Micropolar nanofluid overlying a porous layer: Thermosolutal convection

Abstract

An investigation of the stability of an micropolar nanofluid overlying a sparsely packed porous medium and implanted in a parallel conduit is reviewed. Linear and also nonlinear terms are incorporated for the study. A Darcy-Brinkman-Forchheimer drag force model is deployed. To evaluate nanoscale effects the Buongiorno model is employed. The equations for mass, momentum, angular momentum, energy and nanoparticle species conservation with correlated wall conditions are non-dimensionalized. Modified diffusivity ratio and Lewis number stable the system, the micropolar parameters concentration Rayleigh number destable system for stationary convection. Concentration Rayleigh number, micropolar parameters stabilize and Lewis number destabilizes the system for oscillatory convection. Applications of the study include micro/nano-fluidic devices, nano-doped energy systems and packed beds in chemical engineering.

Introduction

Eringen ¹ explained the concept of micromorphic or “simple microfluids.” This elegant mathematical framework model certain microscopic effects. A simpler division of microfluids was also contributed by Eringen ² which exhibit only micro-rotational effects and possess micro-rotational inertia and such fluids are known as micropolar fluids. Micropolar fluids provides a rheological formulation of balance equations required for microfluids and thereby greatly facilitates the extraction of analytical and numerical solutions in boundary value problems. Micropolar fluids are characterized by a non-symmetric stress tensor and physically represent viscous fluids containing non-deformable (rigid) particles (micro-elements) which can sustain body couples, couple stresses, and possess a rotational filed. Eringen ³ also presented the applications related to micropolar fluids. Micropolar fluids constitute a non-Newtonian fluid dynamics and as such have been deployed to simulate accurately many diverse liquids arising in industrial, medical and environmental applications. Some recent studies featuring micropolar fluid dynamics have examined geophysical convective plumes, ⁴ stratified flows, ⁵ bacterial slime dynamics, ⁶ and stagnation coating flows of engineering constitutional components. The investigations have also demonstrated the exceptional versatility of this rheological model.

In many technological and geophysical systems, thermal instability (also known as Rayleigh-Bénard convection) plays a momentous aspect. The micropolar fluid convection was first advised by Ahmadi. ⁷ Datta and Sastry ⁸ concluded that for Rayleigh numbers to be positive or negative (negative Rayleigh number is unrealistically high for actual physical systems) the convection sets on. Lebon and Perez-Garcia ⁹ observed that for $N_5\ne 0$ the convection sets on for stationary case. Pérez-Garcia et al.^10,11 showed adopting micropolar fluid that, for stationary convection the subcritical stability was not possible. Dhiman et al. ¹² opted a variational finite element technique to compute Rayleigh-Bénard convection using different types of boundary conditions on micropolar fluid. They showed that greater microrotation characteristic accelerate the onset of instability using dissimilar boundaries. They further proved that for the rigid-free wall condition, the critical Rayleigh number is elevated for increasing wave numbers and shows high sensitivity to micropolar effects. Sharma and Gupta ¹³ discussed the stability using micropolar fluids and confirmed that suspended particles and rotation parameter can instigate over-stability in the system. They further concluded that the critical Rayleigh number for stationary and over-stability convection is boosted with increasing rotation parameters whereas it is suppressed with greater micropolar effects by fixing wave number. Further studies have been communicated by Payne and Straughan ¹⁴ (on nonlinear micropolar thermal stability in permeable media), Goyal and Jaimala ¹⁵ (on double-diffusive micropolar convection with cross diffusion effects) and Sharma et al. ¹⁶ (on ferromagnetic thermal convection onset in porous media). All these studies have established the significant impact of micropolar material properties to understand the stability. Although micropolar fluids are in themselves a comprehensive pattern for simulating microstructural effects in non-Newtonian fluids, they have also been analyzed in conjunction with other rheological effects. These include shear-thinning/shear-thickening micropolar boundary layer flows, ¹⁷ viscoplastic micropolar stretching sheet flows ¹⁸ and more recently micropolar viscoelastic enrobing flows on a cone with the Jeffreys model. ¹⁹

The onset of convection using porous matrix is of great interest in modern engineering systems and arises in for example, microfluidic devices, heat exchangers, cooling of electronics components, building insulations, and drying processes. Owing to the high surface area achieved with porous fibers, fluid mixing characteristics, higher thermal conductivity, and facility for deployment, natural convection through porous bed has attracted considerable attention in recent years. A comprehensive review of many studies has been provided by Nield and Bejan ²⁰ and their survey includes wavy surface instability, double-diffusive convection, stratified flows, mixed convection in non-Newtonian sandwiched layers, super-critical convection in geophysics, vapor front instability, phase change etc.

In the last several years, an ingenious method for developing the thermal transfer characteristics by suspending ultra fine metallic particles in common fluids such as water and oil was investigated. The term nanofluid relates to these colloidal fluids engineered at the nanoscale. Nanofluids are applicable in automotive industries, energy saving devices, nuclear reactors, etc. The unique characteristic property of nanofluids is to increase the thermal conductivity. Nanofluids have great potential as coolants due to their enhanced thermal conductivities. Nanoparticles also have biomedical applications including cancer therapy and nano-drug delivery. Eastman et al. ²¹ and Choi et al. ²² presented that thermal conductivity was enlarged by as much as 160% when a less number of nanoparticles or nanotubes was added to the base fluid. Buongiorno ²³ predicted that the sum of the relative (slip) velocity and velocity of the base fluid to be the velocity of the nanoparticles. Nanofluids play a compelling role for thermal management in next-generation microfluidic/nanofluidic devices, energy systems, fuels, lubricants etc. The review papers by Kakaç and Pramuanjaroenkij, ²⁴ Fan and Wang, ²⁵ and Yu and Xie ²⁶ have manifested that the research on nanofluids using CNTs (carbon nanotubes) and metallic/metallic oxide nanoparticles was in large scale. Shekar et al. ²⁷ investigated numerically to study natural convection in a square cavity filled with a nanofluid porous medium. They concluded that the Brownian motion effects reduces the absolute maximum value of streamlines and isoconcentrations. The Brownian motion reduces the skin friction coefficient and Nusselt numbers. The thermophoresis force increases the values of average skin friction coefficient and Nusselt number along with the increase in radiation parameter. Shekar et al. ²⁸ also investigated the natural convection flow in a porous square cavity filled with a nanofluid in the presence of magnetic field and viscous dissipation. They claimed that the Brownian motion reduces the skin friction coefficient and Nusselt numbers. The local skin friction and local Nusselt number decreases with the increase in Eckert number and Brownian motion parameter while it increases with the parameters of thermophoresis, magnetic field, and buoyancy ratio. Finite element analysis of magnetohydrodynamic transient free convection flow of nanofluid over a vertical cone with thermal radiation was performed by Balla and Naikoti. ²⁹ They remarked that the velocity profile decreases with magnetic parameter, Lewis number and Prandtl number and increases with radiation parameter, Brownian motion parameter, thermophoresis parameter, ratio of Grashof numbers and power law index parameter. The combined effect of viscous and Ohmic dissipations on unsteady, laminar magneto convection fully developed flow in a vertical rectangular duct considering the effects of heat source/sink was investigated by Kishan and Shekar. ³⁰

The non-Newtonian nature of nanofluids at certain volume fraction fractions (percentage doping) has also been confirmed experimentally in many studies. These include pseudo plastic (shear thinning) performance copper oxide or alumina nanoparticles dispersed in carboxymethyl cellulose (CMC), ³¹ viscoelasticity, ³² glycol nanofluids, ³³ plug flow in multiwall carbon nanotubes and magnesium oxide nanoparticles suspended in engine oil base fluid ³⁴ and titanium dioxide-bentonite drilling mud fluids. ³⁵ Many non-Newtonian models have been explored to define discrete nanofluids in applications including Carreau shear-thinning models, ³⁶ power-law pseudoplastic/dilatant models,^37,38 Casson viscoplastic models, ³⁹ the Johnson-Segalman viscoelastic model, ⁴⁰ and Oldroyd-B viscoelastic formulation. ⁴¹ Using porous matrix, several analyses of the in Newtonian/ non-Newtonian nanofluid on the stability have also been communicated. These include Bhadauria and Agarwal ⁴² who also examined rotational body forces. Binary nanofluid was chosen by Agarwal et al. ⁴³ to investigate thermosolutal convective instability. Buongiorno two-component model was employed by Sheu ⁴⁴ to research the thermal convection in an Oldroyd-B nanofluid. He established that there is competition between Brownian motion, thermophoretic body force and viscoelastic parameter (Weissenberg number) leading to a dominance of oscillatory over stationary modes. He also identified that with viscoelasticity, oscillatory instability may arise in both bottom- and top-heavy nanoparticle distributions cases. These studies did not however consider microstructural aspect of nanofluids. Several detailed experimental and molecular dynamics computational studies have confirmed that the rotation of nanoparticles and their shape and size can dramatically influence migration.^45–47 These studies have among other aspects identified that the irregular micro-motions of nanoparticles can substantially enhance acceleration which in turn control properties of the thermal conductivity. This may explain other phenomena including clustering and heterogenous thermal distributions recorded in rheological nanofluids. ⁴⁸ The micropolar model therefore provides a logical framework for exploring spin of suspended particles in nanofluids and more elaborate simulation of their behavior. A number of researchers have therefore examined transport phenomena in engineering using micropolar nanofluid. Recent investigations include Abdul Latiff et al. ⁴⁹ who used Maple software to compute the transient slip micropolar nanofluid dynamics in extrusion/contracting sheet flows including micro-organism bioconvection. Rafique et al. ⁵⁰ reported on Hiemenz convection flows of micropolar nanofluids from a tilted surface. Bourantas and Loukopoulos ⁵¹ employed a meshless point collocation method with a velocity-correction algorithm and the Tiwari-Das model to enumerate convection in a tilted square cavity containing micropolar-nanofluid (aluminum oxide-water) at different volume fractions. Prasad et al. ⁵² presented finite difference solutions for axisymmetric thermosolutal micropolar nanofluid along a cylinder using a modified Buongiorno nanoscale model.

In this investigation, a more detailed appraisal of microstructural characteristics for the permeable micropolar nanofluid stability analysis is presented. A non-Darcian model is implemented considering hydrodynamic drag effects. The Buongiorno pattern is utilized to replicate nanoscale effects (Brownian motion and thermophoretic body force). Extensive visualization of solutions and different convection modes (stationary, oscillatory) are presented. Furthermore, new insight into gyro-viscosity micropolar material parameter effects is elucidated via a regional diagram in which variations of critical Rayleigh and wave numbers are delineated into different quadrant regions. The inspection therefore constitutes an important advancement in current understanding of micropolar nanofluid thermosolutal instability simulation of relevance to geothermics, micro/nanofluid systems and materials fabrication.

Formulation and method of solution

The regime to be analyzed comprises a micropolar nanofluid-saturated porous layer, and the top and bottom walls are heated with a temperature difference $\Delta T$ (confined between two infinite horizontal plate boundaries located at z* = 0 and z* = H) The objective model is exposed in Figure 1. Following Eringen^1,3 and Buongiorno, ²³ the mass, momentum, angular momentum (micro-rotation), energy and nanoparticle species conservation equations together with the non-Darcy model incorporating the Boussinesq paradox may be presented as:

$${\nabla }^{*}.{\mathbf{v}}_D^{*}=0$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\rho }_0}{\epsilon }\frac{\partial {\mathbf{v}}_D^{*}}{\partial t^{*}}=-{\nabla }^{*}{p}^{*}-\frac{(\mu +\kappa )}{k_1}{\mathbf{v}}_D^{*}+(\mu +\kappa ){{\nabla }^{*}}^2{\mathbf{v}}_D^{*}}\\ \multicolumn{1}{c}{-\frac{b}{k_1}|{\mathbf{v}}_D^{*}|{\mathbf{v}}_D^{*}+\kappa \nabla \times {\mathbf{\omega }}^{*}-\rho g}\end{array}$$

(2)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_{0}j[\frac{\partial {\mathbf{\omega }}^{*}}{\partial t^{*}}+{\mathbf{v}}_D^{*}·{\nabla }^{*}{\mathbf{\omega }}^{*}]=(\alpha \prime +\beta \prime ){\nabla }^{*}({\nabla }^{*}·{\mathbf{\omega }}^{*})}\\ \multicolumn{1}{c}{+\gamma {{\nabla }^{*}}^2{\mathbf{\omega }}^{*}+\kappa ({\nabla }^{*}\times {\mathbf{v}}_D^{*}-2{\mathbf{\omega }}^{*})}\end{array}$$

(3)

$$\begin{array}{c}{\left(\rho c\right)}_{\mathrm{m}}\frac{\partial T^{*}}{\partial t^{*}}+{\left(\rho c\right)}_{\mathrm{f}}{\mathbf{v}}_{\mathrm{D}}^{*}.{\nabla }^{*}{T}^{*}=k_{\mathrm{m}}{\nabla }^{*2}{T}^{*}\hfill \\ \multicolumn{1}{c}{+\delta ({\nabla }^{*}\times {\mathbf{\omega }}^{*})·{\nabla }^{*}{T}^{*}+}\\ \epsilon {\left(\rho c\right)}_{\mathrm{p}}[D_{\mathrm{B}}{\nabla }^{*}{\varphi }^{*}.\nabla T^{*}+D_{\mathrm{T}}\frac{\nabla T^{*}.\nabla T^{*}}{T^{*}}]\hfill \end{array}$$

(4)

$$\frac{\partial {\varphi }^{*}}{\partial t^{*}}+\frac{1}{\epsilon }{\mathbf{v}}_D^{*}.\nabla {\varphi }^{*}={\nabla }^{*}.[D_B{\nabla }^{*}{\varphi }^{*}+D_T\frac{{\nabla }^{*}{T}^{*}}{T^{*}}]$$

(5)

Figure 1.

Schematic diagram.

where ${\mathbf{v}}_D^{*}$ , ${\mathbf{\omega }}^{*}$ and $T^{*}$ are velocity, spin and temperature; Here, ${\mathbf{v}}_D^{*}$ is the nanofluid Darcy velocity. Writing ${\mathbf{v}}_D^{*}=(u^{*},v^{*},w^{*})$ $\rho$ and ${\rho }_0$ are total density, and reference density, $k_1$ is the medium permeability, $b$ is Forchheimer constant for nonlinear velocity term; ${\varphi }^{*}$ is the nanoparticle volume fraction, $\epsilon$ is the porosity, $T^{*}$ is the temperature, $D_B$ is Brownian diffusion, $D_T$ is the thermophoretic diffusion coefficient, $p,$ pressure, $\mu ,$ viscosity, $g$ gravity and $j$ microinertia; $\alpha \prime ,\beta \prime$ , $\gamma$ and $\kappa$ are Eringen’s micropolar viscosity coefficients. $(\rho c{)}_m$ mixture heat capacity, ${\left(\rho c\right)}_{\mathrm{f}}$ fluid heat capacity, $k_{\mathrm{m}}$ the thermal conductivity and $\delta$ the coefficient denoting the link between thermal and the microelement spin (gyratory motion). The state equation is given by:

$$\rho ={\varphi }^{*}{\rho }_{\mathrm{p}}+(1-{\varphi }^{*}){\rho }_0[1-{\beta }_{\mathrm{T}}(T^{*}-T_0^{*})]$$

(6)

Here $T_0$ is the temperature (reference) at the lower boundary, ${\rho }_{\mathrm{p}}$ is the particle density, and ${\beta }_{\mathrm{T}}$ is the heat expansion coefficient. volumetric fraction of the nanoparticles and temperature are treated to be constant on the boundaries.

Hence the wall settings are (following Buongiorno ²³ ):

$$w^{*}=0,{\mathbf{\omega }}^{*}=0,T^{*}=T_0^{*}+\Delta T^{*},{\varphi }^{*}={\varphi }_0^{*}\mathrm{on}{z}^{*}=0$$

(7)

$$w^{*}=0,{\mathbf{\omega }}^{*}=0,T^{*}=T_0^{*},{\varphi }^{*}={\varphi }_1^{*}\mathrm{on}{z}^{*}=H$$

(8)

Next dimensionless variables invoked are:

$$\begin{array}{c}\multicolumn{1}{c}{(x,y,z)=(x^{*},y^{*},z^{*})/H,t=t^{*}{\alpha }_m/\sigma H^2,}\\ \multicolumn{1}{c}{(u,v,w)=(u^{*},v^{*},w^{*})H/{\alpha }_m,p=p^{*}{H}^2/\mu {\alpha }_m,}\\ \multicolumn{1}{c}{\varphi =\frac{{\varphi }^{*}-{\varphi }_0^{*}}{{\varphi }_1^{*}-{\varphi }_0^{*}},T=\frac{T^{*}-T_0^{*}}{\Delta T^{*}}}\end{array}\}$$

(9)

Here ${\alpha }_m=\frac{k_m}{{(\rho c_p)}_f},\sigma =\frac{{(\rho c_p)}_m}{{(\rho c_p)}_f}$ .

Equations (1)–(8) therefore emerge as:

$$\nabla .\mathbf{v}=0$$

(10)

$$\begin{array}{c}\multicolumn{1}{c}{({\gamma }_a\frac{\partial \mathbf{v}}{\partial t}+\nabla p+\mathrm{Rm}{\hat{e}}_z-R{a}_{T}T{\hat{e}}_z+\mathrm{Rn}\varphi {\hat{e}}_z)}\\ \multicolumn{1}{c}{+\frac{(1+N_1)}{K}-(1+N_1){\nabla }^2\mathbf{v}-N_1(\nabla \times \mathbf{\omega })=0}\end{array}$$

(11)

$$N_2\frac{\partial \mathbf{\omega }}{\partial t}=N_3\nabla (\nabla ·\mathbf{\omega })+N_4{\nabla }^2\mathbf{\omega }+N_1(\nabla \times \mathbf{v}-2\mathbf{\omega })$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial T}{\partial t}+\mathbf{v}.\nabla T={\nabla }^{2}T+N_5(\nabla \times \mathbf{\omega })\nabla T+\frac{N_B}{\mathrm{Ln}}\nabla \varphi .\nabla T}\\ \multicolumn{1}{c}{+\frac{N_A{N}_B}{\mathrm{Ln}}\nabla T.\nabla T}\end{array}$$

(13)

$$\frac{1}{\sigma }\frac{\partial \varphi }{\partial t}+\frac{1}{\epsilon }\mathbf{v}.\nabla \varphi =\frac{1}{\mathrm{Ln}}{\nabla }^2\varphi +\frac{N_A}{\mathrm{Ln}}{\nabla }^{2}T$$

(14)

$$\begin{array}{c}w=0,{\mathbf{\omega }}^{*}=0,T=1,\varphi =0\mathrm{at}z=0,\hfill \\ \multicolumn{1}{c}{w=0,{\mathbf{\omega }}^{*}=0,T=0,\varphi =0\mathrm{at}z=1}\end{array}\}$$

(15)

Here the following definitions apply:

$$\begin{array}{c}{N}_1=\frac{\kappa }{\mu },N_2=\frac{j}{H^2},K=\frac{k_1}{H^2},N_3=\frac{(\alpha +\beta )}{\mu H^2},\hfill \\ \multicolumn{1}{c}{N_4=\frac{\gamma }{\mu H^2},N_5=\frac{\delta }{{\left(\rho c\right)}_f{H}^2},}\\ \multicolumn{1}{c}{{\gamma }_a=\frac{\epsilon }{\sigma \mathrm{Va}},\mathrm{Va}=\frac{{\epsilon }^2⪻}{\mathrm{Da}},⪻=\frac{{\mu }_f}{\rho {\alpha }_m},\mathrm{Da}=\frac{K}{H^2},}\\ \multicolumn{1}{c}{\mathrm{Ln}=\frac{{\alpha }_m}{D_B},R{a}_T=\frac{{\rho }_0\mathrm{gK}(1-{\varphi }_0^{*}){\beta }_{T}H\Delta T^{*}}{{\mu }_f{\alpha }_m},}\\ \multicolumn{1}{c}{\mathrm{Rm}=\frac{[{\rho }_p{\varphi }_0^{*}+{\rho }_0(1-{\varphi }_0^{*})]\mathrm{gKH}}{{\mu }_f{\alpha }_m},\mathrm{Rn}=\frac{({\rho }_p-{\rho }_0)({\varphi }_1^{*}-{\varphi }_0^{*})\mathrm{gKH}}{{\mu }_f{\alpha }_m},}\\ \multicolumn{1}{c}{N_A=\frac{D_T\Delta T^{*}}{D_B{T}_c^{*}({\varphi }_1^{*}-{\varphi }_0^{*})},}\\ \multicolumn{1}{c}{N_B=\frac{{(\rho c)}_p({\varphi }_1^{*}-{\varphi }_0^{*})}{{(\rho c)}_f}}\end{array}\}$$

(16)

The parameter ${\gamma }_a$ is the non-dimensional acceleration coefficient, $\mathrm{Ln}$ is a Lewis number, $R{a}_T$ is the thermal Rayleigh–Darcy number, $\mathrm{Va}$ is a Vadász number, Pr is the Prandtl number, and $\mathrm{Da}$ is the Darcy number. Rm basic-density Rayleigh number, and Rn is concentration Rayleigh number, respectively. $N_B$ is a modified particle-density increment and $N_A$ is a modified diffusivity ratio. Equation (11) has been linearized by not including the product of T and $\phi$ .

Basic solution

The solutions of equations (10)–(15) are considered to be time-independent quiescent varying in the z-direction only and takes the form

$$\mathbf{v}=0,{\mathbf{\omega }}^{*}=0,p=p_b(z),T=T_b(z),\varphi ={\varphi }_b(z)$$

(17)

Equations (11)–(14) reduce to:

$$0=-\frac{d{p}_b}{\mathrm{dz}}-\mathrm{Rm}+R{a}_T{T}_b-\mathrm{Rn}{\varphi }_b$$

(18)

$$\frac{d^2{T}_b}{d{z}^2}+\frac{N_B}{\mathrm{Ln}}\frac{d{\varphi }_b}{\mathrm{dz}}\frac{d{T}_b}{\mathrm{dz}}+\frac{N_A{N}_B}{\mathrm{Ln}}{\left(\frac{d{T}_b}{\mathrm{dz}}\right)}^2=0$$

(19)

$$\frac{d^2{\varphi }_b}{d{z}^2}+N_A\frac{d^2{T}_b}{d{z}^2}=0$$

(20)

According to Buongiorno, ²³ for most nanofluids $\mathrm{Ln}/\mathrm{Ln}({\varphi }_1^{*}-{\varphi }_0^{*})({\varphi }_1^{*}-{\varphi }_0^{*})$ is large, of the order of ${10}^5$ – ${10}^6$ . The volume fraction of the nanoparticles decrement is typically not less than ${10}^3$ and this suggest that Lewis number, Ln is large, of order ${10}^2$ – ${10}^3$ , while $N_A$ is less than 10. Hence the basic solutions become

$$T_b=1-z\mathrm{and}\mathrm{so}{\varphi }_b=z$$

(21)

Perturbation solution

By imposing the perturbations on the basic solution. Using:

$$\mathbf{v}=\mathbf{v}\prime ,\mathbf{\omega }=\mathbf{\omega }\prime ,p=p_b+p\prime ,T=T_b+T\prime ,\varphi ={\varphi }_b+\varphi \prime$$

(22)

Substitution of equation (22) in equations (10)–(14), and linearization via neglecting products of primed quantities, results as after using equation (21) is:

$$\nabla .\mathbf{v}\prime =0$$

(23)

$$\begin{array}{c}\multicolumn{1}{c}{({\gamma }_a\frac{\partial \mathbf{v}\prime }{\partial t}+\nabla p\prime -R{a}_{T}T\prime {\hat{e}}_z+\mathrm{Rn}\varphi \prime {\hat{e}}_z)+\frac{(1+N_1)}{K}}\\ \multicolumn{1}{c}{-(1+N_1){\nabla }^2\mathbf{v}\prime -N_1(\nabla \times \mathbf{\omega }\prime )=0}\end{array}$$

(24)

$$N_2\frac{\partial \mathbf{\omega }\prime }{\partial t}=N_3\nabla (\nabla ·\mathbf{\omega }\prime )+N_4{\nabla }^2\mathbf{\omega }\prime +N_1(\nabla \times \mathbf{v}\prime -2\mathbf{\omega }\prime )$$

(25)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial T\prime }{\partial t}-w\prime ={\nabla }^{2}T\prime -N_5(\nabla \times \mathbf{\omega }\prime )+\frac{N_B}{\mathrm{Ln}}(\frac{\partial T\prime }{\partial z}-\frac{\partial \varphi \prime }{\partial z})}\\ \multicolumn{1}{c}{-\frac{2N_A{N}_B}{\mathrm{Ln}}\frac{\partial T\prime }{\partial z}}\end{array}$$

(26)

$$\frac{1}{\sigma }\frac{\partial \varphi \prime }{\partial t}+\frac{1}{\epsilon }w\prime =\frac{1}{\mathrm{Ln}}{\nabla }^2\varphi \prime +\frac{N_A}{\mathrm{Ln}}{\nabla }^{2}T\prime$$

(27)

$$w\prime =0,\mathbf{\omega }\prime =0,T\prime =0,\varphi \prime =0\mathrm{at}z=0\mathrm{and}\mathrm{at}z=1$$

(28)

$\mathrm{Rm}$ merely quantifies the basic pressure gradient and therefore is not connected in the following analysis.

Operating on equation (24) with ${\hat{e}}_z$ curl curl and applying the relation curl curl $\equiv$ grad div - ${\nabla }^2$ and imposing weak heterogeneity assumption on equation (23), yields the following equation.

$$\begin{array}{c}\multicolumn{1}{c}{(s{\gamma }_a+\frac{(1+N_1)}{K}-(1+N_1){\nabla }^2){\nabla }^2{w}^{\prime }}\\ \multicolumn{1}{c}{=R{a}_T{\nabla }_H^{2}T\prime -\mathrm{Rn}{\nabla }_H^2{\varphi }^{\prime }+N_1{\nabla }^{2}Z\prime }\end{array}$$

(29)

${\nabla }_H^2$ is the two-dimensional Laplacian operator on the horizontal plane.

The differential equations (25), (26), (27), (29) along with (28) constitute a linear boundary-value problem and is determined adopting the method of normal modes. We write:

$$\begin{array}{c}\multicolumn{1}{c}{(w\prime ,T\prime ,\varphi \prime ,Z\prime )=}\\ \multicolumn{1}{c}{[W(z),\mathbf{\Theta }(z),\mathbf{\Phi }(z),Z(z)]\exp (\mathrm{st}+\mathrm{ilx}+\mathrm{imy})}\end{array}$$

(30)

Implementation in the differential equations (25)–(27) and (29) leads to:

$$\begin{array}{c}\multicolumn{1}{c}{(\frac{s{\gamma }_a}{(1+N_1)}+\frac{1}{K}-(D^2-{\alpha }^2))(D^2-{\alpha }^2)}\\ \multicolumn{1}{c}{W=\frac{R{a}_T{\alpha }^2}{(1+N_1)}T-\frac{\mathrm{Rn}{\alpha }^2}{(1+N_1)}{\varphi }^{\prime }+\frac{N_1}{(1+N_1)}(D^2-{\alpha }^2)Z}\end{array}$$

(31)

$$(\frac{N_{2}s}{N_4}+\frac{2N_1}{N_4}-(D^2-{\alpha }^2))Z=-\frac{N_1}{N_4}(D^2-{\alpha }^2)W$$

(32)

$$(s-(D^2-{\alpha }^2))\mathbf{\Theta }=W-N_{5}Z$$

(33)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{\epsilon }W-\frac{N_A}{\mathrm{Ln}}(D^2-{\alpha }^2)\mathbf{\Theta }-(\frac{1}{\mathrm{Ln}}(D^2-{\alpha }^2)-\frac{1}{\sigma }s)\mathbf{\Phi }=0}\end{array}$$

(34)

$$W=0,\mathbf{\Theta }=0,\mathbf{\Phi }=0,Z=0\mathrm{at}z=0\mathrm{and}z=1$$

(35)

where

$$D\equiv \frac{d}{\mathrm{dz}}\mathrm{and}\alpha =(l^2+m^2{)}^{1/2}$$

(36)

Thus $\alpha$ is a dimensionless horizontal wave number.

For neutral stability the real part of s is zero and $s=i\omega$ , where $\omega$ is real denoting the frequency. By the Galerkin method approximate results are established for the equations (31)–(35). The trial functions $W_p,{\mathbf{\Theta }}_p,{\mathbf{\Phi }}_p,Z_p;p=1,2,3......$ are:

$$\begin{array}{c}\multicolumn{1}{c}{W=\sum _{p=1}^N{A}_p{W}_p,\mathbf{\Theta }=\sum _{p=1}^N{B}_p{\mathbf{\Theta }}_P,}\\ \multicolumn{1}{c}{\mathbf{\Phi }=\sum _{p=1}^N{C}_p{\mathbf{\Phi }}_P,Z=\sum _{p=1}^N{D}_p{Z}_P,}\end{array}$$

(37)

A system of 4N linear algebraic equations in 4N unknowns $A_p,B_p,C_p,D_p$ , p = 1, 2, . . . N are obtained by making the residuals orthogonal to the trial functions. The eigen values are extracted by vanishing the determinant of coefficients. $R{a}_T$ become one of the eigenvalue which helps to compute $R{a}_T$ in terms of other parameters. Trial functions satisfying the boundary condition (37) are:

$$W_p={\mathbf{\Theta }}_p={\mathbf{\Phi }}_p,Z_p=\mathrm{Sin}\left(p\pi z\right);p=1,2,3,\dots$$

(38)

The Eigen value equation is:

$$\begin{array}{c}\multicolumn{1}{c}{R{a}_T=\frac{(1+N_1)}{{\alpha }^2}(\frac{N_1{N}_{5}J}{N_4}(\frac{J}{\mathrm{Ln}}+\frac{s}{\sigma })}\\ \multicolumn{1}{c}{-(J+\frac{N_2}{N_4}s+\frac{2N_1}{N4})(\frac{J}{\mathrm{Ln}}+\frac{s}{\sigma }){)}^{-1}}\\ -(J+\frac{N_2}{N_4}s+\frac{2N_1}{N4})(\frac{J}{\mathrm{Ln}}+\frac{s}{\sigma })(J+s)\hfill \\ \multicolumn{1}{c}{(J^2+J(\frac{{\gamma }_{a}s}{(1+N_1)}+\frac{1}{K}))}\\ +\frac{N_1^2{J}^2(J+s)}{(1+N_1)N_4}(\frac{J}{\mathrm{Ln}}+\frac{s}{\sigma })\hfill \\ -\frac{\mathrm{Rn}{\alpha }^2}{(1+N_1)}(\frac{J^2{N}_A{N}_5{N}_1}{\mathrm{Ln}{N}_4}-(J+\frac{N_2}{N_4}s+\frac{2N_1}{N4})\hfill \\ \multicolumn{1}{c}{\frac{J{N}_A}{\mathrm{Ln}}+(J+\frac{s}{\sigma }))}\end{array}$$

(39)

Linear stability analysis

Stationary mode

For steady case we take $s=0$ and $N=1$ as a first approximation. Therefore, the Rayleigh number for steady state is

$$\begin{array}{c}{\mathrm{Ra}}_T^{\mathrm{St}}=\frac{(1+N_1)}{{\alpha }^2}{(\frac{N_1{N}_{5}J}{N_4}\left(\frac{J}{\mathrm{Ln}}\right)-(J+\frac{2N_1}{N4})\left(\frac{J}{\mathrm{Ln}}\right))}^{-1}\hfill \\ -(J+\frac{2N_1}{N4})\left(\frac{J^2}{\mathrm{Ln}}\right)(J^2+J\left(\frac{1}{K}\right))\hfill \\ +\frac{N_1^2{J}^4}{(1+N_1)N_4\mathrm{Ln}}-\frac{\mathrm{Rn}{\alpha }^2}{(1+N_1)}\hfill \\ \multicolumn{1}{c}{(\frac{J^2{N}_A{N}_5{N}_1}{\mathrm{Ln}{N}_4}-(J+\frac{2N_1}{N4})\frac{J{N}_A}{\mathrm{Ln}}+J)}\end{array}$$

(40)

Oscillatory mode

Consider $s=i\omega$ in equation (39) and after removing the complex strings in the denominator we get

$$R{a}_T={\Delta }_1+i\omega {\Delta }_2$$

(41)

Dropping the subscript $i$ and setting ${\Delta }_2=0$ $({\omega }_i\ne 0)$ leads for the relation

$$b_1{\left({\omega }^2\right)}^2+b_2\left({\omega }^2\right)+b_3=0$$

(42)

Now equation (41) with ${\Delta }_2=0$ become

$$R{a_T}^{\mathrm{Osc}}=a_0(a_1+{\omega }^2{a}_2)$$

(43)

and $b_1,b_2,$ $b_3$ , $a_0,a_1,$ $a_2$ , ${\Delta }_1$ , ${\Delta }_2$ are lengthy algebraic expression and omitted for brevity.

If there are no positive solutions for equation (43), then it implies that oscillatory instability does not exist. The oscillatory neutral Rayleigh number is derived by choosing the minimum of the two positive solutions of equation (43) with ${\omega }^2$ (equation (42)). There are possibilities of having two positive solutions for ${\omega }^2$ , however for the values chosen for $\mathrm{Rn},\mathrm{Ln},$ $N_A,\sigma ,{\gamma }_a,N_{\mathrm{TC}},N_{\mathrm{CT}},\mathrm{Rs},N_1,N_2,N_4$ and $N_5$ , end up with only one positive solution of equation (42) which endorse that there exists only one oscillatory neutral solution. Equation (43) is minimized with wave number numerically after substituting ${\omega }^2$ for values greater than zero, from equation (42) for distinct values of dimensionless parameters to create convection in oscillatory mode.

Stability analysis for non-linear terms

Assuming physical quantities to be independent of $y$ and considering only two-dimensional rolls, introducing stream function and eliminating pressure yields

$$\begin{array}{c}\multicolumn{1}{c}{s{\gamma }_a{\nabla }^2\Psi +R{a}_T\frac{\partial T}{\partial x}-\mathrm{Rn}\frac{\partial S}{\partial x}=(1+N_1){\nabla }^4\Psi }\\ \multicolumn{1}{c}{-\left(\frac{1+N_1}{\mathrm{K}}\right){\nabla }^2\Psi +N_1{\nabla }^{2}Z}\end{array}$$

(44)

$$N_{2}sZ=N_3\nabla (\nabla .Z)+N_4{\nabla }^{2}Z+N_1(\nabla \times \mathbf{v}-2\omega )$$

(45)

$$\frac{\partial T}{\partial t}+\frac{\partial \Psi }{\partial x}={\nabla }^{2}T-N_5(\nabla \times Z)+\frac{\partial (\Psi ,T)}{\partial (x,z)}$$

(46)

$$\frac{1}{\sigma }\frac{\partial S}{\partial T}+\frac{1}{\epsilon }\frac{\partial \Psi }{\partial x}=\frac{1}{\mathrm{Ln}}{\nabla }^{2}S+\frac{N_A}{\mathrm{Ln}}{\nabla }^{2}T+\frac{1}{\epsilon }\frac{\partial (\Psi ,S)}{\partial (x,z)}$$

(47)

Equations (49)–(52) are evaluated by subjecting them to stress-free, isothermal, iso-nano-concentration boundary conditions:

$$\psi =\frac{{\partial }^2\psi }{\partial z^2}=T=S=C=Z=0\mathrm{at}z=0,1$$

(48)

To perform a local non-linear stability analysis, the following Fourier expansions are adopted:

$$\begin{array}{c}\psi =\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{A}_{mn}\left(t\right)\mathrm{Sin}\left(m\alpha x\right)\mathrm{Sin}\left(n\pi z\right)\hfill \\ \multicolumn{1}{c}{T=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{B}_{mn}(t)\mathrm{Cos}\left(m\alpha x\right)\mathrm{Sin}\left(n\pi z\right)}\\ \multicolumn{1}{c}{S=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{C}_{mn}(t)\mathrm{Cos}(m\alpha x)\mathrm{Sin}(n\pi z)}\\ \multicolumn{1}{c}{Z=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{D}_{mn}(t)\mathrm{Cos}(m\alpha x)\mathrm{Sin}(n\pi z)}\end{array}\}$$

(49)

Taking (0, 2) modes for temperature, and (1, 1) modes for stream function and nanoparticle concentration gives

$$\begin{array}{c}\psi =A_{11}(t)\mathrm{Sin}(\alpha x)\mathrm{Sin}(\pi z)\hfill \\ T=B_{11}(t)\mathrm{Cos}(\alpha x)\mathrm{Sin}(\pi z)+B_{02}(t)\mathrm{Sin}(2\pi z)\hfill \\ S=C_{11}(t)\mathrm{Cos}(\alpha x)\mathrm{Sin}(\pi z)+C_{02}(t)\mathrm{Sin}(2\pi z)\hfill \\ Z=D_{11}(t)\mathrm{Cos}(\alpha x)\mathrm{Sin}(\pi z)+D_{02}(t)\mathrm{Sin}(2\pi z)\hfill \end{array}$$

(50)

The amplitudes which are functions of time are $A_{11}(t)$ , $B_{11}(t)$ , $B_{02}(t)$ , $C_{11}(t)$ , $C_{02}(t)$ , $D_{11}(t)$ and $D_{02}(t)$ . By the interactions of $\psi ,T$ and $\psi ,S$ and $\psi ,Z$ , the flow fields are distorted. It is evident that $\psi$ is minimally represented, as it is the simplest possible form for satisfying the plate conditions. The amplitude $A_{11}(t)$ is lean on time and should be evaluated. The term $B_{11}(t)\cos (\alpha x)\sin (\pi z)$ is the nominal representation for $\theta$ and is included. The term $B_{02}(t)\sin (2\pi z)$ perform the basic depiction for the distortion of the mean temperature field. The reason for the value 2 in the argument is that the mean temperature field is distorted by the convective term $\psi \theta$ , in the heat equation; since both $\psi$ and $\theta$ have components proportional to $\sin (\pi z)$ , this will force a $\sin (2\pi z)$ dependence on the mean temperature. Similar remarks apply to angular spins and the nanoparticle concentrations.

Adopting orthogonality condition over the Eigen functions we obtain

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{A}_{11}(t)}{\mathrm{dt}}=\frac{1}{{\gamma }_a{\delta }^2}}\\ \multicolumn{1}{c}{\left[\begin{array}{c}\alpha \mathrm{Rn}{C}_{11}(t)-\alpha R{a}_T{B}_{11}(t)-\frac{(1+N_1)}{K}{\delta }^2{A}_{11}(t)+\hfill \\ \multicolumn{1}{c}{(1+N_1){\delta }^4{A}_{11}(t)}\end{array}\right]}\end{array}$$

(51)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{B}_{11}}{\mathrm{dt}}=-}\\ \multicolumn{1}{c}{[\alpha A_{11}(t)+{\delta }^2{B}_{11}(t)+\alpha \pi A_{11}(t)B_{02}(t)+N_5{D}_{11}(t)]}\end{array}$$

(52)

$$\frac{d{B}_{02}}{\mathrm{dt}}=-4{\pi }^2{B}_{02}(t)+\frac{\alpha \pi }{2}{A}_{11}(t)B_{11}(t)-N_5{D}_{02}(t)$$

(53)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{C}_{11}}{\mathrm{dt}}=-\sigma }\\ \multicolumn{1}{c}{[\frac{1}{\epsilon }\alpha A_{11}(t)+{\delta }^2(\frac{C_{11}(t)}{\mathrm{Ln}}+\frac{N_A}{\mathrm{Ln}}{B}_{11}(t))+\frac{1}{\epsilon }\alpha \pi A_{11}(t)C_{02}(t)]}\end{array}$$

(54)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{C}_{02}}{\mathrm{dt}}=-\sigma }\\ \multicolumn{1}{c}{[\frac{1}{\mathrm{Ln}}4{\pi }^2{C}_{02}(t)+4{\pi }^2{B}_{02}(t)\frac{N_A}{\mathrm{Ln}}-\frac{a\pi }{2\epsilon }{A}_{11}(t)C_{11}(t)]}\end{array}$$

(55)

$$\frac{d{D}_{11}}{\mathrm{dt}}=-\frac{1}{N_2}[2N_1{D}_{11}(t)+N_4{\delta }^2{D}_{11}(t)]$$

(56)

$$\frac{d{D}_{02}}{\mathrm{dt}}=-\frac{1}{N_2}[2N_1{D}_{02}(t)+N_4{\delta }^2{D}_{02}(t)]$$

(57)

In case of steady motion $\frac{d\left(\right)}{\mathrm{dt}}=D_i=0$ , (i = 1, 2, .., 7) and writing all $D_i\prime$ s in terms of $A_{11}$ yields:

$$\begin{array}{c}\multicolumn{1}{c}{D_1=\frac{1}{{\gamma }_a{\delta }^2}}\\ \multicolumn{1}{c}{\left[\begin{array}{c}\alpha \mathrm{Rn}{C}_{11}(t)-\alpha R{a}_T{B}_{11}(t)-\frac{(1+N_1)}{K}{\delta }^2{A}_{11}(t)+\hfill \\ \multicolumn{1}{c}{(1+N_1){\delta }^4{A}_{11}(t)}\end{array}\right]}\end{array}$$

(57)

$$D_2=-[\alpha A_{11}(t)+{\delta }^2{B}_{11}(t)+\alpha \pi A_{11}(t)B_{02}(t)+N_5{D}_{11}(t)]$$

(57)

$$D_3=-4{\pi }^2{B}_{02}(t)+\frac{\alpha \pi }{2}{A}_{11}(t)B_{11}(t)-N_5{D}_{02}(t)$$

(57)

$$\begin{array}{c}\multicolumn{1}{c}{D_4=-\sigma }\\ \multicolumn{1}{c}{[\frac{1}{\epsilon }\alpha A_{11}(t)+{\delta }^2(\frac{C_{11}(t)}{\mathrm{Ln}}+\frac{N_A}{\mathrm{Ln}}{B}_{11}(t))+\frac{1}{\epsilon }\alpha \pi A_{11}(t)C_{02}(t)]}\end{array}$$

(57)

$$\begin{array}{c}\multicolumn{1}{c}{D_5=-\sigma }\\ \multicolumn{1}{c}{[\frac{1}{\mathrm{Ln}}4{\pi }^2{C}_{02}(t)+4{\pi }^2{B}_{02}(t)\frac{N_A}{\mathrm{Ln}}-\frac{a\pi }{2\epsilon }{A}_{11}(t)C_{11}(t)]}\end{array}$$

(57)

$$D_6=-\frac{1}{N_2}[2N_1{D}_{11}(t)+N_4{\delta }^2{D}_{11}(t)]$$

(57)

$$D_7=-\frac{1}{N_2}[2N_1{D}_{02}(t)+N_4{\delta }^2{D}_{02}(t)]$$

(57)

Along with

$$D_1=D_2=D_3=D_4=D_5=D_6=D_7=0$$

(57)

Runge-Kutta method is engaged to acquire the solutions of the above system of simultaneous ordinary differential equations.

Nanoparticle transport of concentration and heat transport

The Thermal Nusselt number $\mathrm{NuT}$ is described as (following Malashetty et al. ⁵⁴ )

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{NuT}=\frac{\mathrm{Heat}\mathrm{transport}\mathrm{by}(\mathrm{conduction}+\mathrm{convection})}{\mathrm{Heat}\mathrm{transport}\mathrm{by}\mathrm{conduction}}}\\ \multicolumn{1}{c}{=1+{\left[\frac{\underset{0}{\overset{2\pi }{\int }}\frac{\partial T}{\partial z}\mathrm{dx}}{\underset{0}{\overset{2\pi /a}{\int }}\frac{\partial T_B}{\partial z}\mathrm{dx}}\right]}_{z=0}}\end{array}$$

(57)

Substituting expressions (24) and (50) in $\mathrm{NuT}$ gives:

$$\mathrm{NuT}=1-2\pi B_{02}(t)$$

(57)

$\mathrm{NuF}$ is the thermal Nusselt numbers. $\mathrm{NuF}$ is estimated similar to $\mathrm{NuT}$ and they are:

$$\mathrm{NuF}=(1-2\pi C_{02}(t))+N_A(1-2\pi B_{02}(t))$$

(57)

Results and discussion

Prior to elaborating the computational results, some discussion is warranted regarding the parameter of the micropolar fluid $N_5$ describing the coupling between thermal flux and spin flux. The thermodynamic restriction imposed on the coefficients of micropolar viscosity is

$${(\alpha -\frac{\beta }{T})}^2<\frac{2\kappa }{T}(\gamma -\overline{\beta }).$$

(57)

The sign of $\hat{\beta }=\beta /T$ as suggested by Lebon and Perez-Garcia ⁹ is negative (or $\alpha <0$ and $\beta$ constant given in Datta and Sastry ⁸ will result in a stationary onset for the plates heated either from down or above. However, in Payne and Straughan ¹⁴ $\hat{\beta }>0$ and he showed that oscillatory convection occurs only for heating the upper plate and stationary convection occur only heating the lower plate.

We will discuss the stability characteristics for arbitrary choices of $N_5$ , since the thermodynamic restrictive condition (6) does not require polarity of $\alpha$ or $\beta$ explicitly, nor indeed does $N_5$ . Under the condition $N_5=0$ there is no coupling between the heat flux and spin flux, and one can conjecture that the stability characteristics will be identical to those of classical Rayleigh-Bénard convection. The oscillatory convection is not possible when $N_5=0$ . Heating the lower plate the convection in stationary mode occurs. The micropolar fluid in the presence of porous material is more stable when compared with the fluid without the porous material. The Rayleigh number increases by increasing $1/K$ (progressively more densely packed porous bed). Hence, porous material helps to stabilize the flow system. Porous materials also provide a greater surface area for energy transfer as noted by Bég et al. ⁵³

Figures 2 to 4, are the graphs of the critical Rayleigh and wave numbers for micropolar fluid with suspension of nanoparticles and without the suspension of nanoparticles. Figure 2(a) and (b) exhibit $R_c^{(c)}$ (critical Rayleigh number) and $a_c$ (critical wave number) with different sets $N_1$ (vortex viscosity parameter) and $N_4$ (spin gradient viscosity parameter) decrease as permeability, $K$ increases for the case $N_5=0$ (vanishing micropolar coupling parameter). $R_c^{(c)}$ and $a_c$ approach to the limiting values 719.348/2.219, 664.043/2.221, and 704.363/2.215 with $N_1/N_4$ being 0.1/0.1, 0.01/0.1, and 0.1/0.01, respectively, as $K$ becomes very large, which corresponds to vanishingly small porous media effects. It is also evident from Figure 3 that the Rayleigh number increase with $N_1$ or $N_4$ for $N_5=0$ and $K=1$ . Curve $A$ behaves as a monotonic increasing function of $R^{(c)}$ with $N_1$ while curve $B$ albeit initially increasing with $N_4$ thereafter apparently exhibits independence of $N_4$ when $N_4>0.1$ for $N_1=0.1$ .

Figure 2.

(a) The sketch of ${\mathrm{Ra}}_c^{\left(c\right)}$ versus $K$ for $N_5=0$ and (b) the sketch of $a_c$ versus $K$ for $N_5=0$ .

Figure 3.

The sketch of ${\mathrm{Ra}}_c^{\left(c\right)}$ versus $N_1$ or $N_4$ for $N_5=0$ .

Figure 4.

(a) The sketch of ${\mathrm{Ra}}_c^{\left(c\right)}$ versus $N_5$ and (b) the sketch of $a_c$ versus $N_5$ .

Figure 4(a) and (b) show the diversity of $R_c^{\left(c\right)}$ and $a_c$ with the micropolar coupling parameter, $N_5$ . Inspection of Figure 4(a) flourish that the Rayleigh number is discontinuous with $N_5$ variation. $R_c^{\left(c\right)}$ for lower $K$ is greater than that computed for larger $K$ . For Negative values of $N_5$ , $R^{(c)}>0$ and $R^{(c)}<0$ for positive values of $N_5$ .

Figure 4(b) reproduce that the $a_c$ approaches a limiting value for different $K$ when $N_5$ is large positive or large negative. There also exists a discontinuity for the wave number as $N_5$ is near that threshold. From these sketches it is concluded that the value of $R_c^{\left(c\right)}$ and $a_c$ is lower for only micropolar fluid compared with micropolar nanofluid, that is, the convection is induced earlier in purely micropolar fluids than with micropolar nanofluids. This accord the fact that the thermal conductivity of nanofluids is higher than ordinary fluids, signifying that the existence of nanoparticles influence the initiation of convection.

The thermal Rayleigh number for stationary and oscillatory convections have been defined earlier in equations (40) and (43) respectively. Figure 5 (stationary convection) prove that large values of $\mathrm{Rn}$ destabilize the system as $R{a}_T$ decreases by enlarging $\mathrm{Rn}$ . Figure 5(b) signals that as Lewis number $\mathrm{Ln}$ increases the $R{a}_T$ also increases which indicates that the system gets stabilized for large values of $\mathrm{Ln}$ . To sketch the profiles in Figure 5(b) to (f), the value of $\mathrm{Rn}$ is taken as −0.1. Physically positive values of $\mathrm{Rn}$ leads for top-heavy and bottom-heavy for negative values of $\mathrm{Rn}$ . The values of $R{a}_T$ are advanced by enhancing values of $N_A$ and hence $N_A$ approach the convection to occur as can be viewed in Figure 5(c).

Figure 5.

Trajectories of neutral stability for stationary convection on (a) $\mathrm{Rn}$ , (b) $\mathrm{Ln}$ , (c) $N_A$ , (d) $N_1$ , (e) $N_4$ , and (f) $N_5$ .

The enact of $N_1$ , $N_4$ , and $N_5$ on thermal Rayleigh number are displayed in Figure 5(d) to (f) for micropolar nanofluid and only for micropolar fluid. These figures detail that for only micropolar fluid as $N_1$ and $N_5$ increases $R{a}_T$ increases but it decreases for $N_4$ . Thus $N_1$ and $N_5$ advances and $N_4$ delay the onset of convection. For micropolar nanofluid, $R{a}_T$ decreases with $N_1$ and $N_4$ which cases the system to destabilize and increases with $N_5$ which stabilize the system.

Figure 6(a) to (g) characterize the oscillatory convection with several selected parameters. In Figure 6(a) it is realized that as $\mathrm{Rn}$ raises $R{a}_T$ is also upsurged which will quicken the convection. $\mathrm{Ln}$ reduces the $R{a}_T$ and induces the destabilization (Figure 6(b)). $N_A$ however does not exert any noteworthy impact on the convection (Figure 6(c)) as the values of $R{a}_T$ are not altered significantly. The impact of $N_1$ for nanofluid (Figure 6d) and micropolar fluid interpret that $R{a}_T$ increases for nanofluid and downturn for micropolar fluid with $N_1$ . Hence nanofluid helps to hasten and micropolar fluid lag the convection to set on. The $R{a}_T$ is boosted with $N_4$ (Figure 6(e)) and $N_5$ (Figure 6(f)) and hence $N_4$ and $N_5$ help to stable the system.

Figure 6.

Trajectories of neutral stability for oscillatory convection on (a) $\mathrm{Rn}$ , (b) $\mathrm{Ln}$ , (c) $N_A$ , (d) $N_1$ , (e) $N_1$ , (f) $N_4$ , and (g) $N_5$ .

To understand the heat and mass transfer, nonlinear scrutiny is undertaken in this study. The Nusselt $\mathrm{Nu}$ , and shearwood $\mathrm{Sh}$ numbers are computed in terms of $R{a}_T$ and the impact of $\mathrm{Nu}$ and $\mathrm{Sh}$ are outlined in Figures 7 and 8 respectively. In general from Figures 7 and 8 one can conclude that both $\mathrm{Nu}$ and $\mathrm{Sh}$ posses the value 1 at the conduction state and the maximum bound for $\mathrm{Nu}$ is 3 (similar observation was made by Malashetty et al. ⁵⁴ ) both for micropolar and micropolar nanofluid. Shearwood number attains the upper limit to be 3 for micropolar fluid and it is flexible for micropolar nanofluid (nanofluid upper bound property was similar to Bhadauria and Agarwal ⁴² ). Transfer of mass and heat are sustained by $\mathrm{Rn}$ as narrated in Figures 7(a) and 8(a) because $\mathrm{Nu}$ and $\mathrm{Sh}$ are enhanced with $\mathrm{Rn}$ . The Lewis number $\mathrm{Ln}$ reduces $\mathrm{Nu}$ and increases $\mathrm{Sh}$ as can be detained in Figures 7(b) and 8(b) respectively. The modified diffusivity ration $N_A$ does not play any role on $\mathrm{Nu}$ (Figure 7(c)) whereas it increases $\mathrm{Sh}$ significantly (Figure 7(d), similar to Bhadauria and Agarwal ⁴² ). Micropolar parameter $N_1$ do not disturb much the $\mathrm{Nu}$ (Figure 7(d)) and Sh (Figure 8(d)). The mass and heat transfer are stimulated by the porous parameter $K$ .

Figure 7.

Trajectories of Nu with $R{a}_T/R{a}_{\mathrm{TC}}$ on (a) Rn, (b) Ln, (c) $N_A$ , (d) $N_1$ , and (e) $K$ .

Figure 8.

Trajectories of $\mathrm{Sh}$ with $R{a}_T/R{a}_{\mathrm{TC}}$ on (a) Rn, (b) Ln, (c) $N_A$ , (d) $N_1$ , and (e) $K$ .

The transition from linear to non-linear convection although challenging, reveals delightful information of the fluid dynamics. The unsteady transient property of $\mathrm{Nu}$ and $\mathrm{Sh}$ are sketched in Figures 9 and 10 respectively. Initially when time is small, large oscillations occur for both $\mathrm{Nu}$ and $\mathrm{Sh}$ and these oscillations tends to steady state after a some time. Figures 9 and 10 illustrate the transient property of thermal $\mathrm{NuT}$ and concentration $\mathrm{NuF}$ Nusselt numbers over the key parameters. Figures 9(a) and 10(a) report that as $\mathrm{Rn}$ increases $\mathrm{NuT}$ and $\mathrm{NuF}$ is diminished claiming that transfer of heat is reduced to the boundaries and the similar result was also mentioned in Agarwal et al. ⁴³ The $\mathrm{Ln}$ suppresses $\mathrm{NuT}$ (Figure 9(b)) and it helps to increase $\mathrm{NuF}$ (Figure 10(b)). The modified diffusivity ratio $N_A$ (Figures 9(c) and 10(c)), micropolar parameters $N_1$ , $N_2$ and $N_5$ (Figures 9(d), (e), (g), 10(d), (e) and (g)) portrait that they all activates both $\mathrm{NuT}$ and $\mathrm{NuF}$ . The micropolar parameter $N_4$ (Figures 9(f) and 10(f)) accelerate $\mathrm{NuT}$ and decelerate $\mathrm{NuF}$ .

Figure 9.

Trajectories of $\mathrm{NuT}$ versus $t$ (a) Rn, (b) Ln, (c) $N_A$ , (d) $N_1$ , (e) $N_2$ , (f) $N_4$ , and (g) $N_5$ .

Figure 10.

Trajectories of $\mathrm{NuF}$ versus $t$ on (a) Rn, (b) Ln, (c) $N_A$ , (d) $N_1$ , (e) $N_2$ , (f) $N_4$ , and (g) $N_5$ .

Conclusion

The micropolar nanofluid saturated overlying on the porous bed inserted in a horizontal channel was taken to study stability. The upper plate was cooled and the lower plated was heated importing non-Darcy model and Buongiorno’s nanoscale model. Normal mode method was utilized for linear terms analysis and two term Fourier series was implemented to solve nonlinear terms. The reports are

1. Convection sets in earlier in micropolar fluids compared with micropolar nanofluids.

2. For stationary convection, increasing Lewis number $\mathrm{Ln}$ , modified diffusivity ratio $N_A$ and micropolar parameter $N_5$ helps to stable the system while concentration Rayleigh number $\mathrm{Rn}$ and the micropolar parameters $N_1$ and $N_4$ destabilizes the system.

3. For oscillatory convection, an elevation in the $\mathrm{Rn}$ , $N_1$ , $N_4$ , and $N_5$ leads for the stable system, $\mathrm{Ln}$ destabilizes and $N_A$ will not disturb the status of the system.

4. For steady finite amplitude motions, $\mathrm{Rn}$ and porous parameter $K$ reinforce the mass and heat transfer, $N_1$ do not significantly change the transfer of heat and mass, $\mathrm{Ln}$ and $N_A$ strengthen the mass transfer but condense the heat transfer.

5. The transient Nusselt number and Sherwood number are both accentuated by expanding $N_A$ , $N_1,N_2,N_5$ , while $\mathrm{Rn}$ and $N_4$ rise the heat transfer and hinder the mass transfer and opposite impact was observed with $\mathrm{Ln}$

6. The transient Shearwood and Nusselt numbers were oscillatory for short time and for enormous time they approach steady state.

This work is confined to rigid horizontal boundaries. However wavy boundaries offer key advantages in heat transfer enhancement also^55,56 and these may be explored in future investigations in addition to motile micro-organism bio convection phenomena.