ENGINEERING PHYSICS AND MATHEMATICS
Nonlinear thermal convection in a viscoelastic
nanofluid saturated porous medium under gravity
modulation
Palle Kiran
*Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow 226025, India
Received 3 December 2014; revised 30 April 2015; accepted 2 June 2015 Available online 18 July 2015
KEYWORDS Gravity modulation; Viscoelastic nanofluid; Darcy model; Nonlinear theory
Abstract This paper carried out a nonlinear thermal convection in a porous medium saturated with viscoelastic nanofluid under vibrations. The Darcy model has been used for the porous med-ium, while the nanofluid layer incorporates the effect of Brownian motion along with thermophore-sis. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The non-uniform vertical vibrations of the system, which can be realized by oscillating the system vertically, is considered to vary sinusoidally with time. In order to find the heat and mass transports for unsteady state, a nonlinear analysis, using a minimal representa-tion of the truncated Fourier series of two terms, has been performed. Effect of various parameters has been investigated on heat and mass transport and then presented graphically. It is found that gravity modulation can be used effectively to regulate either heat or mass transports in the system. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
As it is well known fact that thermal conductivity of solids is greater than fluids, in general fluids in heat transfer has appli-cations, such as water, ethylene glycol and engine oil have low
thermal conductivity when compared to thermal conductivity of solids, especially metals. Hence an addition of solid particles in a fluid can increase the conductivity of fluids. Due to the Brownian motion of nano-particles through fluids, better results obtained for heat transport. Brownian motion increases the mode of heat transfer or mass transfer in the system. The word nanofluid to represent the dispersion of nanoparticles and the enhancement of higher thermal conductivity of nanofluids were introduced by Masoud et al. [1] and Choi [2]. Numerous attempts have been made to find the enhanced behavior of high thermal conductivity of nanofluids, some of them are Chen et al.[3], Vada´sz[4,5]. However, a satisfactory explanation has yet to be found as emphasized by Eastman et al.[6]in their recent comprehensive review of the nanofluid literature. This enhanced behavior of thermal conductivity
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implies an enormous potential of nanofluids for device minia-turization and process intensification which could have impacts on many industrial sectors including chemical process-ing, transportation, electronics, medical, energy, and the envi-ronment. The ballistic nature of heat transport within nanoparticles was analyzed by Chen [7]. Further 10–30% increase of the effective thermal conductivity in alumina/water nanofluids with 1–4% of alumina was reported by Das et al. [8]. These reports led Buongiorno and Hu[9] to suggest the possibility of using nanofluids in advanced nuclear systems. Another application of the nanofluid flow is in the delivery of nano-drug as suggested by Kleinstreuer et al.[10].
A comprehensive review of heat transport in nanofluids is done by Eastman et al. [11]. Buongiorno [12] has given an extensive study to account for the unusual behavior of nanofluids based on inertia, Brownian diffusion thermophore-sis, diffusiophorethermophore-sis, Magnus effects, fluid drainage and grav-ity settling, and proposed a model incorporating the effects of Brownian diffusion and the thermophoresis. With the help of these equations, studies were conducted by Kim et al.[13–15], Tzou [16,17] and Nield and Kuznetsov [18]. Employing Darcy model, the Horton–Rogers–Lapwood problem was investigated by Nield and Kuznetsov [19]. Using a three-temperature model Kuznetsov and Nield [20] investigated the effect of local thermal non-equilibrium among the nanoparticle, fluid, and solid-matrix phases. They conclude
that in some circumstances, the effect of LTNE can be signif-icant, but for large Lewis number, the effect was small. Wu and Kao [21] found that engine oil with TiO2 nanoparticle
additive exhibited lower friction force as compared to the original oil. Their experiment showed that a smaller particle size exhibits better friction reduction with particle size rang-ing from 59 to 220 nm. Some other studies related to the nanofluids are given by[22–32] and corresponding introduc-tion therein.
The above literature deals with nanofluids as Newtonian fluids. Non-Newtonian rheological behavior of nanofluids is indicated by many investigators like Chen et al. [34], Schmidt et al.[35]. Thermal convection of non-Newtonian flu-ids in a porous medium received considerable importance in several fields of application such as food processing, oil recov-ery, and the spread of contaminants in the environment, and in various processes in the chemical and materials industries. The onset of convection in a Non-Newtonian nanofluid fluid satu-rated porous medium was briefly discussed by Nield [36]. He noticed that, the HortonRogersLapwood problem becomes singular when a Newtonian fluid is replaced by a standard power-law fluid. This singularity can be removed when the nanofluid effects due to thermophoresis and Brownian motion become independent of the power-law index. The concept of thermal convection in a viscoelastic fluid-saturated porous medium was investigated by many authors given in [37–47]. Nomenclature
Latin symbols
DB Brownian diffusion coefficient
DT thermophoretic diffusion coefficient
PrD Prandtl–Darcy number, PrD¼ md
2
kTK
d dimensional layer depth
kT effective thermal conductivity of porous medium
km thermal diffusivity of porous medium
Le Lewis number, Le¼kT
DB
NA modified diffusivity ratio, NA¼DDBTTðTcð/hTcÞ
1/0Þ
NB modified particle-density increment, NB¼ d1ðqcÞpð/1/0Þ
ðqcÞf
p pressure
~g modulated gravity field
Ra thermal Rayleigh–Darcy number, Ra¼
qg0bKdðThTcÞ
lkT
Rm basic density Rayleigh number, Rm¼
½qp/0þqð1/0Þg0Kd
lkT
Rn concentration Rayleigh number, Rn¼
ðqpqÞð/1/0Þg0Kd
lkT
s time
T temperature
Tc temperature at the upper wall
Th temperature t the lower wall
v nanofluid velocity vD Darcy velocity ðx; y; zÞ Cartesian coordinates Greek symbols e amplitude of modulation X frequency of modulation a horizontal wave number b proportionality factor d1 porosity
l viscosity of the fluid
l effective viscosity of the porous medium qf fluid density
qp nanoparticle mass density ðqcÞf heat capacity of the fluid
ðqcÞm effective heat capacity of the porous medium ðqcÞp effective heat capacity of the nanoparticle material c heat capacity ratioðqcÞm
ðqcÞf
/ nanoparticle volume fraction m kinematic viscosity l=qf e strain retardation time w stream function k stress relaxation time Subscripts b basic solution Superscripts * dimensional variable 0 perturbation variable Operators r2 @2 @x2þ @ 2 @y2þ @ 2 @z2
Here most of the authors investigated onset of thermal convec-tion and nonlinear thermal instability for binary fluid satu-rated porous medium. Some of them are considered rotating porous medium, and effective results obtained for linear and nonlinear studies. But, modulation work has not been evalu-ated in their studies which is important in regulation of con-vective phenomenon in the medium.
The gravity modulation is one consisting of varying acceler-ation term in the gravitacceler-ational Rayleigh number around the gravitational acceleration, i.e., by vertically oscillating a hori-zontal porous layer. This modulation leads to the variable coefficients in the momentum equation and involves the verti-cal time-periodic vibrations of the system. Also, this leads to the appearance of a modified gravity, collinear with actual gravity, in terms of a time-periodic gravitational perturbation and it is known as g-jitter. The modulated gravity field which can be used to modify the momentum equation in order to control the convective phenomenon in the form of amplitude and frequency of modulation is an important phenomenon in thermal and engineering sciences. An application can be seen in materials processing under reduced gravity conditions when convection due to buoyancy forces is strongly reduced. For zero gravity the desired basic state may be set up in a melt when both temperature and concentration gradients are paral-lel. Also for any residual gravity and in particular fluctuations of effective gravity due to orientation changes of the vehicle and on-board activities introduce notable perturbations. The residual acceleration fields on board of a spacecraft are nonsta-tionary and the measured oscillation frequencies are from 102Hz to 100. In crew activity or orbital maneuvers (in
space-flight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft) give rise to time dependent accelerations with high amplitudes and fluctuating direction. In space, the gravity effect is suppressed and hence buoyancy effect also reduces. However, microgravity environment is helpful in suppressing convective flows. The effect of g-jitter which originates from crew motions, mechanical vibrations (motors, pumps, excitations of natural frequencies of space-craft structure), atmospheric drag and the earth’s gravity gra-dient have shown to make it difficult to realize a diffusion controlled growth from melts in micro-gravity. The topology of the neutral curves is more complex than that encountered in constant gravity multiply diffusive layers, leading to new types of behavior not possible in the absence of modulation.
Gresho and Sani[48]was the first to study the gravity mod-ulation on the stability of a heated fluid layer. They studied the impact of the two-dimensional gravity modulation on the con-vective threshold of a stable and an unstable motionless state. Malashetty and Padmavathi[49]investigate the effect of grav-ity modulation on the onset of convection in fluid and porous layers. Recently, Umavathi[50]studied both temperature and gravity modulation of convection in a porous medium satu-rated by a nanofluid by using a linear stability analysis. For nonlinear case of thermal instability in fluid saturated porous medium with vibrations was studied by Bhadauria et al. [51,52]. They show that, the gravity modulation can be used to alter the heat and mass transport in the medium. The effect of nonlinear throughflow on binary viscoelastic fluid saturated porus medium under gravity modulation is investigated by Kiran[53]. He shows that, throughflow plays a dual role, for outflow enhances the heat and mass transfer and inflow dimin-ishes the heat and mass transfer. The modulation frequency diminishes or modulation amplitude increases the heat and mass transfer in the system. He also found that, oscillatory flows strengthen the heat and mass transfer in terms of oscilla-tory frequency than stationary flows. Bhadauria and Kiran [54–57], studied a nonlinear convection under gravity modula-tion for stamodula-tionary and oscillatory modes, they found that, heat or mass transfer rates are better for oscillatory mode than sta-tionary mode of convection.
The onset of thermal convection in a viscoelastic nanofluid saturated porous medium is studied by Shue[58]using modi-fied Darcy model. They derived analytically the onset criterion for stationary and oscillatory convection and found that, the oscillatory case is possible in both bottom and top heavy nanoparticle distributions. They found that, the viscoelasticity and nanofluid properties cause the convection to set in through oscillatory rather than stationary modes. This gives only onset criteria but, missing finite amplitude convection which is an important phenomenon for nonlinear theories. The convection of non-Newtonian fluids in a porous medium has a wide range of applications; some of them mentioned in 2nd paragraph, and in various processes in the chemical and material indus-tries. Since the elastic behavior of viscoelastic fluids and it is inherent in non-Newtonian fluids, oscillatory instabilities can set in before stationary modes. While considering the above lit-erature, very few works are available for nanofluid saturated porous medium under modulation due to [50,51]. But, for
non-Newtonian nanofluid fluid case no study is being reported under vibrating porous medium, where vibrations of the system provides analysis to regulate heat or mass transport in terms of amplitude and frequency of modulation. This moti-vated, to study a nonlinear analysis of thermal instability in a viscoelastic nanofluid saturated porous medium under gravity modulation.
2. Mathematical formulation
An Oldroyd-B nanofluid-saturated horizontal layer of porous medium confined between z¼ 0 and z ¼ d is considered. The physical configuration is represented byFig. 1. Each boundary wall is assumed to be perfectly thermally conducting. The por-ous layer is extended infinitely in x and y-directions, and z-axis is taken vertically upward with the origin at the lower bound-ary. The temperatures at the lower and upper walls are taken to be Th and Tc respectively, the layer is heated from below
and cooled from above. The Darcys law is assumed to hold and the OberbeckBoussinesq approximation is employed. Homogeneity and local thermal equilibrium solid and fluid phases are assumed. The reference temperature is taken to be Tc. For linear theory, the change in temperature of the
nanofluid is assumed to be small in comparison to Tc.
Employing the Oberbeck–Boussinesq approximation, the governing equations to study the thermal instability are given by Sheu[58]: r vD¼ 0; ð1Þ 1þ k@ @s q f d1 @vD @s þ rp ½/qpþ ð1 /Þ fqð1 bðT TcÞÞg~gÞ ¼ l K 1þ e @ @t vD; ð2Þ ðqcÞm@T @sþ ðqcÞfvD rT ¼ kmr2Tþ d1ðqcÞp½DBr/ rT þDT Tc rT rT; ð3Þ @/ @sþ 1 d1 vD r/ ¼ DBr2/þ DT Tc r2T; ð4Þ ~g¼ g0ð1þ cosðXsÞÞ~k ð5Þ
where vDis the Darcy velocity, where DBis the Brownian
dif-fusion coefficient and DTis the thermophoretic diffusion
coef-ficient. The physical variables have their own meanings given in nomenclature. Assuming temperature and volumetric frac-tion of the nanoparticles to be constant at the stress-free boundaries, one may take the boundary conditions on T and / as:
v¼ 0; T ¼ Th; /¼ /0at z¼ 0; ð6Þ
v¼ 0; T ¼ Tc; /¼ /1 at z¼ d; ð7Þ
where /1 is greater than /0. The dimensionless variables are
considered as given below: ðx; y; zÞ ¼ ðx; y; zÞ=d; s¼ sk T=cd2; ðu; v; wÞ ¼ ðu; v; wÞ d=kT; p¼ pK=lkT;/¼///0 1/0and T ¼ TTc ThTc, where kT¼ km ðqcÞf; c¼ðqcpÞm ðqcpÞf; k¼ kkT d2 and e¼ ekT d2. The non-dimensionalized
governing equations are (after dropping the asterisk for simplicity) r v ¼ 0; ð8Þ 1þ k@ @s 1 PrD @v @sþ rp þ gmðRm Ra T þ Rn/Þ ^ez ¼ 1 þ e@ @t v ð9Þ @T @sþ v rT ¼ r 2TþNB Ler/ rT þ NANB Le rT rT; ð10Þ 1 c @/ @sþ v r/ ¼ 1 Ler 2 /þNA Ler 2 T ð11Þ v¼ 0; T ¼ 1; / ¼ 0 at z ¼ 0; and v ¼ 0; T ¼ 0; /¼ 1 at z ¼ 1; ð12Þ
where gm¼ 1 þ cosðXtÞð Þ. The non-dimensionalized parame-ters in the above equations have their usual meanings given in nomenclature, NA is the modified diffusivity ratio, which is
similar to the Soret parameter that arises in cross diffusion in thermal instability. At the basic state, the nanofluid is assumed to be at rest, therefore the quantities at the basic state will vary only in z-direction, and are given by:
v¼ 0; p ¼ pbðzÞ; T ¼ TbðzÞ; / ¼ /bðzÞ: ð13Þ
Substituting the Eq.(13)in Eqs.(10) and (11), one can get: d2Tb dz2 þ NB Le d/b dz dTb dz þ NANB Le dTb dz 2 ¼ 0; ð14Þ d2/b dz2 þ d2Tb dz2 ¼ 0: ð15Þ
According to Buongiorno[12], for most of the nanofluid stud-ies the value of Le=ð/1 /0Þ is large of order 10
5 105
, since the nanoparticle fraction decrement ð/1 /0Þ is typically no
smaller than 103 this means that Le is large of order 102 103, while N
A is no greater then about 10. Using the
above analysis, Tzou[16,17], Nield and Kuznetsov[18]showed that the second and third terms in equation Eq.(14)are small and hence obtain the following:
d2Tb
dz2 ¼ 0; d2/b
dz2 ¼ 0: ð16Þ
The boundary conditions for solving Eq.(16)can be obtained from Eq.(12)as:
Tb¼ 1; /b¼ 0 at z ¼ 0; ð17Þ
Tb¼ 0; /b¼ 1 at z ¼ 1: ð18Þ
Solving the Eq.(16), subject to the above conditions given in Eqs.(17) and (18), obtain the following the solution:
Tb¼ 1 z; ð19Þ
/b¼ z: ð20Þ
3. Nonlinear stability
Now superimpose the perturbations on the basic state as given below:
v¼ v0; p¼ p
bþ p0; T¼ Tbþ T0; /¼ /bþ /0: ð21Þ
Substituting the above Eq.(21)in Eqs.(8)–(11), and using the expressions(19) and (20), eliminating the pressure and intro-ducing the stream function, one can arrive at
1þ k@ @s 1 PrD @ @sðr 2wÞ þ g mRa @T @x gmRn @/ @x ¼ 1 þ e @ @s r2 w; ð22Þ @w @x r 2T¼ @T @sþ @ðw; TÞ @ðx; zÞ; ð23Þ 1 d1 @w @x NA Ler 2T¼ 1 Ler 2/1 c @/ @sþ @ðw; /Þ @ðx; zÞ: ð24Þ
A local nonlinear stability analysis shall be performed and hence consider the following Fourier expressions:
w¼X 1 n¼1 X1 m¼1 AmnðsÞ sinðmaxÞ sinðnpzÞ; ð25Þ T¼X 1 n¼1 X1 m¼1 BmnðsÞ cosðmaxÞ sinðnpzÞ; ð26Þ /¼X 1 n¼1 X1 m¼1 CmnðsÞ cosðmaxÞ sinðnpzÞ: ð27Þ
In general the following modes ð1; 1Þ is for stream function, ð0; 2Þ for temperature and ð1; 1Þ for nanoparticle concentra-tion, which means only two terms have been considered (also see the studies of [27–33]) in order to study heat and mass transfer. The reader may note that, here is first nonlinear effects are accounted and further terms may slightly be addi-tion to the nonlinear effects.
w¼ A11ðsÞ sinðaxÞ sinðpzÞ; ð28Þ
T¼ B11ðsÞ cosðaxÞ sinðpzÞ þ B02ðsÞ sinð2pzÞ; ð29Þ
/¼ C11ðsÞ cosðaxÞ sinðpzÞ þ C02ðsÞ sinð2pzÞ; ð30Þ
where the amplitudes A11ðsÞ; B11ðsÞ; B02ðsÞ; C11ðsÞ and
C02ðsÞ are functions of time and are to be determined.
Substituting the Eqs. (28)–(30)in Eqs.(22)–(24), and taking the orthogonality condition with the eigenfunctions, associated with the considered minimal mode, obtain the following simul-taneous differential equations:
d2A 11ðsÞ ds2 ¼ PrD kd2 agm RnC11ðsÞ þ Rnk dC11ðsÞ ds RaB11ðsÞ Rak dB11ðsÞ ds PrD kd2 d 2A 11ðsÞ þ d2e dA11ðsÞ ds 1 k dA11ðsÞ ds ; ð31Þ dB11ðsÞ ds ¼ ½aA11ðsÞ þ d 2 B11ðsÞ þ paA11ðsÞB02ðsÞ; ð32Þ dB02ðsÞ ds ¼ pa 2A11ðsÞB11ðsÞ 4p 2B 02ðsÞ; ð33Þ 1 c dC11ðsÞ ds ¼ aA11ðsÞ þ 1 Led 2 C11ðsÞ þ pa d1 A11ðsÞC02ðsÞ þ NA Led 2 B11ðsÞ ; ð34Þ 1 c dC02ðsÞ ds ¼ pa 2d1 A11ðsÞC11ðsÞ 4p2 Le½C02ðsÞ þ NAB02ðsÞ: ð35Þ
The above system of simultaneous autonomous ordinary dif-ferential equations can be subsequently solved numerically using NDSolve Mathematic 8.
4. Heat and mass transport
The Nusselt number for heat transport NuðsÞ is defined as
NuðsÞ ¼Heat transport byðconduction þ convectionÞ Heat transport by conduction ¼ 1 þ R2p=ac 0 @T @z dx R2p=ac 0 @Tb @z dx " # z¼0 : ð36Þ
Substituting the Eqs. (19) and (29) in Eq. (36), obtain the Nusselt number
NuðsÞ ¼ 1 2pB02ðsÞ ð37Þ
Figure 2 Nu versus s for different values of PrD.
Figure 3 Sh versus s for different values of PrD.
The Sherwood number, ShðsÞ is defined similar to the Nusselt number, as follows
ShðsÞ ¼Mass transport byðmolecular diffusion þ advectionÞ Mass transfer by molecular diffusion ;
ð38Þ ¼ 1 þ R2p=ac 0 1 Le @/ @zþ NA Le @T @z dx R2p=ac 0 1 Le @/b @zþ NA Le @Tb @z dx " # z¼0 ; ¼ ð1 2pC02ðsÞÞ þ NAð1 2pB02ðsÞÞ: ð39Þ
5. Results and discussion
Since transport phenomena associated with nanofluids have received numerous applications in many fields such as the delivery of nanodrug solar collectors, thermal management, transportation, the environment and national security, nanofluids can be optimized during manufacture using sheet processing. Many superior lubricants as well as thermal work-ing fluids may develop for applications in aerospace, medical engineering, energy systems, etc. Moreover modulated flows provides a way that is external to the system helps us to control heat or mass transfer. More recent motivation for the present
work has been provided by the development of space experi-ments and the use of mechanical vibration in industrial pro-cesses requiring control of convective motions. In this paper the effect of gravity modulation in a horizontal layer of a por-ous medium saturated with a viscoelastic nanofluid is investi-gated. Using Darcy model in the momentum equation a nonlinear stability analysis is performed to study heat and mass transport. A linear theory has been investigated by Umavathi [50] while considering temperature and gravity
Figure 5 Sh versus s for different values of k1.
Figure 6 Nu versus s for different values of k2.
Figure 7 Sh versus s for different values of k2.
Figure 8 Nu versus s for different values of Rn.
modulations for ordinary nanofluid saturated porous medium. Sheu[58]investigated a linear stability analysis for a layer of porous medium saturated with a viscoelastic nanofluid. Keeping in mind of both the papers, the aim of the present paper is to study a nonlinear thermal instability under gravity modulation to control heat and mass transport in the medium. According to Buongiorno [12], for most nanofluids investi-gated so far Le is large, but Bhadauria and Agarwal [31]
considered Le¼ 10 in order to show the parameter effect clearly for nanoparticle concentration Rayleigh number. In this problem the value of Le has taken around 30.
The effect of gravity modulation on heat transport has been depicted in Figs. 2–15. The following parameters PrD; Rn; NA; Le; k; e; c; and X occur in the present study,
and influence the convective heat and mass transport. The amplitude e and frequency X of modulation are chosen to be
Figure 10 Sh versus s for different values of Le.
Figure 11 Sh versus s for different values of Na.
Figure 12 Nu versus s for different values of e.
Figure 13 Sh versus s for different values of e.
Figure 14 Nu versus s for different values of X.
external to the system of controlling convection. Because of small amplitude of modulation, the values of e are considered to be small. Further, the gravity modulation assumed to be of low frequency, as at low range of frequencies, the effect of frequency on onset of convection as well as on heat and mass transport is maximum. The coefficient of heat transport, i.e. Nusselt number and the coefficient of nanoparticle concentration transport, i.e. Sherwood number are calculated as function of time and other parameters of the system. The obtained results are depicted inFigs. 2–15for Nu and Sh ver-sus time s. In the figures the values of Nu and Sh start with 1 and 2 respectively, and remain constant for a quite some time, showing the conduction state. Then the values of Nu and Sh increase as time passes, thus showing that the convection is
taking place. These values oscillate and then approach con-stant values thus showing the steady state.
While keeping the parameter values as ðRa ¼ 104
; PrD¼ 1:0; k1¼ 0:6; k2¼ 0:1; Rn ¼ 5:0; Na ¼ 1:0; Le ¼ 30; c ¼ 1:0;
¼ 0:1Þ and x ¼ 2:0, fixing them, each individual effect of parameter on heat and mass transport is discussed and the results are presented in the graphs. From Figs. 2 and 3, it is found that initially when time s is small the vibra-tions become of high amplitudes as the value of Prandtl– Darcy number PrD, Nusselt and Sherwood numbers
increases as PrD, thus increasing the rate of heat and mass
transport. But, at large values of time s, the vibrations become smaller and subsequently the values of Nu and Sh approach steady state values. The value of PrD can be
taken more than one, in that case the effect of local accel-eration term which appears in momentum equation will be disappeared. Taking PrD¼ 0:5; 1:0; 3:0 one can see that,
there is increment in heat transfer, the same effect can be seen in the case of Sherwood number. The effect of the stress relaxation parameter, k, on heat and mass transport is shown in Figs. 4, 5. The Nu and Sh increase with an increase in the stress relaxation parameter which indicates that the effect of stress relaxation parameter is to advance the onset of convection in viscoelastic nanofluid-saturated porous media and increases heat and mass transport. Figs. 6, 7 depicts the effects of the strain retardation parameter, e, Nu and Sh. It is found that with increasing the value of the strain retardation parameter, both Nu
and Sh increase, indicating that it delays the onset of con-vection in viscoelastic nanofluid-saturated porous media and decrease heat and mass transport. The present results are comparable with the results obtained by Malashetty et al. [39–44], Kumar and Bhadauria [45,46] for ordinary vis-coelastic fluids.
It was also noted that a negative value of Rn indicates a bottom-heavy case, while a positive value indicates a top-heavy case. Further, the influence of the concentration Rayleigh number Rn > 0 on both Nusselt and Sherwood numbers is found to enhance the heat and mass transport as given in Figs. 8 and 9for top-heavy case, while opposite effect can be seen for bottom-heavy case. The same results were obtained by Agarwal et al. [27,28]. On the contrary
in the case of concentration Sherwood number both Le and NA have increasing effect, given in Figs. 10 and 11, and so
the heat and mass transport. One can see that, in Fig. 9 when Le takes 50 there is an increment in nanoparticle con-centration. Also the frequency and magnitude of oscillations increases. The value of Le is taken as 30 for showing the effect of parameters clear in the figures. The reader may note that, the values of Le may consider more than 30 but, for author convenient it is taken as 30. Since NA and
Le do not have significant effect on Nu, the corresponding figures are not presented to avoid recreation of figures. These results for unmodulated case earlier reported by Bhadauria and Agarwal [30–32].
Figs. 12 and 13show that, the effect of amplitude of gravity modulation on heat and mass transport is to increase the val-ues of Nu and Sh and hence transport phenomena in both the cases. The comparison is also made for with or without mod-ulation, modulation case is more in heat and mass transport than in unmodulated case, these results conform the results obtained by Bhadauria and Kiran [51]. For un-modulation case one can see the paper of Agarwal et al. [29]. But, for Newtonian fluid saturated porous medium case Srivastava et al. [60]show the quite opposite results for weak nonlinear convection using Ginzburg–Landau model. Also Kiran [53] shows the same for viscoelastic fluid saturated porous media using complex Ginzburg–Landau equation. The reader may
note that these opposite results are due to non-Newtonian vis-coelastic nanofluid saturated porous media. Similarly in Figs. 14 and 15 show that the effect of frequency of gravity modulation on heat and mass transport is to decrease the val-ues of Nu and Sh, and hence stabilize the system. Thus the classical results are obtained by Gresho and Sani [48]. Also the results correspond to heat and mass transfer for gravity modulation one can see [54–57] for viscoelastic fluids. The effect of heat capacity ration c is to decrease the values of Nu and Sh and porosity d1 is to increase the values of Nu
and Sh these are the results earlier obtained by Bhadauria et al.[32]. In order to avoid more number the figures corre-sponding to the figures of c and d1 have not presented. It is
observed that, in most of the cases there is significant effect of parameters on Nu and Sh at low values of time, but less effect at large time, since vibrations become smaller in magni-tude, and disappear as Nu, Sh reach steady state value. The results of gravity modulation on the system preserve the results obtained by Bhadauria and Kiran [59] for rotational speed modulation.
InFigs. 16–18show the time-dependent fields for different values of w; T; /, at different times. It is clear that with increasing in time, magnitudes of stream lines increases and further in time achieves steady state. But, for the case of isotherms and isohalines as time passes loses their evenness showing the flow of heat and mass transports through conduction to convection. In allFigs. 16–18for w, the sense of motion in the subsequent cells is alternately identical with and opposite to that of the adjoining cell. In case of isotherms, at the starting time, conduction occurs which approaches to convection stage very soon, with the magnitude of isotherms increasing with time. In the intermediate time range, uniform convection cells are observed which change to strong convection with the passage of time. For the isohali-nes, it is observed that the concentration is more near the walls and less in the middle of the system. The particles remain concentrated toward the walls and enhance the convection toward the walls. The trend observed for steady and unsteady streamlines, isotherms, and isohalines is well in agreement with each other.
6. Conclusions
A nonlinear stability of a horizontal layer of porous medium saturated with viscoelastic nanofluid is investigated, which is heated from below and cooled from above, while incorporates the effects of Brownian motion along with thermophoresis. The results have been obtained in terms of Nusselt and Sherwood numbers with the help of finite amplitude equation. The effect of various parameters have been obtained and depicted graphically. The following conclusion are drawn from the above study.
1. It is found that, the Gravity modulation can be used to reg-ulate the heat and mass transports effectively.
2. The effect of viscoelastic parameters has significant effect on heat and mass transport.
3. Increase in concentration Rayleigh number Rn, Modified diffusivity ratio NA and Lewis number Le increases the
effect of gravity modulation.
4. An increment in Prandtl–Darcy number PrD is to increase
the values of Nu and Sh at small values of time s but no effect at large values of time s.
5. The effect of increased nanoparticle concentration Rn pos-itively (indicates a top-heavy nanoparticle distribution) is to enhance the heat and mass transport, but increased nanoparticle concentration Rn negatively (bottom-heavy nanoparticle distribution) is to reduce the heat and mass transport.
6. There is no significant effect of NA and Le on heat
trans-port, but on mass transport.
7. Increasing e, is to increase the both Nu and Sh values, whereas an increase in X decreases the same.
Acknowledgments
The author Palle Kiran would like to thank Prof. B.S. Bhadauria Ph.D. supervisor, Department of Mathematics, Banaras Hindu University and Prof. P.G. Siddheshwar, Department of Mathematics, Bangalore University for their valuable guidance and suggestions. The author would also like to acknowledge the support and encouragement from his father P. Thikkanna and mother P. Sugunamma. This work was done by the author during his stay at home after submis-sion of his Ph.D. thesis (Ph.D. period from 23:08:2012 to 15:12:2014) to the Babasaheb Bhimrao Ambedkar University, Lucknow 226025, India.
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The Author Dr. Palle Kiran received his Ph.D. degree from the Department of Applied Mathematics, Babasaheb Bhimrao Ambedkar Central University, Lucknow, India during (2012Aug–2015April). Also he got his M.Phil and MSc Degrees from Pandicherry Central University, Puducherry, India during 2006– 2010. He worked as Assistant Professor of Mathematics at SGCET, Puducherry during 2010–2012. He has published more than 27 international repute publications and attended more than 26 conferences in India during his Ph.D. His research interest is nonlinear thermal instability.
