Stability analysis of the mixed convective flow of Jeffrey nanofluid through a porous medium

Abstract

Keywords: Horizontal Pressure Gradient; Instability; Jeffrey Fluid; Mixed Convection; Nanofluid; Porous Layer

Introduction

Mixed convective flows of non-Newtonian fluids represent a complex and fascinating area of study in fluid

dynamics. These fluids, which include various types such as polymer solutions, slurries, and biological fluids, exhibit behaviors that deviate from the classic Newtonian fluid model. In mixed convective flows, the fluid motion

*Corresponding author. *E-mail address: chandrashekarag.maths@bmsce.ac.in This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

is influenced by both buoyancy forces due to temperature gradients and external forces or pressures. Understanding the behavior of mixed convective non-Newtonian fluids is crucial in many industrial processes, such as polymer processing, food processing, and oil recovery. Researchers and engineers often employ advanced numerical simulations and experimental techniques to unravel the intricate dynamics of these fluids and optimize processes for efficiency and performance. Understanding mixed convection in porous media is essential for geothermal reservoirs, underground energy storage, nuclear reactors, and solar collector applications. Prasad et al. [1] investigated mixed convection in horizontal porous layers heated from below, shedding light on heat transfer mechanisms in such systems. Sphaier and Barletta [2] explored unstable mixed convection in a heated horizontal porous channel, highlighting the importance of understanding flow instabilities in porous media systems. Ozgen and Varol [3] conducted a numerical study of mixed convection in a channel filled with a porous medium, providing valuable insights into heat transfer phenomena in porous media through computational simulations. These studies collectively enhance our understanding of mixed convection in porous media and its implications for various engineering and environmental applications. Vafai [4] provided a comprehensive reference for researchers, covering various aspects of porous media and its applications. Kim and Vafai [5] studied the use of nanofluids to enhance buoyancy-driven heat transfer in porous enclosures, demonstrating the potential of nanofluids to enhance heat transfer in porous media. Abu-Nada and Chamkha [6] provided insights into the effect of solid boundaries on mixed convection in a lid-driven cavity filled with a fluid-saturated porous medium. Nield and Bejan [7] provided a comprehensive overview of convection in porous media, emphasizing its importance in various engineering applications. Ingham and Pop [8] further advanced the understanding of transport phenomena in porous media, focusing on theoretical and experimental aspects. Nield [9] contributed to the development of boundary conditions for porous media simulations with a porous medium model with the Navier slip boundary condition. A study by Postelnicu [10] examined mixed convective instability due to the effect of a horizontal pressure gradient and temperature difference for the Newtonian fluids on the onset of Darcy-Bénard convection in thermal non-equilibrium conditions, revealing insights into the interplay between pressure gradients and thermal non-equilibrium, thus elucidating the mechanisms governing convective flow in porous media. On the other hand, the significance of nanofluids has gained recognition alongside the research conducted, as evidenced by the following studies. Buongiorno [11] studied convective transport in nanofluids. The study by Choi et al. [12] investigates the unusual increase in thermal conductivity observed in nanotube suspensions, a phenomenon with significant implications for various technological applications. An investigation conducted by Mostafizur

et al. [13] aimed to quantify the thermal conductivity of methanol-based nanofluids (MBNF) and demonstrated that nanoparticle clustering is the primary factor contributing to enhanced thermal conductivity. Tzou [14] investigated the thermal instability of nanofluids in natural convection. Nield and Kuznetsov [15] analyzed thermal instability in a porous medium layer saturated by a nanofluid. Kuznetsov and Nield [16] discussed the effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Bhaduria and Agarwal [17] studied convective transport in a nanofluid-saturated porous layer with a thermal non-equilibrium model. Chand and Rana [18] discussed the oscillating convection of nanofluid in a porous medium. Yadav et al. [19] analyzed the onset of convection in a binary nanofluid-saturated porous layer. Yadav et al. [20] discussed thermal convection in a Kuvshiniski viscoelastic nanofluid-saturated porous layer. Sheu [21] analyzed the linear stability of convection in a viscoelastic nanofluid layer. Chand et al. [22] discussed the thermal instability analysis of an elastic-viscous nanofluid layer. Recent studies have advanced nanofluid behavior comprehension and thermal performance improvement across diverse applications, while additional research has focused on refining analytical and numerical techniques in thermal transport [23-32] and these studies offer insights into thermal performance improvement in engineering. Many studies have explored the behavior of Jeffrey fluid in porous media, some of which are summarized below. Nadeem and Akbar [33] analyzed the peristaltic flow of a Jeffrey fluid with variable viscosity in an asymmetric channel. Nallapu et al. [34] investigated the flow of a Jeffrey fluid through a porous medium in narrow tubes. Yadav [35] examines the influence of anisotropy on Jeffrey fluid convection in horizontal rotary porous layers, enhancing our understanding of convective heat transfer in anisotropic porous media. Naganthran [36] studied the effects of heat generation/absorption in Jeffrey fluids flowing past permeable stretching/shrinking discs, providing valuable insights into viscoelastic fluid behavior in thermal gradients. Numerous studies have been conducted to investigate free convection in Jeffrey nanofluids through porous media. The study of Jeffrey nanofluid convection in porous media has enhanced heat transfer efficiency and energy conservation in diverse applications. By leveraging the unique properties of Jeffrey fluids and the heat transfer characteristics of nanoparticles, these fluids exhibit superior heat transfer performance compared to conventional fluids. They are particularly beneficial in applications where efficient heat transfer is crucial, such as cooling systems, thermal management, and heat exchangers. Moreover, Jeffrey nanofluids can improve energy efficiency, reduce environmental impact, and enhance stability in high-temperature environments. These attributes underscore the promising nature of research in Jeffrey nanofluid convection in porous media, with significant potential for practical applications across

various fields. For instance, Zhang et al. [37] explored the thermal behavior of Jeffrey nanofluid flow in a porous medium with a convective boundary condition. This study Siddiqui et al. [38] investigates the influence of Darcy and Prandtl numbers on the unsteady heat transfer characteristics of Jeffrey nanofluid flowing through a porous medium. Sharma et al. [39] investigated thermal convective instability in a Jeffrey nanofluid saturating a porous medium under rigid-rigid and rigid-free boundary conditions. Recent work by Pushap et al. [40] explored the thermal instability of rotating Jeffrey nanofluids in porous media with variable gravity, highlighting the importance of understanding their stability for optimizing heat transfer processes in engineering applications. Gautam et al. [41] investigate stationary convection in the electrohydrodynamic thermal instability of Jeffrey nanofluid layers saturating porous media under different boundary conditions, furthering our understanding of nanofluid dynamics. Shehzad et al. [42] studied the MHD flow of a Jeffrey nanofluid with convective boundary conditions. Shahzad et al. [43] studied the numerical simulation of magnetohydrodynamic Jeffrey nanofluid flow and heat transfer over a stretching sheet considering Joule heating and viscous dissipation. Sreelakshmi et al. [44] analyzed the homotopy analysis of a Jeffrey nanofluid’s unsteady flow heat transfer over a radially stretching convective surface. Devi et al. [45] explores the behavior of electroconvection in a rotating Jeffrey nanofluid saturating a porous medium under different boundary conditions, with implications for applications in fields like microfluidic devices and geophysics. Promila Devi et al. [46] explore the initiation of thermal instability within a Darcy–Brinkman porous layer containing a Jeffrey nanofluid under rotation, offering a novel perspective on heat transfer processes in non-Newtonian fluids with potential applications in improving industrial system efficiency. Sharma et al. [47] investigate the impact of a magnetic field on thermosolutal convection in a Jeffrey nanofluid with a porous medium, employing linear stability analysis and the Darcy model. Maatoug et al. [48] explores the applications of the thermos-diffusion effect in the context of squeezing flow of Jeffrey nanofluids through a horizontal channel, highlighting the influence of inertial effects, Darcy-Forchheimer flow, viscous dissipation, and activation energy, with a focus on numerical computations and physical flow parameters. Sushma et al. [49] discussed the mixed convection flow of a Jeffrey nanofluid in a vertical channel. Current studies have shed light on the intricate dynamics of mixed convective flows in porous media, particularly focusing on the interplay between pressure gradients and viscoelastic fluid behavior. Additionally, Pallavi et al. [50] explore oscillatory Darcy-Bénard-Poiseuille mixed convection by extending the work Postelnicu in an Oldroyd-B fluid-saturated porous layer. It investigates flow and thermal behavior through theoretical analysis and numerical simulations of pressure gradients. The study contributes to understanding complex fluid dynamics in porous media for practical applications. In a related study to Postelnicu,

Hemanthkumar and Shivakumara [51] studied the thermal instability of an Oldroyd-B fluid-saturated porous layer, incorporating considerations of pressure gradients and LTNE temperatures, thus providing insights into the thermal behavior of viscoelastic fluid flows in porous media and highlighting the role of pressure gradients and temperature differentials in driving mixed convective instabilities. Despite extensive research on the onset of instability in natural, forced, or mixed convection within a horizontal porous layer involving Newtonian or non-Newtonian fluids or nanofluids, there is a significant gap in the literature regarding the contribution of similar works, including horizontal pressure gradient effects, to the study of mixed convection in a porous layer saturated with a Jeffrey nanofluid. This gap has led to novel investigations exploring the effects of horizontal pressure gradient and buoyancy due to temperature differences on mixed convection in such a porous layer. Our study aims to investigate the influence of the Jeffrey parameter and other factors on the instability of stationary and oscillatory convection in a porous layer, focusing on the thermal applications of Jeffrey nanofluids. By examining the linear stability in mixed convection of Jeffrey’s nanoporous fluid, we aim to deepen our understanding of how porosity and nanoparticle characteristics impact the onset of convection. This research is particularly relevant in industries such as food processing and electronics cooling, providing insights into low-volume fraction and low-permeable porous channels, which are crucial for managing and enhancing convection. Using linear stability theory, we identify key factors influencing convective motion, as detailed in the following sections.

Mathematical Formulation

The two-dimensional rectangular coordinate system (x, z) is chosen, where the x-axis is taken along the plates of the horizontal channel and the z-axis is perpendicularly upwards in which gravity ( ) is acting downwards. The plates at z = d and z = 0 are maintained at dissimilar constant temperatures Tc and Th respectively, with Th > Tc a porous matrix in between the plates. It is filled with -nanofluid which is heated from below and cooled from above as illustrated in Figure 1. The free and forced convection flow is due to the buoyant force with temperature difference (ΔT = Th − Tc ) and constant horizontal pressure gradient respectively leads to mixed convection flow.

The following assumptions are made: A fully developed and non-quiescent base flow is assumed with an applied horizontal pressure gradient. · Due to the low permeability of the porous channel, the unsteady Darcy model is adopted. · The Boussinesq approximation ρ = ρ0 (1− β (T − Tc )) is used such that the density varies only with temperature. · The nanoparticle volume fraction is considered to be low and constant [52]. · The study is restricted to linear stability along with normal mode analysis. · The local thermal equilibrium between the fluid and solid phases holds in a porous medium. Based on the aforementioned assumptions, the governing equations are formulated following the approach outlined by Yadav [35]. ·

(3) The corresponding boundary conditions are given by = 0 at z = 0 and z = d,

Thermo-physical Properties of Nanofluids The thermophysical properties of water-based nanofluid are used in the present study. The effective viscosity of the nanofluid, μnf is computed using the base fluid viscosity μbf and a diluted suspension of tiny sphere-shaped nanoparticles, (6) The validation has been confirmed by [53] through experimental investigation utilizing oil-water nanofluids at temperatures ranging from 20 to 50 degrees Celsius. The subsequent experimental findings are in the [54] and discovered that it is completely consistent. The approximate thermal conductivity for the nanofluid is calculated using the [55] model. The model tells us, with the help of suspended nanoparticles how the thermal conductivity of nanofluid increases and is given by,

At the reference temperature, the nanofluids effective density and heat capacitance are computed as [56] and [57] respectively as follows, (8) Following are the formulas for nanofluids volumetric expansion coefficient and thermal diffusivity for nanofluids respectively, (9) A Jeffrey nano-fluid model with single-phase, which is suitable for low-volume quantities of nanoparticles, is used to explain the features of nano-fluids. The thermophysical properties of the several nanoparticles at ambient temperature are shown in Table 1 [52].

Table 1. Thermo-physical properties of nanoparticles Physical properties

Non-dimensional Governing Equations The physical quantities of length, velocity, time, pressure, and temperature of nanofluid in the governing equations (1)– (5) are made non-dimensionalized using the scales

following dimensionless governing equations in the cartesian coordinates are obtained by (10)

where Λ is the nondimensional Jeffrey parameter, is the Darcy-Rayleigh number, is the nano-particle volume fraction ratio of nanofluid to base fluid,

(19) where a is the wavenumber and ω = ωr + iωi is the complex wave speed. The growth rate ωi marks the difference between stability (ωi < 0) and instability (ωi > 0) . Substituting Eq. (19) in Eqs. (17) and (18), we obtain the following equations:

Vadasz number. The corresponding dimensionless boundary conditions are given by

Linear Stability Analysis The linear stability analysis of mixed convection assumes a non-quiescent basic state of fluid flow, influenced by a constant horizontal pressure gradient as described in the following form, (14) The corresponding basic state solutions are given by

The corresponding boundary conditions are given by (22) Growth Rate Analysis The classical integral method, as described by Shankar and Shivakumara [58], is utilized to analyze the thermal instability in the limit as Va approaches infinity. First, we operate (D2 − a2 ) on Eq. (21) to eliminate ψ and obtain the above equation for Θ in the form,

(15) Further, we superimpose infinitesimally small perturbations on the basic state given in (15) in the form:

Multiply the complex conjugate form of Θ and integrate with respect to z over the limit 0 to 1, which yields

(16) The stability equations are derived by following a sequence of operations first we linearize equations (11) and (12) by substituting perturbed quantities given equation (16) then the pressure is eliminated by operating curl on the resultant equation ... and finally substituting stream function in the form

We apply integration by parts to the above equation, we arrive at the following equation,

(18) At this moment for a better understanding of the impact of all parameters on the wave number and frequency of perturbations, the solution of equations (17) and (18) are expressed in the form of normal modes given by

(i) İf then ωi < 0 indicating the system is stable, and (ii) if then ωi > 0 the system becomes unstable. Method of Numerical Solutions Equations (20) and (21) form a complex eigenvalue problem and are solved utilizing WRGM. In this method, the test (weighted) functions are the same as the base (trail) functions. Accordingly, Ψ(z) and Θ(z) are expanded in a finite series of basis functions in the form, (25) where Ai and Bi are constants while Ψ j (z) and Θ j (z) are the basis functions and are generally chosen to satisfy the respective boundary conditions, respectively N represents the number of terms considered in the Galerkin expansion. The basis functions are described by the power series that adhere to the pertinent boundary conditions (26) These series are substituted back into Equations (20) and (21) and the WRGM procedure of demanding that the residues be normal to the basis functions is applied by multiplying the resulting Eq. (20) by Ψj (z) and Eq. (21) by Θj (z) integrating by parts with respect to z between z = 0 and z = 1, and the boundary conditions are used to obtain the following system of algebraic equations: (27) (28) the coefficients of Eji and Jji involve the inner products and are given by

tions (27) and (28) can be written in the matrix form as (30)

Equation (30) forms a generalized eigenvalue problem in which that A and B are 2N × 2N order complex matrices, X is the eigenvector and ω = ωr + iωi is the complex eigenvalue. The integral occurring in the coefficients of E ji and J ji are analytically evaluated to avoid errors during the numerical integration. The main stages of the numerical procedure involved in solving Eq. (30) are as follows i) Among the 2N eigenvalues, we identify the most growing or the least decaying mode having the largest imaginary part of the eigenvalue ω and call that mode simply the most growing mode. ii) The largest value of ωi is now forced to zero by varying RD for a fixed value of wave number a and other governing parameters. The computational software program MATHEMATICA 11.3 (Wolfram Research) is used to provide an ideal platform for the execution of Both stages and the following built-in functions are used: Max[Im[Eigenvalues[A,B]]]= EV (say) and Find Root [EV[RD, a]==0]

Results And Discussion

The study investigates the mixed convective instability of Jeffrey nanofluid flow through a horizontal porous layer using a generalized eigenvalue problem both analytically and numerically. Both one-term and higher-order Galerkin methods are adapted and the obtained results are presented. By assessing the convergence process by increasing the number of terms N in the Galerkin expansion. The process of convergence of WGRM is shown in Table 2-4 for various values of governing parameters involved therein. It is noted that the convergence of the critical Darcy-Rayleigh number, the corresponding critical wave number and the critical frequency is accomplished by considering ten terms in WGRM. We have thoroughly examined and established the convergence and validity of our numerical method. To validate the numerical procedure employed, the results are computed under the limiting case and observe that they are in excellent agreement. A glance at Tables 2-4 shows that there is not much deviation in the values of critical stability parameters between the second (N = 2) and higher order (N =10) (WGRM). Thus it is intuitive to look for the analytical solution for the eigenvalue problem with Ψ = Asinπ z and Θ = Bsinπ z as basis functions for the solutions of equations (20) and (21).

Table 2. Convergence of the WRGM for different values of П with φ = 0.025 for TiO2 nanoparticles α1 = 10, q1 = 0.97, Λ = 0.5, Va = 2 П=2

Table 3. Convergence of the WRGM for different values of П with φ = 0.075 for TiO2 nanoparticles α1 = 9.9075, q1 = 0.9387, Λ = 0.5, Va = 2 П=2

Table 4. Convergence of the WRGM for different values of П with φ = 0.1 for TiO2 nanoparticles α1 = 9.8113, q1 = 0.9182, Λ = 0.5, Va = 1 П=2

By substituting these into Equations (20) and (21), we can express them in the matrix form as follows:

where, . Equation (31) represents a homogeneous system and for a non-trivial solution, we should have (32)

The above expression in the equation (38), with П = 0 and ω = 0, subsequently reducing to (39) This expression matches with [35] and represents the onset of stationary convection limiting the case to LTE. The above expression is in the equation (39) with Λ = 0 which coincides with Horton and Rogers, Lapwood’s Problem. (40)

Solving this determinant for RD, we get (33) After rearranging this expression, it is represented in a complex form as follows, (34) where (35)

Analysis of the Growth Rate The critical wave number and growth rate ωi are plotted in Figures 2-5 for various values of RD, Λ, П and different nanoparticles. The growth rate helps us comprehend the onset of instability in the (a, ωi)plane. The sign of ωi determines the stability of the system: if ωi < 0, the system is stable, and if ωi > 0, it is unstable. Figure 2 shows a plot for TiO2 (α1 = 10, q1 = 0.97) nanoparticles with П = 2, Λ =

Conclusion

In this study, we performed a thorough stability analysis of mixed convective flow with a volume fraction of Jeffrey nanofluid in a porous layer, using both numerical and analytical methods. The study utilized the unsteady Jeffrey-Darcy model, which includes the Jeffrey parameter to address retardation or relaxation effects. This model is particularly well-suited for situations involving low-volume fraction Jeffrey nanoparticles and porous layers with low permeability. Specifically, we utilized the Weighted Residual Galerkin Method (WRGM) for numerical analysis and the one-term Galerkin method for analytical investigation. It is shown that the results are in good agreement with N=8. Also, the growth rate of perturbations is numerically computed over a broad spectrum of governing parameters, revealing a notable change in the growth rate behavior for TiO2 and Al2O3 particles, exhibiting instability, while CuO particles remain stable within the considered parametric range. Some of the important results of this analysis can be outlined as follows:

1. The neutral stability curves exhibit a single minimum,

and we observe that the instability region diminishes with increasing nanoparticle volume fraction in the base fluid. This phenomenon suggests that higher volume fractions result in amplified resistance to flow.

2. Increasing П leads to an expansion of the stability

region, indicating a stabilizing effect. Conversely, an increasing value of the Jeffrey parameter Λ results in a contraction of the stability region for all nanoparticles, implying a destabilizing influence.

3. Increasing the Vadasz number is to decrease the

Darcy-Rayleigh number hence advances the onset of convection.

4. An increase in the Jeffrey parameter Λ tends to decrease

the critical values of RDc and ωic, accelerating the initiation of convective activity. However, ac exhibits a reverse trend, decreasing ac value with an increase of Λ for all nanofluids, indicating a stabilizing effect on convective cell size.

5. We observe that the Jeffrey parameter Λ can act as

both a stabilizer and a destabilizer, depending on its combination with other parameters, highlighting its versatile role in influencing convective stability. These conclusions provide valuable insights into the complex interplay of parameters affecting the stability of mixed convective flow in Jeffrey nanofluids through porous

media, contributing to a deeper understanding of thermal transport phenomena in such systems.

6. The critical Darcy-Rayleigh number for various

nanoparticles exhibits an inequality of the form To develop for further analysis, one challenge is to investigate the non-linear instability analysis, while the paper will also explore extending the present work by considering the solute concentration and heat source.

Nomenclature

Dimensional velocity vector Time Pressure Permeability of porous medium Length of channel Dimensional coordinates Specific heat at constant pressure Gravitational acceleration Temperature Thermal conductivity Darcy-Raleigh Number Vadasz Number Nanoparticle volume fraction parameter.

Greek symbols λ Jeffrey parameter ϕ Nanoparticle volume fraction Reference density of fluid ρ0 ρ Density of fluid β Volumetric expansion coefficient α Thermal diffusivity The ratio of thermal diffusivities. µ ε σ Λ П

Viscosity of the nanofluid Porosity heat capacity ratio Non-dimensional Jeffrey parameter Constant horizontal pressure gradient

Acknowledgment

We extend our appreciation to Anonymous Reviewers for their input and suggestions which have played a crucial role in improving the overall quality of the manuscript also the authors are thankful to the authorities of their respective institutions for encouragement and support in carrying

out this research work. All the authors acknowledge the support of Visvesvaraya Technological University as well.

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 authors 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.

Funding

This research work is funded by BMSCE under the FRP scheme with a project No. R&D/FRPS/2022-23/MAT/02.

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