International Journal of Numerical Methods for Heat & Fluid FlowHeat transfer and fluid flow of a non-Newtonian nanofluid over an unsteadycontracting cylinder employing Buongiorno’s modelA. Mahdy A Chamkha

Article information:To cite this document:A. Mahdy A Chamkha , (2015),"Heat transfer and fluid flow of a non-Newtonian nanofluid over anunsteady contracting cylinder employing Buongiorno’s model", International Journal of NumericalMethods for Heat & Fluid Flow, Vol. 25 Iss 4 pp. 703 - 723Permanent link to this document:http://dx.doi.org/10.1108/HFF-04-2014-0093

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Heat transfer and fluid flow of anon-Newtonian nanofluid over anunsteady contracting cylinderemploying Buongiorno’s model

A. MahdyDepartment of Mathematics, South Valley University, Qena, Egypt, and

A. ChamkhaDepartment of Manufacturing Engineering, College of Technological Studies,

Shuweikh, Kuwait

AbstractPurpose – The purpose of this paper is to discuss a combined similarity-numerical approach that isused to study the unsteady two-dimensional flow of a non-Newtonian nanofluid over a contractingcylinder using Buongiorno’s model and the Casson fluid model that is used to characterize the non-Newtonian fluid behavior.Design/methodology/approach – Similarity transformations are employed to transform theunsteady Navier-Stokes partial differential equations into a system of ordinary differential equations.The transformed equations are then solved numerically by means of the very robust symboliccomputer algebra software MATLAB employing the routine bvpc45.Findings – The effect of increasing values of the Casson parameter is to suppress the velocity field (inabsolute sense), the temperature and concentration decrease as Casson parameter increase. The heatand mass transfer rates decrease with the increase of unsteadiness parameters and Brownian motionparameter. In addition, they increase as the Casson parameter and the thermophoresis parameterincrease.Originality/value – The problem is relatively original and represents a very important contributionto the field of non-Newtonian nanofluids.Keywords Unsteady flow, Buongiorno’s model, Contracting cylinder, Non-Newtonian nanofluidPaper type Research paper

Nomenclaturea(t) radius of cylindera0 positive constantA unsteadiness parameterC nanoparticle concentrationD Brownian diffusion coefficientD̂ thermophoretic diffusion

coefficientf dimensionless stream

functionfw Suction or injection

parameterjm surface mass fluxk thermal conductivity

Le Lewis numberNb Brownian motion parameterNt thermophoresis parameterNu Nusselt numberP pressurePr Prandtl numberqw surface heat fluxSh Sherwood numbert timeT temperature of the fluid(u,w) velocity components of the

fluid(r, z) coordinate axes

International Journal of NumericalMethods for Heat & Fluid Flow

Vol. 25 No. 4, 2015pp. 703-723

©Emerald Group Publishing Limited0961-5539

DOI 10.1108/HFF-04-2014-0093

Received 5 April 2014Revised 18 June 2014

Accepted 12 August 2014

The current issue and full text archive of this journal is available on Emerald Insight at:www.emeraldinsight.com/0961-5539.htm

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Greek symbolsα thermal diffusivityρ density of the fluidc Stream functionv kinematic viscosityg expansion/contraction

strengthθ dimensionless temperature

ϕ dimensionless concentrationβ non-Newtonian Casson

parameterη Similarity variableSubscriptsw Conditions at the wall∞ Conditions in the free

stream

IntroductionConventional heat transfer fluids, including oil, water and ethylene glycol mixture arepoor heat transfer fluids, since the thermal conductivity of these fluids plays animportant role on heat transfer coefficient between the heat transfer medium and theheat transfer surface. Therefore, numerous methods have been taken to improvethe thermal conductivity of these fluids by suspending nano/micro or large-sizedparticle materials in liquids. An innovative technique to improve heat transfer is byusing nano-scale particles in the base fluid. Nanotechnology has been widely used inindustry since materials with sizes of nanometers possess unique physical andchemical properties. Nano-scale particle added fluids are called as nanofluids. The word“nanofluid” coined by Choi (1995) describes a liquid suspension containing ultra-fineparticles (diameter o50 nm). The model for a nanaofluid including the effects ofBrownian motion and thermophoresis, introduced by Buongiorno (2006), was appliedby Kuznetsov and Nield (2010) to the classical problem studied by Pohlhausen, Kuikenand Bejan (Schmidt and Beckmann, 1930; Kuiken, 1968, 1969; Bejan, 1984), namelyconvective boundary layer flow past a vertical plate. In their pioneering paperKuznetsov and Nield (2010) employed boundary conditions on the nanoparticle fractionanalogous to those on the temperature. In this note the problem is revisited and aboundary condition that is more realistic physically is applied. It is no longer assumedthat one can control the value of the nanoparticle fraction at the wall, but rather thatthe nanoparticle fluxes at the wall is zero. This change necessitates a rescaling of theparameters that are involved.

The mechanics of non-Newtonian fluid flows present a special challenge toengineers, physicists and mathematicians. Because of the complexity of these fluids,there is not a single constitutive equation which exhibits all properties of suchnon-Newtonian fluids. In the process, a number of non-Newtonian fluid models havebeen proposed. The vast majority of non-Newtonian fluid are concerned of the types,e.g. like the power-law and Grade 2 or 3 Serdar and Dokuz (2006), Andersson andDandapat (1992), Sadeghy and Sharifi (2004), Hassanien (1996), Sajid et al. (2009). Thesesimple fluid models have the shortcomings that render results that are not inaccordance with the fluid flows in reality. Power-law fluids are by far the most widelyused model to express non-Newtonian behavior in fluids. The model predicts shearthinning and shear thickening behavior. However, it is inadequate in expressingnormal stress behavior as observed in die swelling and rod climbing behavior in somenon-Newtonian fluids. In order to obtain a thorough cognition of non-Newtonian fluidsand their various applications, it is necessary to study their flow behaviors. Due to theirapplication in industry and technology, few problems in fluid mechanics have enjoyedthe attention that has been accorded to the flow which involves non-Newtonianfluids (Cheng, 2012; Ellahi et al., 2012; Esmaeilnejad et al., 2014; Ellahi, 2013).

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The non-linearity can manifest itself in a variety of ways in many fields, such as food,drilling operations and bio-engineering. The Navier-Stokes theory is inadequate forsuch fluids, and no single constitutive equation is available in the literature whichexhibits the properties of all fluids.

Because of the complexity of these fluids, there is not a single constitutive equationwhich exhibits all properties of such non-Newtonian fluids. Thus, a number ofnon-Newtonian fluid models have been proposed. The Casson model is a well-knownrheological model for describing the non-Newtonian flow behavior of fluids with a yieldstress (Casson, 1959). The model was developed for viscous suspensions of cylindricalparticles (Reher et al., 1969). Regardless of the form or type of suspension, somefluids are particularly well described by this model because of their nonlinearyield-stress-pseudoplastic nature. Examples are blood (Cokelet et al., 1963), chocolate(Chevalley, 1991), xanthan gum solutions (Garcia-Ochoa and Casas, 1994). The Cassonmodel fits the flow data better than the more general Herschel-Bulkley model(Kirsanov and Remizov, 1999; Joye, 1998), which is a power-law formulation with yieldstress (Bird et al., 1960; Wilkinson, 1960). For chocolate and blood, the Casson model isthe preferred rheological model. It seems increasingly that the Casson model fits thenonlinear behavior of yield-stress-pseudoplastic fluids rather well and it has thereforegained in popularity since its introduction in 1959. It is relatively simple to use, and it isclosely related to the Bingham model (Bird et al., 1960; Wilkinson, 1960), which isvery widely used to describe the flow of slurries, suspensions, sludge and otherrheologically complex fluids (Churchill, 1988). The Casson fluid model is sometimesstated to fit rheological data better than general viscoplastic models for manymaterials (Mustafa et al., 2011). Boyd et al. (2007) investigated the Casson fluid flowfor the steady and oscillatory blood flow. Boundary layer flow of Casson fluid overdifferent geometries is considered by many authors in recent years. Nadeem et al.(2012) presented MHD flow of a Casson fluid over an exponentially shrinking sheet.Kumari et al. (2011) analyzed peristaltic pumping of a MHD Casson fluid in aninclined channel. Sreenadh et al. (2011) studied the flow of a Casson fluid through aninclined tube of nonuniform cross-section with multiple stenoses. Mernone andMazumdar (2002) discussed the peristaltic transport of a Casson fluid. Porwal andBadshah (2012) work on steady blood flow with Casson fluid along an inclined planeinfluenced by the gravity force. Mukhopadhyay et al. (2013) studied the unsteadytwo-dimensional flow of a non-Newtonian fluid over a stretching surface having aprescribed surface temperature, the Casson fluid model is used to characterize thenon-Newtonian fluid behavior. Rashidi et al. (2012, 2014) and Rashidi and Erfani(2011) considered the case of nanofluid. The present paper focussed on an unsteadyflow of a non-Newtonian nanofluid over a contracting cylinder taking into accountthe effects suction or injection. The Casson fluid model is used to characterize thenon-Newtonian fluid behavior. Similarity transformation is employed, and thereduced ordinary differential equations are solved numerically. The results of thisparametric study are shown graphically and the physical aspects of the problem arehighlighted and discussed.

Analysis of the problemConsider a two-dimensional laminar boundary layer flow of a non-Newtonian nanofluidover an unsteady infinite cylinder in contracting motion as illustrated in Figure 1. It isassumed that the diameter of the cylinder is a function of time with unsteady radiusaðtÞ ¼ a0

ffiffiffiffiffiffiffiffiffiffi1�gt

p, where γ is the constant of the expansion/contracting strength, t is the

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time and a0 is the positive constant. In addition, the rheological equation of state for anisotropic and incompressible flow of a Casson fluid is (Mustafa et al., 2011):

tij ¼2 mB þ Py=

ffiffiffiffiffi2p

p� �eij; p4pc

2 mB þ Py=ffiffiffiffiffiffiffi2pc

p� �eij; popc

(

Here,τij is the (i, j)th component of the stress tensor, τij¼ eijeij and eij are the (i,j)thcomponent of the deformation rate, π is the product of the component of deformationrate with itself, πc is a critical value of this product based on the non-Newtonian model,μB is plastic dynamic viscosity of the non-Newtonian fluid and Py is the yield stress ofthe fluid. So, if a shear stress less than the yield stress is applied to the fluid, it behaveslike a solid, whereas if a shear stress greater than yield stress is applied, it starts tomove. In cylindrical polar coordinates, r and z are measured in the radial and axialdirections, respectively, and based on the axisymmetric flow assumptions, and thatthere is no azimuthal velocity component the four field equations embodying theconservation of mass, momentum, thermal energy and nanoparticle volume friction canbe written as:

1r@ðruÞ@r

þ @w@z

¼ 0 (1)

@u@t

þ u@u@r

þ w@u@z

¼ � 1r@p@r

þ n 1þ 1b

� �@2u@r2

þ 1r@u@r

þ @2u@z2

� ur2

� �(2)

@w@t

þ u@w@r

þ w@w@z

¼ � 1r@p@z

þ n 1 þ 1b

� �@2w@r2

þ 1r@w@r

þ @2w@z2

� �(3)

@T@t

þ u@T@r

þ w@T@z

¼ a@2T@r2

þ 1r@T@r

þ @2T@z2

� �

þ ~t D@C@r

@T@r

þ @C@z

@T@z

þ D̂T1

@T@r

� �2

þ @T@z

� �2 ! !

(4)

a(t )

w

w

u

r

zFigure 1.Physical model andcoordinate system

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@C@t

þ u@C@r

þ w@C@z

¼ D@2C@r2

þ 1r@C@r

þ @2C@z2

� �þ D̂

T1

@2T@r2

þ 1r@T@r

þ @2T@z2

� �(5)

Equations (1)-(5) must be solved subject to the following boundary conditions:

to0: u ¼ w ¼ 0; T ¼ T1; C ¼ C1 for all r; z

t⩾0: u ¼ uw; w ¼ � 1a20

4nz1�gt

; T ¼ Tw; D@C@r

þ D̂T1

@T@r

¼ 0 at r ¼ a tð Þ (6)

w-0; T-T1; C-C1 as r-1here, (u,w) are the velocity components in (r, z) directions, respectively, ρ isthe density of the fluid, b is the non-Newtonian Casson parameter, v is thekinematic viscosity, and α is the thermal diffusivity of the fluid, ~t ¼ ðrcpÞ=ðrcpÞf isthe ratio between the effective heat capacity of the nanoparticle material and heatcapacity of the fluid, Dand D̂ are the Brownian and thermophretic diffusioncoefficients. T is the temperature of the fluid inside the thermal boundary layer,P is the pressure. An unsteady state flow is considered. Equation (6) is a statementthat, with thermophoresis taken into account, the normal flux of nanoparticlesis zero at the boundary (Pakravan and Yaghoubi, 2013; Sheikhzadeh et al.,2013).

A majority of the existing exact solutions in fluid mechanics are similaritysolutions which reduce the number of independent variables by one or more.The methods for generating similarity transformations for equations of physicalinterest are discussed by Ames (1965). Similarity solutions are often asymptoticsolutions to a given problem and may have utility in this area of limitingsolutions. Similarity solutions may be used to gain physical insight into thesedetails of complex fluid flows and these solutions exhibit most of the characteristicas well as the influence of the physical and thermal parameters of the actualproblem. In order to get a similarity solution of the problem we define the followingtransformations:

u ¼ � 1a0

2nffiffiffiffiffiffiffiffiffiffi1�gt

p f ðZÞffiffiffiZ

p ; w ¼ 1a20

4nz1�gt

f 0ðZÞ; Z ¼ ra0

� �2 11�gt

yðZÞ ¼ T�T1Tw�T1

; fðZÞ ¼ C�C1Cw�C1

(7)

It is clear that, Equation (1) is satisfied automatically and since there is no longitudinalpressure gradient. Furthermore, Equations (2)-(5) reduce to the following ordinarydifferential equations:

1 þ 1b

� �Z f 000 þ f 00 þ f f 00�f 02�A f 0 þ Z f 00 ¼ 0 (8)

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1Pr

Zy00 þ y0 þ fy0�AZy0 þ ZNt f0y0 þ Nb y02� � ¼ 0 (9)

Zf00 þ f0� � þ Le ff0�AZf0� � þ NtNb

Zy00 þ y0� � ¼ 0 (10)

and the boundary conditions (6) become:

f ð1Þ ¼ f w; f 0ð1Þ ¼ �1 yð1Þ ¼ 1; Nbf0ð1Þ þ Nty0ð1Þ ¼ 0

Z-1: f 0ðZÞ-0; yðZÞ-0; fðZÞ-0 (11)

the prime represents differentiation with respect to η, and the five dimensionlessparameters in Equations (8)-(10) are:

Pr ¼ na

A ¼ a20g4n

Nb ¼ ~tDðCw�C1Þn

f w ¼ uw

ffiffiffiffiffiffiffiffiffiffi1�gtna0

sLe ¼ n

D

Nt ¼ ~tD̂ðTw�T1ÞnT1

where Pr is the Prandtl number, A is the unsteadiness parameter, Nb is the Brownianmotion parameter, fw is the suction or blowing parameter, Le is the Lewis number andNt is the thermophoresis parameter. The constantA is the unsteadiness parameter for acontracting cylinder, displays the strength of the contraction. The Brownian motionparameter Nb can be observed as random drifting of suspended nanoparticles. Thethermophoresis parameter Nt represents the nanoparticle migration due to imposedtemperature gradient across the fluid. For hot surfaces, due to repelling the submicronsized particles, the thermophoresis tends to blow nanoparticle volume fractionboundary layer away from the surface (Malvandi, 2013). In addition, the pressure canbe obtained from Equation (2) as:

Pr¼ const: þ n 1 þ 1

b

� �@u@r

þ ur

� ��12u2 þ

Z@u@tdr (12)

The physical quantities of interest are the Nusselt number and Sherwood number,which are defined as:

Nu ¼ a0ffiffiffiffiffiffiffiffiffiffi1�gt

pqw

2kðTw�T1Þ; Sh ¼ a0ffiffiffiffiffiffiffiffiffiffi1�gt

pjm

2DðCw�C1Þ (13)

where the surface heat flux qw and the surface mass flux jm are given by:

qw ¼ �k@T@r

� �r¼aðtÞ

¼ �2kðTw�T1Þa0

ffiffiffiffiffiffiffiffiffiffi1�gt

p y0ð1Þ

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jm ¼ �D@C@r

� �r¼aðtÞ

¼ �2DðCw�C1Þa0

ffiffiffiffiffiffiffiffiffiffi1�gt

p f0ð1Þ

with k being the nanofluid thermal conductivity. Using variables (7), we get:

Nu ¼ �y0ð1Þ; Sh ¼ �f0ð1Þ (14)

Numerical techniqueThe set of (8)-(10) is highly nonlinear and coupled and cannot be solved analytically.The nonlinear system considering of (8)-(10) with boundary conditions (11) form a twopoint boundary value problem (BVP) and are solved using the routine bvpc45 of thesymbolic computer algebra software MATLAB, by converting into an initial valueproblem (IVP). In this method we have to choose a finite value of the boundary η→∞,say ηfinite. Care has been taken to choose the suitable value of ηfinite for a given set ofparameters, typically η¼ 8.

We construct the following first order differential equations:

f 0 ¼ p1; p01 ¼ p2;

p02 ¼ p21�f p2 þ Ap1 þ Zp2�1 þ b�1p2=Z1 þ b�1 (15)

y0 ¼ p3; p03 ¼ PrAZp3 � f p3 � ZNtp4p3 þ Nbp23 � p3=Z (16)

f0 ¼ p4; p04 ¼ LeAZp4 � f p4 �NtNb

Zp03 þ p3 � p4

� �=Z (17)

f ð1Þ ¼ f w; p1ð1Þ ¼ �1; yð1Þ ¼ 1; Nbp4ð1Þ þ Ntp3ð1Þ ¼ 0; p1ð1Þ ¼ 0 (18)

To solve (15)-(17) as an IVP we need values of p2(1), p3(1) and p4(1). From theseequations we see that no such values are given. In MATLAB routine bvpc45 we need toguess initial values of to obtain the solution of p2(1), p3(1) and p4(1) satisfying the initialconditions (18).

Results and discussionThe obtained similarity system (8)-(11) is non-linear, coupled, ordinary differentialequations, which possess no closed-form solution. Therefore, the system of Equations(8)-(10), along with the boundary conditions (11) are solved numerically by means thevery robust symbolic computer algebra software MATLAB employing the routinebvpc45. In order to get a clear insight of the behavior of velocity, temperature andconcentration fields for non-Newtonian Casson fluid, a comprehensive numericalcomputation is carried out for various values of the parameters that describe the flowcharacteristics, and the results are reported graphically. For studying anyparameter the other parameters are chosen to be Pr¼ 7.0, Le¼ 3.0, Nb¼ 0.7,Nt¼ 0.5, fw¼ (−0.2, 0.2), β¼ 1.8 and A¼−2.2.

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Figures 2-4 exhibit the velocity, temperature and nanoparticle concentrationdistributions, respectively for several values of unsteadiness parameter A consideringtwo cases namely, suction and blowing. It is observed that the velocity along the tubesurface decreases initially with the increase in unsteadiness parameter A (absolutesense), and this implies an accompanying reduction of the thickness of the momentumboundary layer near the wall. Furthermore, it is noticed that the temperature andconcentration at a particular point is found to increase significantly with increasingvalues of the unsteadiness parameter. This is due to the fact that higher values of Acorrespond to higher cylinder diameters and this will cause the fluid velocity todecrease while the thicknesses of the temperature and concentration profiles increaseyielding enhanced temperature and concentration distribution asA increases. Figures 5and 6 represent the effect of the Brownian motion parameter on the temperature andnanoparticle concentration for the selected values of other parameters. When Nb¼ 0,there is no additional thermal transport due to buoyancy effects created as a result ofnanoparticle concentration gradients. The Brownian motion parameter increases thethermal boundary layer thickness and in consequence decreases the temperaturegradient at the tube surface. This result is consistent with the result illustrated inFigure 7. On the other hand, increases the Brownian motion parameter tends todecrease slightly the nanoparticle concentration and the peak of concentrationnear the tube surface, Figure 6. Moreover, the concentration gradient (masstransfer rate) (in absolute sense) decreases as Nb increases as it is observed fromFigure 8.

1 2 3 4 5 6 7 8−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

η

f′

A = −2.8, −2.7, −2.6, −2.5, −2.4, −2.3, −2.2, −2.1, −2.0, −1.9

Notes: ⎯, Suction; ..., blowing

Figure 2.Effect of A ondimensionlessvelocity

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1 1.2 1.4 1.6 1.8 2 2.2 2.4−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

η

φ

A = −2.8, −2.7, −2.6, −2.5, −2.4, −2.3, −2.2, −2.1, −2.0, −1.9

Notes: ⎯, Suction; ..., blowing

Figure 4.Effect of A ondimensionlessnanoparticleconcentration

1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ

A = −2.8, −2.7, −2.6, −2.5, −2.4, −2.3, −2.2, −2.1, −2.0, −1.9

Notes: ⎯, Suction; ..., blowing

Figure 3.Effect of A ondimensionlesstemperature

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1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θNb = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2

Notes: ⎯, Suction; ..., blowing

Figure 5.Effect of Nb ondimensionlesstemperature

1 1.2 1.4 1.6 1.8 2 2.2 2.4−0.1

0

0.1

0.2

0.3

0.4

0.5

η

φ

Nb = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2

Notes: ⎯, Suction; ..., blowing

Figure 6.Effect of Nb ondimensionlessnanoparticleconcentration

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Variation of non-dimensional temperature, nanoparticle concentration distributionagainst the similarity variable η, is represented, respectively, in Figures 9 and 10 fordifferent values of thermophoresis parameter Nt. From these figures we observe thatthe temperature distribution decreases with the increase of thermphoresis parameterFigure 9, whereas the nanoparticle concentration increases, Figure 10. Influences ofnon-Newtonian Casson parameter β on non-dimensional velocity, temperature andconcentration distributions for unsteady motion are clearly depicted in Figures 11-13,respectively, considering wall mass suction and wall mass injection effects (i.e. fw¼ 0.2,−0.2). The same type of behavior of velocity with increasing β is noted. The effect ofincreasing values of β is to reduce the velocity (in absolute sense), and hence, theboundary layer thickness decreases. The increasing values of the Casson parameter,

−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.20

5

10

15

20

25

30

A

−θ′ (

1)

Nb = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2

Notes: ⎯, Suction; ..., blowing

Figure 7.Effect of Nb ondimensionless

heat transfer rates

−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2−35

−30

−25

−20

−15

−10

−5

0

A

Nb = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2

Notes: ⎯, Suction; ..., blowing

−φ′ (

1)

Figure 8.Effect of Nb ondimensionlessconcentrationtransfer rates

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1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ

Nt = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0

Notes: ⎯, Suction; ..., blowing

Figure 9.Effect of Nt ondimensionlesstemperature

1 1.5 2 2.5 3

−0.1

0

0.1

0.2

0.3

0.4

0.5

η

φ

Nt = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0

Notes: ⎯, Suction; ..., blowing

Figure 10.Effect of Nt ondimensionlessnanoparticleconcentration

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1 2 3 4 5 6

−1

−0.8

−0.6

−0.4

−0.2

0

η

f′ β = 0.4, 0.7, 1.0,1.6, 2.5, 3.0, 4.0, 5.0

Notes: ⎯, Suction; ..., blowing

Figure 11.Effect of β ondimensionless

velocity

1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

1.143 1.144 1.145 1.146 1.147 1.148

0.432

0.433

0.434

0.435

0.436

0.437

0.438

0.439

η

θ

1.152 1.154 1.156 1.158 1.16 1.162

0.456

0.457

0.458

0.459

0.46

0.461

0.462

0.463

0.464

0.465

η

η

θ

β = 0.4, 0.7, 1.0,1.6, 2.5, 3.0, 4.0, 5.0

Notes: ⎯, Suction; ..., blowing

Figure 12.Effect of β ondimensionlesstemperature

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i.e. the decreasing yield stress (the fluid behaves as Newtonian fluid as Cassonparameter becomes large) suppress the velocity field. It is observed that f'(η) and theassociated boundary layer thickness are decreasing function of β. The velocity curvesin Figure 11 show that the rate of transport is considerably reduced with the increase ofβ. The effect of increasing β leads to decrease the temperature and concentration fieldsfor unsteady motion (Figures 12 and 13). This effect is more pronounced for steadymotion. The thickening of the thermal boundary layer occurs due to decrease in theelasticity stress parameter.

The variation of the temperature distributions for the variation of Prandtl number isillustrated in Figure 14. Prandtl number signifies the ratio of momentum diffusivity tothermal diffusivity. It is seen that the temperature decreases with increasing Pr.Furthermore, the thermal boundary layer thickness decreases sharply by increasingPrandtl number. The temperature gradient at surface is negative for all values ofPrandtl number as seen from Figure 15 which means that the heat is alwaystransferred from the surface to the ambient fluid. Fluids with lower Prandtlnumber will possess higher thermal conductivities (and thicker thermal boundarylayer structures), so that heat can diffuse from the surface faster than for higher Prfluids (thinner boundary layers). Physically, if Pr increases the thermal diffusivitydecreases, this phenomenon leads to decrease energy transfer ability that reduces thethermal boundary layer. The effect of Lewis number on nanoparticle concentration issimilar to the effect of Prandtl number on temperature profiles as appears in Figure 16,that is concentration profiles decreases as the Lewis number increases. Furthermore,Figure 17 shows that an increase in Lewis number tends to decrease the mass

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8−0.05

0

0.05

0.1

0.15

0.2

0.25

η

φ

β = 0.4, 0.7, 1.0,1.6, 2.5, 3.0, 4.0, 5.0

Notes: ⎯, Suction; ..., blowing

Figure 13.Effect of β ondimensionlessnanoparticleconcentration

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transfer rate (absolute sense). The effects of suction/blowing parameter fw on velocityfield are presented in Figure 18. With increasing suction (fwW0), fluid velocityat a point is found to decrease. That is, the effect of suction is to decrease thefluid velocity in the boundary layer and in turn, the wall shear stress decreases.

1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ

Pr = 0.3, 0.72, 1.3, 2.0, 6.2, 7.0, 10.0, 20.0

Notes: ⎯, Suction; ..., blowing

Figure 14.Effect of Pr ondimensionlesstemperature

−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.21

2

3

4

5

6

7

8

9

A

−θ′ (

1)

Pr = 0.3, 0.72, 1.3, 2.0, 6.2, 7.0, 10.0

Notes: ⎯, Suction; ..., blowing

Figure 15.Effect of Pr on

dimensionless heattransfer rates

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The increase in suction causes thinning of the boundary-layer. However, the injection(fwo0) has the opposite effect. The variation in dimensionless heat and masstransfer rates vs. unsteadiness parameter is shown in Figures 19 and 20 forvarious values of Nt parameter. The thermophoresis parameter Nt represents the

1 1.5 2 2.5 3

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

η

φ

Le = 1, 2, 3, 4, 6, 8, 10, 20, 50

Notes: ⎯, Suction; ..., blowing

Figure 16.Effect of Le ondimensionlessnanoparticleconcentration

−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

A

−φ′ (

1)

Le = 1.0, 2.0, 3.0, 4.0, 6.0, 8.0, 10.0, 20.0, 50.0

Notes: ⎯, Suction; ..., blowing

Figure 17.Effect of Le ondimensionlessconcentrationtransfer rates

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nanoparticle migration due to imposed temperature gradient across the fluid. As it isobserved, both of heat and mass transfer rates increase with increasing Nt parameter(in absolute sense). The same effect of Casson parameter occurs as displayed inFigures 21 and 22.

1 2 3 4 5 6

−1

−0.8

−0.6

−0.4

−0.2

0

η

f′

S = 0.0, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7

fw = −0.7, −0.6, −0.5, −0.4, −0.3, −0.2

Notes: ⎯, Suction; ..., blowing

Figure 18.Effect of fw ondimensionless

velocity

−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.20

5

10

15

20

25

30

35

40

45

A

Nt = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0

Notes: ⎯, Suction; ..., blowing

−θ′ (

1)

Figure 19.Effect of Nt on

dimensionless heattransfer rates

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ConclusionsA numerical investigation is carried out to analyze the mechanical and thermalproperties of unsteady boundary layer flow of a non-Newtonian Casson nanofluid pasta contracting cylinder taking into account the wall mass suction or injection. With thehelp of appropriate similarity transformations, the governing time-dependentboundary layer equations for momentum, thermal energy and nanoparticle volume

−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2−70

−60

−50

−40

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0

A

Nt = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0

Notes: ⎯, Suction; ..., blowing

−φ′

(1)

Figure 20.Effect of Nt ondimensionlessconcentrationtransfer rates

−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.24

4.5

5

5.5

6

6.5

7

7.5

8

A

−θ′ (

1) −2.9

4

−2.9

35−2

.93

−2.9

25−2

.92

6.78

6.8

6.82

6.84

6.86

β = 0.4, 0.7, 1.0, 1.6, 2.5, 3.0, 4.0, 5.0 10

Notes: ⎯, Suction; ..., blowing

Figure 21.Effect of β ondimensionless heattransfer rates

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friction are reduced to coupled non-linear ordinary differential equations which arethen solved numerically. As a summary, we can conclude that:

• The fluid velocity decreases initially due to the increase in the unsteadinessparameter; whereas the temperature and the concentration increase significantlyin this case.

• The effect of increasing values of the Casson parameter is found to suppress thevelocity field (in absolute sense), the temperature and concentration decrease asthe Casson parameter increases.

• Both of −θ'(1) and −ϕ'(1) (in absolute sense) decrease with the increase of theunsteadiness parameters and the Brownian motion parameter. In addition, −θ'(1)and −ϕ'(1) (in absolute sense) increase as the Casson parameter and thethermophoresis parameter increase.

• The Prandtl number can be used to increase the rate of cooling in conductingflows.

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−3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2−5.5

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−4.5

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−3.5

−3

A

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Corresponding authorProfessor A. Chamkha can be contacted at: achamkha@yahoo.com

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