Entropy Analysis in Double-Diffusive Convection in Nanofluids Through

entropy

Article Entropy Analysis in Double-Diﬀusive Convection in Nanoﬂuids through Electro-Osmotically Induced Peristaltic Microchannel

Saima Noreen 1,2, Sadia Waheed 2, Abid Hussanan 3,4,* and Dianchen Lu 1

1 Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang 212013, China; [email protected] (S.N.); [email protected] (D.L.) 2 Department of Mathematics, COMSATS University Islamabad 45550, Tarlai Kalan Park Road, Islamabad 44000, Pakistan; [email protected] 3 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 4 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam * Correspondence: [email protected]

Received: 30 July 2019; Accepted: 7 October 2019; Published: 10 October 2019

Abstract: A theoretical study is presented to examine entropy generation in double-diffusive convection in an Electro-osmotic flow (EOF) of nanofluids via a peristaltic microchannel. Buoyancy effects due to change in temperature, solute concentration and nanoparticle volume fraction are also considered. This study was performed under lubrication and Debye-Hückel linearization approximation. The governing equations are solved exactly. The effect of dominant hydrodynamic parameters (thermophoresis, Brownian motion, Soret and Dufour), Grashof numbers (thermal, concentration and nanoparticle) and electro-osmotic parameters on double-diffusive convective flow are discussed. Moreover, trapping, pumping, entropy generation number, Bejan number and heat transfer rate were also examined under the influence of pertinent parameters such as the thermophoresis parameter, the Brownian motion parameter, the Soret parameter, the Dufour parameter, the thermal Grashof number, the solutal Grashof number, the nanoparticle Grashof number, the electro-osmotic parameter and Helmholtz–Smoluchowski velocity. The electro-osmotic parameter powerfully affected the velocity profile. The magnitude of total entropy generation increased as the thermophoresis parameter and Brownian motion parameter increased. Soret and the Dufour parameter had a strong tendency to control the temperature profile and Bejan number. The findings of the present analysis can be used in clinical purposes such as cell therapy, drug delivery systems, pharmaco-dynamic pumps and particles filtration.

Keywords: entropy generation; double-diffusive convection; electro-osmosis; nanofluid; bio-microfluidics; exact solution

1. Introduction In 1809, Resuss [1] illustrated an electrokinetic phenomenon known as electroosmosis or Electro-osmotic flow (EOF). It is defined as the flow of fluid in any conduit (e.g., microchannel, capillary tube) under the effect of the external electric field. EOF has many applications in micro-flow injection analysis, micro-liquid chromatography and micro-energy system. Patankar et al. [2] investigated the behavior of electro-osmotic flow through a microchannel. Yang et al. [3] studied the EOF in microfluidics. Tang et al. [4] scrutinized the EOF for a Power-law model. Later, Miller et al. [5] predicted EOF through a carbon nanotube membrane.

Entropy 2019, 21, 986; doi:10.3390/e21100986 www.mdpi.com/journal/entropy Entropy 2019, 21, 986 2 of 21

Peristaltic transport of a fluid is a special type of fluid transport that occurs due to the propagation of progressive waves of area contraction/expansion. This is a natural process of transport in living beings. The literature reveals that different studies have explained this phenomenon under various impacts. Some recent readings are cited in [6–11]. Nanofluids have been found to exhibit enhanced thermophysical properties (thermal conductivity, thermal diffusivity, the viscosity of convective heat transport coefficient) compared to base fluids. Nanofluids dynamics have shown tremendous potential applications in many fields. The endoscopic impact of nanofluid in peristalsis was analyzed by Akbar et al. [12]. Noreen [13] investigated nanofluids under induced magnetic field and mixed convection. Furthermore, Tripathi et al. [14] studied the peristaltic motion of nanofluid. Reddy et al. [15] recommended the peristaltic flow of nanofluid through complaint walls. Ebaid et al. [16] examined the peristaltic motion of nanofluid with convective conditions. Shao et al. [17] studied a reference solution for double-diffusion convection. Double-diffusive convection is a fluid dynamics phenomenon that describes a form of convection driven by two different density gradients (temperature and concentration) that have different diffusion rates. If the temperature difference is held constant, thermal diffusion produces a concentration gradient. Considering the wide applications of double-diffusive convection, it was studied in a nanofluidic flow model with peristaltic pumping. Huppert et al. [18] discussed the applications of double-diffusion convection. Similarly, Bég et al. [19] described the peristaltic pumping of nanofluids through double-diffusive convection. Kefayati [20] explained double-diffusion convection for pseudoplastic fluids. The peristaltic motion of MHD flow through double-diffusive convection was studied by Rout et al. [21]. Gaffar et al. [22] observed the double-diffusive convection under the effect of heat absorption. Furthermore, Mohan et al. [23] also described the influence of double-diffusive convection in a lid-driven cavity. Peristaltic flow and EOF have guaranteed biomedical and engineering applications. Today, the collective effect of peristalsis and electroosmosis is of supreme importance in many applications, such as biomicrofluidic devices and biomimetic pumping. Some mathematical models with various physical applications and various flow regimes have been studied. Bandopadhyay et al. [24] inspected peristaltic motion of viscous fluid in the microchannel. Unsteady electrokinetic transport through peristaltic microchannel was studied by Tripathi et al. [25]. Tripathi et al. [26] explored the peristaltic motion of the couple-stress fluid through the microchannel. Prakash et al. [27] examined the peristaltic motion of Williamson ionic nanofluid in the tapered microchannel. Tripathi et al. [28] also studied blood flow modulated by electroosmosis. Moreover, Tripathi et al. [29] worked on the combined effect of electroosmosis and peristalsis through the microchannel. In addition, Tripathi et al. [30] revealed the peristaltic transport of nanofluids with buoyancy effect via a microchannel. The impact of heat and mass transport for the combined effect of electroosmosis and peristalsis was studied by Waheed et al. [31]. Noreen et al. [32] scrutinized the effect of heat on EOF through peristaltic pumping. In another study, Noreen et al. [33] recommended the transport of MHD nanofluid through the peristaltic microchannel. Entropy generation in peristaltically induced microchannels is a rapidly emerging area of interest. Entropy production is related to thermodynamic irreversibility, which exists in all types of heat and mass transfer processes. The study of entropy generation within the system is important because it helps to track the sources of the available energy. Different sources of irreversibility include heat flow through the finite temperature gradient, convective heat transfer characteristics, viscosity and diffusion effects. Entropy can be minimized to preserve the energy quality for the optimal design of any thermal system. At present, the research topic of minimizing entropy generation has gained special status amongst scientists worldwide. Limited studies have investigated the entropy production in electro-osmotically induced peristaltic microchannels. Kefayati [34] explained entropy production for Non-Newtonian nanofluids through a porous cavity. Kefayati [35] has also described entropy production and double-diffusive convection for power-law fluids. In his study [36], the entropy EntropyEnt ro py2019 2019, ,21 21, 986, x FOR P EER R EVIEW 33 of of 2120 power-law fluids. In his study [36], the entropy production in double-diffusive convection in an productionopen cavity in was double-di explainedffusive. He convection also explained in an openentropy cavity production was explained. for EOF He through also explained FDLBM entropysimulation production [37]. Ranjit for EOFet al. through [38] explained FDLBM entropy simulation production [37]. Ranjit for etEOF al. regulated [38] explained by peristaltic entropy productionpumping. forBhatti EOF et regulated al. [39] byanalyzed peristaltic entropy pumping. production Bhatti et for al. [nanofluid39] analyzed through entropy a productionmicrochannel for. nanofluidKefayati through[40] presented a microchannel. the simulation Kefayati of [ 40entropy] presented production the simulation using ofBuongiorno’s entropy production mathematical using Buongiorno’smodel. Kefayati mathematical et al. [41,42 model.] discovered Kefayati entropy et al. [production41,42] discovered for MHD entropy flow productionand inclined for channel MHD flowflow and. Ranjit inclined et al. channel[43] demonstrated flow. Ranjit an et al.entropy [43] demonstrated analysis for EMHD an entropy flow analysis through for an EMHD asymmetric flow throughmicrochannel an asymmetric. Moreover, microchannel. Noreen et al. Moreover, [44] studied Noreen the entropy et al. [44 production] studied the in entropyperistaltically production flowing in peristalticallyviscous fluid. flowing viscous fluid. AllAll thethe above-citedabove-cited workwork waswas obtainedobtained throughthrough twotwo solutionssolutions approaches,approaches, i.e.,i.e., analyticalanalytical andand numerical.numerical. TheThe authorsauthors computedcomputed thethe exactexact solutions,solutions, asas theythey areare errorerror freefree andand moremore eefficientfficient than than numericalnumerical solutions. solutions This. Th studyis study was was motivated motivated by applications by applications in novel in nanofluidnovel nanofluid drug delivery drug systemsdelivery wheresystems a smallwhere volume a small of volume drugs canof drugs be transported can be transported in the diseased in the diseased portion ofportion physiological of physiological vessels withvessels the helpwithof th organ-on-chipe help of organ devices.-on-chip Thermal devices. enhancement Thermal mechanismenhancement analysis mechanism of double-di analysisffusive of convectiondouble-diffusive in nanofluids convection transported in nanofluids by peristaltic transported microchannel by peristal hastic not microchannel been discussed has previously. not been Therefore,discussed thepreviously current.investigation Therefore, the aimed current to fillinvestigation this gap. The aimed findings to fill of this the gap. present The analysis findings can of the be usedpresent in clinical analysis purposes can be such used as in cell clinical therapy, purposes drug delivery such systems,as cell therapy, pharmaco-dynamic drug delivery pumps systems, and particlespharmaco filtration.-dynamic In addition,pumps and in order particles to deepen filtration. our understanding In addition, ofin thermosolutal order to deepen convection, our specialunderstanding attention of was thermosolutal paid to study convection, entropy generation.special atten Thetion detailedwas paid mathematical to study entropy formulation generation. is givenThe d andetailed solution mathematical expressions formulation are also reported. is given and solution expressions are also reported. 2. Mathematical Formulation 2. Mathematical Formulation 2.1.2.1. FlowFlow RegimeRegime

TheThe physicalphysical modelmodel (shown(shown inin FigureFigure1 )1) consists consists of of a a two-dimensional two-dimensional EOF EOF of of nanofluid nanofluid with with double-didouble-diffusiveffusive convection convection through through a a microfluidic microfluidic channel. channel.

FigureFigure 1. 1. SchematicSchematic of of the the geometry. geometry. Entropy 2019, 21, 986 4 of 21

The medium is induced by propagating a peristaltic wave at velocity c along the microchannel. The temperature, solutal concentration and nanoparticle fraction for the lower channel wall are taken as T0, C0 and F0, respectively. Mathematically, microchannel wall’s geometry is expressed as [23]    Xe cet    eh Xe,et = ea +eb sin2π −  (1)  λ  whereeh represents the transverse vibration of the wall, ea is the half width of channel,eb is the amplitude of wave, Xe represents the axial coordinate, c is the speed of the wave, λ represents the wavelength and et is the time.

2.2. Governing Equations The equations governing the physical flow problem for nanofluid with double-diffusive convection in electrohydrodynamic (EHD) are [19]

∂Ue ∂Ve + = 0. (2) ∂Xe ∂Ye ∂Ue ∂Ue ∂Ue ∂Pe ∂2Ue ∂2Ue n o ρ f + Ue + Ve = + µ + + g (1 F0)ρ0 βt eT T0 + βc Ce C0 ρp ρ0 eF F0 + ρeEx, (3) ∂et ∂Xe ∂Ye − ∂Xe ∂Xe2 ∂Ye2 − − − − − − ∂Ve ∂Ve ∂Ve ∂Pe ∂2Ve ∂2Ve n o ρ f + Ue + Ve = + µ + + g (1 F0)ρ0 βt eT T0 + βc Ce C0 ρp ρ0 eF F0 . (4) ∂et ∂Xe ∂Ye − ∂Ye ∂Xe2 ∂Ye2 − − − − − − ( !) 2 2 ∂eT ∂eT ∂eT ∂2eT ∂2eT ∂2Ce ∂2Ce ∂eF∂eT ∂eF∂eT Dt ∂eT ∂eT ρCp + Ue + Ve = k + + Dtc + + ρCp Db + + + . (5) f ∂et ∂Xe ∂Ye ∂Xe2 ∂Ye2 ∂Xe2 ∂Ye2 p ∂Xe∂Xe ∂Ye∂Ye T0 ∂Xe ∂Ye

   2 2   2 2  ∂Ce ∂Ce ∂Ce  ∂ Ce ∂ Ce  ∂ eT ∂ eT   + Ue + Ve  = Ds +  + Dct + . (6)  ∂et ∂Xe ∂Ye ∂Xe2 ∂Ye2  ∂Xe2 ∂Ye2        ∂eF ∂eF ∂eF   ∂2eF ∂2eF  D  ∂2eT ∂2eT   + Ue + Ve  = D  +  + t  +    b   . (7) ∂et ∂Xe ∂Ye ∂Xe2 ∂Ye2 T0 ∂Xe2 ∂Ye2

Here, ρe is the electrical charge density, ρ f is the fluid density, ρp is the nanoparticle mass density, ρ0 is the nanofluid density at reference temperature (T0), Ex is the applied electric field, Ue represents the velocity along the Xe direction, Ve represents the velocity along the Ye direction, eT is the temperature, Ce is the solutal concentration, eF is the nanoparticle volume fraction, Pe is the pressure, µ is the fluid viscosity, g is the acceleration due to gravity, βt is the volumetric thermal expansion coefficient of fluid, ( ) ( ) βc is the volumetric solutal expansion coefficient of fluid, ρcp f is the heat capacity of fluid, ρcp p is the effective heat capacity of nanoparticle, k is the thermal conductivity, Dtc is the Dufour diffusivity, Dct is the Soret diffusivity, Db is Brownian diffusion coefficient, Dt is Thermophoretic diffusion coefficient and Ds is solutal diffusivity. The Poisson–Boltzmann equation [26] in a microchannel is defined as

ρe 2φe = . (8) ∇ − ∈ + Here, ρe(= zve(en− en )) is the total charge density, is the dielectric permittivity, φe is the − − ∈ ( ezv φe) + TavK electric potential, en− is the anion, en is the cation and deﬁned as en± = en0e ± B . Moreover, n0, zv, e, KB and Tav are the bulk concentration, charge balance, electronic charge, Boltzmann constant and average temperature, respectively. Entropy 2019, 21, 986 5 of 21

The translational transformation between ﬁxed coordinate system Xe, Ye and moving coordinate system (ex, ey) is [8] ex = Xe cet, ey = Ye, eu(ex, ey) = Ue Xe, Ye,et c, ev(ex, ey) = Ve Xe, Ye,et , − − (9) ep(ex, ey) = Pe Xe, Ye,et , T(ex, ey) = eT Xe, Ye,et .

Using the transformations (9) in Equations (2)–(7), after introducing the dimensionless parameters:

ex ey cet eu ev ea2 ea eh en ea eb x = , y = , t = , u = , v = , p = ep, α = , h = , n = , me = , ε = , λ ea λ c αc λcµ λ ea n0 λd ea

eT T0 Ce C0 eF F0 KBT0 ezv cea ρ f cea θ = − , Ω = − , γ = − , λ 2 = ∈ , φ = φe, P = , R = , T C F d 2 T K e D e 0 0 0 2n0(ezv) av B µ µ ρcp 2 f TavKBEx ∂ψ ∂ψ ρ0 gea βtT0(1 F0) q Pr = , Uhs = ∈, u = , v = , Grt = − , f = , (10) ρ f k − ezvcµ ∂y − ∂x cµ cea 2 2 ( ) ga C (1 F ) gea ρp ρ0 F0 Dt ρcp p ρ0 e βc 0 0 − cea cea Grc = − , Gr f = , Nt = , Ss = , Sb = , cµ cµ k Ds Db Db(ρcp) F0 p DctT0 DtcC0 Nb = , Nct = , Ntc = , k DsC0 kT0 where x, y, me, α, λd, p, Uhs, ε, θ, Ω, γ, Pe, Re, Pr, Grt, Grc, Gr f , Nt, Nb, Nct, Ntc, f and Θ are non-dimensional axial coordinate, non-dimensional transverse coordinate, electro-osmotic parameter, peristaltic wave number, Debye length, non-dimensional pressure, Helmholtz-Smoluchowski velocity, amplitude ratio, dimensionless temperature, dimensionless species concentration, dimensionless nanoparticle volume fraction, ionic Peclet number, Reynolds number, Prandtl number, thermal Grashof number, solutal Grashof number, nanoparticle Grashof number, thermophoresis parameter, Brownian motion parameter, Soret parameter, Dufour parameter, dimensionless volume flow rate in moving frame and dimensionless volume flow rate in fixed frame, respectively. .. Applying Debye-Huckel linearization approximation [27] and Equation (10), Equation (8) becomes

2 d φ 2 = me φ, (11) dy2 where me is the electro-osmotic parameter. The analytical solution of Equation (11) subject to boundary ∂φ = = φ = = ( ) condition ∂y 0, at y 0 and 1, at y h x yeild

cosh(me y) φ(y) = . (12) cosh(meh)

Using Equation (10), followed by long wavelength approximation [8], Equations (3)–(7) can be reduced to following form:

2 ∂p ∂ u 2 = + Grtθ + GrcΩ Gr f γ + me Uhsφ, (13) ∂x ∂y2 −

∂p = 0, (14) ∂y !2 ∂2θ ∂2Ω ∂θ ∂γ ∂θ + Ntc + Nb + Nt = 0, (15) ∂y2 ∂y2 ∂y ∂y ∂y Entropy 2019, 21, 986 6 of 21

! ∂2Ω ∂2θ + Nct = 0, (16) ∂y2 ∂y2 ! ∂2γ N ∂2θ + t = 0. (17) 2 2 ∂y Nb ∂y The dimensionless boundary conditions are:

∂u = 0, θ = 0, Ω = 0, γ = 0, at y = 0, (18a) ∂y

u = 0, θ = 1, Ω = 1, γ = 1, at y = h(x) = 1 + ε sin x, (18b)

3. Solution Procedure

Analytical Solution The analytical solutions [17] of temperature, solutal concentration and nanoparticle volume fraction ﬁeld are obtained after solving simultaneous Equations (15)–(17) subject to boundary condition (18a,b):

e N0 y 1 θ = − − , (19) e N0h 1 − − ! ! N e N0 y 1 1 N Ω = t − + 1 + t y, (20) −N h Nb 1 e 0 h Nb − − N y ! e− 0 1 1 γ = Nct − + (1 + Nct)y. (21) 1 e N0h h − − Using Equations (19)–(21) in Equation (13) and using boundary condition (18a) and (18b), the analytical solution of axial velocity is obtained as: p cosh(m y) 1 ∂ 2 2 1 2 2 1 N0 y N0h 1 N2 3 3 e u = y h N1 y h 2 e− e− (y h) y h + Uhs 1 , (22) 2 ∂x − − 2 − − N0 − − N0 − − 6 − − cosh(meh)

where 1 Nt + Nb N0 = , h 1 NctNtc − ! 1 N N = G G ΩN + t G , 1 N h rt rc ct r f 1 e− 0 − Nb − ! ! 1 Nt N2 = Grc(1 + Nct) 1 + Gr f . h − Nb The volume ﬂow rate is given by Z h Q = udy, (23) 0 Using Equation (22) after rearranging the terms, pressure gradient is obtained as

( N h ) ! ∂p 3Q 3e 0 3 3 3N 3U sinh(meh) = N − (1 + N h) + 1 + 2 h + hs h . (24) − 3 1 3 3 0 3 3 3 ∂x h − N0 h − N0 h 2N0h − 8 h − me cosh(meh)

The pressure rise across one wavelength ∆P is as follows: Z 1 ∂p ∆P = dx. (25) 0 ∂x Entropy 2019, 21, 986 7 of 21

The volume ﬂow rate in ﬁxed frame is given by

Z h Q = Ue Xe, Ye,et dYe. (26) 0

Using Equation (9) in Equation (26) and after integration, we obtain

Q = q + ch. (27)

The time average ﬂow rate over one time period is deﬁned as

Z 1 Qe = Qdt. (28) 0

Using Equations (10) and (27) in Equation (28), we obtain

Θ = f + 1. (29) ∂ψ ∂ψ The stream function u = , v = in wave frame takes the following form: ∂y − ∂x ( ! ) 1 ∂p1 11 1 1 1 1 1 ψ = y3 h2 y N y3 h2 y + e N0 y + ye N0h + y2 hy 1 2 − − 2 ∂x 3 − − 2 3 − N0 N0 N0 − N0 2 − ! (30) N2 1 4 3 sinh(meh) y yh y + Uhs y . − 6 4 − − − me cosh(meh)

The heat transfer coeﬃcient Z at the wall (y = h(x)) is as follows:

∂θ Z = hx = . (31) ∂y y h

4. Entropy Generation Analysis The dimensional volumetric entropy generation during nanofluid flow with double diffusive convection is given by " ! 2 2 (ρCp) 2 2 k ∂eT ∂eT Dtc ∂2 ∂2 p ∂eF∂eT ∂eF∂eT Dt ∂eT ∂eT Sgen = ( ) + ( ) ) + + Ce+ Db + + (( ) + ( ) ] (32) eT2 ∂Xe ∂Ye eT ∂Xe2 ∂Ye2 eT { ∂Xe∂Xe ∂Ye∂Ye T0 ∂Xe ∂Ye }

Equation (32) comprises two parts: one part consists of entropy generation due to heat transfer and the second part consists of entropy generation due to solutal concentration. The characteristics entropy generation is given as

k Sg = . (33) ea2 Using Equations (10) and (33), the non-dimensional volumetric entropy generation can be expressed as   S 2 2 !2 gen 1 ∂ θ 1  ∂ Ω ∂θ ∂γ ∂θ  Ns = = + Ntc + Nb + Nt , (34) Sg (1 + θ)2 ∂y2 1 + θ ∂y2 ∂y ∂y ∂y  Entropy 2019, 21, 986 8 of 21

The total entropy generation or entropy generation number Ns in Equation (34) can be written as sum of Ns = NHT + NDC. (35)

Here, NHT is the dimensionless entropy generation due to heat transfer and NDC represent the dimensionless entropy generation due to solutal concentration. The thermal irreversibility of the system is deﬁned as the Bejan number Be [38]:

1 ∂2θ N N (1+θ)2 ∂y2 B = HT = HT = . (36) e + " 2# NHT NDC Ns 1 ∂2θ 1 ∂2Ω ∂θ ∂γ ∂θ 2 2 + Ntc 2 + Nb + Nt (1+θ) ∂y 1+θ ∂y ∂y ∂y ∂y

5. Results and Discussion The main objective of this section is to explain the effects of various parameters (i.e., thermophoresis parameter Nt, Brownian motion parameter Nb, Soret parameter Nct, Dufour parameter Ntc, thermal Grashof number Grt, solutal Grashof number Grc, nanoparticle Grashof number Gr f , electro-osmotic parameter me and Helmholtz-Smoluchowski velocity Uhs) on the velocity profile u, pressure rise ∆P, pressure gradient dp/dx, temperature profile θ, solutal concentration profile Ω, nanoparticle volume faction profile γ, entropy generation number Ns, Bejan number Be and heat transfer coefficient Z. Moreover, the trapping phenomenon is also illustrated. The graphical results are shown in Figures2–17.

5.1. Flow Characteristics Figure2a–i is displayed to show the advancements of physical parameters in velocity across the microchannel. Figure2a,b shows that the magnitude of axial velocity u decreased for increasing values of Nt and Nb. As a result, the deposition of thermophoretic particles was reduced and the central axes of the micro channel had an increased particle accumulation. This sedimentation of the nanoparticles can be adjusted by controlling the thermal gradient, applied externally to the fluid. This is significant as nanoparticle suspension in the nanofluid blocks the fluid flow which affects the system’s performance. Consequently, monitoring the deposition of nanoparticles in the fluid is the main requirement. Figure2c demonstrates that the magnitude of u was reduced for increasing values of Nct, although in Figure2d, a contrary response is observed for Ntc. Figure2e,f portrays that the magnitude of u increased with the increase in Grt and Grc. However, it declined for increasing values of Gr f , as shown in Figure2g. This demonstrates that the change in concentration of nanoparticles reduced velocity while with a changing temperature and solutal concentration, velocity progresses. Figure2h predicts that me will enhance the magnitude of axial velocity u. Since me is the ratio of the channel height to the Debye length λd, it indicates that λd is inversely proportional to electric double layer thickness. Hence, more fluid flows in the central region. Figure2i visualizes that Uhs develops the magnitude of u. That means velocity of the fluid can be improved by increasing the magnitude of external electric field. Entropy 2019, 21, x FOR PEER REVIEW 8 of 20 the system’s performance. Consequently, monitoring the deposition of nanoparticles in the fluid is the main requirement. Figure 2c demonstrates that the magnitude of 푢 was reduced for increasing

values of 푁푐푡, although in Figure 2d, a contrary response is observed for 푁푡푐. Figure 2e,f portrays that the magnitude of 푢 increased with the increase in 퐺푟푡 and 퐺푟푐 . However, it declined for increasing values of 퐺푟푓, as shown in Figure 2g. This demonstrates that the change in concentration of nanoparticles reduced velocity while with a changing temperature and solutal concentration,

velocity progresses. Figure 2h predicts that 푚푒 will enhance the magnitude of axial velocity 푢. Since 푚푒 is the ratio of the channel height to the Debye length 휆푑 , it indicates that 휆푑 is inversely proportional to electric double layer thickness. Hence, more fluid flows in the central region. Figure

Entropy2i visualizes2019, 21, 986 that 푈ℎ푠 develops the magnitude of 푢 . That means velocity of the fluid can 9 be of 21 improved by increasing the magnitude of external electric field.

0.8 Nt 0.3 0.8 Nb 1.0 Nct 0.0 1.5 Nt 0.5 Nb 2.0 Nct 1.0 Nt 0.7 0.6 0.6 Nb 3.0 Nct 2.0 1.0

u 0.4 0.4 u

0.5 0.2 0.2

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y y (a) (b) (c)

1.0 Ntc 1.0 Grt 0.0 1.0 Grc 0.0 0.5 Ntc 2.0 Grt 1.0 Grc 1.0 0.8 0.8 Ntc 3.0 0.4 Grt 2.0 Grc 2.0 0.6 0.6

u u 0.3

0.4 0.2 0.4

0.2 0.1 0.2

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y y (d) (e) (f) 1.0 1.0 Grf 0.0 m 0.0 Uhs 0.0 e 1.5 Grf 1.0 0.8 me 3.0 Uhs 1.0 0.8 G 2.0 rf m 5.0 Uhs 2.0 0.6 e 0.6 1.0

u 0.4 0.4 0.5 0.2 0.2

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y y (g) (h) (i)

FigureFigure 2. 2.Axial Axial velocity velocity u proﬁle u profile for ( afor) N t(,(a)b N) Nt, b(,(b)c )NNbct, (,(c)d )NNcttc, ,((de)) NGtcrt,,( (ef)) GGrcrt,(, (gf) GGrfrc,(, (hg))m Ge,(rf, i()hU) hs, ∂p 휕푝 me, (i) Uhs, while other parameters= = are == 1, 푥 ==1, 휀 = 0.01=, Θ = 0=.9, 푁 ==1, 푁 = =1, 푁 == while other parameters are ∂x 1, x 1, ε휕푥 0.01, Θ 0.9, Nt 1, Nb 1, N푡 ct 1,푏Ntc 2,푐푡Grt 5,1G, 푁rc푡푐==1,2G, 퐺r f푟푡==1,5m, 퐺e푟푐==3,1U, 퐺hs푟푓==1.1, 푚푒 = 3, 푈ℎ푠 = 1.

5.2.5.2. PumpingPumping CharacteristicsCharacteristics FigureFigure3 a–i3a–i shows shows the the variation variation of of pressure pressure di differencefference across across one one wavelength wavelength Δ∆푃 Pagainstagainst the the averageaverage flow flow rate rateΘ Θ.. These figuresfigures show the linear linear relationship relationship between Δ∆푃P andand ΘΘ. The. The pumping pumping phenomenonphenomenon waswas divideddivided into four segments: pumping pumping region region (adverse (adverse pressure pressure gradient gradient Δ푃∆>P 0>, 0, PositivePositive pumping pumping ΘΘ >> 00),), backward/ backward/retrograderetrograde pumping pumping ( (Δ∆푃P>>00, ,ΘΘ<<0)0, ), augmented augmented pumping pumping (favorable(favorable pressure pressure gradient gradien∆t ΔP푃<<0,0 Positive, Positive pumping pumpingΘ Θ>>0)0 and) and free free pumping pumping (∆ P (Δ=푃 0=).0 Figure). Figure3a,b represents3a,b represent∆P fors Δ increasing푃 for increasing values ofvaluesNt and of N푁b푡. and This 푁 shows푏 . This that show pumpings that decreased pumping decreasethroughoutd thethroughout whole region. the whole This isregion. very helpfulThis is invery nanofluid helpful drugin nanofluid delivery drug systems delivery [11]. systems Figure3 c–i[11]. highlights Figure that3c–i pumping highlights increased that pumping as the increase Soret parameterd as the SoretNct, theparameter Dufour 푁 parameter푐푡, the DufourNtc, theparameter thermal 푁 Grashof푡푐, the numberthermalG Grashofrt, the solutal number Grashof 퐺푟푡, the number solutalG Grashofrc, the nanoparticle number 퐺푟푐, Grashof the nanoparticle number G Grashofr f , the electro-osmotic number 퐺푟푓, parameterthe electrom-osmotice and Helmholtz–Smoluchowski parameter 푚푒 and Helmholtz velocity–SmoluchowskiUhs increased. velocity Here, ∆푈Pℎ푠is increase in a lineard. Here, relationship Δ푃 is within a flowlinear rate relationshipΘ. with flow rate Θ. The variation of pressure gradient dp/dx is presented in Figure4a–i for di fferent physical flow parameters. Figure4a,b highlights that the magnitude of dp/dx decreased as Nt and Nb increased. Figure4c–f shows that the Soret parameter Nct, the Dufour parameter Ntc, the thermal Grashof number Grt and the solutal Grashof number Grc distinctly enhanced the magnitude of dp/dx. Conversely, nanoparticle Grashof number Gr f acted to strongly reduce dp/dx values, as shown in Figure4g. This demonstrates that dp/dx can be controlled by adjusting the Gr f . However, Figure4h,i shows that the values of dp/dx were enhanced for increasing values of electro-osmotic parameter me and the Helmholtz–Smoluchowski velocity Uhs. Entropy 2019, 21, x FOR PEER REVIEW 9 of 20

4 4 N 1.0 4 N 1.0 3 t Nb 1.0 ct N 2.0 3 N 2.0 t Nb 2.0 3 ct 2 N 3.0 Nt 3.0 2 Nb 3.0 ct 2 1

P P 1 0 1 0 1 0 1 2 1 2 Entropy 20191.0 , 21, 9861.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.010 of 21 1.0 1.5 2.0 2.5 3.0 Entropy 2019, 21, x( aFOR) PEER REVIEW (b) (c) 9 of 20 5 4 3 4 Ntc 1.0 4 Grt 0.0 4 Grc 0.0 4 NNt 2.01.0 G 1.0 Nct 1.0 33 tc 2 Nrt b 1.0 Grc 1.0 3 3 Nt 2.0 G 2.0 Nct 2.0 Ntc 3.0 Nrtb 2.0 3 Grc 2.0 22 1 N 3.0 Nt 3.0 2 Nb 3.0 2 ct

P P 2 1

P P 1 0

P 1 1 1 00 1 0 0 1 10 1 2 1 2 21 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 2 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 (d) (e) (f) (a) (b) (c) 4 4 Grf 0.0 me 0.0 15 Uhs 0.0 4 3 3 GNrftc 1.01.0 3 mGert 3.00.0 UGhs 1.00.0 04 rc GNrf 2.02.0 Gm 5.01.0 U 2.0 3 tc 2 rte Grchs 1.0 2 2 3 Ntc 3.0 Grt 2.0 1 P G 2.0

P P rc 12 1 1 2

P 2 01 0 1 0 3 10 1 0 1 4 1 21 2 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 2 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 (g) (h) (i) (d) (e) (f)

4 Figure 3. Pressure rise Δ푃 profile4 for (a) Nt, (b) Nb, (c) Nct, (d) Ntc, (e) Grt, (f) Grc, (g) Grf, (h) Grf 0.0 me 0.0 1 Uhs 0.0 3 e hs Grf 1.0 3 푦 = 0, 휀 =me 03.0.06, 푁 = 1, 푁 = 1, 푁 = 1, 푁 =Uhs 1.0 m , (i) U , while the other parameters are 푡 0 푏 푐푡 푡푐 Grf 2.0 me 5.0 Uhs 2.0 2 2 2, 퐺푟푡 = 5, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 = 3, 푈ℎ푠 = 1. 1

P 1 1 2 0 The variation of pressure gradient0 푑푝⁄푑푥 is presented in Figure 4a–3 i for different physical flow 1 1 푑푝⁄푑푥 4 푁 푁 parameters.2 Figure 4a,b highlights that the magnitude of decreased as 푡 and 푏 increased. 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 Figure 4c–f shows that the Soret parameter 푁푐푡, the Dufour parameter 푁푡푐, the thermal Grashof number 퐺푟푡 and(g ) the solutal Grashof number 퐺(푟푐h) distinctly enhanced the magnitude(i) of 푑푝⁄푑푥. Conversely, nanoparticle Grashof number 퐺푟푓 acted to strongly reduce 푑푝⁄푑푥 values, as shown in Figure 3. Pressure rise Δ푃 profile for (a) Nt, (b) Nb, (c) Nct, (d) Ntc, (e) Grt, (f) Grc, (g) Grf, (h) FFigureigure 3.4g.Pressure This demonstrates rise ∆P proﬁle that for 푑푝 (a⁄)푑푥Nt ,(canb) beNb ,( controlledc) Nct,(d) byNtc adjusting,(e) Grt,(f ) theGrc 퐺,(푟푓g.) However,Grf,(h) me ,( Figurei) Uhs , me, (i) Uhs, while the other parameters are 푦 = 0, 휀 = 0.06, 푁푡 = 1, 푁푏 = 1, 푁푐푡 = 1, 푁푡푐 = 4h,iwhile show the others that parameters the values are y of= 푑푝0,⁄ε푑푥= were0.06, Nenhancet = 1, Ndb =for1, N increasingct = 1, Ntc values= 2, Grt of= 5, electroGrc =-osmotic1, Gr f = 2, 퐺푟푡 = 5, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 = 3, 푈ℎ푠 = 1. parameter1, me = 3, 푚Uhs푒 and= 1. the Helmholtz–Smoluchowski velocity 푈ℎ푠. The variation of pressure gradient 푑푝⁄푑푥 is presented in Figure 4a–i for different physical flow 6.0 Nt 1.0 Nb 1.0 Nct 1.0 parameters. Figure 4a,b highlights 6.0that the magnitude of 푑푝N ⁄2.0푑푥 decreased as 푁 and 푁 increased. Nt 2.0 b 6.5 푡 푏 Nct 2.0 Nt 3.0 Nb 3.0 N 3.0 Figure5.5 4c–f shows that the Soret parameter 푁푐푡, the Dufour parameter 푁푡푐, the thermal Grashofct

dx 5.5 dx 6.0

dp ⁄

dp number5.0 퐺푟푡 and the solutal Grashof number 퐺푟푐 distinctly enhancedp d the magnitude of 푑푝 푑푥.

Conversely, nanoparticle Grashof number 퐺푟푓 acted to strongly reduce5.5 푑푝⁄푑푥 values, as shown in 4.5 5.0 Figure 4g. This demonstrates that 푑푝⁄푑푥 can be controlled by adjusting the 퐺푟푓. However, Figure 4.0 5.0 4h,i 0 show5s that 10 the values15 20 of 푑푝⁄푑푥0 were5 enhance10 d 15for increasing20 0 values5 of electro10 -15osmotic20 x x x parameter 푚푒 (anda) the Helmholtz–Smoluchowski( bvelocity) 푈ℎ푠. (c)

N 1.0 7.0 tc Grt 0.0 Grc 0.0 6.5 5.5 Nt 1.02.0 Nb 1.0 N 1.0 6.0 tc Grt 1.0 6.5 Grcct 1.0 6.0 Nb 2.0 6.5 NNttc 2.03.0 Nct 2.0 Grt 2.0 Grc 2.0 N 3.0 5.0 Nb 3.0 dx 6.0 t

dx N 3.0 5.5 dx 6.0 ct

dx dx 6.0 5.5 5.5

dp 5.0 dp 5.5 4.5 5.0 5.5 4.5 5.0 4.0 5.0 4.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 4.0 5.0 0 5 10x 15 20 0 5 10x 15 20 0 5 x10 15 20 Entropy 2019, 21, x( FORd) x PEER REVIEW (e) x (f) x 10 of 20 (a) (b) (c) 3.6

GN 0.01.0 me 0.0 7.0 Uhs 0.0 6.5 rftc Grt 0.0 3.4 Grc 0.0 6.5 6.55.5 U 1.0 GN 1.02.0 me 3.0 hs rftc Grt 1.0 6.5 Grc 1.0 m 5.0 3.2 Uhs 2.0 GNrftc 2.03.0 e 6.0 Grt 2.0 Grc 2.0

dx 6.0 dx dx 5.0 dx 6.0

dx dx 3.06.0

dp 5.5 dp 2.85.5 5.5 5.54.5 2.6 5.0 5.0

4.0 2.4 5.0 5.0 4.5 00 55 1010 1515 2020 00 55 1010 1515 2020 00 55 1010 1515 2020 xx xx xx ((gd)) ((he) ((if))

Figure 4. Pressure gradient 푑푝⁄푑푥 profile for (a) Nt, (b) Nb, (c) Nct, (d) Ntc, (e) Grt, (f) Grc, (g) Figure 4. Pressure gradient dp/dx proﬁle for (a) Nt,(b) Nb,(c) Nct,(d) Ntc,(e) Grt,(f) Grc,(g) Grf,(h) me, Grf, (h) me, (i) Uhs, while other parameters are 푦 = 0, 휀 = 0.06, Θ = 0.9, 푁 = 1, 푁 = 1, 푁 = (i) Uhs, while other parameters are y = 0, ε = 0.06, Θ = 0.9, Nt = 1, Nb = 1,푡 Nct = 푏1, Ntc =푐푡2, Grt = 1, 푁푡푐 = 2, 퐺푟푡 = 5, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 = 3, 푈ℎ푠 = 1. 5, Grc = 1, Gr f = 1, me = 3, Uhs = 1. 5.3. Trapping Characteristics

Trapping for different values of the thermophoresis parameter 푁푡, Brownian motion parameter 푁푏 , Soret parameter 푁푐푡 , Dufour parameter 푁푡푐 , thermal Grashof number 퐺푟푡 , solutal Grashof number 퐺푟푐 , nanoparticle Grashof number 퐺푟푓 , electro-osmotic parameter 푚푒 and Helmholtz- Smoluchowski velocity 푈ℎ푠 are shown in Figures 5 − 11 . Figure 5(푎 − 푐) and 6(푎 − 푐) display streamline structure for different values of 푁푡 and 푁푏. These reveal that the number and size of trapped bolus increased as 푁푡 and 푁푏 increases. Figure 7(푎 − 푐) shows that the size of trapped bolus expanded with increasing 퐺푟푡 . However, a reverse trend for solutal Grashof number 퐺푟푐 , nanoparticle Grashof number 퐺푟푓 and electro-osmotic parameter 푚푒 (shown in Figures (8–10)). Furthermore, Figure 11(푎 − 푐) discloses that the number of trapped bolus increased for dissimilar

values of 푈ℎ푠.

4 4 4

3 3 3

y 2 y 2 2

1 1 1

0 0 0 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 x x x (a) (b) (c)

Figure 5. Streamline distribution for (a) 푁푡 = 1.0, (b) 푁푡 = 1.5 , (c) 푁푡 = 2.0, while other parameters are 휀 = 0.06, Θ = 0.9, 푁푏 = 1, 푁푐푡 = 1, 푁푡푐 = 0.1, 퐺푟푡 = 0.1, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 =

0.8, 푈ℎ푠 = −1.

4 4 4

3 3 3

y 2 y 2 2

1 1 1

1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 x x x (a) (b) (c) Entropy 2019, 21, x FOR PEER REVIEW 10 of 20 Entropy 2019, 21, x FOR PEER REVIEW 10 of 20 3.6 m 0.0 Uhs 0.0 6.5 Grf 0.0 e 6.5 3.63.4 Uhs 1.0 Grf 1.0 me 3.0 G 0.0 m 0.0 Uhs 0.0 6.5 rf me 5.0 3.2 Uhs 2.0 Grf 2.0 6.5 e 3.4 6.0 Uhs 1.0 Grf 1.0 me 3.0

dx 6.0 dx dx 3.0 m 5.0 3.2 Uhs 2.0 Grf 2.0 e

dp 6.0 dp 5.5

dx 6.0 dx dx 3.02.8 5.5

dp 5.5 dp 2.6 5.0 2.8 5.5 2.62.4 5.0 5.0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2.4 x 5.0 x x 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 (g)x (h)x (i) x (g) (h) (i) Figure 4. Pressure gradient 푑푝⁄푑푥 profile for (a) Nt, (b) Nb, (c) Nct, (d) Ntc, (e) Grt, (f) Grc, (g)

FigureGrf, (h) 4.m ePressure, (i) Uhs, whilegradient other 푑푝 ⁄parameters푑푥 profile forare ( a푦) =Nt0, ,(휀b)= N0b,.06 (c), ΘN=ct, (0d.9) ,N푁tc푡 ,= (e1) ,G푁rt푏, =(f)1 G, 푁rc,푐푡 (g=) G1,rf푁, 푡푐(h=) m2e,,퐺 (푟푡i) =Uhs5,, while퐺푟푐 = 1other, 퐺푟푓 =parameters1, 푚푒 = 3, are푈ℎ푠 푦==1.0 , 휀 = 0.06, Θ = 0.9, 푁푡 = 1, 푁푏 = 1, 푁푐푡 = Entropy 2019, 21, 986 11 of 21 1, 푁푡푐 = 2, 퐺푟푡 = 5, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 = 3, 푈ℎ푠 = 1. 5.3. Trapping Characteristics 5.3. Trapping Characteristics 5.3. TrappingTrapping Characteristics for different values of the thermophoresis parameter 푁푡, Brownian motion parameter 푁푏 , SoretTrapping parameter for different 푁푐푡 , Dufourvalues of parameter the thermophoresis 푁푡푐 , thermal parameter Grashof 푁 푡, number Brownian 퐺푟푡 motion, solutal parameter Grashof Trapping for diﬀerent values of the thermophoresis parameter Nt, Brownian motion parameter 푁number푏 , Soret 퐺 푟푐 parameter, nanoparticle 푁푐푡 , Dufour Grashof parameter number 퐺푁푟푓푡푐,, electro thermal-osmotic Grashof parameter number 퐺푚푟푡푒, and solutal Helmholtz Grashof- Nb, Soret parameter Nct, Dufour parameter Ntc, thermal Grashof number Grt, solutal Grashof number numberSmoluchowski 퐺푟푐 , nanoparticle velocity 푈ℎ 푠 Grashof are shown number in Fig 퐺푟푓ure, s electro 5 − 11-osmotic. Figure parameter 5(푎 − 푐) and푚푒 and6(푎 − Helmholtz푐) display- Grc, nanoparticle Grashof number Gr f , electro-osmotic parameter me and Helmholtz-Smoluchowski Smoluchowskistreamline structur velocitye for 푈differentℎ푠 are shown values in of Fig푁푡 andures 푁 5푏.− These11 . Figure reveal 5that(푎 − the푐) andnumber 6(푎 − and푐) display size of velocity Uhs are shown in Figures5–11. Figures5a–c and6a–c display streamline structure for di ﬀerent streamlinetrapped bolus structur increasee ford asdifferent 푁푡 and 푁values푏 increases. of 푁푡 andFigure 푁푏 .7 (These푎 − 푐 ) revealshows thatthat the sizenumber of trapped and size bolus of values of Nt and Nb. These reveal that the number and size of trapped bolus increased as Nt and Nb trappedexpanded bolus with increase increasingd as 푁 푡퐺 and푟푡 . 푁H푏owever, increases. a Figurereverse 7 (푎 trend− 푐) shows for solutal that the Grashof size of trapped number b olus퐺푟푐 , increases. Figure7a–c shows that the size of trapped bolus expanded with increasing Grt. However, a expandnanoparticleed with Grashof increasing number 퐺푟푡 퐺.푟푓 H andowever, electro a -osmoticreverse parameter trend for 푚solutal푒 (shown Grashof in Fig numberures (8– 10)퐺푟푐),. reverse trend for solutal Grashof number G , nanoparticle Grashof number G and electro-osmotic nanoparticleFurthermore, GrashofFigure 11 number(푎 − 푐) discloses퐺푟푓 and electrothatrc the-osmotic number parameter of trapped 푚 bolus푒 (shown increaser f in Figd forures dissimilar (8–10)). parameter m (shown in Figures8–10). Furthermore, Figure 11a–c discloses that the number of trapped Furthermore,values of 푈ℎe푠. Figure 11(푎 − 푐) discloses that the number of trapped bolus increased for dissimilar bolus increased for dissimilar values of U . values of 푈ℎ푠. hs 4 4 4

4 4 4

3 3 3

y 2 y 2 2

1 1 1

0 0 0 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 x 0 x 0 x 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 (a)x (b)x (c)x (a) (b) (c) Figure 5. Streamline distribution for (a) 푁푡 = 1.0, (b) 푁푡 = 1.5 , (c) 푁푡 = 2.0, while other

FigureFigureparameters 5. 5.Streamline Streamline are 휀 = distribution0 .06 distribution, Θ = 0 for.9, 푁 ( a푏 for) N= t (1=a,)푁 1.0푁푐푡푡,(==b11),.N0푁,t푡푐 = (b=1.5) 0푁.푡1,(,=c퐺)푟푡1N.5=t =, 0 (.2.0c1), 푁퐺,푟푐 while푡 ==21.0 other, 퐺, 푟푓 while= parameters1 , other푚푒 = areparameters0.8ε, 푈=ℎ0.06,푠 = − Θ1 are. = 0.9,휀 =N0.06=,1,ΘN=ct0=.91,, 푁N푏 tc==1,0.1,푁푐푡G=rt 1=, 푁0.1,푡푐 =Grc0.=1,1,퐺푟푡G ==0.11,, 퐺m푟푐e == 0.8,1, 퐺푟푓U ==1, 푚1.푒 = b r f hs − 0.8, 푈ℎ푠 = −1. 4 4 4

4 4 4

3 3 3

y 2 y 2 2

1 1 1

1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 x x x 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 (a)x (b)x (c)x (a) (b) (c) Figure 6. Streamline distribution for (a) Nt = 1.0,(b) Nt = 1.5 ,(c) Nt = 2.0, while other parameters are ε = 0.06, Θ = 0.9, Nt = 1, Nct = 1, Ntc = 0.1, Grt = 0.1, Grc = 1, G = 1, me = 0.8, U = 1. r f hs − Entropy 2019, 21, x FOR PEER REVIEW 11 of 20 Entropy 2019, 21, x FOR PEER REVIEW 11 of 20 EntropyFigure 2019, 21 6., x FORStreamline PEER REVIEW distribution for (a) 푁푡 = 1.0, (b) 푁푡 = 1.5 , (c) 푁푡 = 2.0, while other 11 of 20 parametersFigure 6. Streamline are 휀 = 0 .06distribution, Θ = 0.9, 푁 for푡 = 1 (a,)푁 푁푐푡푡 ==11,.0푁,푡푐 (b=) 0푁.푡1,=퐺푟푡1.5= , 0 (.c1), 푁퐺푟푐푡 ==21.0, 퐺, 푟푓 while= 1 , other푚푒 = Figure 6. Streamline distribution for (a) 푁푡 = 1.0, (b) 푁푡 = 1.5 , (c) 푁푡 = 2.0, while other 0parameters.8, 푈ℎ푠 = − 1 are. 휀 = 0.06, Θ = 0.9, 푁푡 = 1, 푁푐푡 = 1, 푁푡푐 = 0.1, 퐺푟푡 = 0.1, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 = Entropyparameters2019, 21, 986 are 휀 = 0.06, Θ = 0.9, 푁 = 1, 푁 = 1, 푁 = 0.1, 퐺 = 0.1, 퐺 = 1, 퐺 = 1, 푚 = 12 of 21 0.8, 푈ℎ푠 = −1. 푡 푐푡 푡푐 푟푡 푟푐 푟푓 푒 4 4 4 0.8, 푈ℎ푠 = −1. 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3

y 2 2 y 2

y 2 2 y 2 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 x x x 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 1 0 1(a)x 2 3 4 1 0 1(b)x 2 3 4 1 0 1(c)x 2 3 4 (a)x (b)x (c)x Figure 7. (Streamlinea) distribution for (a) 퐺푟푡(b=) 0.0, (b) 퐺푟푡 = 0.1, (c) 퐺푟푡 = 0.2, (whilec) other parametersFigure 7. Streamline are 휀 = distribution0.06, Θ = 0. 9for, 푁 푡(=a) 1퐺,푟푡푁푏==0.10,, 푁(푐푡b) =퐺푟푡1,=푁푡푐0.=1, 0(c.1) ,퐺퐺푟푡푟푐 ==01.,2퐺, 푟푓while= 1 , other푚푒 = Figure 7. Streamline distribution for (a=) 퐺푟푡 = 0.0, (=b) 퐺푟푡 = 0.1,= (c) 퐺푟푡 = 0.2, while other Figure0parameters.8, 푈ℎ 7.푠 =Streamline− 1. are distribution휀 = 0.06, Θ = for0 (.9a), 푁G푡rt= 10.0, 푁,(푏 b=) 1G,rt푁푐푡 0.1= 1,(, c푁)푡푐Grt= 0.10.2, 퐺,푟푐 while= 1, other퐺푟푓 = parameters1, 푚푒 = parameters are 휀 = 0.06, Θ = 0.9, 푁 = 1, 푁 = 1, 푁 = 1, 푁 = 0.1, 퐺 = 1, 퐺 = 1, 푚 = are0.8ε, 푈=ℎ0.06,푠 = −Θ1. = 0.9, Nt = 1, Nb = 1, Nct 푡= 1, Ntc 푏= 0.1, G푐푡rc = 1, Gr푡푐 f = 1, me 푟푐= 0.8, Uhs푟푓 = 1. 푒 4 4 4 − 0.8, 푈ℎ푠 = −1. 4 4 4 4 4 4 3 3 3

3 3 3 3 3 3

y 2 2 y 2

y 2 2 y 2 1 1 1

1 1 1 1 1 1 0 0 0 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 x x x 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 x x 1 0 1(a)x 2 3 4 1 0 1(b) 2 3 4 1 0 1(c) 2 3 4 (a)x (b)x (c)x Figure 8. (Streamlinea) distribution for (a) 퐺푟푐(b=) 1.0, (b) 퐺푟푐 = 1.5, (c) 퐺푟푐 = 2.0,( cwhile) other FigureparametersFigure 8. 8.Streamline Streamline are 휀 distribution= 0distribution.06, Θ = for0.9 ( a,for)푁푡G =(rca)=1 ,퐺푁1.0푟푐푏 ,(=b1).,0G푁, rc푐푡(b==) 1.5퐺1푟푐, 푁,(=푡푐c) 1=G.5rc0, .1=(c,)퐺2.0 푟푡퐺,푟푐= while=0.21.,0 other퐺,푟푓 while= parameters1, other푚푒 = Figure 8. Streamline distribution for (a) 퐺푟푐 = 1.0, (b) 퐺푟푐 = 1.5, (c) 퐺푟푐 = 2.0, while other are0parameters.8ε, 푈=ℎ0.06,푠 = −Θ 1 are. = 0.9,휀 =N0t .=061,, ΘN=b =0.1,9,N푁ct푡 = 11,, 푁N푏tc ==10.1,, 푁푐푡Grt==1,0.1,푁푡푐G=r f 0=.11,, 퐺푟푡me==00.8,.1, 퐺U푟푓hs==1, 푚1.푒 = parameters are 휀 = 0.06, Θ = 0.9, 푁 = 1, 푁 = 1, 푁 = 1, 푁 = 0.1, 퐺 = 0.1, 퐺 = 1−, 푚 = 0.8, 푈ℎ푠 = −1. 푡 푏 푐푡 푡푐 푟푡 푟푓 푒 4 4 4 0.8, 푈ℎ푠 = −1. 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3

y 2 2 y 2

y 2 2 y 2 1 1 1

1 1 1 1 1 1 0 0 0 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 x x x 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 1 0 1(a)x 2 3 4 1 0 1(b)x 2 3 4 1 0 1(c)x 2 3 4 (a)x (b)x (c)x 퐺 = 1.0 퐺 = 1.5 퐺 = 2.0 FigureFigure 9. 9.Streamline (Streamlinea) distribution distribution for ( afor) G r( fa=) 1.0푟푓(,(b)b ) G,r f(b=) 1.5푟푓,(c) Gr f, =(c)2.0 푟푓, while other,( cwhile) parameters other 퐺 = 1.0 퐺 = 1.5 퐺 = 2.0 areparametersFigureε = 0.06, 9. StreamlineΘ are= 0.9, 휀 =N t0distribution=.061,, ΘNb==0.1,9 ,Nfor푁ct푡 =(a)1,1 ,N푁푟푓tc푏 = 10.1,, 푁, 푐푡G(brt=) =1푟푓0.1,, 푁푡푐G=rc 0=, .1(1,c,)퐺 m푟푡푟푓e == 00.8,.1, 퐺,U 푟푐whilehs==1, 1.other푚푒 = Figure 9. Streamline distribution for (a) 퐺푟푓 = 1.0, (b) 퐺푟푓 = 1.5, (c) 퐺푟푓 = 2.0, while− other 0parameters.8, 푈ℎ푠 = − 1 are. 휀 = 0.06, Θ = 0.9, 푁푡 = 1, 푁푏 = 1, 푁푐푡 = 1, 푁푡푐 = 0.1, 퐺푟푡 = 0.1, 퐺푟푐 = 1, 푚푒 = parameters are 휀 = 0.06, Θ = 0.9, 푁 = 1, 푁 = 1, 푁 = 1, 푁 = 0.1, 퐺 = 0.1, 퐺 = 1, 푚 = 0.8, 푈ℎ푠 = −1. 푡 푏 푐푡 푡푐 푟푡 푟푐 푒 0.8, 푈ℎ푠 = −1. Entropy 2019, 21, 986 13 of 21 Entropy 2019, 21, x FOR PEER REVIEW 12 of 20 Entropy 2019, 21, x FOR PEER REVIEW 12 of 20 4 4 4

4 4 4

3 3 3

y y 2 2 2

1 1 1

0 0 0 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 x x x 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 (a)x (b)x (c)x (a) (b) (c) Figure 10. Streamline distribution for (a) 푚푒 → 0.0, (b) 푚푒 = 1.0, (c) 푚푒 = 1.5, while other FigureparametersFigure 10. 10.Streamline Streamline are 휀 = distribution0 .distribution06, Θ = 0 for.9, (푁 afor푡) =m (ea1), 푁푚0.0푏푒=→,(1b0,)푁.0m푐푡, (eb==) 1 1.0푚, 푁푒,(푡푐=c=)1m.00e.,1= (,c퐺1.5)푟푡 푚,=푒 while=0.1,.5 other퐺,푟푐 while= parameters1, 퐺other푟푓 = → are1parameters, ε푈= 0.06,= −1 Θ. are= 0.9,휀 =N0t .=061,, ΘN==0.91,,N푁푡ct ==11,, 푁N푏tc==10.1,, 푁푐푡Grt==1,0.1,푁푡푐G=rc0=.11,, 퐺G푟푡 ==01,.1U, 퐺푟푐== 11., 퐺푟푓 = ℎ푠 b r f hs − 1, 푈ℎ푠 = −1. 4 4 4

4 4 4

3 3 3

y 2 2 y 2

1 1 1

0 0 0 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 0 0 0 x x x 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 (a)x (b)x (c) x (a) (b) (c) 푈 = −3.0 푈 = −2.0 푈 = 1.0 FigureFigure 11. 11.Streamline Streamline distribution distribution for for (a) U(ahs) =ℎ푠 3.0,(b,) U (bhs) =ℎ푠 2.0,(c), U (chs) =ℎ푠1.0, while, while other Figure 11. Streamline distribution휀 = 0.06, Θ for= 0 (.9a,)푁 푈 =−=1,−푁3.0=, 1 (b, 푁) 푈 =−=1,−푁2.0=, (0c.)1 푈, 퐺 ==10.0.1, ,while퐺 = parametersother parameters are ε = 0.06, are Θ = 0.9, Nt = 1, Nb 푡ℎ=푠 1, Nct푏 = 1, N푐푡tcℎ푠= 0.1,푡푐Grt = 0.1,ℎ푟푡푠 Grc = 1, 푟푐Gr f = 1, 퐺 = 1, 푚 = 0.8. 휀 = 0.06, Θ = 0.9, 푁 = 1, 푁 = 1, 푁 = 1, 푁 = 0.1, 퐺 = 0.1, 퐺 = 1,otherme푟푓= parameters0.8. 푒 are 푡 푏 푐푡 푡푐 푟푡 푟푐 1, 퐺푟푓 = 1, 푚푒 = 0.8. 5.4.5.4 Temperature Characteristics 5.4 Temperature Characteristics TheThe eeffectffectss ofof thermophoresisthermophoresis parameterparameter 푁N푡t,, Brownian Brownian motion motion parameter parameter 푁N푏, bSoret, Soret parameter parameter N푁ct푐푡and andThe DufourDufour effects parameter parameterof thermophoresis N푁푡푐tc were parameterexamined 푁for for푡, hBrownian heateat profile profile motion 휃θ. .Figure Figure parameter 12a,b 12a,b de demonstrates푁monstrates푏, Soret parameter that that the the magnitudemagnitude푁푐푡 and Dufour of of temperaturetemperature parameter 푁 distribution distribution푡푐 were examined 휃θ declinesdeclines for h foreat for profilevalues values 휃 of of. Figure푁N푡 tandand 12a,b푁N푏. bIn. Inde additionmonstrates addition,, Figure Figure that 12cthe 12 c displaysdisplaysmagnitudethat that of thethe temperature magnitudemagnitude distribution of temperature 휃 declines distribution distribution for values 휃θ waswas ofenhance enhanced 푁푡 andd 푁 for푏 for. In more more addition values values, Figure of of 푁푐푡N. ctAs,12c. As, SoretSoretdisplays parameter parameter that theN 푁magnitudect푐푡is is the the ratio ratio of oftemperature temperatureof temperature distribution to concentration,to concentration, 휃 was enhance hence, hence,d bigger for bigger moreNct standsvalues푁푐푡 stands forof 푁 a푐푡 for higher. As, a temperature.higherSoret parametertemperature. Figure 푁푐푡 12isFigure d the shows ratio 12d that ofshows temperature the magnitude that the to magnitude concentration, of temperature of temperature h distributionence, bigger distribution θ푁푐푡was stands boosted 휃 for was a for 휃 higherboosthighered values temperature.for higher of Ntc .valuesSince Figure Noftc 푁 shows12d푡푐. Since shows the 푁푡푐 contribution shows that the the magnitudecontribution of concentration of of temperature concentration gradient to distribution thermalgradient energy to was flux 푁 . 푁 inboost the flow,ed for it higher is evident values that of the푡푐 increase Since 푡푐 in showsNtc caused the contribution a rise in temperature. of concentration gradient to 1.0 1.0

1.0 Nt 1.0 1.0 Nb 1.0 5.5. Concentration0.8 Characteristics 0.8 Ntt 2.01.0 Nb 2.01.0 0.8 0.8 NNtt 3.02.0 NNbb 3.02.0 The0.6 inﬂuence of thermophoresis parameter Nt, Brownian0.6 motion parameter Nb, Soret parameter Nt 3.0 Nb 3.0 Nct and0.6 Dufour parameter Ntc are observed for concentration0.6 proﬁle Ω. Figure 13a,b concludes that 0.4 0.4 the magnitude of Ω increased for more values of Nt and Nb. Moreover, Figure 13c displays that 0.4 0.4 the magnitude0.2 of Ω decayed for more values of Nct. In0.2 addition, Soret parameter Nct is the ratio of temperature0.2 to concentration. Hence, higher values of0.2Nct lead to decay in concentration. Figure 13d shows that0.0 the magnitude of Ω declined for higher values0.0 of N . 0.0 0.2 0.4 0.6 0.8 1.0 0.0 tc0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.0 0.2 0.4 y 0.6 0.8 1.0 0.0 0.2 0.4 y 0.6 0.8 1.0 (a) y (b) y (a) (b) Entropy 2019, 21, x FOR PEER REVIEW 13 of 20 displays that the magnitude of temperature distribution was enhanced for more values of . As, Soret parameter is the ratio of temperature to concentration, hence, bigger stands for a higher temperature. Figure 12d shows that the magnitude of temperature distribution was boostEntropyed2019 for, 21 higher, 986 values of . Since shows the contribution of concentration gradient14 of to 21 thermal energy flux in the flow, it is evident that the increase in caused a rise in temperature.

Entropy 20191.0 , 21, x FOR PEER REVIEW 1.0 13 of 20  Nb 1.0 Nt 1.0 Nb 1.0 0.8 0.81.0 0.81.0  0.8 Nb 2.0 Nt 2.0 Nb 2.0 Nct 1.0 Nbtc3.01.0 Nt 3.0 Nb3.0 0.60.8 0.60.8 N 2.0 Nct 2.0 tc     N 3.0 Nct 3.0 tc 0.40.6 0.40.6

0.20.4 0.20.4

0.00.2 0.00.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 y 0.0 y 0.0 0.2 0.4(a) 0.6 0.8 1.0 0.0 0.2 0.4(b) 0.6 0.8 1.0 (a) y (b) y 1.0 1.0  (c) Ntc 1.0 (d) Nct 1.0 Ntc 1.0 0.8 0.8 0.8 N 2.0 0.8 Ntc 2.0 Figure 12.N Temperaturect 2.0 휃 profile for (a) Nt, (b) Nb, (c) Nct, tc(d) Ntc, while other parameters N 3.0 Nct3.0 Ntc3.0 are0.6 푥 = 1,ct휀 = 0.01, Θ = 0.9, 푁푡 = 1, 푁푏 = 1, 푁푐푡 =0.61, 푁푡푐 = 2, 퐺푟푡 = 5, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 =    3 , 푈ℎ푠 = 1. 0.4 0.4

5.5. Concentration0.2 Characteristics 0.2 푁 푁 The0.0 influence of thermophoresis parameter 0.0푡 , Brownian motion parameter 푏 , Soret 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 parameter 푁푐푡 and Dufour parameter 푁푡푐 are observed0.0 for concentration0.2 0.4 profile0.6 훺. 0.8Figure 1.013a,b y y concludes that the magnitude of 훺 increased for more values of 푁푡 and 푁푏. Moreover, Figure 13c (c) (d) displays that the magnitude of 훺 decayed for more values of 푁푐푡. In addition, Soret parameter 푁푐푡 is Figure 12. Temperature profile for (a) Nt, (b) Nb, (c) Nct, (d) Ntc, while other parameters the FigureratioFigure of12. 12. temperatureTemperatureTemperature toθ profileproﬁle concentration. forfor ((aa)) NNt ,(t, Hence,(bb))N Nb,(b, (chigherc))N Nctct,(, (dd) values)N Ntctc,, whilewhile of 푁 other other푐푡 lead parameters parameters to decay are in = 1, = 0.01, Θ = 0.9, = 1, = 1, = 1, = 2, = 5, = 1, = 1, = concentrationarex = 1,ε==. 1Figure0.01,, =Θ0 13d=.010.9,, Θshows=Nt0=.9 that1,, N b =the=11, ,magnitudeNct==11,, Ntc of== 1훺2,, Gdeclinert ==5,2,dG rcfor== higher1,5G, r f= values=1,1m, e = of=3, 푁1U푡푐,hs. = =1.

3, = 1.

N 1.0 Nb 1.0 5.5. Concentration1.0 t Characteristics 1.0 5.5. ConcentrationNt Characteristics2.0 Nb 2.0

0.8 Nt 3.0 0.8 Nb 3.0 The influence of thermophoresis parameter , Brownian motion parameter , Soret 0.6 0.6 parameter and Dufour parameter are observed for concentration profile . Figure 13a,b conclude0.4s that the magnitude of increased for more0.4 values of and . Moreover, Figure 13c displays that the magnitude of decayed for more values of . In addition, Soret parameter is 0.2 0.2 the ratio of temperature to concentration. Hence, higher values of lead to decay in concentration0.0 . Figure 13d shows that the magnitude of0.0 declined for higher values of . 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y Nt 1.0 (a) Nb 1.0 (b) 1.0 t 1.0 b 1.0  1.0 N 2.0 Nt 2.0 Nb 2.0 0.8 Nct 1.0 0.8 NNtc3.01.0 0.8 Nt3.0 0.8 Nb3.0 0.8 0.8 Nct 2.0 Ntc 2.0 0.6

0.6   0.6

0.6   Nct 3.0 Ntc 3.0 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2

0.00.2 0.00.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 y 0.0 y 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (a) (b) (a) y (b) y (c) (d)

FigureFigure 13. 13. ConcentrationConcentration Ω훺 profileproﬁle forfor ((aa))N Nt,(t, (bb))N Nb,(b, c(c))N Nctct,(, d(d))N Ntctc,, whilewhile other other parameters parameters are arex = 1,푥ε==10.01,, 휀 =Θ0=.010.9,, Θ =Nt0=.91,, 푁N푡 b==11,, 푁N푏ct==11,, 푁N푐푡tc == 12,, G푁rt푡푐 ==5,2G, 퐺rc푟푡==1,5G, 퐺r f푟푐==1,1m, 퐺e 푟푓= 3,= 1U,hs푚=푒 =1. 3, 푈ℎ푠 = 1.

5.6. Nanoparticle Volume Fraction Characteristics

The influence of thermophoresis parameter 푁푡 , Brownian motion parameter 푁푏 , Soret parameter 푁푐푡 and Dufour parameter 푁푡푐 is noticed for nanoparticle volume fraction profile 훾. Entropy 2019, 21, 986 15 of 21

5.6. Nanoparticle Volume Fraction Characteristics

Entropy 2019, 21, x FOR PEER REVIEW 14 of 20 The inﬂuence of thermophoresis parameter Nt, Brownian motion parameter Nb, Soret parameter Nct and Dufour parameter Ntc is noticed for nanoparticle volume fraction proﬁle γ. Figure 14a Figuredemonstrates 14a demonstrates that the that magnitude the magnitude of γ increased of 훾 for increase more valuesd for ofmoreNt. However,values of Figure 푁푡. However, 14b–d shows Figure 14b–dthat shows the magnitudethat the magnitude of γ declined of for 훾 decline higher valuesd for higher of Brownian values motion of Brownian parameter motionNb, Soret parameter parameter 푁푏, Soret Nparamct andeter Dufour 푁푐푡 parameterand DufourNtc parameter. 푁푡푐.

1.0 Nt 1.0 Nb 1.0 1.5 Nt 2.0 0.8 Nb 2.0 Nt 3.0 Nb 3.0 0.6 1.0 0.4

0.5 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y (a) (b) 1.0 1.0 Nct 1.0 Ntc 1.0 0.8 0.8 Nct 2.0 Ntc 2.0 N 3.0 Nct 3.0 tc 0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y (c) (d)

FigureFigure 14. 14.NanoparticleNanoparticle fraction fractionγ proﬁle 훾 profile for ( a) forNt ,((ba) NNb,(t, c ()bN)ct ,(Ndb,) N (ctc), whileNct, ( otherd) N parameterstc, while are other x = 1, ε = 0.01, Θ = 0.9, N = 1, N = 1, N = 1, N = 2, G = 5, G = 1, G = 1, m = 3, U = 1. parameters are 푥 = 1t , 휀 = 0b.01, Θ ct= 0.9, 푁tc푡 = 1, 푁rt 푏 = 1,rc푁푐푡 = r1 f, 푁푡푐 =e2, 퐺푟푡 =hs 5, 퐺푟푐 = 15.7., 퐺푟푓 Entropy= 1, 푚 Production푒 = 3, 푈ℎ푠 = 1.

5.7. EntropyFigure Production 15a–d illustrates the impact of thermophoresis parameter Nt, Brownian motion parameter Nb, Soret parameter Nct and Dufour parameter Ntc on entropy generation number Ns. Figure 15a,b Figuredemonstrates 15a–d thatillustrates the magnitude the impact of Ns increased of thermophoresis for added values parameter of Nt and 푁N푡 ,b . BrownianThis was due motion to parameterthermal 푁푏 irreversibility,, Soret parameter i.e., for 푁 higher푐푡 and value Dufour of Nt and parameterNb, thermal 푁푡푐 irreversibilityon entropy increasedgeneration very number quickly, 푁푠 . Figurenear 15a,b the wall.demonstrates Moreover, that Figure the 15 magnitudec displays that of the 푁푠 magnitudeincreased offorN saddeddecayed values for more of 푁 values푡 and of 푁N푏.ct This. Figure 15d shows that the magnitude of N increased for higher values of N . was due to thermal irreversibility, i.e., fors higher value of 푁푡 and 푁tc푏, thermal irreversibility Figure 16a–d demonstrates the eﬀect of thermophoresis parameter Nt, Brownian motion parameter increased very quickly, near the wall. Moreover, Figure 15c displays that the magnitude of 푁푠 Nb, Soret parameter Nct and Dufour parameter Ntc on Bejan number Be. Figure 16a,b implies that decayed for more values of 푁푐푡. Figure 15d shows that the magnitude of 푁푠 increased for higher the magnitude of Be declined for more values of Nt and Nb. In addition, Figure 16c displays that the values of 푁푡푐. magnitude of Be enhanced for more values of Nct near y = 0. Figure 16d shows that the magnitude of N y = N 4.5s ascended near 0 for higher values of tc. 4.5 Nt 0.1 Nb 0.1 4.0 4.0 Nt 0.2 Nb 0.2 3.5 3.5 Nt 0.3 Nb 0.3

s s 3.0 3.0

N 2.5 2.5

2.0 2.0

1.5 1.5

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y (a) (b) Entropy 2019, 21, x FOR PEER REVIEW 14 of 20

Figure 14a demonstrates that the magnitude of 훾 increased for more values of 푁푡. However, Figure 14b–d shows that the magnitude of 훾 declined for higher values of Brownian motion parameter 푁푏, Soret parameter 푁푐푡 and Dufour parameter 푁푡푐.

1.0 Nt 1.0 Nb 1.0 1.5 Nt 2.0 0.8 Nb 2.0 Nt 3.0 Nb 3.0 0.6 1.0 0.4

0.5 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y (a) (b) 1.0 1.0 Nct 1.0 Ntc 1.0 0.8 0.8 Nct 2.0 Ntc 2.0 N 3.0 Nct 3.0 tc 0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y (c) (d)

Figure 14. Nanoparticle fraction 훾 profile for (a) Nt, (b) Nb, (c) Nct, (d) Ntc, while other

parameters are 푥 = 1, 휀 = 0.01, Θ = 0.9, 푁푡 = 1, 푁푏 = 1, 푁푐푡 = 1, 푁푡푐 = 2, 퐺푟푡 = 5, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 = 3, 푈ℎ푠 = 1.

5.7. Entropy Production

Figure 15a–d illustrates the impact of thermophoresis parameter 푁푡 , Brownian motion parameter 푁푏, Soret parameter 푁푐푡 and Dufour parameter 푁푡푐 on entropy generation number 푁푠 . Figure 15a,b demonstrates that the magnitude of 푁푠 increased for added values of 푁푡 and 푁푏. This was due to thermal irreversibility, i.e., for higher value of 푁푡 and 푁푏, thermal irreversibility increased very quickly, near the wall. Moreover, Figure 15c displays that the magnitude of 푁푠 decayEntropyed2019 for, 21 , more 986 values of 푁푐푡. Figure 15d shows that the magnitude of 푁푠 increased for higher16 of 21 values of 푁푡푐.

4.5 4.5 Nt 0.1 Nb 0.1 4.0 4.0 Nt 0.2 Nb 0.2 Entropy 20193.5 , 21, x FOR PEER REVIEW 3.5 15 of 20 Nt 0.3 Nb 0.3

s s 3.0 3.0

N N 2.5 3.5 2.5 2.5 Nct 0.1 Ntc 2.0

2.0 Nct 0.2 2.03.0 Ntc 2.2 2.0 1.5 Nct 0.3 1.5 Ntc 2.4

s 2.5

N N 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.5 y y 2.0 Entropy 2019, 21, x FOR PEER REVIEW(a) (b) 15 of 20 1.0 2.5 1.5 3.5 Nct 0.1 Ntc 2.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Nct 0.2 3.0 y Ntc 2.2 2.0 y (c) Nct 0.3 (d) Ntc 2.4

s 2.5

N Figure1.5 15. Heat generation number 푁푠 profile for (a) Nt, (b) Nb, (c) Nct, (d) Ntc, while other 2.0 parameters are 푥 = 1, 휀 = 0.02, Θ = 0.5, 푁푡 = 1, 푁푏 = 1, 푁푐푡 = 1, 푁푡푐 = 2, 퐺푟푡 = 5, 퐺푟푐 = 1, 퐺푟푓 = 1, 푚푒 = 3, 푈ℎ푠 = 1. 1.0 1.5

0.0 0.2 0.4 0.6 0.8 1.0 Figure0.0 16a–0.2d demonstrates0.4 0.6 the 0.8 effect of1.0 thermophoresis parameter 푁푡 , Brownian motion y y parameter 푁푏 , Soret parameter 푁푐푡 and Dufour parameter 푁푡푐 on Bejan number 퐵푒 . Figure 16a,b (c) (d) implies that the magnitude of 퐵푒 declined for more values of 푁푡 and 푁푏. In addition, Figure 16c displaysFigureFigure that 15.15. theHeat Heat magnitude generation generation number of 퐵number푒 enhanceNs proﬁle 푁푠 dprofile for for (a )moreN fort,(b ( )a valuesN) bN,(tc, )(Nb ofct) ,(N 푁db푐푡), N( nearctc) ,N whilect 푦, (=d other)0 N. tcFi, parameters gurewhile 16d other areshows that theparametersx = magnitude1, ε = 0.02, Θ of are= 푁 0.5,푠 ascen푥N=t =ded11,, 휀N =nearb =0.021, 푦N,=Θct 0== for01,.5N higher, tc푁푡==2,1 Gvalues, rt푁푏==5, 1ofG,rc푁 푁푐푡=푡푐=.1, G1r, f푁=푡푐 1,= m2,e퐺=푟푡3,=U5hs, 퐺=푟푐1.= 1, 퐺푟푓 = 1, 푚푒 = 3, 푈ℎ푠 = 1. 0.35 0.35 Nt 1.0 N 1.0 0.30 0.30 b Figure 16a–d demonstrates the effect of thermophoresis parameter 푁푡 , Brownian motion Nt 2.0 Nb 2.0 0.25 0.25 parameter 푁푏 , Soret parameter 푁푐푡 and Dufour parameter 푁푡푐 on Bejan number 퐵푒 . Figure 16a,b Nt 3.0 Nb 3.0 0.20 0.20

e impliese that the magnitude of 퐵푒 declined for more values of 푁푡 and 푁푏. In addition, Figure 16c

B B 0.15 displays0.15 that the magnitude of 퐵푒 enhanced for more values of 푁푐푡 near 푦 = 0. Figure 16d shows that the0.10 magnitude of 푁푠 ascended near 푦 = 0 for higher0.10 values of 푁푡푐. 0.05 0.05 0.35 0.35 0.00 0.00 Nt 1.0 N 1.0 0.300.0 0.2 0.4 0.6 0.8 1.0 0.300.0 0.2b 0.4 0.6 0.8 1.0 Nt 2.0 y Nb 2.0 y 0.25 0.25 N 3.0 t (a) Nb 3.0 (b) 0.20 0.20

e e 0.35

B B 0.30 0.15 N 1.0 0.15 N 1.0 ct 0.30 tc 0.25 0.10 Nct 2.0 0.10 Ntc 2.0 0.25 0.20 0.05 Nct 3.0 0.05 Ntc 3.0 0.20

B 0.150.00 B 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.150.0 0.2 0.4 0.6 0.8 1.0 0.10 y 0.10 y 0.05 (a) 0.05 (b) 0.35 0.000.30 N 1.0 0.00 N 1.0 0.0 0.2ct 0.4 0.6 0.8 1.0 0.300.0 0.2tc 0.4 0.6 0.8 1.0 0.25 Nct 2.0 y N 2.0 y 0.25 tc 0.20 Nct 3.0 (c) Ntc 3.0 (d) 0.20

B 0.15 B FigureFigure 16. BejanBejan number number 퐵푒B eforfor (a ()a N) tN, (tb,() bN)b,N (cb,() Nc0.15ct), N(dct),( Ndtc), whileNtc, while other otherparameters parameters are 푥 are= 0.10 1x,=휀 =1, ε0.=020.02,, Θ =Θ0=.8,0.8,푁푡 =Nt1=, 푁1,푏 N=b 1=, 푁1,푐푡N=ct =1,1,푁푡푐Ntc==20.102,, 퐺G푟푡rt==55,, 퐺G푟푐rc = 1,1,G퐺r푟푓 f ==1,1,m푚e푒==3,3,U푈hsℎ푠==1. 1. 0.05 0.05

0.00 0.00 5.8. Heat Transfer0.0 0.2Coefficient0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y y Figure 17a–d reveal(cs) the changes in heat transfer coefficient (푍d) for different values of thermophoresis parameter 푁푡 , Brownian motion parameter 푁푏 , Soret parameter 푁푐푡 and Dufour 퐵 t b ct tc 푥 = parameterFigure 푁 16.. Bejan This infersnumber that 푒 for for (variousa) N , (b )values N , (c) N of ,푁 (d and) N ,푁 while, the other magnitude parameters of heat are transfer 1, 휀 = 0.푡푐02, Θ = 0.8, 푁 = 1, 푁 = 1, 푁 = 1, 푁 = 2, 퐺 푡 = 5, 퐺푏 = 1, 퐺 = 1, 푚 = 3, 푈 = coefficient 푍 increased. However,푡 푏 the magnitude푐푡 푡푐 of heat푟푡 transfer푟푐 coefficient푟푓 푍 decline푒 d ℎfor푠 bigger 1. values of 푁푐푡 and 푁푡푐, respectively. 5.8. Heat Transfer Coefficient Figure 17a–d reveals the changes in heat transfer coefficient 푍 for different values of thermophoresis parameter 푁푡 , Brownian motion parameter 푁푏 , Soret parameter 푁푐푡 and Dufour parameter 푁푡푐 . This infers that for various values of 푁푡 and 푁푏 , the magnitude of heat transfer coefficient 푍 increased. However, the magnitude of heat transfer coefficient 푍 declined for bigger values of 푁푐푡 and 푁푡푐, respectively. Entropy 2019, 21, 986 17 of 21

5.8. Heat Transfer Coefficient Figure 17a–d reveals the changes in heat transfer coefficient Z for different values of thermophoresis parameter Nt, Brownian motion parameter Nb, Soret parameter Nct and Dufour parameter Ntc. This infers that for various values of Nt and Nb, the magnitude of heat transfer coefficient Z increased. However, the magnitude of heat transfer coefficient Z declined for bigger values of Nct and Ntc, respectively. Ent ro py 2019, 21, x FOR P EER R EVIEW 16 of 20

(a) (b)

Figure 17.17. HeatHeat transfer raterate Z for (a)) Ntt,(, (b) Nb,,( (cc)) NNctct, ,((dd)) NNtctc, ,while while other other parameters parameters are are x ==1,1,ε ==0.01,0.01Θ, ==0.9,0.9N, t ==1,1N,b = =1, N1ct, = 1,=N1tc, = 2,=Grt2,= 5, G=rc5=, 1, G=r f1=, 1, m=e =1, 3, U=hs = 1. 3, = 1. 𝑍𝑍 6. Concluding𝑥𝑥 𝜀𝜀 RemarksΘ 𝑁𝑁𝑡𝑡 𝑁𝑁𝑏𝑏 𝑁𝑁𝑐𝑐𝑐𝑐 𝑁𝑁𝑡𝑡𝑡𝑡 𝐺𝐺𝑟𝑟𝑟𝑟 𝐺𝐺𝑟𝑟𝑟𝑟 𝐺𝐺𝑟𝑟𝑟𝑟 𝑚𝑚𝑒𝑒 𝑈𝑈ℎ𝑠𝑠 6. ConcluA theoreticalding Remarks model for the entropy generation in electro-osmotic peristaltic flow of nanofluid withA double-di theoreticalffusive model convection for the throughentropy microchannelgeneration in iselectro reported.-osmotic The exactperistaltic solutions flow areof nanofluid presented forwith quantities double-diffusive of interest. convection The results through of this studymicrochannel are similar is toreport the resultsed. The of exact [11], whensolutionUhss =are0. Moreover,presented resultsfor quantities of this studyof interest are comparable. The results to of the this results study ofare [6 ],similar when toGrt the= Gresultsrc = G ofr f [11],= U hswhen= 0. From= the0. Moreover, current analysis, results weof this conclude studythat are comparable to the results of [6], when = = = = 0. From the current analysis, we conclude that 𝑈𝑈ℎ𝑠𝑠 The magnitude of total entropy generation increased as the thermophoresis𝐺𝐺𝑟𝑟𝑟𝑟 parameter𝐺𝐺𝑟𝑟𝑟𝑟 𝐺𝐺𝑟𝑟 and𝑓𝑓 • •𝑈𝑈 ℎ𝑠𝑠 BrownianThe magnitude motion of parameter total entropy increases. generation increased as the thermophoresis parameter and SoretBrownian parameter motion and parameter Dufour parameterincreases. Ntc strongly controlld the temperature profile and Bejan • • numberSoret parameter profile. and Dufour parameter strongly controlld the temperature profile and TheBejan velocity, number pressure profile. difference, pressure rise, temperature and Bejan number profile decreased • 𝑁𝑁𝑡𝑡𝑡𝑡 • asThe thermophoresis velocity, pressure parameter difference, and Brownian pressure motion rise, parametertemperature increased. and Bejan number profile Electro-osmoticdecreased as thermophoresis parameter strongly parameter affected and the Brownian velocity motion profile. parameter increased. • TheElectro magnitude-osmotic parameter of pressure strongly difference affected and the pressure velocity profile. gradient enhances in the pumping • regionThe magnitude with the of increase pressure in difference Soret parameter, and pressure Dufour gradient parameter, enhances thermal in the Grashof pumping number, region solutalwith the Grashof increase number, in Soret nanoparticleparameter, Dufour Grashof parameter, number, thermal electro-osmotic Grashof number, parameter solutal and Helmholtz-SmoluchowskiGrashof number, nanoparticle velocity. Grashof number, electro-osmotic parameter and Helmholtz- Smoluchowski velocity. • The volume and number of trapped bolus increased as the thermophoresis parameter ,

Brownian motion parameter, thermal Grashof number and Helmholtz-Smoluchowski velocity𝑡𝑡 increases. 𝑁𝑁 • Heat transfer coefficient, entropy generation number and nanoparticles volume fraction strongly surged with the thermophoresis parameter and the Brownian motion parameter increased.

Nomenclature Half width of channel [L]

𝑎𝑎� Entropy 2019, 21, 986 18 of 21

The volume and number of trapped bolus increased as the thermophoresis parameter Nt, Brownian • motion parameter, thermal Grashof number and Helmholtz-Smoluchowski velocity increases. Heat transfer coeﬃcient, entropy generation number and nanoparticles volume fraction strongly • surged with the thermophoresis parameter and the Brownian motion parameter increased.

Author Contributions: Conceptualization, S.N. and D.L.; methodology, S.W.; software, S.W.; validation, S.N., D.L. and A.H.; formal analysis, S.N.; investigation, S.W.; resources, D.L.; writing—original draft preparation, S.N. and S.W.; writing—review and editing, S.N.; visualization, S.N.; supervision, S.N.; project administration, D.L.; funding acquisition, A.H. Funding: This research received no external funding. The APC was given by Ton Duc Thang University, Ho Chi Minh City, Vietnam. However, no grant number is available from source. Acknowledgments: The third author would like to thank Ton Duc Thang University, Ho Chi Minh City, Vietnam for the financial support. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

ea Half width of channel [L] eb Amplitude of wave [L] Be Bejan number [-] c Wave speed [L/T] C Solutal concentration C0 Solutal concentration for lower wall h 2 2 i c f Specific heat ML /T K h 2 2 i cp Heat capacity of fluid ML /T K h 2 i Db Brownian diffusion coefficient L /T h 2 i Dt Thermophoretic diffusion coefficient L /T h 2 i Ds Solutal diffusivity L /T h 2 i Dct Soret diffusivity L /TK h 3i Dtc Dufour diffusivity ML/T e Electron charge [C] h 3 i Ex Electric field M/T A h i f Dimensionless volume flow rate in fixed frame L3/T eF Nanoparticle volume fraction F Nanoparticle volume fraction for lower wall 0 h i g Acceleration due to gravity L2/T Grt Thermal Grashof number [-] Grc Solutal Grashof number [-] Gr f Nanoparticle Grashof number [-] eh Transverse vibration of wall [L] K Boltzmann constant B h i k Thermal conductivity ML/T3K me Electroosmotic parameter [-] n± Positive, negative ions + n0 Average number of n or n− ions Nb Brownian motion parameter [-] Nt Thermophoresis parameter [-] Nct Soret parameter [-] Ntc Dufour parameter [-] Ns Entropy generation number [-] NHT Dimensionless entropy generation due to heat transfer [-] Entropy 2019, 21, 986 19 of 21

NDC Dimensionless entropy generation due to solutal concentration [-] Pe Pressure field [ML/T2] p Pressure field [-] Pr Prandtl number [-] Pe Peclet number [-] Re Reynolds number [-] Sc Schmidt number [-] et Dimensional time [T] t Dimensionless time [-] eT Temperature field [K] T0 Temperature of fluid at lower wall [K] Tav Average temperature [K] Uhs Helmholtz-Smoluchowski velocity Ue, Ve Dimensional velocity components in stationary frame [L/T] u, v Non-dimensional velocity components in wave frame [-] Xe, Ye Dimensional coordinates in stationary frame [L] x, y Non-dimensional coordinates in wave frame [-] zv Valence of ions Z Heat transfer coefficient [-] α Wave number [-] βt Volumetric thermal expansion coefficient of fluid [1/K] βc Volumetric solutal expansion coefficient of fluid [-] γ Dimensionless nanoparticle volume fraction [-] Dielectric permittivity [-] ∈ ε Amplitude ratio [-] Θ Non-dimensional volume flow rate [-] θ Dimensionless temperature [-] λ Wavelength of peristaltic wave [L] λd Debye length [-] µ Fluid viscosity [M/LT] 3 ρ0 Nanofluid density at reference temperature T0 [M/L ] 3 ρp Nanoparticle mass density [M/L ] 3 ρ f Density of fluid [M/L ] 3 ρe Net ionic charge density [M/L ] 2 2 (ρc) f Heat capacity of fluid [ML /T K] 2 2 (ρc)p Effective heat capacity of nanoparticle [ML /T K] φe Dimensional electric potential distribution [ML2/T2I] φ Non-dimensional electric potential distribution [-] ψ Dimensional Stream function [L2/T] Ω Non-dimensional concentration field [-]

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