SORET EFFECT ON STAGNATION-POINT FLOW PAST A … · 3Department of Mathematics, Aswan University, Faculty of Science, Aswan, 81528, Egypt 4 Mechanical Engineering Department, Prince

Special Topics & Reviews in Porous Media — An International Journal, 7 (3): 229–243 (2016)

SORET EFFECT ON STAGNATION-POINT FLOWPAST A STRETCHING/SHRINKING SHEET IN ANANOFLUID-SATURATED NON-DARCY POROUSMEDIUM

Ch. RamReddy,1,∗ P. V. S. N. Murthy,2 A. M. Rashad,3 &Ali J. Chamkha4,5

1Department of Mathematics, National Institute of Technology Warangal-506004, India2Department of Mathematics, Indian Institute of Technology, Kharagpur-721 302, India3Department of Mathematics, Aswan University, Faculty of Science, Aswan, 81528, Egypt4Mechanical Engineering Department, Prince Mohammad Bin FahdUniversity (PMU), Al-Khobar 31952, Kingdom of Saudi Arabia

5Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin FahdUniversity, Al-Khobar 31952, Kingdom of Saudi Arabia

∗Address all correspondence to: Ch. RamReddy, E-mail: [email protected]

Original Manuscript Submitted: 10/10/2014; Final Draft Received: 9/8/2016

The significance of the Soret effect on the boundary-layer stagnation-point flow past a stretching/shrinking sheet in ananofluid-saturated non-Darcy porous medium is investigated in this study. The nanofluid-saturated porous mediumis considered by incorporating the Brownian motion and thermophoresis effects. A similarity transformation is usedto reduce the governing fluid flow equations into a set of differential equations and then solved numerically by anaccurate implicit finite-difference method. The flow; temperature; concentration and nanoparticle concentration fields;skin friction coefficient; and heat, mass, and nanoparticle mass transfer rates are affected by the complex interactionsamong the various physical parameters involved in the analysis. These profiles are illustrated graphically in order toreveal interesting phenomena.

KEY WORDS: Soret effect, stagnation-point flow, stretching/shrinking sheet, nanofluid, non-Darcyporous medium

1. INTRODUCTION

Nanotechnology is considered by many to be one of thesignificant forces that will drive the next major indus-trial revolution of this century. It represents the most rele-vant technological cutting edge currently being explored.Nanofluid heat transfer is an innovative technology thatcan be used to enhance heat transfer. Heat transfer is oneof the most important processes in many industrial andconsumer products. The inherently poor thermal conduc-

tivity of conventional fluids puts a fundamental limit onheat transfer. Therefore, for more than a century sincethe research done by Maxwell (1873), scientists and en-gineers have made great efforts to break this fundamentallimit by dispersing millimeter- or micrometer-sized parti-cles in liquids. However, the major problem with the useof such large particles is the rapid settling of these par-ticles in fluids. Because extended surface technology hasalready been adapted to its limits in the designs of ther-mal management systems, once again technologies with

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230 RamReddy et al.

the potential to improve a fluid’s thermal properties are ofgreat interest. The concept and emergence of nanofluidsis related directly to trends in miniaturization and nan-otechnology. Maxwell’s concept is old, but what is newand innovative in the concept of nanofluids is the idea thatparticle size is of primary importance in developing sta-ble and highly conductive nanofluids. Nanomaterials haveunique mechanical, optical, electrical, magnetic, and ther-mal properties. Nanofluids are engineered by suspendingnanoparticles with average sizes below 50 nm in conven-tional heat transfer fluids such as water, oil, and ethyleneglycol. A very small amount of guest nanoparticles, whendispersed uniformly and suspended stably in host fluids,can provide dramatic improvements in the thermal prop-erties of host fluids. The term nanofluids (nanoparticlefluid suspensions) was first coined by Choi (1995) to de-scribe this new class of nanotechnology-based heat trans-fer fluids that exhibit thermal properties superior to thoseof their host fluids or conventional particle fluid suspen-sions. The goal of nanofluids is to achieve the highest pos-sible thermal properties at the smallest possible concen-trations by uniform dispersion and stable suspension ofnanoparticles in host fluids. To achieve this goal it is im-portant to understand how nanoparticles enhance energytransport in liquids. Cooling is one of the top technicalchallenges facing high-tech industries such as microelec-tronics, transportation, manufacturing, metrology, and de-fense. For example, the electronics industry has providedcomputers with faster speeds, smaller sizes, and expandedfeatures, leading to ever-increasing heat loads, heat fluxes,and localized hot spots at the chip and package levels.These thermal problems are also found in power electron-ics or optoelectronic devices. Air cooling is the most basicmethod for cooling electronic systems. The detailed intro-duction and applications of nanofluids can be found in thebook by Das et al. (2007).

Problems involving fluid flow over a stretching orshrinking surface can be found in many manufacturingprocesses. These processes include polymer extrusion,wire and fiber coating, food stuff processing, etc. Theflow field past a stretching surface was first found byCrane (1970), who presented a closed-form solution to theNavier–Stokes equations. A similarity solution to naturalconvective heat transfer from a vertical stretching sheetwith surface mass transfer was presented by Gorla andSidawi (1994). Takhar et al. (2001) considered the un-steady laminar boundary-layer flow of a viscous electri-cally conducting fluid induced by the impulsive stretch-ing of a flat surface in two lateral directions through anotherwise quiescent fluid. Similarity and local similarity

solutions for the boundary-layer flow on a linearly stretch-ing permeable vertical surface were obtained by Ali andAl-Yousef (2002) when the buoyancy force assists or op-poses the flow and the boundary-layer equations are sub-jected to power-law temperature and velocity variations.Mohammadein et al. (2007) presented a boundary-layeranalysis to study the influence of radiation and buoyancyon the heat and mass transfer characteristics of contin-uous surfaces having a prescribed variable surface tem-perature and stretched with rapidly decreasing power-lawvelocities. Khedr et al. (2009) investigated the magneto-hydrodynamic (MHD) flow of a micropolar fluid past astretched permeable surface with heat generation or ab-sorption.

In fluid flow system, the Soret effect (thermal-diffusionprocess) is a thermodynamic phenomenon in which dif-ferent particles respond in different ways to varyingtemperatures. In a simple way, we can say that massfluxes can be created by temperature gradients known asthe Soret or thermodiffusion effect. In 1879, the Swisschemist Charles Soret first applied this effect to examinein detail Earth’s gravity. Furthermore, Soret noticed that asalt solution contained in a tube with two ends at differenttemperatures did not remain uniform in composition andthe salt was more concentrated near the cold end than nearthe hot end of the tube. In particular, the strength of the ef-fect is characterized by the Soret coefficient, which takesvalues of about 10−4 K−1 for molecular mixtures. In sus-pensions it can be about two to three orders of magnitudehigher, making the effect a phenomenon of technical im-portance. For all commonly used fluids, the Soret coef-ficient has been found to be a material constant. It playsan important role in the operation of solar ponds, biologi-cal systems, and the microstructure of oceans. In biologi-cal systems, mass transport across biological membranesinduced by small thermal gradients in living matter is animportant factor. Different aspects of the Soret effect havebeen reviewed by Platten (2006) and Rahman and Saghir(2014).

Furthermore, the Soret effect may become significantwhen large density differences exist in a flow regime. Forexample, the Soret effect can be significant when speciesare introduced at a surface in a fluid domain with a den-sity lower than the surrounding fluid. In view of theseimportant applications, several researchers have been ex-ploring the usefulness of this effect on different fluidsby considering various surface geometries. Abd El-Aziz(2008) studied the influence of the Dufour and Soret ef-fects on the heat and mass transfer characteristics of freeconvection past a continuously stretching permeable sur-

Special Topics & Reviews in Porous Media — An International Journal

Soret Effect on Stagnation-Point Flow 231

face in the presence of a magnetic field, blowing/suction,and radiation. Beg et al. (2009) examined the cross-diffusion effects and porous impedance on the laminarMHD mixed-convection heat and mass transfer of an elec-trically conducting fluid from a vertical stretching sur-face in a Darcian porous medium under a uniform trans-verse magnetic field. The effect of cross diffusion on thedouble-diffusive free-convection flow in a porous mediumwas presented by Magyari and Postelnicu (2011). Srini-vasacharya and Swamy Reddy (2012) investigated similareffects in a shear thinning fluid–saturated Darcy porousmedium (pseudo plastic;n < 1) and a shear thickeningfluid–saturated Darcy porous medium (dilatants;n > 1).A mathematical model was given by Murthy et al. (2013)to investigate the thermal-diffusion effect on an inclinedplate in a nanofluid-saturated non-Darcy porous medium.RamReddy et al. (2013) stated that the Soret number ac-counts for the additional mass diffusion due to the tem-perature gradients and showed that the mixed-convectionflow on a semi-infinite vertical flat plate in a nanofluid un-der the convective boundary conditions is appreciably in-fluenced by the Soret parameter. Also, they concluded thatthe Soret effect enhanced the skin friction, heat, nanopar-ticle mass, and regular mass transfer rates in the medium.

In recent years, several authors extended the studyof boundary-layer stagnation-point flow over a stretch-ing surface to base fluids with suspended nanoparticles(nanofluids) under different physical conditions. The nu-merical solution for laminar fluid flow, which resultsfrom stretching a flat surface in a nanofluid, was pre-sented by Khan and Pop (2010). A similarity solutionof the steady boundary-layer flow near the stagnation-point flow on a permeable stretching sheet in a porousmedium saturated with a nanofluid under the influenceof internal heat generation/absorption was theoreticallyinvestigated by Hamad and Pop (2011). The problemof convective flow and heat transfer of an incompress-ible Newtonian nanofluid along a semi-infinite verticalstretching sheet under the effect of a magnetic fieldwas solved analytically by Hamad (2011). Bachok et al.(2011) carried out an analysis to study the steady two-dimensional stagnation-point flow of a nanofluid pasta stretching/shrinking sheet in its own plane. Chamkhaand Aly (2011) focused on the numerical solution ofthe steady natural convection boundary-layer flow of ananofluid consisting of a pure fluid with nanoparticlesalong a permeable vertical plate in the presence of mag-netic field, heat generation or absorption, and suctionor injection effects. A boundary-layer analysis was pre-sented by Gorla and Chamkha (2011) for the natural con-

vection past a horizontal plate in a porous medium satu-rated with a nanofluid. Hamad and Ferdows (2012) car-ried out heat and mass transfer analysis for boundary-layer stagnation-point flow over a stretching sheet in aporous medium saturated by a nanofluid with internal heatgeneration/absorption and suction/blowing. The steadyMHD two-dimensional stagnation-point flow of an in-compressible nanofluid toward a stretching surface sub-jected to convective boundary conditions in the presenceof radiation was investigated numerically by Akbar etal. (2013). Stagnation-point flow and heat transfer dueto nanofluid toward a stretching sheet in the presenceof a magnetic field was investigated by Ibrahim et al.(2013). Kameswaran et al. (2013) discussed the effects ofhomogeneous–heterogeneous reactions on the stagnation-point flow of a nanofluid over a stretching or shrink-ing sheet. Makinde et al. (2013) analyzed the combinedeffects of buoyancy force, convective heating, Brown-ian motion, thermophoresis, and magnetic field on thestagnation-point flow and heat transfer due to nanofluidflow toward a stretching sheet. Unsteady two-dimensionalstagnation-point flow of a nanofluid over a stretchingsheet was investigated numerically by Malvandi et al.(2014), in which Navier’s slip condition was applied incontrast to the traditional no-slip condition at the surface.Nandy and Pop (2014) reported a numerical solution tothe problem of steady two-dimensional MHD stagnation-point flow and heat transfer, with thermal radiation, of ananofluid past a shrinking sheet.

Convective transport in porous media has been a sub-ject of great importance and interest in recent years ow-ing to its broad range of applications in civil, chemi-cal, and mechanical engineering. Some studies on con-vection heat and mass transfer in a porous medium satu-rated with a nanofluid have been reported in the literatureover the past few years. In many practical situations theporous medium is bounded by an impermeable wall (seeKhidir and Sibanda, 2014; Uddin et al., 2015), has higherflow rates, and reveals a non-uniform porosity distribu-tion near the wall region, making Darcy’s law inappli-cable. Therefore, to better model the real physical situa-tion, it is necessary to include the non-Darcy terms (eitherBrinkman or Forchheimer porous medium) in the analysisof convective transport in a porous medium. Non-Darcymodels are extensions of the classical Darcy formula-tion, which incorporate inertial drag effects, vorticity dif-fusion, and combinations of these effects (see Chamkhaet al., 2014; Chand and Rana, 2014; Srinivasacharya andVijay Kumar, 2015, and citations therein). To the bestof our knowledge, no studies have been reported in the

Volume 7, Issue 3, 2016

232 RamReddy et al.

literature analyzing the Soret effect on the stagnation-point flow of a nanofluid past a vertical linearly stretch-ing/shrinking sheet in a non-Darcy porous medium. Moti-vated by the aforementioned investigations, we have madean attempt to investigate the effects of the Soret effect onthe boundary-layer stagnation-point flow past a stretch-ing/shrinking sheet in a nanofluid-saturated non-Darcyporous medium. The transformed nonlinear conservationequations are solved numerically.

2. MATHEMATICAL FORMULATION

Consider a steady, incompressible two-dimensionalboundary-layer flow near the stagnation point in a non-Darcy porous medium saturated with nanofluids. TheCartesian coordinatesx andy are taken along the surfaceand normal to the surface, respectively, andu andv arethe respective velocity components. The flow is generateddue to stretching or shrinking of the sheet caused by thesimultaneous application of two equal forces along thex-axis. Keeping the origin fixed, it is assumed that the sur-face is stretched/shrunk with linear velocityuw (x) = bxas shown in Fig. 1, whereb is a constant withb > 0for a stretching sheet,b < 0 for a shrinking sheet, andb = 0 for a static sheet. It is also assumed that the ambi-ent fluid moves with velocityue (x) = ax, wherea is aconstant. The sheet is maintained at uniform wall temper-atureTw, nanoparticle concentrationϕw, and concentra-tion Cw. The free stream temperature, nanoparticle vol-ume fraction, and concentration areT = T∞, ϕ = ϕ∞,andC = C∞, respectively. In addition, the Soret effect istaken into consideration.

FIG. 1: Physical configuration and coordinate system ofthe problem

Under these assumptions and after making use of theBoussinesq approximation, the equations governing theflow are given by

∂u

∂x+

∂v

∂y= 0 (1)

u∂u

∂x+ v

∂u

∂y= ue (x)

∂ue (x)

∂x+ ν

∂2u

∂y2+εν

KP

× [ue (x)− u] +ε2b

KP

[u2e (x)− u2

]+ (1− ϕ∞)

× g [βT (T − T∞) + βC (C − C∞)]

− (ρP − ρf∞) g

ρf∞(ϕ− ϕ∞) (2)

u∂T

∂x+ v

∂T

∂y= αm

∂2T

∂y2+ J

[DB

∂ϕ

∂y

∂T

∂y

+DT

T∞

(∂T

∂y

)2 ](3)

u∂ϕ

∂x+ v

∂ϕ

∂y= DB

∂2ϕ

∂y2+

DT

T∞

∂2T

∂y2(4)

u∂C

∂x+ v

∂C

∂y= DS

∂2C

∂y2+DCT

∂2T

∂y2(5)

whereu andv are the intrinsic velocity components in thex andy directions, respectively;T is the temperature;Cis the concentration;ϕ is the nanoparticle concentration;g is the acceleration due to gravity;KP is the permeabil-ity; b is the empirical constant associated with the Forch-heimer porous inertia term;σ is the electrical conductivityof the fluid;ε is the porosity;αm = k/(ρc)f andDS arethe thermal and solutal diffusivities of the fluid, respec-tively; ν = µ/ρf∞ is the kinematic viscosity coefficient;andJ = ϕ(ρc)p/(ρc)f . Furthermore,ρf∞ is the den-sity of the base fluid;ρ, µ, k, βT , andβC are the den-sity, viscosity, thermal conductivity, and volumetric ther-mal and solutal expansion coefficients of the nanofluid,respectively;ρp is the density of the nanoparticles; (ρc)fis the heat capacity of the fluid; and (ρc)p is the effec-tive heat capacity of the nanoparticle material. The coef-ficients that appear in Eqs. (3)–(5) are the Brownian dif-fusion coefficientDB ; the thermophoretic diffusion coef-ficientDT ; and the last term in Eq. (5) is due to the Soreteffect (see RamReddy, 2013, and citations therein).

The associated boundary conditions are

u = uw (x) , v = 0, T = Tw, ϕ = ϕw,

C = Cw at y = 0 (6a)



u = ue (x) , T = T∞, ϕ = ϕ∞, C = C∞

as y → ∞ (6b)

where subscriptsw and∞ indicate the conditions at thewall and outer edge of the boundary layer, respectively;uw (x) = bx; andue (x) = ax.

The following non-dimensional transformations are in-troduced:

η =

√a

νy, ψ (x,η) =

√aνxf (η) ,

θ (η) =T − T∞

Tw − T∞, γ (η) =

ϕ− ϕ∞

ϕw − ϕ∞,

S (η) =C − C∞

Cw − C∞(7)

wheref is the dimensionless stream function. In view ofcontinuity Eq. (1), we introduce the stream functionψ

u =∂ψ

∂y, v = −∂ψ

∂x(8)

Substituting Eq. (8) in Eqs. (2)–(5) and then using non-dimensional transformations (7), we obtain the followingsystem of non-dimensional equations:

f ′′′ + ff ′′ − f ′2 + 1 + Ri [θ+NcS −Nrγ]

+ε

Da · Re(1− f ′) +

ε2Fs

Da

(1− f ′2) = 0 (9)

1

Prθ′′ + fθ′ +Nb · θ′γ′ +Nt · θ′2 = 0 (10)

γ′′ + Le · fγ′ + Nt

Nbθ′′ = 0 (11)

1

ScS′′ + fS′ + Srθ′′ = 0 (12)

where the primes indicate partial differentiation with re-spect toη alone; Gr =

[(1− ϕ∞) gβ (Tw − T∞)x3

]/ν2

is the local Grashof number; Re =[ue (x)x]/ν is the lo-cal Reynolds number; Ri = Gr/Re2 is the mixed-convection parameter;Fs = b/x is the local Forch-heimer number; Da =KP /x

2 is the local Darcy number;α = ν/αm is the Prandtl number; Sc =ν/DS is theSchmidt number; Le =αm/ϕDB is the Lewis number;Nc = [βC (Cw − C∞)]/[βT (Tw − T∞)] is the regularbuoyancy ration;Nr = [(ρp − ρf∞) (ϕw − ϕ∞)]/[ρf∞βT (1− ϕ∞) (Tw − T∞)] is the nanoparticlebuoyancy parameter;Nb = [JDB (ϕw − ϕ∞)]/ν is theBrownian motion parameter;Nt = (JDT /νT∞)× (Tw − T∞) is the thermophoresis parameter; and Sr =

[DCT (Tw − T∞)]/[ν (Cw − C∞)] is the Soret number.Thus, boundary conditions (6) in terms off , θ, γ, andSbecome

η = 0 : f (0) = 0, f ′(0) = λC , θ (0) = 1,

γ (0) = 1, S (0) = 1 (13a)

η→ ∞ : f ′ (∞) → 1, θ (∞) → 0, γ (∞) → 0,

S (∞) → 0 (13b)

whereλC = b/a is the stretching (>0) and shrinking(<0) parameter.

3. HEAT AND MASS TRANSFER COEFFICIENTS

The wall shear stresses acting on a surface in contactwith an ambient fluid of constant density are given byτw = µ (∂u/∂y)y=0 and the non-dimensional skin fric-tionCf = 2τw/ρu

2e is given by

Cf

√Rex = 2f ′′ (0) (14)

where Rex = [ue (x)x]/ν is the local Reynolds numberbased on the surface velocity. The local heat, nanoparticlemass, and regular mass fluxes from the vertical plate canbe obtained from

qw = −k

(∂T

∂y

)y=0

, qn = −DB

(∂ϕ

∂y

)y=0

,

qm = −DS

(∂C

∂y

)y=0

(15)

The dimensionless local Nusselt number Nux =qwx/[km (Tw − T∞)], local nanoparticle Sherwood num-ber NShx = qnx/[DB (ϕw − ϕ∞)], and regular Sher-wood number Shx = qmx/[DS (Cw − C∞)] are givenby

Nu

Re1/2x

= −θ′ (0) , NShxRe1/2x

= −γ′ (0) ,

ShxRe1/2x

= −S′ (0) (16)

The effects of the various parameters involved in the in-vestigation on these coefficients are discussed in Sec-tion 4.

4. RESULTS AND DISCUSSION

4.1 Numerical Method

Reduced flow governing Eqs. (9)–(12) are nonlinear,coupled, ordinary differential equations that possess no


234 RamReddy et al.

closed-form solution. Therefore, they must be solved nu-merically subject to the boundary conditions given byEq. (13). The implicit, iterative finite-difference methoddiscussed by Blottner (1970) has proven to be adequatefor the solution of these types of equations. For this rea-son, this method is employed in the present study. Equa-tions (9)–(12) are discretized using three-point central-difference quotients. This converts the differential equa-tions into linear sets of algebraic equations, which can bereadily solved by the well-known Thomas algorithm. Onthe other hand, Eqs. (9)–(12) are discretized and solvedsubject to the appropriate boundary conditions by thetrapezoidal rule. The computational domain in theη-direction was made up of 196 non-uniform grid points. Itis expected that most changes in the dependent variablesoccur in the region close to the plate where viscous ef-fects dominate. However, small changes in the dependentvariables are expected far away from the plate surface.For these reasons, variable step sizes in theη-direction areemployed. The initial step size∆η1 and growth factorK∗

employed, such that∆ηi+1 = K∗∆ηi (where subscripti indicates the grid location), were 10−3 and 1.0375, re-spectively. These values were found (by performing many

numerical experimentations) to give accurate and grid-independent solutions. The solution convergence criterionemployed in the present study was based on the differencebetween the values of the dependent variables at the cur-rent and previous iterations. When this difference reached10−5, the solution was assumed converged and the itera-tion process was terminated.

4.2 Numerical Validation

With Nb → 0, Nt = 0, ε = 0, Ri = 0, Le = 0, andS(η) → 0 (i.e., for a regular Newtonian fluid), Eqs. (9)–(12) governing the present investigation of a nanofluid-saturated non-Darcy porous medium reduces to the limit-ing case in Mahapatra and Gupta (2002), who investigatedthe boundary-layer stagnation-point flow past a stretch-ing/shrinking sheet in a viscous fluid in the absence ofSoret and non-Darcy effects. Also, the results found inthis study have been compared with Mahapatra and Gupta(2002) and other results and are found to be in good agree-ment, as shown in Tables 1 and 2. Therefore, the devel-oped code can be used with great confidence to study theproblem considered in this paper.

TABLE 1: Comparison of dimensionless similarity function−θ′(0) for the boundary-layer stagnation-point flow past astretching/shrinking sheet in a viscous fluid with the Mahap-atra and Gupta (2002) study by takingNb → 0, Nt = 0, ε =0, Ri= 0, Le= 0, andS(η) → 0

Pr−θ(0)

Mahapatra and Gupta (2002) Present Study0.05 0.178 0.1780.50 0.563 0.5631.00 0.796 0.7961.50 0.974 0.974

TABLE 2: Comparison of dimensionless similarity functionf ′′ (0) for the boundary-layer stagnation-point flow past a stretching/shrinking sheet in a viscous fluid with variousstudies by takingNb → 0,Nt = 0, ε = 0, Ri= 0, Le= 0, andS(η) → 0

λ

f ′′ (0)Mahapatra and Gupta Pop et al. Nazar et al. Layek et al. Present

(2002) (2004) (2004) (2007) Study0.1 −0.9694 −0.9694 −0.9694 −0.9696 −0.96940.2 −0.9181 −0.9181 −0.9181 −0.9182 −0.91810.5 −0.6673 −0.6673 −0.6673 −0.6672 −0.66732 0.20175 0.20174 0.20176 0.20175 0.201753 4.7293 4.729 4.7296 4.7293 4.7293



4.3 Effects of the Physical Parameters

4.3.1 Brownian Motion Parameter

The nanoparticle volume fraction is a key parameter inconvection in nanofluid-saturated porous media, and itwill have a significant effect on the flow field, temper-ature, and concentration distributions. Figure 2 repre-sents the effect of the Brownian motion parameterNb onthe velocity, temperature, concentration, and nanoparticlevolume fraction distributions for both aiding and oppos-ing flows. WhenNb = 0, there is no additional thermaltransport due to buoyancy effects created as a result ofnanoparticle concentration gradients. It is interesting to

note that an increase in the intensity of Brownian mo-tion parameterNb produces significant enhancement inthe fluid velocity within the momentum boundary layer,thus enhancing the fluid flow, as shown in Fig. 2(a) in thecase of aiding flow, and the reverse trend is observed inthe case of opposing flow. It is interesting to note that theBrownian motion of nanoparticles at the molecular andnanoscale levels is a key nanoscale mechanism governingtheir thermal behaviors. In nanofluid systems, due to thesize of the nanoparticles, Brownian motion takes place,which can affect the heat transfer properties. As the parti-cle size scale approaches the nanometer scale, the particleBrownian motion and its effect on the surrounding liquids

(a) (b)

(c) (d)

FIG. 2: Variation of the(a) velocity, (b) temperature,(c) concentration, and(d) nanoparticle volume fraction profileswith the Brownian motion parameter


236 RamReddy et al.

play an important role in the heat transfer. From Fig. 2(b),with the rise in the value of the Brownian motion parame-ter, enhancement in the temperature is observed, whereasa decrease in the concentration field is noticed, as de-picted in Fig. 2(c) for both aiding and opposing flows.From Fig. 2(d), we notice that the nanoparticle volumefraction is slightly decreased, with a gain in the values ofNb for both the aiding and opposing flow cases.

4.3.2 Thermophoresis Parameter

The effect of the thermophoresis parameterNt on the ve-locity, temperature, concentration, and nanoparticle vol-

ume fraction distributions is demonstrated in Fig. 3 forboth aiding and opposing flows. Figure 3(a) reveals thatfor aiding flow the momentum boundary-layer thicknessincreases with escalating values ofNt, whereas the op-posite orientation is noticed in the case of opposing flow.A similar trend is observed for the temperature, con-centration, and nanoparticle volume fraction profiles forboth the aiding and opposing flow cases when the ther-mophoresis parameter is increased. With an increase invalues of the thermophoresis parameter, an increase in thetemperature and concentration fields is noticed, as por-trayed in Figs. 3(b) and 3(c), respectively. Figure 3(d)shows that with an increase inNt, a depletion in the

(a) (b)

(c) (d)

FIG. 3: Variation of the(a) velocity, (b) temperature,(c) concentration, and(d) nanoparticle volume fraction profileswith the thermophoresis parameter



nanoparticle volume fraction distribution is noted. Here,Nt > 0 designates a cold surface, whileNt < 0 cor-responds to a hot surface. In the case of a hot surface,thermophoresis tends to blow the nanoparticle volumefraction away from the surface since a hot surface repelssubmicron-sized particles from it, thereby forming a rela-tive particle-free layer near the surface.

4.3.3 Varying the Stretching/Shrinking Parameter

The variation of the velocity, temperature, concentration,and nanoparticle volume fraction profiles for various val-ues of the stretching/shrinking parameter is demonstrated

in Fig. 4 for both aiding and opposing flows. In both theaiding and opposing flow cases, a similar trend is ob-served with increasing values of the stretching/shrinkingparameter. From Fig. 4(a), it is evident that an increasein the stretching or shrinking parameter significantly en-hances the momentum boundary-layer thickness. Thisis because in the presence of suction, a heated fluid ispushed toward the wall where the buoyancy forces canact to retard the fluid due to the high influence of theBrownian motion effects. This effect acts to decrease thewall shear stress. An increase in values of the stretch-ing/shrinking parameter resulting in a lower temperature,concentration, and nanoparticle volume fraction is shownin Figs. 4(b)–4(d), respectively.

(a) (b)

(c) (d)

FIG. 4: Variation of the(a) velocity, (b) temperature,(c) concentration, and(d) nanoparticle volume fraction profileswith the stretching/shrinking parameter


238 RamReddy et al.

4.3.4 Varying the Soret Number

Figure 5 shows the velocity, temperature, concentration,and nanoparticle volume fraction profiles for various val-ues of Soret number Sr for both the aiding and oppos-ing flow cases. The Soret number accounts for the ad-ditional mass diffusion due to the temperature gradients.Figure 5(a) show that higher Sr values result in a highervelocity field for aiding flow, but a reverse trend is ob-served for opposing flow. An increase in the Soret pa-rameter causes a significant decrease in the temperatureand concentration fields for aiding flow (increases foropposing flow) within the boundary layer, as shown inFigs. 5(b) and 5(c), respectively. Figure 5(d) reveals thatthe nanoparticle volume fraction is enhanced with an in-

crease in the Soret parameter for both aiding and opposingflows. The present analysis indicates that the flow field isappreciably influenced by the Soret parameter. This anal-ysis is in line with the observation made in RamReddy etal. (2013).

4.3.5 Varying the Forchheimer Parameter

The variation of velocity, temperature, concentration, andnanoparticle volume fraction profiles for different valuesof Forchheimer parameterFs is presented in Fig. 6 forboth aiding and opposing flows. It can be observed fromFig. 6(a) that the velocity of the fluid decreases for aid-ing flow (increases in the opposing flow case) with anincrease in the Forchheimer number. Since the Forch-

(a) (b)

(c) (d)

FIG. 5: Variation of the(a) velocity, (b) temperature,(c) concentration, and(d) nanoparticle volume fraction profileswith the Soret parameter



(a) (b)

(c) (d)

FIG. 6: Variation of the(a) velocity, (b) temperature,(c) concentration, and(d) nanoparticle volume fraction profileswith the non-Darcy parameter

heimer number represents the inertial drag, an increasein theFs value increases the resistance to the flow; there-fore, a decrease in the fluid velocity ensues. Here,Fs =0 represents the case where the flow is Darcian. The ve-locity is at its maximum in this case due to the total ab-sence of inertial drag. It can be noticed from Fig. 6(b)that the temperature of the fluid increases for the aidingflow (decreases for the opposing flow) with an increasein the Forchheimer number. An increase in theFs valueincreases the temperature values because as the fluid isdecelerated, energy is dissipated as heat and this servesto enhance the temperature in the boundary layer. Theconcentration and nanoparticle volume fraction distribu-tions increase for aiding flow (decrease for opposing flow)with an increase in the Forchheimer number, as shown in

Figs. 6(c) and 6(d), respectively. An increase in the non-Darcy parameter reduces the intensity of the flow but en-hances the thermal, concentration, and nanoparticle vol-ume fraction boundary-layer thicknesses.

4.3.6 Analysis of the Skin Friction, Heat, Mass, andNanoparticle Transfer Coefficients for DifferentValues of the Physical Parameters

The variation in the profiles of the local skin friction, heat,mass, and nanoparticle transfer coefficients against thestretching/shrinking parameter for different values of per-tinent parameters is presented graphically in Figs. 7–10for both the aiding and opposing flows.


240 RamReddy et al.

(a) (b)

FIG. 7: Variation of the local(a) skin friction and heat transfer coefficients and(b) mass and nanoparticle transfercoefficients with the Brownian motion parameter

(a) (b)

FIG. 8: Variation of the local(a) skin friction and heat transfer coefficients and(b) mass and nanoparticle transfercoefficients with the thermophoresis parameter

(a) (b)

FIG. 9: Variation of the local(a) skin friction and heat transfer coefficients and(b) mass and nanoparticle transfercoefficients with the Soret parameter



(a) (b)

FIG. 10: Variation of the local(a) skin friction and heat transfer coefficients and(b) mass and nanoparticle transfercoefficients with the non-Darcy parameter

The local skin friction, heat, mass, and nanoparticletransfer coefficients for various values of Brownian mo-tion parameterNb are presented in Fig. 7. From Fig. 7(a),we notice that the local skin friction coefficient is in-creased for aiding flow (decreased for opposing flow)with an increase inNb. From Fig. 7(a), it can be seenthat the local heat transfer coefficient is diminished witha hike in the values of the Brownian motion parameter forboth aiding and opposing flows. The higher values of theBrownian motion parameter results in higher local massand nanoparticle transfer coefficients for both aiding andopposing flows, as depicted in Fig. 7(b).

The variation of the local skin friction, heat, mass,and nanoparticle transfer coefficients for various valuesof thermophoresis parameterNt are presented in Fig. 8.From Fig. 8(a), we notice that the local skin friction co-efficient is enhanced for aiding flow (diminished for op-posing flow) with the rise ofNt. From Fig. 8(a), weobserve that the local heat transfer coefficient is dimin-ished with escalating values of the thermophoresis param-eter for both aiding and opposing flows. An increase inthermophoresis parameter significantly enhances the localmass transfer coefficient but reduces the local nanoparti-cle transfer coefficient for both aiding and opposing flows,as displayed in Fig. 8(b).

Figure 9 depicts the effect of the Soret parameter onthe local skin friction, heat, mass, and nanoparticle coef-ficients transfer rates for both aiding and opposing flows.For aiding flow, an increase in the values of the Soret pa-rameter enhances (reduces for opposing flow) the localskin friction, heat, and mass transfer coefficients but re-duces the local nanoparticle transfer coefficient for bothaiding and opposing flows, as shown in Figs. 9(a) and9(b), respectively.

The influence of the Forchheimer parameter on localskin friction, heat, mass, and nanoparticle transfer coef-ficients is depicted in Fig. 10. The local skin friction,heat, and mass transfer coefficients are increased for aid-ing flow (reduced for opposing flow) with the rising val-ues of the Forchheimer parameter, as shown in Figs. 10(a)and 10(b), respectively. HigherFs values results in higherlocal nanoparticle transfer coefficients for both aiding andopposing flows, as shown in Fig. 10(b).

5. CONCLUSIONS

The boundary-layer stagnation-point flow past a stretch-ing/shrinking sheet in a non-Darcy porous medium sat-urated with a nanofluid in the presence of the Soret ef-fect is analyzed. The wall is subjected to uniform tem-perature, concentration, and nanoparticle volume fractionconditions. Using dimensionless variables, the governingequations are transformed into a set of nonlinear parabolicequations, where a numerical solution has been presentedusing the implicit, iterative finite-difference method dis-cussed in Blottner (1970) for a wide range of parameters.Some important observations are given as follows:

• Higher values of the Soret parameter result in highervelocity distribution, skin friction, heat, and masstransfer rates for the aiding flow but a reverse trendis noticed for the opposing flow. Higher values ofthe Soret parameter result in a higher nanoparticlevolume fraction distribution but a lower nanoparti-cle mass transfer rate in both aiding and opposingflows. Furthermore, the effect of the Soret param-eter on the temperature and concentration distribu-tions within the boundary layers is negligible.


242 RamReddy et al.

• In the aiding flow case, an increase in non-Darcy pa-rameterFs decreases the velocity distribution andnanoparticle mass transfer rates, whereas it increasesthe temperature, concentration, nanoparticle volumefraction, skin friction, heat, and mass transfer rates.The opposite nature is found in the case of opposingflow.

• The velocity distribution, heat, mass, and nanopar-ticle mass transfer rates are higher but the temper-ature, concentration, nanoparticle volume fraction,and skin friction are lower in the stretching parame-ter case when compared to the shrinking parametercase in both aiding and opposing flows.

• The results also indicate that the presence of theSoret parameter on the stagnation-point flow pasta stretching/shrinking sheet in a non-Darcy porousmedium saturated with a nanofluid influences theflow, heat, mass, and nanoparticle volume fractionof the fluid flow.

ACKNOWLEDGMENT

The authors thank the anonymous reviewers for theirvaluable comments, which have improved the paper.

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SORET EFFECT ON STAGNATION-POINT FLOW PAST A … · 3Department of Mathematics, Aswan University, Faculty of Science, Aswan, 81528, Egypt 4 Mechanical Engineering Department, Prince