Magneto-double-diffusive convection in a hybrid nanofluid

Abstract

Keywords: Brownian Motion; External Magnetic Field; Hybrid Nanofluid; Rayleigh Number; Stability Analysis

Introduction

Fluids that are frequently employed as efficient heat transfer media include water, ethylene glycol, and motor oil. These fluids are not very suitable as heat transfer fluids because of their lower thermal conductivity. It is, however, naturally

preferable to mix metals with fluids to create a heat transfer medium that functions like a fluid but has the thermal conductivity of a metal because metals have three times the thermal conductivity of fluids [1]. In nanofluid, which [2] initially described, non-micrometer-sized particles are scattered

*Corresponding author. *E-mail address: atulshaswat@gmail.com, aksrivastava@rnc.amity.edu This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey © Author. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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throughout the host (base) fluid. A larger thermal conductivity enhancement can be obtained if a base fluid with lower thermal conductivity is utilized, as demonstrated by [3]. The physical and thermal properties of nano-metal particles and the base liquid have a significant impact on the behavior of nanofluid. Four types of nanofluid are distinguished based on the type of nanoparticles: metal, metal oxide, carbon, and metal hybrid. The host (base) fluids that are commonly used to produce nanofluids are ethylene glycol, methanol, water, and oils. Nanofluids have a wide range of applications in the automotive sector (including hybrid and internal combustion engines), air conditioning, solar and power plant cooling, nuclear reactors, space and defense systems, and pharmaceutical processes. As a result, in recent years, interest in oil-based nanofluids, including carbon nanotubes, TiO2, CuO, Al2O3, AlN, and SiO2, for industrial and engineering applications has increased [4-6]. Using the two-phase model [7, 8], quantitatively investigated natural and mixed convection heat transfers in a nanofluid within a square cavity. They found that the nanoparticle distribution is non-uniform for low Rayleigh and high Richardson numbers and that thermophoretic effects are minimal for nanoparticles with high thermal conductivity. They also concluded that the use of homogeneous and single-phase models is suitable in such circumstances. By taking into account Newtonian and non-Newtonian nanoliquids with and without dusty particles, [9, 10] were able to determine the prerequisites for the existence of stationary and oscillatory convections in such nanoliquids in the presence or absence of a porous medium. There has been a lot of research done on nanofluids in recent years, including research on their preparation, characterization, modeling, convective and boiling heat transfer, and applications. These are due to their unique properties, which include excellent load-carrying capacity, extreme pressure, and friction-reducing qualities [11]. However, hybrid nanofluids are a new class of nanofluids that can be made by suspending (i) different types of nanoparticles (two or more types) in base fluid and (ii) hybrid (composite) nanoparticles in base fluid. A hybrid material is something that simultaneously combines the physical and chemical properties of many materials and offers these properties in a homogeneous form. Amazing physicochemical properties that do not exist in the separate components can be found in synthetic hybrid nanomaterials. The properties of these composites have been extensively studied [12], and hybrid materials containing carbon nanotubes (CNTs) have been applied in nanocatalysts, electrochemical sensors, biosensors, and other applications [13]. Ahmad, [14] imposed Cu-Al2O3 nanoparticles and water as the base fluid to study the mixed convection flow of the hybrid nanofluid over a nonlinear stretching sheet. This research shows that the temperature of the hybrid nanofluid rises in response to thermal radiation, but its velocity falls in the presence of a magnetic field. However, the application of these hybrid nanomaterials in nanofluids has not yet been identified. There has been very little work done on hybrid nanofluids, and there is still a lot of research being done.

Double diffusive convection of hybrid nanofluids is examined by [15]. Double-diffusive convection in flow has various applications, and it is widely employed in many natural and experimental systems where mass and heat diffusion occur simultaneously. The fluid motion is produced by the simultaneous diffusion of two factors, namely temperature and concentration variations, which result in buoyancies. As a result, this phenomenon in flow is known as double-diffusive convection [16]. The fluid density is significantly influenced by the solute gradient and heat, and as a result, these linked phenomena can create unique flow behavior even in fluids with a uniform density distribution. Double diffusion in the naturally convective flow of water-based fluid saturated in a corrugated enclosure and containing hybrid nanoparticles composed of copper (Cu) and alumina (Al2O3) is examined by [17]. Sharma and Gupta [18] took into account the heat and solute transport in a rotating nanofluid with a porous, below-heated layer. According to their research, the coriolis force term caused by rotation stabilizes the system; however, the Darcy number causes the layer to become unstable or stable depending on the wave number, which can be small or big. In their study, Pranesh et al. [19] looked at the convection that occurs in water when three diffusing components are added. They discovered in their investigation that the best estimates of the thermophysical properties of the aqueous solutions serve as the foundation for the critical values of the Rayleigh, Nusselt, and Sherwood numbers. [20-22] evaluated the coupled effect of Hall and heat transport on the MHD flow in different fluids. The Hall impact on a vertical porous Jeffrey fluid flow with a magnetic field was also explored by [20]. What happens when we look at how an external magnetic field interacts with double diffusive convection in hybrid nanoliquid? Due to its engineering applications in crystal formation in liquids, geophysics, geothermal reservoirs, etc., convective heat transfer associated with magnetic field impact has gained interest recently [23, 24]. Numerous studies have looked at how different configurations and circumstances affect convective heat transport when subjected to external magnetic fields. Using the shooting method, [25] investigated magnetohydrodynamic Casson fluid flow over a deformable porous channel and found that the influence of the magnetic field and Casson parameters reduced the fluid velocity and particle displacement. By using the symmetrical finite volume method, [26] investigated magnetized nanofluid flow in a symmetrical E-shaped chamber in the presence of an inclined magnetic field. They discovered that the increase in Richardson number increased the heat transfer rate, while the magnetic field effect decreased flow circulation. In the presence of a magnetic field applied at an angle to the horizontal axis, Roy, [27] investigated the natural convective flow and heat transfer of a hybrid nanofluid confined in an enclosure with several heat sources at the bottom wall. A magnetized Ag-MgO water hybrid nanofluidic stream inside a cube-shaped enclosure with an inner circular cylinder installed inside the

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cube is computationally investigated by [28]. According to this study, the magnetic factor lowers energy transmission, whereas the Rayleigh number increases it. High heat transference is revealed by the cold mode of the cylinder, whereas high velocity magnitudes within the cube are explored by the hot mode. For steady two-dimensional and incompressible flow with a convective boundary condition in a curved-coordinate porous system with ohmic heating, Rashad [29] examined the variation of heat on the MHD Williamson hybrid nanofluid (Ag-TiO2/H2O) model. This study found that while convective boundary conditions and thermal radiation increase surface friction, growing permeability, Biot, and Eckert numbers enhance temperature profiles while reducing heat transfer. In 2024, Anee et al. [30] presented the study for analysing the magnetohydrodynamic convective heat transport of hybrid nanofluids in a chamber with multiple heaters (such as hot microchips). They used the multiple-relaxation-time, lattice Boltzmann method and graphics processing unit computing for analysing their work. The radiative magnetohydrodynamic (MHD) mixed convective alumina-copper/water hybrid nanofluid flow via an angled shrinking plate was also studied by Wahid et al. [31]. Their research shed important light on basic transport phenomena as the movement of mass, momentum, or heat. Taking motivation from Pundir’s recent work [15], we are interested in understanding the influence of magnetic field on hybrid nanofluid double diffusive convection since the aforementioned study does not account for magnetic field, which is presently having a lot of practical applications. The outline of the paper is as follows: We provide a brief mathematical model of the physical situation in Section 2. In Section 3, linear stability analysis for both realistic and non-realistic boundary conditions is provided. In Section 4, findings and discussion are presented. The analysis ends with a few key points in Section 5.

Mathematical Formulation Of The Problem

An unbounded horizontal layer of hybrid nanofluid of thickness d is considered between the perfectly insulating planes located at z* = 0 and z* = d under the influence of an externally imposed constant magnetic field. At the lower plane (z* = 0) and higher plane (z* =d), the temperature is assumed T0* + ∆T* and T0* respectively with ∆T* ≪ T0*. Also, at the lower plane (z* = 0) and, higher plane (z* = d), the concentrations are assumed C0* + ∆C* and C0* respectively. In this analysis, astrisks represent the dimensional variables, and non-dimensional variables are exclusive of asterisks. Here, the gravitational force is taken as . The temperature and concentration changes in nanofluid are assumed to be very small. It is assumed the nanofluid mixture is uniform and in a local thermal symmetrical state. To linearize the mathematical equations, the OberbeckBoussinesq hypothesis has been used. Also, for the mathematical formulation, we have taken into account that (i) Newtonian flow is regarded as laminar, and the nanofluid under consideration is incompressible. (ii) The nanoparticles are being perched in the base liquid using either surface charge technology or surfactant, preventing the hoard so that the nanoliquids are stable. (iii) During the convection, there is no chemical reactions take place. The base liquid phase and both types of nanoparticles are symmetrical with respect to temperature. Also, the thermo-physical ability of nanofluid is fixed (Boussineq Hypothesis). (iv) Size matters in making nanofluids with novel properties, so in this study, the spherical nano-particles are measured. Radioactive heat transport is not considered as it is exceptionally small. The principle equations of conservation that characterize the flow of a hybrid nanofluid layer in dimensional form are as follows [15] and [32] :-

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Here d stands for the width of the layer. For the case of rigid boundaries, the parameters λ1 and λ2 taken to have a zero value, and for the case of a free boundary, the values of the parameters are infinity. Now, introduce the non-dimensional variables (x, y, z), ϕ1, ϕ2, p, T, (u, v, w), t and C is defined as

Here ρp1, ρp2 and ρf are the nanoparticles mass densities and base fluid density. We obtain

(15) Since the volumetric fraction of nanoparticles is small, taking the density of the base fluid as that of the density of nanofluid i.e. ρ≅ρf0 as adopted by [32], the specific weight becomes (9) Here, the velocity of the nanofluid layer is taken as * u = (u*, v*, w*); T*(°K) is the nanofluid temperature; βT(°K) is the thermal volumetric coefficient; ϕ*1, ϕ*2 are the nanoparticle volume fractions; H* = (Hx*, Hy*, Hz*) is Magnetic field;; μ(N s/m2) is the viscosity; DB1, DB2 (m2/s) are the Brownian diffusion coefficients; DT1, DT2 (m2/s) are the thermophoretic diffusion coefficients; Ds (m2/s) is the solutal diffusivity. Also, we have assumed that the solute concentration and nanoparticle volume fractions are fixed at the boundaries. Now the boundary conditions are defined as:

(19) The dimensionless boundary conditions take the form:(20) (21)

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Ln1 are large and of order 102 - 103 and NA1 and NA2 will not be greater than 105 and 106. Using this approximation, the basic solution of the present problem is;

Perturbed State The basic state now considered under the influence of small perturbations. So, the perturbation parameters are

containing the product of ϕ1 and ϕ2 with T are neglected to linearization the equations. The resultant linear governing equations then become:-

Linear Stability Analysis

Basic State The basic state of present problem expressed as:-

(36) (25) (37) (26) (38) (27) (28) Boundary conditions becomes:-

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And for Rigid-Rigid boundaries:(40) The parameter Rm represents the pressure gradient in static state. The parameters Rn, NA1, NA2, NB1 and NB2 do not appear in the equation for regular binary fluid. After eliminating p’ from the above equations, we obtain:(41) Combining Eqn. (34) and (41), we get

(42) The differential Eqn. (35)-(38) with (42) along with boundary conditions (39) and (40) comprise a linear BVP. Now to analyse the perturbations into normal modes, we take (43) Here, the wave numbers in x direction, y direction are . Also, n (comkx and ky respectively while plex) is the frequency. Using Eqn. (43) in Eqn. (35)-(38) and (42), we get

A Comparative Study

We solve the aforementioned eigenvalue problem under various boundary conditions. For every boundary condition in the stationary case (n=0), we compare the outcomes. The aforementioned set of equations has exact solutions for non-realistic boundary conditions (F/F), while numerical solutions are obtained for realistic boundary conditions (R/R, F/R, and R/F). Realistic Boundary Conditions In this case, the eigenvalue problem for R/R boundary conditions (51) R/F boundary conditions (52) (53) and F/R boundary conditions (54) (55)

are solved. For the three types of boundary conditions mentioned above, we solve the associated eigenvalue problem using the Galerkin technique. The test (weighted) functions in this method are identical to the base (trial) functions. Initially, the variables are written as linear combination of basis functions as:-

where Ai, Bi, Ci, Di and Ei are constants and the basis function Wi, Θi, Γi, Φ1i, Φ2i, are represented by power series satisfy the corresponding boundary conditions. Substituting Eqn. (56) into (44)-(48), multiplying the resulting equations by Wj (z), Θj (z), Γj (z), Φj1 (z), and Φj2 (z), performing the integration by parts with respect to z between z = 0 and z = 1 and using the boundary conditions, leads to the following system of linear homogeneous algebraic equations

To study the linear stability analysis, the corresponding boundary conditions for Free-Free boundaries are:-

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(63) for R/F boundaries, trial functions are:(64) and for F/R boundaries conditions, trial functions are:(65) Here (i=1,2,3…n). From a rigorous numerical experiment, it has been found that the order of the polynomial, n = 5 is sufficient to reach the required accuracy, which is shown in Table 1. Non Realistic Boundary Conditions We take the solution of Eqn. (44)-(48) satisfying the boundary condition for Free-Free (F/F) case:(66) Substituting Eqn. (66) in the above equations, we get the matrix equation as:(67) Here

Here the inner product ⟨...⟩ is defined as ⟨...⟩ = ∫01(…)dz. The above set of homogeneous algebraic equations can have a non-trivial solution if and only if

Table 1. Comparison of results for different orders of Galerkin approximation for Ln2=100 for different values Q. n=2 RaT

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The non-trivial solution of the above homogeneous system of equations require that

Non Oscillatory Convection When n = 0. The stationary Rayleigh number is given by:-

After neglecting the solute Rayleigh number (Rs) and Chandrashekhar number (Q), Eqn. (69) gives (70) The same result has been obtained by [33].

Results And Discussion

The effect of the main controlling parameters on the onset of convection in hybrid nanofluid is examined with the objective of understanding the control over convection. It has been observed that the thermal Rayleigh numbers of hybrid nanofluid are unaffected by the modified particle density parameter increment for non-oscillatory convection. Linear Stability Analysis Linear stability analysis is divided into one subsection: It is dedicated to a comparative study of stationary or non-oscillatory cases for different boundary conditions. Comparative Study (Non-Oscillatory Case) Here, we have plotted the graphs for the main controlling parameters, such as Chandrashekhar number (Q), which represents the magnetic field effect, thermo-nanofluid Lewis numbers (Ln1, Ln2), nanoparticle concentration numbers (Rn1, Rn2) in the (Rs, RaC) plane for FreeFree(F/F), Free-Rigid (F/R), Rigid-Free (R/F) and RigidRigid(R/R) boundaries. The effect of the Chandrashekhar number (Q) is depicted in Figure 2 for various boundaries in the (Rs, RaC) plane when other parameters have fixed values. In

Figure 2. Variation of RaC as function for different values of (Q) on (Rs, RaC) plane.

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Figure 3. Variation of RaC as function for different values of (Ln1) on (Rs, RaC) plane.

Figure 4. Variation of RaC as function for different values of (Ln2) on (Rs, RaC) plane.

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each of the four cases, F/F, F/R, R/F, and R/R, the stability of the system increases as the value of Q increases. This shows that the system stabilizes with an increasing value of Q, indicating the onset of convection delay. This phenomenon can be explained by the fact that viscosity is introduced into the fluid, and convection distorts the magnetic lines when a magnetic field penetrates the medium with a significant degree of strength. Consequently, the development of disturbances is impeded by these magnetic lines, delaying the onset of instability. Also from the graphs in Figure 2, one can conclude that the stability of the system is most dominant for F/R boundaries and least for F/F boundaries. This result is similar to the findings of [34]. Figure 3 illustrates how the stability of a system under different boundaries is affected by the change in the value of the thermo-nanofluid Lewis number (Ln1). The stability of the system reduces when the value of Ln1 increases in all scenarios, i.e., F/F, F/R, R/F, and R/R. The stability of the system is described as: F/R>R/R>F/R>F/F i.e. the system is most stable at F/R and least stable at F/F boundaries.

The impact of the thermo-nanofluid Lewis number (Ln2) for different boundaries is depicted in Figure 4. We observed a complex nature with the change in the value of Ln2. As Ln2 increases, the stability of the system falls in the cases of F/F and R/F, whereas it increases for the F/R and R/R boundaries. Stability is greatest for F/R boundaries and lowest for F/F boundaries. Similar results were obtained by [33]. Figure 5 shows the behavior of stability criteria for different types of boundaries for changes in the value of nanoparticle concentration number (Rn1). In all cases, i.e., F/F, F/R, R/F, and R/R, the system’s stability decreases with an increasing value of Rn1. Figure 6 depicts the impact of nanoparticle concentration number (Rn2) on the stability of the system for all four boundaries combinations. In all cases i.e. F/F, F/R, R/F, R/R, stability of the system decreases as the value of Rn2 increases. The behavior of nanoparticles concentration Rayleigh numbers is clear and consistent with what Chand and Rana reported [35, 36] . In general, if both Rn1 and Rn2 take the negative values, the hybrid nanofluid layer will be more stable.

Figure 5. Variation of RaC as function for different values of (Rn1) on (Rs, RaC) plane.

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Figure 6. Variation of RaC as function for different values of (Rn2) on (Rs, RaC) plane.

Acknowledgements

We have tried to understand how hybrid nanofluids and magnetic fields can influence the onset of double diffusive convection. We carried out a comparison analysis based on various boundary conditions using linear stability analysis. Four different velocity boundary conditions, namely R/R, R/F, F/R, and F/F have been analyzed. The Galerkin technique is used to solve the eigenvalue problem for the R/R, R/F, and F/R boundaries, whereas exact solutions are obtained for the F/F boundaries. In the non-oscillatory condition, we examine the outcomes among four boundary conditions. The conclusions are as follows:Ø For each of the four boundary surfaces, the stability of the system rises as Q grows. Ø With rising values of Ln1, Rn1, and Rn2 for every boundary, the system’s stability diminishes. Ø With an increasing value of Ln2, an intriguing stability phenomenon in the system was found. With an increasing value of Ln2, the system becomes unstable in two combinations (F/F and R/F), but it has the opposite effect on the other two combinations (R/R and F/R). Ø Furthermore, based on the studies mentioned above, we can deduce that the system’s stability is stated in the following order: F/R > R/R > R/F > F/F. This means that the system is most stable for F/R boundaries and least stable for F/F boundaries.

Author Monal Bharty gratefully acknowledges the financial assistance from Central university of Jharkhand as a research fellowship.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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Nomenclature

a c C C0 DB1, DB2 DT1, DT2 Ds g H Le Ln1, Ln2 n NA1, NA2 NB1, NB2 Pr T T0 Ra Rs Rm

Wave number, m Specific heat of nanofluid, , kJ / kg oC Concentration of the solute, mol/m3 Upper layer concentration, Brownian diffusion coefficients of nanoparticles, m2/s Thermophoretic diffusion coefficients Solutal diffusivity, m2/s Acceleration due to gravity, m/s2 Magnetic field, T Concentration based Lewis number Themo-nanofluid Lewis number Growth rate Modified diffusivity ratio Modified particle density increment Prandtl number Temperature, K Temperature of upper layer Temperature based Rayleigh number Concentration based Rayleigh number Density based Rayleigh number -1

Greek symbols Fluid’s thermal diffusivity, m2/s αf Temperature based volumetric coefficient, K-1 βT βC Solutal Volumetric fraction κ Thermal conductivity of nanofluid, W / m oC µ The viscosity of the nanofluid, Pa s Nanoparticle volume fraction ϕ*1, ϕ*2 Nanoparticles mass densities, kg/m3 ρp1, ρp2

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