J. Nig. Soc. Phys. Sci. 5 (2023) 1366 Journal of the Nigerian Society of Physical Sciences Thermal instability of rotating Jeffrey nanofluids in porous media with variable gravity Pushap Lata Sharmaa, Deepak Bainsb, Pankaj Thakurc,∗ aDepartment of Mathematics & Statistics, Himachal Pradesh University, Summer Hill, Shimla, India bDepartment of Mathematics & Statistics, Himachal Pradesh University, Summer Hill, Shimla, India cFaculty of Science and Technology, ICFAI University, Baddi, Solan, India Abstract It is investigated how changes in gravity affect the thermal instability rotating Jeffrey nanofluids in porous media. Along with the Galerkin method and normal mode approach, the Darcy model is used. The distinct variable gravity parameters taken in this paper are: h(z) = z2 − 2z, h(z) = −z2, h(z) = −z and h(z) = z and their effects on the Jeffrey parameter, Taylor number, moderated diffusivity ratio, porosity of porous media, Lewis number and nanoparticle Rayleigh number on stationary convection have been scrutinized and graphically shown. Our finding demonstrates that varying gravity parameter h(z) = z2 − 2z has more stabilising impact on stationary convection. We have also discovered the necessary condition for overstability in the instance of oscillatory convection for this problem. DOI:10.46481/jnsps.2023.1366 Keywords: Jeffrey nanofluid; variable gravity; porous medium; Galerkin method; rotation Article History : Received: 23 January 2023 Received in revised form: 06 April 2023 Accepted for publication: 10 April 2023 Published: 20 May 2023 c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: J. Ndam 1. Introduction Choi [1] devised the term “nanofluid”, which was defined as a liquid having a dispersion of submicronic solid particles (nanoparticles). Convective transport in nanofluids was an issue that Buongiorno [2] examined. He advanced Choi [1] work by including mathematical terms. Different uses for nanofluid were introduced by Tzeng et al. [3], Kim et al. [4], Routbort et al. [5] and Donzelli et al. [6]. The theoretical and experimental findings in Chandrasekhar [7] are based on the Newtonian fluid’s capacity to convect steadily ∗Corresponding author tel. no: +918570975865 Email addresses: pl maths@yahoo.in (Pushap Lata Sharma), deepakbains123@gmail.com (Deepak Bains) in the absence of a porous medium while subject to rotation and a magnetic field. Papamarkos et al. [8] described a method based upon octagonally symmetric design and IIR digital fil- terations. A Study of Non-Newtonian Nanofluids like Rivlin- Ericksen, Maxwellian and Modified Darcy-Maxwell Model for S.C. is employed by Rana et al. [9], Chand [10] and Singh et al. [11], respectively. Linearised stability theory was used by Lap- wood [12] to investigate convective flow in a porous material. Nield et al. [13] introduced convection in porous media. Con- vection with internal heating in a porous material saturated by a nanofluid was examined by Nield et al. [14]. The results reveal that the inclusion of nanofluid particles increases the system’s instability. Later, Nield et al. [15] provided brief introduction to the book Nield et al. [13]. Tzou [16, 17] investigated how 1 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 2 natural convection affected nanofluids’ thermal instability. Sev- eral researchers Nield et al. [18, 19, 20], Sheu [21] and Chand et al. [22, 23, 24] used the Buongiorno [2] model to investigate nanofluids’ thermal instability in porous media. Ramanuja et al. [25] also used porous medium in their problem. The development of objects in an astrophysical plasma en- vironment is caused by thermal instability, which is studied by Kaothekar [26] for partly ionised thermal plasma. This plasma has a relation to astrophysical condensations. Chand et al. [27] studied T.I. effect on Oldroydian nanofluid by considering real- istic boundary conditions. Nield et al. [20], Sharma [28], Yadav et al. [29], Chand et al. [30, 31], Govender [32] and Chand et al. [33] examined the of nanofluid’s thermal instability in rota- tion. Some of them examined rotation’s interactions with sus- pended particles, many non-newtonian fluids, couple-stress ro- tation’s interactions with porous media, and rotation’s interac- tions with itself. They discovered that a system’s thermal insta- bility depends heavily on rotation. Yadav et al. [34] employed magneto-convection in rotatory layer of nanofluid and electro- thermo-convection in a horizontal layer of rotating nanofluid is examined by Chand et al. [35]. Additionally, several of them created rotation-based industrial applications, including those found in nuclear reactors, power plants, the petroleum sector, geophysics etc. Nanofluid oscillating convection in a porous media was explored by Chand et al. [36]. Gautam et al. [37] established the concepts of free-free, rigid-free, and rigid- rigid boundary conditions for the electrohydrodynamic T.I. of a Jeffrey nanofluid layer saturating a porous medium and con- cluded that the rotation parameter stabilises the system for bot- tom and top-heavy layouts. A porous-medium-saturated Jeffrey nanofluid flow was studied by Rana [38] for the effects of ro- tation. For both bottom and top-heavy arrangements and pro- vided evidence that the rotation parameter stabilises the sys- tem. Sharma et al. [39] studies the electrohydro dynamics convection in dielectric rotating Oldroydian nanofluid in porous medium. The idealisation of uniform gravity used in theoretical re- search, while appropriate for lab applications, is seldom war- ranted for large-scale convection events happening in the Earth’s atmosphere, ocean, or mantle. Gravity must thus be viewed as a changeable quantity that changes with distance from a surface or other reference point. Pradhan et al. [40] investigated the thermal instability of a fluid layer in a changeable gravitational field and discovered that boosting the gravitational field verti- cally accelerates the commencement of convection. A porous media with an internal heat source and an inclined temperature gradient was studied by Alex et al. [41] to see how changing gravity affected thermal instability. Straughan [42] used both linear theory and nonlinear energy theory to analyse the issue for the case of stiff boundaries in a spatially changing gravita- tional field and discovered that the nonlinear conclusions were remarkably similar to the linear ones. Chand et al. [43] looked into how changing gravity would affect a layer of nanofluid in a porous medium and found that the gravity parameter had a big impact on fluid stability. Theoretically and visually, Chand [10] investigates thermal instability of Maxwell non-Newtonian fluid with varying gravity. Using a higher order Galerkin method, Yadav [44] investigated the joint effects of variable gravity fields and throughflow on the beginning of convective motion in a porous medium layer. The results showed that both the through- flow and gravity variation parameters serve to delay the mo- tion’s onset. Mahajan et al. [45] analyses the effects of several fundamental temperature and concentration gradients on a layer of reactive fluid in a varied gravity field utilising both linear and non-linear analysis. Surya et al. [46] examine the thermal in- stability of a horizontal layer of liquid heated from below that is contained between thermally conductive porous walls under the influence of a fluctuating gravitational field. As limiting exam- ples of the permeability parameters of the borders, the impact of the gravity variation growing vertically upward for various particular situations of the boundary conditions is derived and graphically depicted. In a layer of porous media, the effects of rotation and varying gravitational strengths on the beginning of heat convection were computed by Yadav [47]. The findings demonstrate how the gravity variation parameter and the rota- tion parameter both delay convection’s arrival. With increased rotation and gravity variation parameters, the measurement of the convection cells diminishes. Shekhar et al. [48] investigates numerically how varying gravity affects rotational convection in a porous material that is poorly packed. The linear, parabolic, cubic, and exponential functions are taken into account for vari- ations in gravitational force. While the Darcy number increases convection cell size, convection cell size falls when the variable gravity parameter and rotation parameter are increased. Addi- tionally, it has been found that the system is more stable for exponential gravity functions than for cubic gravity functions. Chand et al. [49] investigated the impact of variable gravity on the thermal instability of rotating nanofluids in porous me- dia and discovered that, in the presence of rotation and also for nanoparticle Rayleigh numbers, decreasing the gravity parame- ter has a stabilising effect while increasing it has a destabilising effect. By taking into account its numerous applications in vari- ous fields like geophysics, astrophysics, food processing, oil reservoir modeling, building of thermal insulations and nuclear reactors etc. This brief review of the literature leads one to be- lieve that such a problem was nonexistent; hence, the current problem of thermal instability of rotating Jeffrey nanofluids in porous media with variable gravity was chosen. 2. Mathematical Formulation Here, we examine a rotating horizontal Jeffrey nanofluid layer heated from below in a porous medium with medium per- meability k1 and porosity ε and angular velocity Ω(0, 0, Ω) bor- dered by plane z = 0 and z = d, working upward under the influence of variavle gravity. Furthermore, it is assumed from Nield et al. [18] and Chand et al. [49] that there is variable gravity along z -direction i.e. g = (1+δh(z))g, where δh(z) is the variable gravity parameter. When the top boundary layer is at z = d, the temperature T and volumetric fraction ϕ of nanopar- ticles are assumed to be T1 and ϕ1, respectively, with T0 > T1 and ϕ0 > ϕ1. 2 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 3 Figure 1: Physical Configuration For the sake of simplicity, Oberbeck-Boussinesq approxi- mation is used and Darcy’s law is taken to be true by Nield et al. [18] and Chand et al. [49]. Thus from Buongiorno [2], Chandrasekhar [7], Nield et al. [18] and Chand et al. [33, 49] the pertinent governing equations for the study of spinning Jef- frey nanofluid in porous medium are ∇.q = 0 (1) 0 = −∇p + ( ϕρp + (1 −ϕ) {ρ0 (1 −α (T − T1))} ) g − µ k1(1 + λ) q + 2ρ0 ε (q × Ω) (2) For nanoparticle, the continuity equation is given by (Buon- giorno [2]) ∂ϕ ∂t + 1 ε q.∇ϕ = DB∇2ϕ + DT T1 ∇ 2T (3) For the nanofluid, the equation of thermal energy is given by (Buongiorno [2] and Chand et al. [49]) (ρc)m ∂T ∂t + (ρc) f q.∇T = km∇ 2T +ε (ρc)p [ DB∇ϕ.∇T + DT T1 ∇T.∇T ] (4) where q is the fluid velocity, p is the pressure, ρ0 is nanofluid density at z = 0, ρP is nanoparticles density, ϕ is the volume fraction of the nanoparticles, T is temperature, T1 is the refer- ence temperature, α is thermal expansion coefficient, g is grav- itational acceleration and k1 is medium fluid permeability, µ is coefficient of viscosity, ε is the porosity of the porous me- dia, λ = λ1 λ2 the Jeffrey parameter (which is the ratio of stress- relaxation-time parameter, λ1 to strain-retardation-time param- eter, λ2), the fluid’s heat capacity in porous medium is (ρc)m, (ρc)P stands for heat capacity of nanoparticles, (ρc) f stands for fluid’s heat capacity, km is the fluid’s thermal conductivity, the Brownian diffusion coefficient is DB and DT is nanoparticles’ the thermophoretic diffusion coefficient (Chand et al. [49]). We presumed nanoparticles’ temperature and volumetric frac- tion as constant. Thus, boundary conditions (Chandrasekhar [7] and Nield et al. [18]) are{ w = 0, T = T0, ϕ = ϕ0 at z = 0 w = 0, T = T1, ϕ = ϕ1 at z = d (5) On introducing non-dimensional variables as (Chandrasekhar [7]) (x∗, y∗, z∗) = (x, y, z) d , q∗ = q d κm , t∗ = tκm σd2 p∗ = pk1 µκm , ϕ∗ = ϕ−ϕ0 ϕ1 −ϕ0 , T∗ = T − T1 T0 − T1 where κm = km (ρc) f , σ = (ρc)m(ρc) f are fluid’s thermal diffusivity and thermal capacity ratio, respectively. Relaxing the star (∗) for 3 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 4 simplification. The reduced non-dimensional form of equations 1,2,3,4 are: ∇.q = 0 (6) 0 = −∇p − 1 1 + λ q − Rm(1 + δh(z))k̂ +RD(1 + δh(z))T k̂ − Rn(1 + δh(z))ϕk̂ + √ Ta ( q × k̂ ) (7) 1 σ ∂ϕ ∂t + 1 ε q.∇ϕ = 1 Ln ∇ 2ϕ + NA Ln ∇ 2T (8) ∂T ∂t + q.∇T = ∇2T + NB Ln ∇ϕ.∇T + NA NB Ln ∇T.∇T (9) where dimensionless parameters are Rm = (ρPϕ0 +ρ0 (1−ϕ0 ))gk1 d µκm is density Rayleigh number, RD = ρ0α(T0−T1 )gk1 d µκm is Rayleigh Darcy Number Rn = (ρp−ρ0 )(ϕ1−ϕ0 )gk1 d µκm is nanoparticle Rayleigh number, Ta = ( 2Ωρd2 µ )2 is Taylor number, Ln = κm DB is Lewis number, NA = DT (T0−T1 ) DB T1 (ϕ1−ϕ0 ) is nanofluid modified diffusivity ratio, NB = ε(ρc)p (ϕ1−ϕ0 ) (ρc) f is modified nanoparticle-density increment. The reduced non-dimensional boundary conditions are:{ w = 0, T = 1, ϕ = 0 at z = 0 w = 0, T = 0, ϕ = 1 at z = 1 (10) 3. Basic States and it’s Solutions The time independent basic states for nanofluid are expressed as (Nield et al. [18, 19] and Chand et al. [49]):{ q(u, v, w) = 0 ⇒ u = v = w = 0, p = pb(z), T = Tb(z), ϕ = ϕb(z) (11) The basic variable represented by subscript b. Using equation (11) in 6), (7,8), (9), these equations reduce to 0 = − d dz pb(z) − Rm(1 + δh(z)) + RD(1 + δh(z))Tb(z) −Rn(1 + δh(z))ϕb(z) (12) d2 dz2 ϕb(z) + NA d2 dz2 Tb(z) = 0 (13) d2 dz2 Tb(z)+ NB Ln d dz ϕb(z) d dz Tb(z)+ NA NB Ln ( d dz Tb(z) )2 = 0(14) Solving equation 13 with boundary conditions equation 10, we get ϕb(z) = (1 − NA)z + (1 − Tb)NA (15) Using (15) in equation (14), we have d2 dz2 Tb(z) + (1 − NA)NB Ln d dz Tb(z) + NA NB Ln ( d dz Tb(z) )2 = 0 Neglecting the higher order term, we have d2 dz2 (Tb(z)) + (1 − NA)NB Ln d dz (Tb(z)) = 0 (16) Using boundary conditions (10), the solution of differential equa- tion 16 is Tb(z) = e− (1−NA )NB Ln z [ 1 − e− (1−NA )NB Ln (1−z) ] 1 − e− (1−NA )NB Ln (17) According to Buongiorno [2] hypothesis, the approximated so- lutions for equations 15 and 17 are given as Tb = 1 − z, and ϕb = z (18) These approximated solutions 18 agrees well with the results obtained by Nield et al. [18, 19, 20], Sheu [21] and Chand et al. [49]. 4. Perturbation Solutions superimposing infinitesimal perturbation on the basic states in ordered to examine the stability of the system, the basic states equation 11 are written in following form (Nield et al. [18, 19, 20] and Chand et al. [49]){ q(u, v, w) = 0 + q′(u, v, w), p = pb + p′ T = Tb + T′ = (1 − z) + T′, ϕ = ϕb + ϕ′ = z + ϕ′ (19) Using (19) in equations (6, 7,8,9), and linearize by ignoring the products of primes and for convenience discarding primes (′) . We obtain the reduced equations (6,7,8,9) as ∇.q = 0 (20) 0 = −∇p − 1 1 + λ q − Rn(1 + δh(z))ϕk̂ +RD(1 + δh(z))T k̂ + √ Ta ( q × k̂ ) (21) 1 σ ∂ϕ ∂t + 1 ε w = NA Ln ∇ 2T + 1 Ln ∇ 2ϕ (22) ∂T ∂t − w = ∇2T − 2 NA NB Ln ∂T ∂z + NB Ln ( ∂T ∂z − ∂ϕ ∂z ) (23) and Boundary Conditions are ϕ = 0, T = 0, w = 0 at z = 0 and z = 1 (24) It should be noted that Rm is unrelated in equations 21,22 and 23, it is simply the basic static pressure gradient measurement. Operating equation 21 with k̂.curl.curl, we get (i.e. Mak- ing use of result curl.curl = grad.div −∇2) 1 1 + λ ∇ 2w = −Rn(1 + δh(z))∇ 2 Hϕ +RD(1 + δh(z))∇ 2 H T − √ Ta ∂ξ ∂z (25) 4 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 5 Now eliminating p from equation (21), i.e. by operating it with i ∂ ∂y and further with − j ∂ ∂x , respectively and further solving, we get ξ = (1 + λ) √ Ta ∂w ∂z (26) Now, using (26) in equation (25), we have 1 1 + λ [ 1 1 + λ ∇ 2w + Rn(1 + δh(z))∇ 2 Hϕ −RD(1 + δh(z))∇ 2 H T ] + Ta ∂2w ∂z2 = 0 (27) 5. Stability Analysis by Normal Mode The disturbances analysing by normal mode analysis as fol- low (Chandrasekhar [7]):[ w, T,ϕ ] = [W(z), Θ(z), Φ(z)] ex p(ikx x + ikyy + nt) (28) where growth rate is represented as n and the wave number along x and y directions are kx and ky, respectively. Using equa- tion 28 in equations 22, 23 and 27, we get 1 1 + λ [ 1 1 + λ ( D2 − a2 ) W + RD(1 + δh(z))a 2 Θ −Rn(1 + δh(z))a 2 Φ ] + Ta D 2W = 0 (29) 1 ε .W − NA Ln (D2 − a2)Θ + [ n σ − (D2 − a2) Ln ] Φ = 0 (30) W+ [ (D2 − a2) + NB Ln D − 2 NA NB Ln D − n ] Θ− NB Ln DΦ = 0(31) where D = ddz and −a 2 = k2x + k 2 y = ∂2 ∂x2 + ∂2 ∂y2 , ∇ 2 = d 2 dz2 − a 2 = D2−a2. The a is the dimensionless resulting wave number. The boundary conditions by considering normal mode are written as Chandrasekhar [7] (free- free Boundary Condition) W = D2W = Θ = Φ = 0 at z = 0 and z = 1 (32) Assume that the solutions for W, Θ and Φ are of the form (Chandrasekhar [7]) W = W0 sin(πz), Θ = Θ0 sin(πz), Φ = Φ0 sin(πz) (33) These solutions in (33) satisfy the boundary conditions (32). Substituting solution (33) into equations (29,30,31) and inte- grating each equations individually within limits z = 0 to z = 1, we gain the following matrix equation  J 1+λ + (1 + λ)π 2Ta − a2RD(1 + δh(z)) a2Rn(1 + δh(z)) 1 −(J + n) 0 1 ε NA Ln J JLn + n σ   W0 Θ0 Φ0  =  0 0 0  (34) where J = π2 + a2 is the entire wave number. The eigen- value to the system of linear equation 34 is given as RD = [ J 1 + λ + (1 + λ)π2Ta ] (J + n) a2(1 + δh(z)) − [ NA Ln J + (J+n) ε ] J Ln + n σ Rn (35) 6. Stationary Convection For steady state, put n = 0 in equation 35, we obtain RD = (π2 + a2)2 a2(1 + λ)(1 + δh(z)) + (π2 + a2)(1 + λ)π2Ta a2(1 + δh(z)) − ( NA + Ln ε ) Rn (36) The Rayleigh Darcy Number for stationary convection re- veal by the equation 36 is a function of a, λ, δh(z), Ta, NA, Ln, ε, Rn. In non-appearance of Jeffrey’s nanofluid (λ = 0), the equation 36 reduces to RD = (π2 + a2)2 a2(1 + δh(z)) + (π2 + a2)π2Ta a2(1 + δh(z)) − ( NA + Ln ε ) Rn(37) Equation 37 agrees well with the results obtained by Chand et al. [49] for stationary convection. In non-appearance of Jeffrey’s nanofluid (λ = 0) and rotation (Ta = 0), the equation 36 reduces to RD = (π2 + a2)2 a2(1 + δh(z)) − ( NA + Ln ε ) Rn (38) Equation 38, agrees well with the results obtained by Pradhan et al. [40]. In non-appearance of Jeffrey’s nanofluid (λ = 0), rotation (Ta = 0) and constant gravity (δh(z) = 0), then the equation 36 reduces to RD = (π2 + a2)2 a2 − ( NA + Ln ε ) Rn (39) Equation 39, agrees well with the results obtained by Nield et al. [18] and Chand et al. [49]. According to Nield et al. [18], the critical value of equation 36 is accomplished at a = π, so 5 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 6 Table 1: On the onset of Stationary Convection (RD)c Variation of Constant Variables Variable Gravity’s impact on Stationary Convection z in graphs λ Ta Ln Rn ε NA δ h(z) = z2 − 2z h(z) = −z2 h(z) = −z h(z) = z 0.3 λ 0.6 0 - 1 100 500 -1 0.6 5 0.5 Stabilising Stabilising Stabilising Destabilising 0.9 100 Ta 200 0 - 1 0.6 500 -1 0.6 5 0.5 Stabilising Stabilising Stabilising Destabilising 300 100 Ln 500 0 - 1 0.6 100 -1 0.6 5 0.5 Stabilising Stabilising Stabilising Destabilising 1000 -1 Rn -0.5 0 - 1 0.6 100 500 0.6 5 0.5 Destabilising Destabilising Destabilising Destabilising -0.1 0.3 ε 0.6 0 - 1 0.6 100 500 -1 5 0.5 Destabilising Destabilising Destabilising Destabilising 0.9 1 NA 5 0 - 1 0.6 100 500 -1 0.6 0.5 Stabilising Stabilising Stabilising Destabilising 10 Figure 2: Variability of (RD)c w.r.t. z for distinct values of h(z) by taking distinct values of λ for stationary convection the critical Rayleigh-Darcy Number is specified as (RD)c = 4π2 (1 + λ)(1 + δh(z)) + 2π2(1 + λ)Ta 1 + δh(z) − ( NA + Ln ε ) Rn (40) In non-appearance of rotation (Ta = 0), Jeffrey’s nanofluid (λ = 0), nanoparticles and at constant gravity (δh(z) = 0), we Figure 3: Variability of (RD)c w.r.t. z for distinct values of h(z) by taking distinct values of Ta obtained the Rayleigh Darcy Number given as (RD)c = 4π 2 This agrees well with the results obtained by Lapwood [12] for regular field. 7. Oscillatory Convection Here, possibility for oscillatory convection is considered. For oscillatory convection, put n = ini in equation 35, we have RD = [ J 1 + λ + (1 + λ)π2Ta ] (J + ini) a2(1 + δh(z)) 6 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 7 Figure 4: Variability of (RD)c w.r.t. z for distinct values of h(z) by taking distinct values of Ln Figure 5: Variability of (RD)c w.r.t. z for distinct values of h(z) by taking different values of Rn − [ NA Ln J + (J+ini ) ε ] J Ln + ini σ Rn (41) By equating the real and imaginary components of equation 41, we get a2 JRD(1 + δh(z)) Ln + ( NA Ln + 1 ε ) JRna 2(1 + δh(z)) = [ J 1 + λ + (1 + λ)π2Ta ] J2 Ln − [ J 1 + λ + (1 + λ)π2Ta ] n2i σ (42) and RD σ + Rn ε − J a2(1 + δh(z)) × [ J 1 + λ + (1 + λ)π2Ta ] ( 1 σ + 1 Ln ) = 0 (43) Figure 6: Variability of (RD)c w.r.t. z for distinct values of h(z) by taking different values of ε Figure 7: Variability of (RD)c w.r.t. z for distinct values of h(z) by taking different values of NA where J = π2 + a2. The frequency of the oscillatory mode is calculated as follows n2i Ln a2σ = J2 a2 − JRD(1 + δh(z))[ J 1+λ + (1 + λ)π 2Ta ] − J(1 + δh(z))[ J 1+λ + (1 + λ)π 2Ta ] (NA + Ln ε ) Rn (44) In order for ni to be real it is necessary that J(1 + δh(z))[ J 1+λ + (1 + λ)π 2Ta ] [RD + (NA + Ln ε ) Rn ] 6 J2 a2 (45) where J = π2 + a2. The equations 43,44,45 becomes as the absence of the Jeffrey nanofluid (λ = 0), rotation (Ta = 0) and constant gravity (δh(z) = 0) RD σ + Rn ε − (π2 + a2)2 a2 ( 1 Ln + 1 σ ) = 0 (46) 7 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 8 n2i Ln a2σ = (π2 + a2)2 a2 − RD − ( NA + Ln ε ) Rn (47) and RD + ( NA + Ln ε ) Rn 6 (π2 + a2)2 a2 (48) The above result obtained in 46,47,48 are good agreement of results obtained by Nield et al. [18] and Chand et al. [36, 49]. According to Nield et al. [18], the critical value of the wave number is accomplished at a = π , therefore, by setting a = π in equations 46,47,48, we obtain the result for the stability boundary case as RD σ + Rn ε = 4π2 ( 1 Ln + 1 σ ) (49) n2i Ln a2σ = 4π2 − [ RD + ( NA + Ln ε ) Rn ] (50) and [ RD + ( NA + Ln ε ) Rn ] 6 4π2 (51) These results obtained in equations 49,50,51 are same as that of obtained by Nield et al. [18] for particular case. 8. Results and Discussion Variable gravity factors’ impacts on density Rayleigh num- ber, nanoparticle Rayleigh number, Lewis number, porosity of porous media, modified diffusivity ratio, and rotation on sta- tionary convection have been graphed, and their stabilising or destabilising effect has been explored below. The variable grav- ity parameters are as follow: h(z) = z2 − 2z, h(z) = −z2, h(z) = −z and h(z) = z. Figure 2 shows the graph for (RD)c with respect to z for distinct values of λ = 0.3, 0.6, 0.9 by fixing other parameters as Ln = 500, NA = 5,ε = 0.6,δ = 0.5, Ta = 100, Rn = −1. It is discovered that when the gravity parameter changes, such as when it becomes h(z) = z2 − 2z, h(z) = −z2, h(z) = −z it stabilises, however when it becomes h(z) = z, it destabilises. These match those in Straughan [42] for the variable gravity parameter. Figure 3 depicts the graph for (RD)c with respect to z for various values of Ta = 100, 200, 300 setting other parameters like Ln = 500, NA = 5,ε = 0.6,δ = 0.5, Rn = −1,λ = 0.6. It is found that Ta has a stabilising impact when the gravity parame- ters are h(z) = z2 −2z, h(z) = −z2, h(z) = −z, but a destabilising effect when the gravity parameter is h(z) = z. This is in good ac- cord with the finding reported by Chand et al. [49], which states that reducing the gravity parameter has a stabilising impact on stationary convection while raising the gravity parameter has a destabilising effect. Figure 4 depicts the curve for (RD)c with respect to z for distinct values of Ln = 100, 500, 1000 while holding other pa- rameters constant like NA = 5,ε = 0.6,δ = 0.5, Ta = 100, Rn = −1,λ = 0.6. It is found that Ln has a stabilising impact when the gravity parameters are h(z) = z2 −2z, h(z) = −z2, h(z) = −z, but a destabilising effect when the gravity parameter is h(z) = z. This is in excellent accord with the finding from Chand et al. [49] that reducing the gravity parameter stabilises stationary convection while raising the gravity parameter destabilises it. Figure 5 shows that (RD)c decreases with increase in Rn (as = −1,−0.5,−0.1). Thus Rn has destabilizing effect for all variable gravity parameter on stationary convection. Figure 6 shows that (RD)c decreases with increase in ε (as = 0.3, 0.6, 0.9). Thus ε has destabilizing effect for all variable gravity parameter on stationary convection. Figure 7 depicts the graph for (RD)c with respect to z for various values of NA = 1, 5, 10 setting other parameters like Ln = 500,ε = 0.6,δ = 0.5, Ta = 100, Rn = −1,λ = 0.6. It is found that NA has a stabilising impact when the gravity parame- ters are h(z) = z2 −2z, h(z) = −z2, h(z) = −z, but a destabilising effect when the gravity parameter is h(z) = z. 9. Conclusion This article investigates the thermal instability of spinning Jeffrey nanofluids in porous media with changing gravity. The problem is examined for free-free boundary conditions using Galerkin technique and normal mode analysis. Equation 40 is the essential Rayleigh-Darcy number for stationary convection, and it has been studied whether this number stabilises or desta- bilises stationary convection with regard to changing gravity. Equation 45 yields the adequate condition for the oscillatory mode’s frequency, while equation 44 also finds the oscillatory mode’s frequency. In Table 1, the varied gravity’s effects in stationary convection are illustrated visually by changing one parameter at a time while keeping the other parameters constant by assigning them certain constant values. The main conclusions from Table 1 are: 1. Jeffrey parameter (λ), Taylor number (Ta), modified dif- fusivity ratio (NA) and Lewis number (Ln) have stabiliz- ing effect on stationary convection when variable gravity parameters varies as h(z) = z2 − 2z, h(z) = −z2, h(z) = −z whereas have destabilizing effect when variable gravity parameter varies as h(z) = z. In other words, stationary convection has a stabilising impact for lowering the vari- able gravity parameters and destabilising the system for raising the variable gravity parameters. 2. When the variable gravity parameter fluctuates as: h(z) = z2 − 2z, h(z) = −z2, h(z) = −z and h(z) = z, the nanopar- ticle Rayleigh number (Rn), porosity of porous medium (ε) destabilise the system. 3. Variable gravity perameters h(z) = z2 − 2z, h(z) = −z2, h(z) = −z delay the motion’s onset. 4. NB has no effect on (RD)c. 5. The variable gravity parameter h(z) = z2 − 2z has more stabilizing impact on stationary convection rather than other variable gravity parameters taken in this paper. 8 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 9 6. The sufficient condition for the oscillatory mode’s fre- quency is obtained and is represented by equation 45. Acknowledgments The second author gratefully acknowledges the financial as- sistance of UGC for NFSC. References [1] S.U. Choi & J. A. Eastman, “Enhancing thermal conductivity of fluids with nanoparticles”, Argonne National Lab.(ANL) (1995). [2] J. Buongiorno, “Convective transport in nanofluids”, Transactions of the ASME 128 (2006) 240. [3] S. C. Tzeng, C. W. Lin & K. Huang, “Heat transfer enhancement of nanofluids in rotary blade coupling of four-wheel-drive vehicles”, Acta Mechanica 179 (2005) 11. [4] S. Kim, I. C. Bang, J. Buongiorno & L. Hu, “Study of pool boiling and critical heat flux enhancement in nanofluids”, Bulletin of the Polish Academy of Sciences: Technical Sciences 55 (2007) 211. [5] J. Routbort, et al., “Argonne national lab, michellin north america, st.”, Gobain Corp (2009). [6] G. Donzelli, R. Cerbino & A. Vailati, “Bistable heat transfer in a nanofluid”, Physical review letters 102 (2009) 104503. [7] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability”, Dover Publications, Inc. (2013). [8] N. Papamarkos, B. Mertzios & G. Vachtsevanos, “On the optimum design of symmetric two-dimensional iir digital filters with coefficients of finite word length”, International journal of circuit theory and applications 14 (1986) 339. [9] G. Rana & R. Chand, “Stability analysis of double-diffusive convection of rivlin-ericksen elastico-viscous nanofluid saturating a porous medium: a revised model”, Forschung im Ingenieurwesen 79 (2015) 87. [10] R. Chand, “Nanofluid Technologies and Thermal Convection Tech- niques”, IGI Global (2017). [11] R. Singh, V. K. Tyagi & J. Bishnoi, “A study of non-newtonian nanofluid saturated in a porous medium based on modified darcy-maxwell model”, Cognitive Informatics and Soft Computing: Proceeding of CISC (2022) 241. [12] E. Lapwood, “Convection of a fluid in a porous medium”, Mathematical Proceedings of the Cambridge Philosophical Society 44 (1948) 508. [13] D. A. Nield & A. Bejan, “Convection in Porous Media”, Springer 3 (2006). [14] D. Nield & A. Kuznetsov, “Onset of convection with internal heating in a porous medium saturated by a nanofluid”, Transport in porous media 99 (2013) 73. [15] D. Nield & C. T. Simmons, “A brief introduction to convection in porous media”, Transport in Porous Media 130 (2019) 237. [16] D. Y. Tzou, “Thermal instability of nanofluids in natural convection”, In- ternational Journal of Heat and Mass Transfer 51 (2008) 2967. [17] D. Tzou, “Instability of nanofluids in natural convection”, Journal of Heat Transfer 130 (2008). [18] D. Nield & A. V. Kuznetsov, “Thermal instability in a porous medium layer saturated by a nanofluid”, International Journal of Heat and Mass Transfer 52 (2009) 5796. [19] D. Nield & A. V. Kuznetsov, “The onset of convection in a horizontal nanofluid layer of finite depth”, European Journal of Mechanics-B/Fluids 29 (2010) 217. [20] A. Kuznetsov & D. Nield, “Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model”, Transport in Porous Media 81 (2010) 409. [21] L. J. Sheu, “Thermal instability in a porous medium layer saturated with a viscoelastic nanofluid”, Transport in Porous Media 88 (2011) 461. [22] R. Chand & G. Rana, “Thermal instability of rivlin–ericksen elastico- viscous nanofluid saturated by a porous medium”, Journal of fluids engi- neering 134 (2012). [23] R. Chand & G. C. Rana, “Hall effect on thermal instability in a horizontal layer of nanofluid saturated in a porous medium”, International Journal of Theoretical and Applied Multiscale Mechanics 3 (2014) 58. [24] R. Chand & D. Puigjaner, “Thermal instability analysis of an elastico- viscous nanofluid layer”, Engineering Transactions 66 (2018) 301. [25] M.Ramanuja, J. Kavitha, A. Sudhakar & N. Radhika, “Study of MHD SWCNT-Blood Nanofluid Flow in Presence of Viscous Dissipation and Radiation Effects through Porous Medium”, Journal of the Nigerian So- ciety of Physical Sciences (2023) 1054. [26] S. Kaothekar, “Thermal instability of partially ionized viscous plasma with hall effect flr corrections flowing through porous medium”, Journal of Porous Media 21 (2018). [27] R. Chand & S. Kango, “Thermal instability of oldroydian visco-elastic nanofluid in a porous medium for more realistic boundary conditions”, Special Topics & Reviews in Porous Media: An International Journal 12 (2021). [28] R. Sharma, “Effect of rotation on thermal instability of a viscoelastic fluid”, Acta Physica Academiae Scientiarum Hungaricae 40 (1976) 11. [29] D. Yadav, G. Agrawal & R. Bhargava, “Thermal instability of rotating nanofluid layer”, International Journal of Engineering Science 49 (2011) 1171. [30] R. Chand & G. Rana, “On the onset of thermal convection in rotating nanofluid layer saturating a darcy–brinkman porous medium”, Interna- tional Journal of Heat and Mass Transfer 55 (2012) 5417. [31] R. Chand, G. Rana, A. Kumar & V. Sharma, “Thermal instability in a layer of nanofluid subjected to rotation and suspended particles”, Re- search Journal of Science and Technology 5 (2013) 2. [32] S. Govender, “Thermal instability in a rotating vertical porous layer satu- rated by a nanofluid”, Journal of Heat Transfer 138 (2016). [33] R. Chand, D. Yadav & G. C. Rana, “Thermal instability of couple-stress nanofluid with vertical rotation in a porous medium”, Journal of Porous Media 20 (2017). [34] D. Yadav, R. Bhargava, G. Agrawal, G. S. Hwang, J. Lee & M. Kim, “Magneto-convection in a rotating layer of nanofluid”, Asia-Pacific Jour- nal of Chemical Engineering 9 (2014) 663. [35] R. Chand, D. Yadav & G. Rana, “Electrothermo convection in a horizon- tal layer of rotating nanofluid”, International Journal of Nanoparticles 8 (2015) 241. [36] R. Chand & G. Rana, “Oscillating convection of nanofluid in porous medium”, Transport in Porous Media 95 (2012) 269. [37] P. K. Gautam, G. C. Rana & H. Saxena, “Stationary convection in the electrohydrodynamic thermal instability of jeffrey nanofluid layer satu- rating a porous medium: free-free, rigid-free, and rigid-rigid boundary conditions”, Journal of Porous Media 23 (2020). [38] G. C. Rana, “Effects of rotation on jeffrey nanofluid flow saturated by a porous medium”, Journal of Applied Mathematics and Computational Mechanics 20 (2021) 17. [39] P.L. Sharma, M. Kapalta, A. Kumar, D. Bains, S. Gupta, & P. Thakur. ”Electrohydro dynamics convection in dielectric rotating Oldroydian nanofluid in porous medium”, Journal of the Nigerian Society of Phys- ical Sciences (2023) 1231. [40] G. Pradhan, P. Samal & U. Tripathy, “Thermal stability of a fluid layer in a variable gravitational field”, Indian J. pure appl. Math 20 (1989) 736. [41] S. M. Alex, P. R. Patil & K. Venkatakrishnan, “Variable gravity effects on thermal instability in a porous medium with internal heat source and inclined temperature gradient”, Fluid Dynamics Research 29 (2001) 1. [42] B. Straughan, “The Energy Method, Stability, and Nonlinear Convec- tion”, Springer Science & Business Media 91 (2013). [43] R. Chand, G. Rana & S. Kumar, “Variable gravity effects on thermal insta- bility of nanofluid in anisotropic porous medium”, International Journal of Applied Mechanics and Engineering 18 (2013) 631. [44] D. Yadav, “Numerical investigation of the combined impact of variable gravity field and throughflow on the onset of convective motion in a porous medium layer”, International communications in heat and mass transfer 108 (2019) 104274. [45] A. Mahajan & V. K. Tripathi, “Effects of spatially varying gravity, tem- perature and concentration fields on the stability of a chemically reacting fluid layer”, Journal of Engineering Mathematics 125 (2020) 23. [46] D. Surya & A. Gupta, “Thermal instability in a liquid layer with perme- able boundaries under the influence of variable gravity”, European Jour- nal of Mechanics-B/Fluids 91 (2022) 219. [47] D. Yadav, “Effects of rotation and varying gravity on the onset of con- vection in a porous medium layer: a numerical study”, World Journal of Engineering 17 (2020) 785. 9 Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1366 10 [48] S. Shekhar, R. Ragoju & D. Yadav, “The effect of variable gravity on rotating rayleigh–Bénard convection in a sparsely packed porous layer”, Heat Transfer 51 (2022) 4187. [49] R. Chand, G. Rana & S. Kango, “Effect of variable gravity on thermal instability of rotating nanofluid in porous medium”, FME Transactions 43 (2015) 62. 10