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- Springer Journals
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- Copyright © The Author(s) 2020
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- 2213-9621
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- 10.1186/s40486-020-00123-y
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Abstract
The extraordinary enhancement in heat transfer efficiency of nanofluids at extremely low volume fractions has attracted a lot of attention in identifying the governing mechanisms. The nanoscale effects, Brownian motion (ran- dom motion of particles inside the base fluid) and thermophoresis (diffusion of particles due to temperature gradient) are found to be important slip mechanisms in nanofluids. Based on these findings, a set of partial differential equa- tions for conservation laws for nanofluids was formed. Since then, a large number of mathematical studies on convec- tive heat transfer in nanofluids became feasible. The present paper summarizes the studies pertaining to instability of a horizontal nanofluid layer under the impact of various parameters such as rotation, magnetic field, Hall currents and LTNE effects in both porous and non-porous medium. Initially, investigations were made using the model consider - ing fixed initial and boundary conditions on the layer, gradually the model was revised in the light of more practical boundary conditions and recently it has been modified to get new and more interesting results. The exhaustive analy- sis of instability problems is presented in the paper and prospects for future research are also identified. Keywords: Nanofluids, Thermal and thermosolutal convection, Rotation, Magnetic field, Hall currents, LTNE effects, Porous medium Introduction is the heat transfer rate, h is the convective heat trans- With the advancement in industrial sector, effective fer coefficient, A is the surface area and ∆T is the tem - cooling techniques have become the significant require - perature that varies across which the transfer of thermal ment for many industrial processes. Efficient transfer of energy occurs. It has been always the pursuit of the ther- energy in the form of heat from one body to another is mal engineers to maximize q for given ∆T or A which commonly needed in most industries. There are many can be done by increasing h. Heat transfer coefficient is examples related to successful production and safety that a complex function of the fluid property, velocity and hinge upon effective transfer of heat namely, thermal surface geometry. From various fluid properties, ther - and nuclear power plant, refrigeration and air condition- mal conductivity influences the heat transfer coefficient ing system, chemical and processing plants, electronic in the most direct way as this is the property that regu- devices, space shuttles and rocket-launching vehicles. lates the thermal transport at the micro-scale level. Heat Often a fluid is chosen as a medium for transferring heat transfer by conduction through solid is much larger than and consequently the mode of heat transfer is convec- as compared to the conductive or convective heat trans- tion. The amount of heat transposition in convection fer through a fluid. For example, when copper is kept is explained by a clear easy seeming connection, that is at room temperature, its conductivity is approximately known as Newton’s law of cooling; q = h A ∆T, where q 700 times more than that of water and almost 3000- folds greater in comparison to engine oil. Regular fluids like water, ethylene glycol and oil were used initially for *Correspondence: jyoti.maths@gmail.com such procedures but due to their restricted heat transfer University Institute of Engineering and Technology, Panjab University, properties, they were not able to serve the purpose com- Chandigarh, India Full list of author information is available at the end of the article pletely. On the contrary, metal’s thermal conductivity is © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://crea- tivecommons.org/licenses/by/4.0/. Ahuja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 2 of 15 very high in comparison to the regular fluids. These facts systems and advanced nuclear systems, respectively. elicited and caught the eye of research workers to club Novel projected applications of nanofluids include sen - both to generate a heat transfer mode which has attrib- sor and diagnostics that instantly detect chemical warfare utes of fluid as well as metal. Keeping this perspective in agents in water or water-or food borne contamination; consideration theoretical as well as practical work has biomedical applications such as cooling medical devices, been done considerably to enhance the thermal prop- detect unhealthy substance in the blood, cancer treat- erties of fluids by suspending solid particles. Almost a ment, or drug delivery; and development of advanced century ago, Maxwell [1] initiated the theoretical work technologies such as advanced vapour compression on thermal conductivity enhancement by addition of refrigeration systems. The present paper largely summa - micrometer and millimetre sized particles and gradually rizes the mathematical findings related to convective heat nanoparticles were suspended in the fluids termed as the transfer in nanofluids under effects of rotation, magnetic nanofluids [2]. field, hall currents and local thermal non-equilibrium in Afterwards, nanofluids have emerged as an exciting porous and non-porous medium. The partial differential area for advance research and development. The thermal equations for nanofluids based on conservation laws are conductivity enhancement in nanofluids was explored by studied by various researchers to establish significant and Masuda et al. [3], Eastman et al. [4], Das et al. [5] and oth- interesting results which are presented and analysed in ers. They alleged an increment varying 10–30% in ther - the subsequent sections. mal conductivity with the use of nanofluids at very low concentration. Buongiorno [6] formulated a mathemati- Instability of nanofluids cal model to study nanofluid instability phenomenon for Thermal instability the first time. He made an observation that the velocity A horizontal fluid layer is heated underside with main - of nanoparticles can be perceived as a sum of base fluid tained temperature difference across its boundaries leads and relative (slip) velocities. To prosecute his research, to convection currents in the fluid. At the onset of insta - he considered seven slip mechanisms; inactivity, magnus bility, the temperature difference exceeds a certain value effect, Brownian motion, diffusiophoresis, thermophore - was observed first time by Bénard [24] in 1900. He found sis, gravitational settling and fluid drainage. Throughout that the fluid at the bottom becomes lighter and rises up his investigation, he agreed that out of all the seven tech- while the fluid density higher on the top makes the sys - niques, Brownian diffusion and thermophoresis have a tem top heavy. Further, Bénard [25] carried out an exper- significant role in the absence of turbulent effects. Choi iment using metallic plate and a thin non-volatile liquid et al. [7] established that the highest thermal conductiv- layer which is maintained under constant temperature. ity enhancement in fluids is with the addition of carbon He found that fluid layer was decomposed into number nanotubes. A lot of analytical and experimental work has of cells at the onset of instability called Bénard cells. been done on thermal conductivity of nanofluids in the Lord Rayleigh [26] explored the phenomenon ana- past [8–11]. Das and Choi [12], Ding et al. [13] and Das lytically in detail. The work carried out by Rayleigh and et al. [14] studied convective heat transfer in nanofluids Bénard to study thermal instability of fluids is known as extensively. Chen [15] derived heat conduction equations Rayleigh Bénard convection. The schematic representa - from Boltzmann equation. The presence of nanoparticles tion of Rayleigh–Bénard convection is shown in Fig. 1. enhances the conductivity of base fluids [4, 7, 16] and rate They found that at the onset of convection, Rayleigh of heat transfer [17–19]. A small amount of nanoparticle number, given by Ra = gβ d �T/ν α ; exceeds a cer- volume fractions (< 0.1%) leads to conductivity enhance- tain critical value; where β is the volumetric coefficient ment up to 40% [8] and it rises with rise in temperature [5] and nanoparticles [16]. The results of Choi et al. [7] established the unexpected non-linear character of meas- ured thermal conductivity with nanotube loadings at low concentration while all theoretical studies concluded a linear relationship. Also, it was discovered that thermal conductivity strongly depends on temperature [5] and particle size [20]. Pak and Cho [21] considered alumin- ium and titanium nanoparticles in circular tubes to study turbulent flow of nanofluids and found 30% increase in the value of Nusselt number in comparison to base fluid. Kleinstreuer et al. [22] and Buongiorno and Hu [23] Fig. 1 A schematic representation of Rayleigh–Bénard convection found the applications of nanofluids in drug delivery A huja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 3 of 15 of thermal expansion, g is the acceleration due to grav- ρ = φρ + (1 − φ)ρ φρ + (1 − φ) ρ (1 − β(T − T )) . p f p f 0 0 ity, T is the temperature difference between bounda - (5) ries of the layer, α is the thermal diffusivity of fluid, d where ρ is the base-fluid’s density, T is the reference f 0 is depth of the layer, and ν is the kinematic viscosity. For temperature and ρ is the fluid density at reference tem - f 0 the stabilizing viscous force, Ra parameter gives the force perature. The partial differential Eqs. (1–5) along with of destabilizing buoyancy. Chandra [27] explained that momentum equation for different instability problems the instability of the fluid layer depends on its depth by are considered to study the convective motions in the conducting an experiment on the layer in air. Spiegel and fluid. The momentum equation based on conservation Veronis [28] simplified the partial differential equations of mass is redefined by researchers in each case to inves - for the fluid flow by taking depth of the layer to be very tigate the different hydrodynamic and hydromagnetic small as compared to the height and equations for porous problems. The conservation equations were non-dimen - medium were derived by Joseph [29] using Boussinesq sionalized to get new parameters and further the expres- approximation. The thermal convection of a fluid layer sion for thermal Rayleigh number was found to study using different assumptions of rotation and magnetic the various instability problems. Tzou [33, 34] analyti- field has been considered in detail by Chandrasekhar cally solved the conservation equations of nanofluids for [30]. Kim et al. [31] showed that heat capacity and den- convective situations and established that the presence sity of nanoparticles influence the convective motions of nanoparticles hastens the onset of instability of the directly while conductivity has adverse impact. Hwang fluid layer significantly. Nield and Kuznetsov [35] con - et al. [32] found that the presence of alumina nanopar- sidered the nanofluid layer heated from below as shown ticles enhances the stability of the base fluid which rises in Fig. 2 and same geometry was further used by many with the volume fraction of nanoparticles while decreases researchers to study problems. They solved the conser - with the size of nanoparticles. Buongiorno [6] initiated vation Eqs. (1–5) by using Galerkin method and normal the analytical treatment on nanofluid convection by mode technique for free-free, rigid-free and rigid-rigid deriving the conservation equations of nanofluids based boundaries. on nano effects (Brownian and thermophoretic diffusion) For free-free boundaries, the expression of thermal Ray- as follows: leigh number for stationary motions was obtained by Nield and Kuznetsov [35] as: ∇.v = 0 Continuity equation , (1) 2 2 π + α (6) Ra = − Rn Le + Na , [ ] ∂φ ∇T + v.∇φ =∇. D ∇φ + D B T ∂t T where Rn, Le, Na represent the concentration Ray- (Nanoparticle conservation equation), (2) leigh number, the Lewis number and the modified dif - fusivity ratio, respectively which are non-dimensional ∂v parameters. This result is complementary to the result ρ + v.∇v = −∇p + μ∇ v ∂t of Tzou [21, 22] as the reduction of critical Rayleigh + ρg (Momentum equation), number for bottom-heavy case was established [21, 22] (3) whereas Nield and Kuznetsov [35] claimed the increase in the value of the critical Rayleigh number for non- ∂T (ρc) + v.∇T = (k∇ T ) + (ρc) f p oscillatory instability. Also, Nield and Kuznetsov [35] ∂t presented the impact of nanofluid parameters and con - ∇T .∇T D ∇φ.∇T + D Thermal Energy equation . B T cluded that Rn, Le, Na destabilize the system for bottom heavy case. Yadav et al. [36] also performed the analyti- (4) cally investigation on thermal instability of nanofluids by where v = (u, v, w) is the nanofluid velocity, φ is the nan- carrying the conservation Eqs. (1–5). In addition to the oparticles volume fraction, ρ is the nanoparticle mass nanofluid parameters they also examined the impact of density, D is the Brownian diffusion coefficient, D is B T temperature gradient and found that temperature gradi- the thermophoretic diffusion coefficient, µ is the viscos- ent postpones the convective motions and nano-effects ity of the fluid, t is the time, (ρc) is the heat capacity of destabilize the layer significantly. Further, the nanofluid fluid, (ρc) is the heat capacity of nanoparticle, k is the convection problem was revisited by Sharma and Gupta thermal conductivity of the medium, T is the tempera- [37] to explore the problem in detail without combining ture and the nanofluid’s density ρ is given by: the terms at any stage and the expression of thermal Ray- leigh number was found in terms of physical properties Ahuja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 4 of 15 and Kuznetsov [45] using Darcy model and Kuznetsov and Nield [46] further extended the problem in porous medium using Brinkman model. In Darcy model, porous medium is assumed to have porosity ε and permeability K. The Darcy velocity is denoted by v = εv . Then the conservation Eqs. (1–5) for Darcy model were modified as [45]: ∇. = 0 Continuity equation , (7) ∂φ v ∇T + .∇φ =∇. D ∇φ + D B T ∂t ε T (Nanoparticle conservation equation), (8) Fig.2 Geometry for Rayleigh–Bénard convection problems 0 = −∇p + µ ∇ v − v + ρg (Momentum equation), D D (9) of nanofluids. Recently, both experimental and analytical studies were carried out by Kumar et al. [38] to investi- ∂T (ρc) + v .∇T = (k∇ T ) + ε(ρc) f p gate Rayleigh–Bénard instability in nanofluids. Silver ∂t and selenium nanoparticles were synthesized using plant ∇T .∇T extract and base fluid was taken to be water to study the D ∇φ.∇T + D Thermal Energy equation . B T onset of convection. It was observed that the presence of (10) nanoparticles delay the onset of instability in the fluid. On solving Eqs. (7–10), Nield and Kuznetsov [45] A few additional complexities crop up due to the inter- obtained an expression of thermal Rayleigh number in actions between fluid and porous material. In the past porous medium as: investigation of thermal instability of fluids in porous medium became prominent due to its large applications. 2 2 π + α Le The impact of strong magnetic field of earth on the sta - Ra = − Rn + Na , (11) α ε bility of this flow is a key area of interest in geophysics. It becomes more prominent while studying earth’s core and concluded that Rn, Le, Na destabilize the system of where earth’s mantle conducts like a porous medium nanofluid layer for bottom heavy distribution of nano - comprising of conducting fluids. A great amount of work particles whereas porosity stabilizes it. It was found that on the convection problem for Newtonian/non-New- the critical thermal Rayleigh number has a substantial tonian fluids in a porous medium has been accounted change in its value depending on whether the basic nano- by Lapwood [39], Wooding [40], MacDonald et al. [41], particle distribution is top-heavy or bottom-heavy, by the Ingham and Pop [42], Vafai and Hadim [43], and Nield presence of the nanoparticles. They claimed that oscilla - and Bejan [44]. Owing to the applications of convection tory instability is possible only for bottom-heavy nano- in porous media and keeping in mind the thermal prop- particle distribution. Kuznetsov and Nield [46] further erties of nanofluids, convection problem for nanofluids extended their work in porous medium by incorporat- in porous medium has also been given due attention in ing Brinkman model. For Brinkman model conservation the research work. The investigations in porous medium equation of momentum changes to: were started with Darcy model and further it has been extended to develop as Darcy–Brinkman model. By tak- ρ ∂v µ = −∇p +˜µ ∇ v − v + ρg, D D (12) ing Darcy resistance term into consideration, Lapwood ε ∂t K [39] and Wooding [40] examined the stability of a flow where µ ˜ is the effective viscosity. Thus, the set of Eqs. (7, of the fluid saturating porous medium. Following Ray - 8, 10 and 12) constitutes the governing equations of the leigh’s procedure, they have shown that the value of criti- system for Brinkman model. Kuznetov and Nield [46] cal Rayleigh number for the convective flow in porous processed this system for analysis by using normal mode medium is 4π . A detailed and thorough review of the technique and obtained the expression of thermal Ray- work related to convection of fluids in porous medium leigh number as: has been published in a book by Nield and Bejan [44]. Lapwood problem for nanofluids was solved by Nield A huja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 5 of 15 3 2 process. The field has also broadened considerably, with 2 2 2 2 Da π + α + π + α Le Ra = − Rn + Na , new applications becoming apparent in addition to those α ε outlined by Turner [52]. The double-diffusive concepts (13) are mainly applied in large-scale engineering applications where Da is Darcy number which got introduced in and can be observed in solar ponds, shallow artificial Brinkman model. They established that for a typical lakes etc. A direct analogue of thermosolutal convec- nanofluid (having larger Lewis number) buoyancy forces tion has been used to describe the properties of large along with the conservation of nanoparticles has a pri- stars with a helium-rich core which is heated from below. marily effect on the system while the concentration of Spiegel [56] has shown that the helium/hydrogen ratio nanoparticles has a second-order effect. They found the has significant impact on density gradient and can lim - critical value of thermal Rayleigh number with Darcy its the helium transport by double diffusive convection. number effect and concluded that for large values of Another example of double diffusive convection process Darcy number critical Rayleigh number is 3% greater is solidification of metals. than the classical result of Chandrasekhar while in the Kuznetsov and Nield [57] initiated the mathematical absence of Darcy number it is 11% greater than the clas- work on double diffusive instability in a nanofluid layer sical result. Chand and Rana [47] also examined the saturating porous medium under Darcy model. They oscillating convection of nanofluids in porous medium classified the investigated problem as triple diffusion- and questioned the validity of principle of exchange of type process due to involvement of the heat, the nano- stability for the problem and also derived the condition of particles and the solute. The complex equations were non-existence of oscillatory motions. simplified by analytical expressions for non-oscillatory and oscillatory cases. The results predicted that the non- Thermosolutal instability oscillatory mode is expected for top heavy distribution Melvin Stern [48] was first to consider the case of linear of nanoparticles, a situation which corresponds to the opposing gradients of two properties between horizon- fact that the existence of oscillations requires two of the tal boundaries at fixed concentrations. He revealed that buoyancy forces acting in opposite directions. Further, the interesting effect in binary convection is due to sharp Kuznetsov and Nield [58] studied the companion paper difference between diffusivities of heat and solute. Since in non-porous medium. The momentum, thermal energy then many more researchers, including Veronis [49] and and solute conservation equations were redefined for a Nield [50] have developed the idea. The problem of ther - horizontal binary nanofluid layer which is heated and sol - mosolutal convection in a layer of fluid under a stable sol - uted from below as: ute gradient which is heated from below has been studied ∂v by Veronis [49]. Linear calculations for the problem have 2 ρ + v.∇v = −∇p + μ∇ v ∂t been made for a variety of boundary conditions by Nield [50]. It has shown by Turner [51, 52] that the convec- + ρg (Momentum equation), (14) tive motions depend on the component having higher or where ρ = φρ + (1 − φ)ρ φρ + (1 − φ) ρ (1− p f p f 0 lower diffusivity leading to driving forces. When lighter β(T − T ) − β (C − C ) , 0 0 fluid layer is placed over denser of different diffusivi - ties, two types of convective motions crop up; diffusive ∂T (ρc) + v.∇T = (k∇ T ) + (ρc) [D ∇φ.∇T and finger configurations. The excellent review works B f p ∂t on double diffusive system were given by Huppert and ∇T .∇T Turner [53] and Turner [54]. The interference of multi- +D + ρcD ∇ C Thermal energy equation , T TC components transport processes produce the cross-dif- (15) fusion (Soret and Dufour) effects. The mass flux due to ∂C 2 2 temperature gradient is defined as Soret effect and the + v.∇C = D ∇ C + D ∇ T S CT Dufour effect refers to heat flux due to solute gradient. ∂t Insignificant role of the Soret and Dufour effects allows (Solute conservation equation), (16) ignoring their presence in simple models of coupled heat where β is the solutal volumetric coefficient, D is the and mass transfer [44]. Mc Dougall [55] has made an in- diffusivity of solute, D is the Dufour type diffusiv - TC depth study of double diffusive convection considering ity, D is the Soret type diffusivity and C is the solute CT both the solutal effects (Soret and Dufour). The existence concentration. The one term Galerkin approximation of these ideas has observed in the field of oceanography method was used to analyze the stability and the expres- and the role of theoreticians, laboratory experiments and sion for the Rayleigh number was found as sea-going oceanographers became vital to explore this Ahuja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 6 of 15 3 3 2 2 2 2 π + α π + α (1 − N N L ) CT TC S (19) Ra = + Le Rn Ra = − Rn Na[Le + 1]. 2 2 (1 − L N ) α α S TC 1 − N ( ) CT The concentration Rayleigh number was involved with − Rs − Na Rn. (1 − L N ) a new scaling and a major difference was that the sign of S TC (17) concentration Rayleigh number cannot be negative and The expression contains four additional nano-dimen - hence oscillatory convection was ruled out, in contrast sional solute numbers; N Soret parameter, N Dufour CT TC to the conclusion in Nield and Kuznetsov [35] and Nield parameter, L Solute Lewis number and Rs solute Rayleigh and Kuznetsov [45]. Realizing the fact that original and number. The stability boundaries were approximated using revised models mentioned so far were not sensitive to the single term Galerkin approximation which produced the conductivity of nanoparticles; Sharma et al. [67] modified critical Rayleigh number about 5% higher than the true the model by assuming initial constant nanoparticle vol- value. The analytical results for oscillatory instability were ume fraction in the fluid layer and derived the expression established by simplifying complex expressions with the for Rayleigh number (in the absence of solute param- assumptions of large Prandtl number and large nanopar- eters) as: ticle Lewis number. Same problem of binary nanofluid convection was revisited by Gupta et al. [59] to show the 2 2 π + α (20) Ra = − Rn Na, existence of oscillatory motions. They analyzed the impact of different parameters on onset of thermosolutal convec - tion in a nanofluid layer in detail using the software Math - which was obtained to be independent of Lewis num- ematica. Further, Yadav et al. [60] investigated the problem ber and hence established the sensitivity of Ra for both of binary nanofluid layer using a Darcy–Brinkman model. density and conductivity of nanoparticles. It was found The numerical results on the onset of convection were that density of nanoparticles hastens the onset of con- derived using alumina-water nanofluid. Thermosolutal vection in the fluid whereas increase in conductiv - natural instability boundary layer nanofluid flow past a ity delays the same. The stability pattern followed by vertical plate was investigated by Kuznetsov and Nield non-metals is: alumina–water > silica–water > > copper [61]. In this paper, numerical calculations were performed oxide–water > titanium oxide–water and metals is: alu- in order to obtain the terms. Agarwal et al. [62] studied minium–water > copper–water > silver–water > > iron– non-linear convection in binary nanofluid layer saturat - water are shown in Figs 3 and 4 [67]. ing porous medium in terms of Nusselt number and found that initially the effect of time on Nusselt number is oscil - Eec ff ts of different parameters on instability latory while it becomes steady as the time increases. Yadav of nanofluids et al. [63] explored the thermal conductivity and viscosity Eec ff t of rotation variations effects on binary nanofluid convection in porous When a fluid spreads under gravity in a rotating system, medium. Further, Umavathi [64] conducted the studies to motions normal to the rotation vector induce Coriolis analyze the impact of variable viscosity and conductivity on forces that tend to oppose the spreading. In the absence linear and nonlinear stability analysis of binary convection of boundaries intersecting isopotential surfaces and of in a porous medium layer saturated in a Maxwell nanofluid. instability or viscous dissipation, the flow approaches In all the above studies it was assumed that nanoparti- a state of geotropic equilibrium in which buoyancy cle flux can be controlled across the boundary as the tem - and Coriolis forces are in balance. The rotation has an perature thereat. Further, it turned out that these boundary important impact on the onset of convective motions conditions are hard to achieve physically so need was felt in the fluid. Such problem has an application in ocean - for more realistic boundary conditions. Nield and Kuznet- ography, limnology and engineering processes where sov [65, 66] came out with new conditions on boundaries thermal instability of rotating fluid is needed to exam - of the layer and assumed that nanoparticle flux across the ine. It defines some new parameters in fluid dynam - boundaries is zero written as (which is more realistic than ics, and its outcomes are surprising, like the function top heavy/bottom heavy configuration of nanoparticles). of viscosity is reversed [30]. Impact of rotation on the system of a nanofluid layer has been analyzed by Yadav ∂φ D ∂T et al. [68] and Chand [69]. For non-porous medium D + = 0 at z = 0, d. B (18) ∂z T ∂z conservation equation of momentum in the presence of vertical rotation was defined as [68, 69]. The expression of Rayleigh number for revised bound - ary conditions was found to be: A huja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 7 of 15 Fig. 4 Eec ff t of metals on Rayleigh number [67] Fig. 3 Eec ff t of non-metals on Rayleigh number [67] porous medium and obtained the expression for bottom ∂v heavy configuration of nanoparticles as: ρ + v.∇v = −∇p + µ ∇ v + 2ρ(v × ) + ρg , ∂t 2 4 2 2 (21) 1 + DaJ J + Taπ J Le Ra = + Rn − Na , 2 2 where = (0, 0, �) is the angular velocity and the term α 1 + DaJ ε 2ρ(v × ) represents the Coriolis force term which was (25) introduced due to the presence of rotation while for 2 2 2 where J = π + α , and Da is Darcy number which got porous medium momentum equation due to rotation introduced due to Brinkman model and was found to was modified as [70–72]. have stabilizing effect for the stationary mode of convec - tion along with other nanofluid parameters effect except 0 =−∇p +˜µ ∇ v − v D D Na which has a destabilizing effect on the system. They 2 also found the expression of Rayleigh number for oscil- + (v × �) + ρg Darcy model , latory motions (bottom heavy distribution of nano- δ (22) particles). Bhadauria and Agarwal [70] also dealt with nonlinear study of instability of rotating nanofluid layer ρ ∂v µ 2 f D =−∇p +˜ µ ∇ v − v + (v × �) D D D in porous medium. With a Brinkman model in porous ε ∂t K δ medium, they used minimal representation of the trun- + ρg Darcy Brinkman model . cated Fourier series analysis for non-linear. In their anal- (23) ysis, Nusselt number got introduced that represent the Chand [69] considered the top-heavy configuration rate of heat transfer and found that with the rise of Ray- of nanoparticles in non-porous medium and performed leigh number, the Nusselt number also rises, thus the rate the numerical calculations by using Normal mode of heat transfer increases. But for large values of Rayleigh technique. The obtained expression of thermal Ray - number, the Nusselt number tends to a fixed value and leigh number for stationary mode of convection was becomes constant thus the rate of heat transfer becomes obtained as: constant. They also showed that rate of mass transfer of nanoparticles increases with the increase of Darcy num- 2 2 2 π + α + Taπ (24) ber and modified diffusivity ratio. Further, Chand and Ra = − Rn[Le + Na], Rana [71] also employed the Brinkman model but for top heavy configuration of nanoparticles and obtained Ra as: where Ta is Taylor number representing the effect of rotation. He claimed that rotating nanofluid is more sta - 3 2 2 2 2 2 Da π + α + π + α ble than non-rotating layer. Also, in the stationary con- Ra = vection it is found that Taylor number Ta (rotation), 2 2 2 1 Taπ π + α Le Lewis number Le, have stabilizing effect on the system + − Na + Rn. 2 2 while concentration Rayleigh number Rn and modified α Da π + α + 1 ε diffusivity ratio Na have destabilizing effect on the sys - (26) tem. Bhadauria and Agarwal [70] considered Brinkman They found that porosity and concentration Rayleigh model to investigate the instability of nanofluid layer in number decrease the stability of the system while Darcy Ahuja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 8 of 15 Eec ff t of magnetic field number was found to have dual character both stabiliz- When an electrically conducting fluid comes under ing/destabilizing effect for the stationary mode of con - the influence of a uniform magnetic field, two kind of vection depending on the value of Taylor number. In electromagnetic effects are observed within the fluid. the absence of rotation, the Darcy number has stabiliz- Firstly, the currents are induced in the fluid due to its ing effect on the system. Agarwal et al. [72] used Darcy motion across the magnetic field which tends to modify model to investigate the effect of rotation on a nano - the existing fields. Secondly, electric current transverse fluid layer in anistropic porous medium. Their outcomes to the magnetic lines of forces within the fluid exert were that bottom-heavy and top-heavy arrangement forces that adds up to the existing fields. This twofold favour oscillatory and stationary convections, respec- interaction among the fluid motions and magnetic tively. Rotation aids either of the two in this trend. For fields causing unexpected patterns of behaviour are both the arrangements, rotation parameter (Taylor num- depicted and well contained in Maxwell’s equations. ber) gives an enhancement in the stability of the system. Hydrodynamic equations are modified in more suitable Yadav et al. [73] solved the thermal instability problem of way by considering Maxwell’s equations [30]. Thomson rotating nanofluid layer numerically. Six-term Galerkin [83] modified the theory of slow thermal convection method has been adopted to solve the Eigen-value equa- proposed by Rayleigh [26] and Jeffrey [84] by adding tion for rigid-free and rigid-rigid boundary conditions. the Lorentz force which is induced by the interaction of The results for two different types of boundary condi - magnetic field and conducting fluid. The result of such tions were compared and found that system with both interaction has also been concluded by Fermi [85] and rigid boundaries is more stable than rigid-free bounda- Alfvén [86]. Riley [87] carried out further investigation ries at small Taylor number domain however stress-free on Rayleigh–Bénard convection under the influence boundaries offer more stability than rigid boundaries of vertical magnetic field called magneto-convection. when its values are higher. Rana et al. [74] and Rana and Ghasemi et al. [88] and Hamada et al. [89] considered Agarwal [75] investigated the effect of rotation on dou - water based nanofluids with copper, alumina and sil - ble diffusive nanofluid convection saturating a porous ver nanoparticles, to investigate thermal instability for medium. The stabilizing impact of rotation parameter numerical computations. Ghasemi et al. [88] investi- was established in their work. Agarwal [76], Rana and gated the impact of both magnetic field and nanofluids Chand [77] and Yadav et al. [78] re-explored the prob- on natural convection in square cavity while Mahmoudi lem of convective motions in a nanofluid layer subjected et al. [90] investigated the same impact for rectangu- to rotation with new boundary conditions (nanoparticle lar cavity. They argued that the magnetic field resulted flux is zero across the boundaries) for porous and non- in the decrease of convective circulating flows within porous medium. Yadav et al. [78] solved the eigenvalue the enclosures which resulted in the reduction of heat problem numerically using 6-term Galerkin method for transfer rate. The work (magneto-convection) has water based nanofluid with alumina and copper nano - contributed in the field of engineering in the form of particles. Stability of alumina–water nanofluid was com - various applications such as crystal growth in liquids, pared with that copper water nanofluid and observed cooling of rods in nuclear reactor, cooling of micro- that with these new boundary conditions alumina–water chips in electronics and microelectronic devices, solar nanofluid shows more destabilizing effect under the con - technology etc. By conceptualizing the utility aspect stant nanoparticle boundary conditions, while reverse of applying magnetic field, Heris et al. [91] studied the trend was observed for copper–water nanofluid. This is impact of both magnetic field and nanofluid on two because the modified diffusivity ratio has a significant phases closed thermosyphon and found that with the effect for zero nanoparticles flux on boundaries and its increase in magnetic field strength as well as nanopar - value is higher for alumina–water nanofluid than cop - ticle concentration; thermal efficiency of thermosy - per–water nanofluid. The stabilizing impact of rotation phon has significantly increased. Nemati et al. [92] on binary nanofluid convection was analyzed by Sharma in their theoretical study investigated the impact of et al. [79]. Oscillatory motions come into existence for magnetic field on nanofluid convection in a rectangu - bottom heavy arrangement of nanoparticles in the fluid lar cavity by considering the Lattice Boltzman model. layer saturating porous medium. The stabilizing effect of They concluded that increase in magnetic field reduces Taylor number for stationary as well as oscillatory mode the convective heat transfer rate while conductive heat of convection is shown in Fig. 5 and mode of convection transfer rate becomes dominant. Gupta et al. [93] and is found to be oscillatory [80]. Yadav et al. [94] considered the magneto-convection of Further, the onset of thermosolutal convection in a a nanofluid layer for bottom heavy and top-heavy dis - rotating porous nanofluid layer was investigated in many tributions of nanoparticles, respectively. By applying works [81, 82] using Darcy and Darcy Brinkman model. A huja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 9 of 15 number increases with the increase of Darcy number and the magnetic field parameter. Gupta et al. [96] and Ahuja et al. [97, 98] carried out their research of hydromagnetic stability by comparing thermal instabilities of Al O – 2 3 water and CuO–water nanofluids in non-porous medium and in porous medium. They interpreted that magnetic field parameter stabilizes the system for all types of nano - fluids. Further, nanofluid with alumina nanoparticles is found to exhibit more stability than the nanofluid con - taining copper-oxide nanoparticles. In porous medium, analysis is done for three different boundaries free-free, rigid-free and rigid-rigid using Brinkman model. For Fig. 5 Eec ff t of rotation parameter on Rayleigh number [80] free-free boundaries, the expression of Rayleigh number was obtained as [98]: magnetic field on a nanofluid layer, Lorentz force is 1 Qπ 2 2 2 2 Ra = π + α + π + α induced which combines with the element of thermal α ε buoyancy. Thus, the system of conservation equations Le + Rn − Na for Darcy model, in the presence of magnetic field includes Eqs. (1 –3) (31) along with, ∂v µ e 3 2 2 1 ρ + v.∇v = −∇p + µ ∇ v + ρg + ∇× h × H , 2 2 2 2 ( ) Ra = Da π + α + π + α ∂t 4π (27) Qπ 2 2 and Maxwell’s equations + π + α dh Le = (H .∇)v + η∇ h, (28) − Rn + N for Brinkman model. dt (32) In the presence of magnetic field stability of Cu–water ∇.h = 0. (29) nanofluid and Ag–water nanofluid was compared and it where h = (0, 0, h) is the magnetic field that is applied was found that Cu–water nanofluid is more stable than in vertical direction and (∇× h) × H represents the 4π Ag–water nanofluid for top heavy configuration of nano - Lorentz force term which was introduced due to applied particles. The system with both-rigid boundaries is found magnetic field. System of Eqs. (1–3 and 27–29) were ana - to have more stability as compared to rigid-free bounda- lyzed to examine the effect of magnetic field for bottom ries which in turn are more stable than free-free bounda- heavy distribution of nanoparticles and the expression ries. In porous medium they also examined the effect of for stationary convection [93, 94] was obtained as: volume fraction of nanoparticles and temperature differ - ence across the boundaries on stability of the system and 2 2 π + α 2 2 2 Ra = π + α + Qπ + Rn[Le − Na]. found that temperature difference stabilizes the nano - fluid layer appreciably, whereas the volume fraction of (30) nanoparticles and porosity destabilize the layer. Chand Expression for oscillatory convection was also found. and Rana [99] found the solution of the nanofluid layer Due to the presence of magnetic field Chandrasekhar for more realistic boundary conditions in the presence of number Q came into existence. The authors found that uniform vertical magnetic field in a porous medium. They Chandrasekhar number delays the onset of convection derived the stability criterion for stationary and oscil- and the oscillatory mode of heat transfer was established latory convection in the presence of magnetic field and for bottom heavy distribution whereas it was found to be depicted that oscillatory motions do not occur. Sharma through stationary convection for top heavy arrangement et al. [100] and Gupta et al. [101] established the stabiliz- of nanoparticles. Shaw and Sibanda [95] used Brinkman ing impact of vertical magnetic field on binary nanofluid model to investigate the hydromagnetic instability of convection in a horizontal fluid layer in porous and non- a nanofluid layer in Darcy porous medium using con - porous medium, respectively. The stabilizing influence of vective boundary condition. It has been shown that for magnetic field parameter is shown in Fig. 6 for both sta- the case of stationary convection the critical Rayleigh tionary and oscillatory convection [101]. Ahuja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 10 of 15 ∇.h = 0. (35) u Th s Eqs. (1, 2, 4) along with (33–35) form the system of conservation equations in the presence of Hall cur- rents. For bottom heavy distribution of nanoparticles, Gupta et al. [107, 108] got the expression 2 2 π + α Ra = + Rn(Le − Na) 2 2 2 2 2 2 Qπ π + α π + α + Qπ + , 2 2 2 2 2 2 2 α Mπ π + α + π + α + Qπ Fig. 6 Eec ff t of magnetic field parameter on Rayleigh number [101] (36) where additional Hall current parameter M was found to exist. Effect of Hall currents is to hasten the convection Eec ff t of Hall currents (Fig. 7) while magnetic field delays it. It was also estab - When an applied electric and magnetic field are both lished that stability of alumina is more than copper nano- perpendicular to each other, the current does not flow in particles in water in the presence of Hall currents (Fig. 8). the direction of electric field. So, when an electric current The mode of heat transfer is found to be through station - pass through a conducting fluid in the presence of mag - ary convection for top heavy configuration of nanoparti - netic field, transverse force is exerted by the magnetic cles. Further, for porous medium velocity is replaced by field which produces a measurable voltage across the two Darcy velocity in the Eqs. (1, 2, 4) alongwith (33–35) to sides of a conducting fluid. The presence of this meas - get the conservation equations of nanofluid in the pres - urable transverse voltage under the effect of magnetic ence of Hall currents for porous medium. Yadav and field due to which electric current tends to flow across Lee [109] and Yadav et al. [110] modified the convective an electric field is called Hall effect. Thus, Lorentz forces boundary conditions and presented a more realistic feasi- acting on the charges in the current induced the Hall ble system of nanofluid layer in the presence of Hall cur - effect. Gupta [102] studied the effect of Hall currents and rents for non-porous and porous medium, respectively. described that these currents hasten the onset of thermal They examined the stability of a nanofluid layer with convection under the presence of uniform magnetic field. large magnetic fields and obtained the expression of Ray - A considerable work has been done by many research- leigh number [110] in porous medium as: ers in the past [103–105] on the effects of magnetic field/ 2 2 π + α Hall currents on Newtonian/non-Newtonian (viscoelas- 2 2 2 2 2 2 Ra = α ε + α ε + M π 4 2 2 2 2 2 α ε + α ε + M π tic) fluids, and associated problems. It was shown that a vertical component of vorticity induced by Hall currents 1 1 2 2 2 × 1 + Da π + α + επ Q − − Rn Le Na. is one of the possible reasons for destabilizing effect of ε Le Hall currents. Gupta and Sharma [106] further studied (37) the impact of Hall currents and rotation on the double- diffusive convection of Rivlin–Erickson elastic–viscous fluid. Gupta et al. [107, 108] considered the Hall effect on thermal stability of a nanofluid layer in porous and non- porous medium. Due to the presence of Hall currents, conservation equations in non-porous medium were modified as: ∂v µ ρ + v.∇v = −∇p + µ ∇ v + ρg + (∇× h) × H , ∂t 4π (33) along with Maxwell equation dh 1 = (H .∇)v + η∇ h − ∇× [(∇× h) × H ], dt 4πNe (34) Fig. 7 Eec ff t of Hall current parameter on Rayleigh number [107] A huja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 11 of 15 influence of thermally non-equilibrium phases two tem - perature model has been used which was described by Nield and Kuznetsov [115] as follows: ∂T h f fp (ρc) + v.∇T =(k ∇ T ) + T − T f f f p f ∂t 1 − φ ∇T .∇T + (ρc) D ∇φ.∇T + D , B T (38) ∂T φ (ρc) + v.∇T = φ (k ∇ T ) + h T − T . 0 p p 0 p p fp f p ∂t (39) Fig. 8 Eec ff t of alumina and copper nanoparticles on Rayleigh Due to LTNE effects additional variables number in the presence of Hall effects [108] k , k , T , T , h got introduced in which k , k denote f p f p fp f p respectively the effective thermal conductivity of the fluid and particle phase, T , T denote the temperature of fluid f p and particle phase and h is the interphase heat transfer fp According to their result, for small values of the Hall coefficient between the fluid/particle phases. While for current parameter, it has a destabilizing effect on the porous medium a three-temperature model suited well system while for its large values no significant effect is to analyse the thermal lagging among fluid phase, particle observed on the system. On the same way, magnetic phase and solid matrix phase [113] was given as: field parameter is found to delay the onset of convection ∂T v appreciably, for small values of the Hall current param- ε(1 − φ )(ρc) + .∇T = ε(1 − φ ) 0 f 0 ∂t ε eter while for large values of Hall current parameter it ∇T .∇T has no effect on the system. They also observed that the 2 (k ∇ T ) + ε(1 − φ )(ρc) D ∇φ.∇T + D 0 B T f f p size of convection cells depends on the magnetic field parameter and the Hall current parameter for small val- + h T − T + h T − T , fp p f fs s f ues of the Hall current parameter while for large values (40) of the Hall current parameter roll of magnetic field and ∂T v Hall current become insignificant. The conditions for the p D εφ (ρc) + .∇T = εφ (k ∇ T ) + h T − T , 0 p p 0 p p fp f p ∂t ε instability through stationary convection is also found (41) and showed that the oscillatory convection cannot occur with the new boundary conditions. In porous medium, ∂T (1 − ε)(ρc) = (1 − ε)(k ∇ T ) + h T − T , s s s fs f s Hall current parameter and nanoparticles parameters ∂t are found to accelerate the onset of convection, while (42) the Darcy number, magnetic Darcy number and porosity u Th s, for non-porous medium Eqs. (38, 39) along with parameter delay the onset of instability in the fluid layer. Eqs. (1–3) form a system of conservation equations for LTNE model while for porous medium set of Eqs. (7–9) along with (40–42) constitute the system of equations. Eec ff t of LTNE Expression of Rayleigh number [115] for non porous All the above-mentioned studies are based on local ther- medium was: mal equilibrium (LTE) where temperature gradient is assumed to be negligible between the fluid and particle γ + 1 N Ra 1 + phases but Vadasz [111, 112] clarified that there is always 2 2 δ π + α a thermal lagging among the fluid and particle phases if (γ + δ)Le + (γ + 1)N N A H the thermal conductivity is increased. Kuznetsov and + Rn Le + N + 2 2 δ π + α Nield [113, 114] and Nield and Kuznetsov [115] explored 2 2 the impact of this thermal lagging named as local ther- π + α γ + δ N = 1 + , mal non-equilibrium model (LTNE) for the thermal 2 2 2 α δ π + α instability of a nanofluid layer for both porous and non- (43) porous medium. In non-porous medium to account the Ahuja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 12 of 15 where additional parameters Nield number, N , modi- fied thermal capacity ratio γ and modified thermal dif - fusivity δ ratio came into existence in the process of non dimensionalization due to LTNE model while for porous medium these parameters came into existence for par- ticle phase and solid phase separately and effect of all these parameters is also analysed. Here, modified ther - mal capacity ratio γ, and modified thermal diffusivity δ increases the stability of the system while Nield number N tends to reduce it. They found that impact of LTNE is significant in case of non-oscillatory stability but insig - nificant for typical dilute nanofluids. Further, the thermal Fig. 9 Comparison of LTNE and LTE model [120] instability in porous medium for both linear and non-lin- ear conditions using LTNE model is investigated by Bha- dauria and Agarwal [116]. Convection in LTNE is found 2 2 (Qπ + J ) (J + N )(δJ + N γ ) − N γ H H Ra = to set earlier as compared to LTE. For linear conditions α (δJ + N γ + N ) H H Bhadauria and Agarwal [116] obtained the expression as: Taπ J (J + N )(δJ + N γ ) − N γ H H 2 2 2 ε RnLeα α (Qπ + J )(δJ + N γ + N ) 2 2 H H Ra = J (1 + DaJ ) + 2 2 α J ε 2 (J + N )(εJ + N γ ) − N γ Le H H − Rn + Na . ε J + γ N p p HP (δJ + N γ + N )J 2 H H J + N + N HP HS ε J + 1 + γ N p p HP (45) They found that Taylor number, Chandrasekhar num - γ N (γ N ) p HP s HS − − ber, modified thermal diffusivity ratio and modified 2 2 ε J + 1 + γ N ε J + (1 + γ )N s s HS p p HP thermal capacity ratio enhance the stability of the sys- − Rn Na, tem while concentration Rayleigh number, Nield num- (44) ber, modified diffusivity ratio and Lewis number hasten 2 2 2 where J = π + α and α = π 2, N , N are the onset of thermal convection for top heavy distribu- c HP HS tion of nanoparticles in LTNE. Further, Yadav et al. [119] interface heat transfer parameters and γ , γ are modified p s used zero nanoparticle flux boundary condition to study thermal capacity ratios and ε , ε are modified thermal p s the effect of local thermal non-equilibrium on the onset capacity ratios. With the increase in concentration Ray- of nanofluid convection in a porous layer subjected to leigh number, Nield number and modified diffusivity rotation. For porous medium, Brinkman model was ratio, the decrease in Nusselt number is observed thus employed. The influence of double-diffusion and LTNE diminishing the heat transfer rate. While it increases on on the onset of convection in porous medium was con- increasing the values of modified thermal capacity ratio, sidered by Nield and Kuznetsov [120]. They found that thus the rate of heat transfer is increased. On the other the system with LTNE exhibits lesser stability than LTE hand, for solid-matrix phase an unsteady rate of heat model as shown in Fig. 9. transfer is observed initially and with the passage of time It is worthwhile to mention that all the studies assume it approaches to a constant value. Agarwal and Bhadauria that the nanoparticles volume fractions are constant [117] studied the thermal instability of a rotating nano- along the boundaries of the layer which is very difficult fluid layer in non-equilibrium conditions. In addition to to achieve practically. As a result, this model is revised the above results, they noted a slight variation in critical by considering zero nanoparticle volume fractions at the Rayleigh number for small values of nanoparticle concen- boundaries. Most of the problems were revisited by mak- tration Rayleigh number, Lewis number and Taylor’s ing use of revised model. In both the models (Original number and then rises steadily with an increase in the and Revised), nanoparticle volume fractions are assumed value of these parameters whereas for modified diffusivity to vary in horizontal direction only and the model is ratio, an opposite trend was observed. Ahuja and Gupta recently modified by taking constant value of nanoparti - [118] examined the MHD effects of rotating nanofluid cles at the basic state which established more effectively layer using LTNE model. One term Galerkin approxima- the contribution of metallic and non-metallic nanoparti- tion has been used to analyse the stability. For top heavy cles on the convection in the layer. By considering these distribution of nanoparticles they got the expression as: facts, it is concluded that different alternations can be A huja and Sharma Micro and Nano Syst Lett (2020) 8:21 Page 13 of 15 Authors’ contributions made on the applied models to study convective motions JA surveyed the literature on thermal instability of nanofluids. JS surveyed the which altogether could make a significant difference. literature on thermosolutal instability of nanofluids. Both authors wrote the final manuscript. Both authors read and approved the final manuscript. Concluding remarks and scope for future work Funding The paper presents an overview of various instability Not applicable. problems for nanofluids under the effects of different Availability of data and materials hydrodynamic and hydromagnetic parameters. The sig - Not applicable. nificant heat transfer enhancement of convective fluids at very low nanoparticles concentration has been estab- Competing interests The authors declare that they have no competing interests. lished by many researchers and related literature has been reviewed in detail. As a consequence, mathematical Author details investigations to explore the related mechanisms were Department of Mathematics, Post Graduate Government College, Chan- digarh, India. University Institute of Engineering and Technology, Panjab initiated and the effects produced due to the presence of University, Chandigarh, India. nanoparticles lead to new set of equations based on con- servation laws which further encouraged theorists to for- Received: 6 April 2020 Accepted: 1 November 2020 mulate the instability problems for nanofluids. 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Micro and Nano Systems Letters – Springer Journals
Published: Nov 13, 2020