }{d^2}$, the stress relaxation time and ${\lambda }_2=\frac{\overline{{\lambda }_2}\alpha }{d^2}$, strain retardation time, respectively, are the dimensionless parameters.

Writing Equation (16) in the cartesian form and eliminating pressure term by applying some operators, we get. (20) $\begin{array}{rl}& \left[\left(1+{\lambda }_2\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right){\mathrm{\nabla }}^4-\frac{1}{P_r}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(1+{\lambda }_1\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right){\mathrm{\nabla }}^2\right]{\mathit{w}}^{\mathbf{\prime }}\\ & +(R_a{\mathrm{\nabla }}_1^2{T}^{\mathrm{\prime }}-{\sqrt{T}}_a\frac{\mathrm{\partial }\xi }{\mathrm{\partial }z}+R_e{\mathrm{\nabla }}_1^2{T}^{\mathrm{\prime }}\\ & -R_n{\mathrm{\nabla }}_1^2{\varphi }^{\mathrm{\prime }}+R_e{\mathrm{\nabla }}_1^2\frac{\mathrm{\partial }{\phi }^{\mathrm{\prime }}}{\mathrm{\partial }z})\\ & \times \left(1+{\lambda }_1\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)=0\end{array}$(20) (21) $\begin{array}{rl}& \left[\left(1+{\lambda }_2\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right){\mathrm{\nabla }}^4-\frac{1}{P_r}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(1+{\lambda }_1\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right){\mathrm{\nabla }}^2\right]\xi \\ & =-\left(\frac{\mathrm{\partial }{w}^{\mathrm{\prime }}}{\mathrm{\partial }z}\right){\sqrt{T}}_a,\end{array}$(21) where ${\mathrm{\nabla }}_1^2=\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\right),{\mathrm{\nabla }}^2=\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}\right)$, and $\xi$ are the two-dimensional, three-dimensional Laplacian operators and the vorticity(z-component) respectively.

Eliminating $\xi$ from Equations (20) and (21), we get. (22) $\begin{array}{rl}& \left[\left(1+{\lambda }_2\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right){\mathrm{\nabla }}^4-\frac{1}{P_r}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(1+{\lambda }_1\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right){\mathrm{\nabla }}^2\right]\\ & \times \left[\begin{array}{c}[\left(1+{\lambda }_2{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}\right){\mathrm{\nabla }}^4-{\displaystyle \frac{1}{P_r}}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}\left(1+{\lambda }_1{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}\right){\mathrm{\nabla }}^2]{\mathrm{\nabla }}^2{w}^{\mathrm{\prime }}+R_n{\mathrm{\nabla }}_1^2{\varphi }^{\mathrm{\prime }}-R_a{\mathrm{\nabla }}_1^2{T}^{\mathrm{\prime }}+R_e{\mathrm{\nabla }}_1^2\left(T^{\mathrm{\prime }}-{\displaystyle \frac{\mathrm{\partial }{\phi }^{\mathrm{\prime }}}{\mathrm{\partial }z}}\right)\end{array}\right]\\ & -T_a\frac{{\mathrm{\partial }}^2{w}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^2}=0\end{array}$(22)

3.3. Normal mode method

The coupled differential Equations (17)–(19) and (22) are solved by analysing the perturbed physical quantities as normal modes and are described as. (23) $\begin{array}{r}(w^{\mathrm{\prime }},T^{\mathrm{\prime }},{\varphi }^{\mathrm{\prime }},{\phi }^{\mathrm{\prime }})=(W,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })(z)\times e^{\{i(k_{x}x+k_{y}y)+\mathrm{\sigma t}\}},\end{array}$(23) where $k_x,k_y$, are wave numbers along the x- and y-axes and $\sigma$ is the complex growth rate. $a=(k_x^2+k_x^2{)}^{1/2}$ denotes the resultant wave number.

Using expression (23) in Equations (17)–(19) and (22), we obtain. (24) $\begin{array}{rl}& (D^2-a^2){\left[(1+{\lambda }_2\sigma )(D^2-a^2)-\frac{\sigma }{P_r}(1+{\lambda }_1\sigma )\right]}^{2}W\\ & -\left[(1+{\lambda }_2\sigma )(D^2-a^2)-\frac{\sigma }{P_r}(1+{\lambda }_1\sigma )\right]\\ & \times [a^2(R_a+R_e)\mathrm{\Theta }-a^2{R}_n\mathrm{\Phi }+a^2{R}_{e}D\mathrm{\Psi }]\\ & -T_a{D}^{2}W=0,\end{array}$(24) (25) $\begin{array}{rl}& W+\left[-\sigma +(D^2-a^2)+\frac{N_B}{L_n}D-\frac{2N_A{N}_B}{L_n}\right]\mathrm{\Theta }\\ & -\left(\frac{N_B}{L_n}\right)D\mathrm{\Psi }=0,\end{array}$(25) (26) $\begin{array}{rl}& W+\left(-\frac{N_A}{L_n}\right)(D^2-a^2)\mathrm{\Theta }\\ & +\left[-\frac{1}{L_n}(D^2-a^2)-\sigma \right]\mathrm{\Phi }=0,\end{array}$(26) (27) $\begin{array}{rl}& D\mathrm{\Theta }+(D^2-a^2)\mathrm{\Psi }=0\end{array}$(27)

After utilizing (20), the boundaries (10) yield (28) $\begin{array}{r}W=\mathrm{\Theta }=\mathrm{\Phi }=D^{2}W=D\mathrm{\Psi }=0\mathrm{at}z=0\mathrm{and}z=1\end{array}$(28) The trial functions of Equations (24)–(27) using GWRM satisfying boundary conditions (28) are taken as. (29) $\begin{array}{rl}& W=A_1\sin (\mathrm{\pi z}),\mathrm{\Theta }=B_1\sin (\mathrm{\pi z}),\\ & \mathrm{\Phi }=C_1\sin (\mathrm{\pi z}),\mathrm{\Psi }=D_1\cos (\mathrm{\pi z})\end{array}$(29) here $A_1,B_1,C_1$ and $D_1$ represent unknown constants.

Using these solutions (29) in Equations (24)–(27) and integrating each equation between the limits 0–1 i.e. (0 < z < 1) individually and using boundary conditions (28), one obtains (30) $\begin{array}{rl}& [\begin{array}{ccccc}\left(J{K}_2+{\displaystyle \frac{\sigma }{P_r}}{K}_1\right)J-{\displaystyle \frac{T_a{\pi }^2{K_1}^2}{\left(J{K}_2+{\displaystyle \frac{\sigma }{P_r}}{K}_1\right)}}& -a^2(R_a+R_e)K_{1}1& -J-\sigma 1& {\displaystyle \frac{N_A}{L_n}}J0& -\pi \end{array}\\ & \begin{array}{cc}{a}^2{K}_1{R}_n& -a^2{K}_1{R}_e\pi \\ 0& 0\\ \sigma +{\displaystyle \frac{J}{L_n}}& 0\\ 0& -J\end{array}]\left[\begin{array}{c}{A}_1{B}_1{C}_1{D}_1\end{array}\right]=0,\end{array}$(30) where $J=(a^2+{\pi }^2),K_1=(1+{\lambda }_1\sigma )$ and $K_2=(1+{\lambda }_2\sigma ).$

Equation (30) gives $R_a$ as. (31) $\begin{array}{rl}{R}_a& =\frac{{\pi }^2{R}_e}{J}-R_e-\frac{\{(J+\sigma )L_n+J{N}_A\}{R}_n}{(\sigma L_n+J)}\\ & +\frac{(J+\sigma )P_r\{{\pi }^2{T}_a{(1+\sigma {\lambda }_1)}^2+J^3{(1+\sigma {\lambda }_2)}^2\}}{a^2(1+\sigma {\lambda }_1)\{(1+\sigma {\lambda }_1)\sigma +J(1+\sigma {\lambda }_2)P_r\}}\\ & +\frac{\begin{array}{c}\mathrm{J\sigma }(1+\sigma {\lambda }_2)\{2J{P}_r(1+\sigma {\lambda }_1)\\ (J+\sigma )+\sigma (1+\sigma {\lambda }_2)\}\end{array}}{a^2(1+\sigma {\lambda }_1)P_r\{\sigma (1+\sigma {\lambda }_1)+J{P}_r(1+\sigma {\lambda }_2)\}}.\end{array}$(31)

4. Mathematical analysis

4.1 Oscillatory convection

For oscillatory modes ${\sigma }_r=0$, now putting $\sigma =\mathrm{\iota \omega }$ in Equation (31), we obtain (32) $\begin{array}{r}{R}_a={\mathrm{\Delta }}_1+i{\mathrm{\Delta }}_2\omega \end{array}$(32) Comparing real and imaginary parts of Equation (32), gives oscillatory thermal Rayleigh number (33) $\begin{array}{rl}{R}_a^{\mathrm{osc}}& =\frac{\begin{array}{c}{\omega }^2\{J^2{L}_n{R}_n+{\omega }^2{L}_n^2(R_n+R_e)\\ +2J(J{N}_A{R}_n+(J+{\pi }^2)R_e)\}\\ (-1+2P_r{\lambda }_1-{\omega }^2{\lambda }_1^2)\end{array}}{\begin{array}{c}(J^2+{\omega }^2{L}_n^2)\{{\omega }^2+{(P_r-{\omega }^2{\lambda }_1)}^2\\ +2{\omega }^2{P}_r{\lambda }_2+{\omega }^2{P}_r^2{\lambda }_2^2\}\end{array}}+\\ & -\frac{\begin{array}{c}{a}^2{J}^3{L}_n{P}_r{R}_n{J}_1+{\omega }^2{L}_n^2\{J^2{\omega }^2{J}_1-J^2{P}_r(J+{\omega }^2{\lambda }_1)\\ +a^2{P}_{r}J{R}_n{J}_1+a^2{P}_r(J+{\pi }^2)R_n{J}_1\}\end{array}}{a^{2}J(J^2+{\omega }^2{L}_n^2)P_r{J}_1}\\ & +\frac{J^2{\omega }^2{\lambda }_2(-1+J{\lambda }_1)(J^2+{\omega }^2{L}_n^2)}{a^{2}J(J^2+{\omega }^2{L}_n^2)J_1}\\ & +\frac{\begin{array}{c}{P}_r(J^2{\omega }^2+3J{\omega }^4{\lambda }_1)-J{\omega }^4{J}_1\\ +P_r{\lambda }_2(-3J{\omega }^4+J^2{\omega }^4{\lambda }_1)\end{array}}{\begin{array}{c}{a}^2\{{\omega }^2+{(P_r-{\omega }^2{\lambda }_1)}^2\\ +2{\omega }^2{P}_r{\lambda }_2+{\omega }^2{P}_r^2{\lambda }_2^2\}\end{array}}\\ & +\frac{{\pi }^2{T}_a{P}_r^3{J}_1\{J+{\omega }^2\{{\lambda }_2+{\lambda }_1(-1+J{\lambda }_2)\}\}}{\begin{array}{c}{a}^2{P}_r{J}_1\{{\omega }^2+{(P_r-{\omega }^2{\lambda }_1)}^2\\ +2{\omega }^2{P}_r{\lambda }_2+{\omega }^2{P}_r^2{\lambda }_2^2\}\end{array}}\\ & -\frac{J{\omega }^2\{J+{\omega }^2\{-3{\lambda }_2+{\lambda }_1(3+J{\lambda }_2)\}\}}{a^2\{{\omega }^2+{(P_r-{\omega }^2{\lambda }_1)}^2+2{\omega }^2{P}_r{\lambda }_2+{\omega }^2{P}_r^2{\lambda }_2^2\}}\\ & +\frac{\begin{array}{c}{J}^2\{J^2{\omega }^2{J}_1+-J^2{P}_r(J+{\omega }^2{\lambda }_1)+a^2{P}_{r}J{N}_A{R}_n{J}_1\\ +a^2{P}_r(J+{\pi }^2)R_n{J}_1\}\end{array}}{a^{2}J(J^2+{\omega }^2{L}_n^2)P_r{J}_1}\\ & +\frac{J{J}_1{\omega }^4+J{J}_1{\omega }^6{\lambda }_1^2+{\pi }^2{T}_a{\omega }^2{P}_r^2{J^2}_1}{\begin{array}{c}{a}^2{P}_r{J}_1\{{\omega }^2+{(P_r-{\omega }^2{\lambda }_1)}^2\\ +2{\omega }^2{P}_r{\lambda }_2+{\omega }^2{P}_r^2{\lambda }_2^2\}\end{array}},\end{array}$(33) where $J_1=(1+{\omega }^2{{\lambda }_1}^2)$, and (34) $\begin{array}{r}{a}_1{\omega }^8+a_2{\omega }^6+a_3{\omega }^4+a_4{\omega }^2+a_5=0\end{array}$(34) where $a_1=J^2{L}_n^2{{\lambda }^3}_1({\lambda }_1+P_r{\lambda }_2),$ $\begin{array}{rl}{a}_1& =J^2{L}_n^2{{\lambda }^3}_1({\lambda }_1+P_r{\lambda }_2),\\ a_2& =a^2{L}_{n}J(-1+N_A)P_r{R}_n{\lambda }_1^4+J^4{{\lambda }_1}^3({\lambda }_1+P_r{\lambda }_2)\\ & +J{\lambda }_1{L}_n^2\{P_r(-{\pi }^2{T}_a{P}_r+a^2{R}_n){\lambda }_1^3\\ & +J{P}_r{\lambda }_2{(1+J{P}_r{\lambda }_2)}^2\}+P_r{{\lambda }_1}^{3}J\{-3J^2+{\pi }^2{T}_a{P}_r^2{\lambda }_2\}\\ & +J^2{{\lambda }^2}_1\{2+P_r\{1+J{\lambda }_2(3+P_r(-2+J{\lambda }_2))\}\},\\ a_3& =a^2{P}_r{L}_n{R}_n{\lambda }_1^{2}J(-1+N_A)\{2+J{P}_r\{-2{\lambda }_1\\ & +{\lambda }_2(2+J{P}_r{\lambda }_2)\}\}-{\pi }^2{J}^3{T}_a{P}_r^2{\lambda }_1^4\\ & +J^4{\lambda }_1{P}_r{\lambda }_2(1+J{P}_r{\lambda }_2{)}^2\\ & +P_r{J}^3{\lambda }_1^3\{-3J^2+{\pi }^2{T}_a{P}_r^2{\lambda }_2\}\\ & +J^4{{\lambda }^2}_1\{2+P_r\{1+J{\lambda }_2\{3+P_r(-2+J{\lambda }_2)\}\}\}\\ & +{L^2}_n{J}^2+J^2{L^2}_n{P}_r+J^2{L^2}_n{P^2}_r\{{\pi }^2{T}_a{P}_r-2a^2{R}_n\}{\lambda }_1^3\\ & +{\lambda }_1{P}_r{J}^2{L^2}_n\{-3J^2-2J^2{P}_r\{1+2J{\lambda }_2\}\\ & +P_r^2{\lambda }_2\{J^3+{\pi }^2{T}_a-J^4{\lambda }_2\}\}\\ & +2a^2{R}_n{\lambda }_1^2{L^2}_{n}J{P}_r+{\lambda }_1^2{L^2}_{n}J{P^2}_r\{3J^3-2{\pi }^2{T}_a\\ & +2a^2\{J{R}_n{\lambda }_2\}+P_r^2\{{\pi }^2{T}_a+J{\lambda }_2\{-{\pi }^2{T}_a+a^{2}J{R}_n{\lambda }_2\}\}\}\\ & +J^3{L^2}_n{\lambda }_2{\lambda }_1^2{P^2}_r\{3+P_r\{2+J{\lambda }_2\{3+P_r(1+J{\lambda }_2)\}\}\}.\\ a_4& =J^4+J^4{P}_r+a^2{L}_n{P}_{r}J(-1+N_A)\\ & \times R_n\{1+2J{P}_r(-{\lambda }_1+{\lambda }_2)+J^2{P}_r^2({\lambda }_1^2+{\lambda }_2^2)\}\\ & +a^2{R}_n{L}_n^2{P}_{r}J+J{P^2}_r{L}_n^2(J^3-{\pi }^2{T}_a\\ & +2a^{2}J{R}_n(-{\lambda }_1+{\lambda }_2))+P_r^3{L}_n^{2}J\{J^3+{\pi }^2{T}_a\\ & +{\lambda }_{1}J\{-J^3+{\pi }^2{T}_a+a^{2}J{R}_n{\lambda }_1\}\}\\ & +P_r^3{L}_n^{2}J\{J^2\{J^3-{\pi }^2{T}_a\}{\lambda }_2+a^2{J}^3{R}_n{\lambda }_2^2\}\\ & -J^3{P}_r\{-{\pi }^{2}J{T}_a{P}_r^2{\lambda }_1^3\}\\ & +{P_r}^{2}J{\lambda }_1^2\{-3J^3+2{\pi }^2{T}_a+{\pi }^2{T}_a{P}_r(-1+J{\lambda }_2)\}\\ & \times {\lambda }_1{P}_{r}J\{J^2(3+2P_r)\\ & -P_r\{-4J^3+\{J^3+{\pi }^2{T}_a\}{P}_r\}{\lambda }_2+J^4{P}_r^2{\lambda }_2^2\}\\ & +J^3{\lambda }_2{P}_r\{-3+P_r\{-2-J{\lambda }_2\{3+P_r(1+J){\lambda }_2\}\}\},\\ a_5& =J^3{P}_r^2\{J^3-{\pi }^2{T}_a+P_r\{J^3+{\pi }^2{T}_a+a^2{L}_n^2{R}_n\\ & +a^2{L}_n(-1+N_A)R_n-J\{J^3-{\pi }^2{T}_a\}({\lambda }_1-{\lambda }_2)\}\},\end{array}$

4.2. Stationary convection

Stationary convection sets in the marginal state are characterized by substituting $\sigma =0$ in Equation (31), the thermal Rayleigh of stationary convection, ${R_a}^s$, given by (35) $\begin{array}{r}{R_a}^s=\frac{{(a^2+{\pi }^2)}^3}{a^2}-\frac{a^2{R}_e}{(a^2+{\pi }^2)}-{\mathrm{R}}_n(N_A+L_n)+\frac{{\pi }^2{T}_a}{a^2}.\end{array}$(35) From expression (35), it is apparent that ${R_a}^s$ depends on non-dimensional parameters, namely $a,R_e,L_n,N_A$ and $R_n,T_a$.

Special cases:

Case I: In the absence of electric field, i.e. $R_e=0,$ Equation (35) reduces to (36) $\begin{array}{r}{R_a}^s=\frac{{(a^2+{\pi }^2)}^3}{a^2}-{\mathrm{R}}_n(N_A+L_n)+\frac{{\pi }^2{T}_a}{a^2},\end{array}$(36) which is identical to the result derived by Chand and Rana [19].

Case II: For a non-rotating nanofluid and without electric field i.e. $T_a=R_e=0,$ Equation (35) reduces to. (37) $\begin{array}{r}{R_a}^s=\frac{{(a^2+{\pi }^2)}^3}{a^2}-{\mathrm{R}}_n(N_A+L_n).\end{array}$(37) Equation (37) concides with the result derived by Nield and Kuznetsov [17] and Sheu [18].

Case III: For rheological regular fluid, i.e. ${\mathrm{R}}_n=N_A=0$, Equation (35) further shrinks to (38) $\begin{array}{r}{R}_a^s=\frac{{(a^2+{\pi }^2)}^3}{a^2},\end{array}$(38) which coincides with the prior result of Chandrashekhar [2].

Equation (35), gives (39) $\begin{array}{rl}\frac{d{R}_a^s}{d{T}_a}& =\frac{{\pi }^2}{a^2},\end{array}$(39) (40) $\begin{array}{rl}\frac{d{R}_a^s}{d{R}_e}& =-\frac{a^2}{(a^2+{\pi }^2)},\end{array}$(40) (41) $\begin{array}{rl}\frac{d{R}_a^s}{d{L}_n}& =-R_n,\end{array}$(41) (42) $\begin{array}{rl}\frac{d{R}_a^s}{d{N}_A}& =-R_n,\end{array}$(42) (43) $\begin{array}{rl}\frac{d{R}_a^s}{d{R}_n}& =-(N_A+L_n),\end{array}$(43)

In Equation (39), it is evident that the sign of $\frac{d{R}_a^s}{d{T}_a}$ is always positive for all wave numbers, indicating that the rotation parameter significantly stabilizes the system which verifies the result obtained by Chand and Rana [19]. Conversely, in Equation (40), $\frac{d{R}_a^s}{d{R}_e}$ is negative, indicating that the AC electric field destabilizes the physical system.

In Equations (41) and (42), the positive values of $\frac{d{R}_a^s}{d{L}_n}$ and $\frac{d{R}_a^s}{d{N}_A}$ imply that the Lewis number and modified diffusivity ratio respectively destabilize/stabilize the physical system in the bottom-/top-heavy nanoparticle configurations, when the conditions $(R_n\le 0,N_A<0)/(R_n\ge 0,N_A>0)$ are satisfied.

Furthermore, it is found from Equations (43) that $\frac{d{R^s}_a}{d{R}_n}$ is never positive for $(N_A+L_n)>0$, implying thereby the concentration Rayleigh number destabilizes a system.

5. Numerical discussion

The expressions for thermal Rayleigh number of stationary and oscillatory modes for stress-free boundaries are encapsulated in Equations (35) and (33), respectively. These values are evaluated numerically by applying the software MATHEMATICA −12. Control in the values of $R_n$ depends on the concentration of nanoparticles dispersed at the bounding surfaces and the gap between these surfaces. The permissible values of all the involved parameters used in the range ${10}^2\le L_n\le {10}^3,-1\le \mathrm{Rn}\le 1,{10}^2\le {\mathrm{R}}_e\le {10}^3,{10}^{-1}\le P_r\le 1,{10}^2\le T_a\le {10}^3{10}^{-1}\le {\lambda }_1\le 1,{10}^{-1}\le {\lambda }_2\le 1$ (Roberts [3], Sheu [17], Chand and Rana [19] and Gautam et al. [33]).

To understand the stability of the convective phenomena of the nanofluid layer completely, stationary and oscillatory curves are plotted in $(R_a,a)$ plane.

Figure 2(a,b) shows that a rise in $R_a$ with an augmentation in the rotation parameter $T_a$ is also depicted. Thus, $T_a$ has stabilized a system substantially for both stationary and oscillatory convection. Hence, the influence of the Coriolis force, induced by rotation, serves as a suppressing factor in the development of convection within a nanofluid layer. Consequently, the vertical velocity of the fluid diminishes, effectively minimizing the occurrence of thermal convection.

Figure 2. (a) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of Taylor number $(T_a)$ for bottom – heavy case. (b) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of Taylor number $(T_a)$ for top – heavy case.

Figure 3(a,b) indicates a decrement in $R_a$ with a rise in electric Rayleigh number, which postpones the onset of convection for oscillatory and stationary modes. Hence, the destabilization of a system is depicted by the electric Rayleigh number,$R_e$. This occurs because the physical system’s electrostatic energy increases as a result of a greater electric field, rendering it less stable for both $R_n<0,N_A<0$ and $R_n>0,N_A>0$.

Figure 3. (a) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of electric Rayleigh number $(R_e)$ for bottom case. (b) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of electric Rayleigh number $(R_e)$ for top – heavy case.

The concentration Rayleigh number, R_nis influenced by top-/bottom-heavy nanoparticle arrangements in addition to the distance between both the boundaries The stationary and oscillatory thermal Rayleigh numbers have been plotted versus wavenumber in Figure 4(a,b) for various concentration Rayleigh numbers for both top-/bottom-heavy nanoparticles distribution, respectively. As expected, $R_n$ has destabilizing/stabilizing effect for the top-/ bottom-heavy nanoparticles arrangement on both the stationary/oscillatory modes.

Figure 4. (a) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of concentration Rayleigh number $(R_n)$ for bottom – heavy case. (b) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of concentration Rayleigh number $(R_n)$ for top – heavy case.

Figure 5(a,b) represents the plot of $R_a$ versus $a$ for distinct values of nanofluid Lewis number, in bottom-/top-heavy arrangements, respectively. It has been depicted from the curves that as $L_n$ rises/falls, the thermal Rayleigh number of stationary modes gets reduced/enhanced. Hence, $L_n$ has destabilizing/stabilizing impact on stationary convection in top-/bottom-heavy arrangements, whilst reversing the trend for oscillatory thermal Rayleigh number.

Figure 5. (a) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of nanofluid Lewis number $(L_n)$ for bottom – heavy case. (b) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of nanofluid Lewis number $(L_n)$ for top – heavy case.

The impact of various stress relaxation parameters on the onset of oscillatory convection in top-/bottom-heavy arrangements is illustrated in Figure 6(a,b) and the oscillatory thermal Rayleigh number is found to fall as the stress relaxation parameter increases. As a result, ${\lambda }_1$ has contracted the stability regime.

Figure 6. (a) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of stress relaxation parameter $({\lambda }_1)$ for bottom – heavy case. (b) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of stress relaxation parameter $({\lambda }_1)$ for top – heavy case.

The effect of various stain retardation parameters on oscillatory modes is assessed inFigure 7(a,b) for both bottom-/top-heavy nanoparticles configuration. The regime of oscillatory modes gets expanded with an increment in ${\lambda }_2$, leading to the stability of a system.

Figure 7. (a) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of stain retardation parameter $({\lambda }_2)$ for bottom – heavy case. (b) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of stain retardation parameter $({\lambda }_2)$ for top – heavy case.

Figure 8(a,b) depicts the destabilizing impact of the Prandtl number on the emergence of oscillatory modes. Notably, for bottom-/top-heavy distributions, the critical wave number falls as the value of $P_r$ rises, implying thereby that the thermal Prandtl number has sped up the onset of oscillatory convection. All the results obtained from Figures 2–8(b) are in good agreement with the result derived by Roberts [3], Sheu [17], Chand and Rana [19] and Gautam et al. [33].

Figure 8. (a) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of thermal Prandtl number $(P_r)$ for bottom – heavy case. (b) Graph of Rayleigh number $(R_a)$ versus wave number $(a)$ as a function of thermal Prandtl number $(P_r)$ for top – heavy case.

6. Conclusions

The combined effect of rotation and an AC electric field in the rheological nanofluid layer is investigated by employing the Oldroyd model for both kinds of nanoparticle distribution (top-/bottom-heavy). The main conclusions include:

• The angular velocity due to rotation and rheology described by strain retardation viscoelastic parameter postpones the start of convection for bottom-/top-heavy configurations, whilst the electric field reverses the trend.

• The stress relaxation parameter and thermal Prandtl number advance the start of oscillatory convection thereby destabilizing a physical system.

• The modified diffusivity ratio contracts/expands the region of stability and the concentration Rayleigh number expands/shrinks under oscillatory and stationary modes, respectively for top-/bottom- heavy nanoparticles distributions.

• The presence of nanoparticles, characterized by the Lewis number, has contrasting effects on the initiation of convection. It causes a delay in the onset of stationary convection while promoting the advancement of oscillatory convection.

• The bottom-/top-heavy nanofluids are found to be more/less stable.

For future investigations, it is advisable to expand the research by incorporating nonlinear stability analysis and to study the effect of magnetic fields.

Acknowledgement

The authors would like to thank the editor and reviewers for their grammatical corrections, and critical comments which enhanced the quality and clarity of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).