Scientia Iranica B (2017) 24(2), 637{647

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

MHD mixed convection boundary layer ow of aCasson uid bounded by permeable shrinking sheetwith exponential variation

S.S.P.M. Isaa;1;�, N.M. Ari�nb, R. Nazarc, N. Bachokb, F.M. Alib and I. Popd

a. Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia.b. Department of Mathematics, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia.c. School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi,

Selangor, Malaysia.d. Department of Mathematics, Babe�s-Bolyai University, R-400084 Cluj-Napoca, Romania.

Received 20 April 2015; received in revised form 14 December 2015; accepted 9 May 2016

KEYWORDSCasson uid;MHD;Mixed convection;Exponentiallyshrinking sheet.

Abstract. A review was carried out on the exponentially permeable shrinking sheetand how it in uenced the magnetohydrodynamic (MHD) mixed convection boundary layer ow of a Casson uid. The boundary layer equations in the form of partial di�erentialequations were transformed into the ordinary di�erential equations by using the similaritytransformation. Subsequently, shooting technique is used to provide solutions for theordinary di�erential equations. Di�erent factors related to the ow and heat are indicatedby the attained results as well as graphs. Moreover, 4 solutions are presented graphically.Also, the numerical calculations exhibit that the Casson uid parameter, ", buoyancyparameter, �, and suction parameter, s, would signi�cantly a�ect the characteristics of ow and thermal boundary layers of a Casson uid.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

There is great bene�t to the industry by exploringthe convection features of non-Newtonian uids, suchas liquid-liquid extractors [1], catalytic reactors [2],blood plasmaphosresis devices [3], and the �ltrationdevices [4]. The Casson uid is commoly known in themidst of non-Newtonian uids, which was introducedby Casson [5]. This uid is a good approximationof many substances, such as blood, molten chocolate,foams, and cosmetics, among others. As a consequence,numerous researchers have investigated the features

1. Present address: Centre of Foundation Studies forAgricultural Science, Universiti Putra Malaysia, 43400UPM Serdang, Selangor, Malaysia.

*. Corresponding author. Tel.: +603-8946 6998;Fax: +603-8946 6997E-mail address: suzi isa@yahoo.com (S.S.P.M. Isa)

of Casson uid ow ([6-9] etc.). Besides, Boyd etal. [10] studied two types of ow, namely, steadyand oscillatory ows, in straight and curved pipegeometries. In their research, Boyd et al. [10] usedthe Carreau-Yasuda and Casson uids.

There are various bene�ts to the engineering pro-cess by studying boundary layer ow and transmissionof heat instigated by a shrinking sheet, for exampleglass casting, wire drawing, glass �bre manufacturing,and removal of polymer sheet. Goldstein [11] men-tioned that the ow of uid caused by coiling sheetwas fundamentally a rearward ow, which was thereverse of the ow caused by an extending sheet. Astudy by Miklav�ci�c and Wang [12] found that analyticalsolution to two-dimensional viscous ows was dueto the shrinking sheet with suction; the researchersreported that the ow over the shrinking sheet wassustained by the suction. Mahapatra and Nandy [13]

638 S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647

agreed with study results of Miklav�ci�c and Wang [12]who came to the conclusion that the steady ow asa result of the shrinking sheet was unable to avoidthe impact of suction. An evaluative study examinedthe two-dimensional ow together with heat transfer ofviscous uid towards a shrinking sheet was performedby Wang [14]. He proved that the solutions obtainedwere non-unique for insigni�cant value of the shrinkingfactor.

Magyari and Keller [15] were the pioneers whostudied the ow and thermal boundary layers con�nedby an exponential stretching sheet. However, in thecase related to a shrinking sheet with exponential veloc-ity, Bhattacharyya [16] was the �rst person to explorethe related �eld. His study results demonstrated thepresence of twofold solutions in the suction. Someof the latest dual nature studies have involved expo-nentially stretching/shrinking sheet [17,18]. However,recent studies have been reported on the boundarylayer ow and the transfer of heat in non-Newtonian uid towards a stretching/shrinking sheet, which in-volves the concept of magnetohydrodynamic (MHD)stagnation-point ow [19,20] and unsteady state [21-23].

Recently, the presence of multiple solutions hasbeen con�rmed by certain researchers [24-29]. A recentstudy by Rosca and Pop [29] came up with engagingfacts to provide evidence for the assumption of dualnumerical solutions. The researchers [29] carried out astability examination and reported that the solutionsfound on the upper branch (primary solution) werelinearly stable, whereas those on the lower branch (sec-ondary solution) were linearly unstable. The physicallyrelevant solution was related to the steady solution.The discovery of triple solutions is quite new and ofinterest. Therefore, in the event that Newtonian uidis able to pass through the shrinking sheet with anincreasingly exponential velocity, there exists forma-tion of the triple solution because of the impact of the ow's buoyancy force along with traits associated withthe ow and heat transfer [30]. Subsequently, Rohniet al. [31] proved that triple solutions also occur whenthe stagnation point is implemented in the system ofNewtonian uid. Recently, an extension of the workby Rohni et al. [30] was carried out by Isa et al. [32].Accordingly, the researchers [32] were able to get thequad solutions because of the addition of a magnetic�eld in the uid ow's system developed by Rohni etal. [30].

The theoretical study in the Casson uid modelis reportedly based on the features of the ow and heattransmission. A unique solution is illustrated for linearstretching sheet case, whereas dual solutions exist forthe problem of magnetohydrodynamic linear shrinkingsheet [33-36]. MHD Casson uid ows in a system ofthree-dimensional coordinates related to the stretching

sheet in linear velocity, as reported by Nadeem etal. [37,38]. A numerical evaluation was carried out onthe challenges associated with using the Casson uid ow in a non-Darcy porous medium commencing at ahorizontal circular cylinder surrounded by partial slipscondition [39-40]. The Casson uid ow over a sheet,which is stretched with exponential variation, is takeninto account by Pramanik [41], Raju et al. [42], andAnimasaun et al. [43]. The permeable shrinking sheet,which has an exponential velocity and contributes asigni�cant in uence on the characteristics of MHDCasson uid ow, has been analyzed by Nadeem etal. [44]. However, they only analyzed the featuresof Casson ow, and heat transfer was not taken intoaccount. Later, Haq et al. [45] used the same modelas that of Nadeem et al. [44], but with the additionale�ect of a convective boundary condition.

Inspired by the previous research, this studyexpands the work by Isa et al. [32] in an e�ort too�er replacement of the viscous uid with the non-Newtonian Casson uid. This study is also an exten-sion of the study by Nadeem et al. [44] by includingan analysis of heat transfer characteristics in Casson ow over a sheet, which has an exponential shrinkingrate. The results obtained by the present study revealthat the temperature and ow �elds are controlled byCasson uid parameter, ", mixed convection/buoyancyparameter, �, and suction parameter, s.

2. Mathematical formulation

Figure 1 illustrates the steady, laminar, two-dimensi-onal boundary layer ow of an incompressible Cas-son uid over an exponentially shrinking sheet. Theuniform temperature of the ambient uid is T1 andthe surface temperature is Tw. Assisting ow case is

Figure 1. Physical model and coordinate system of aCasson uid.

S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647 639

denoted by Tw > T1, whereas Tw < T1 correspondsto the opposing ow case. There is application of atransverse magnetic �eld, B(x), in a tangential lineperpendicular to the plane y = 0. Nakamura andSawada [46] studied the status of an isotropic Casson uid along with an incompressible one in a rheologicalequation. This equation is shown below:

�ij =

8>><>>:2��B + pyp

2�

�eij ; � > �c

2��B + pyp

2�c

�eij ; � < �c

(1)

where �B is plastic dynamic viscosity of the non-Newtonian uid, py is the yield stress of uid, � =eijeij and eij are the (i; j)ih components of the defor-mation rate, � is the square of the component of thedeformation rate, and �c is a critical value of � basedon the non-Newtonian model. From Eq. (1), plasticdynamic viscosity is �B = 1=2(�ij=eij) � py=

p2�,

kinematic viscosity is � = �B=� where � is the uiddensity, and Casson parameter is formulated as =�B(p

2�c=py).The governing boundary layer equations can be

written as:

@u@x

+@v@y

= 0; (2)

u@u@x

+ v@u@y

=��

1 +1

�@2u@y2 + g� (T � T1)

� �B2(x)�

u; (3)

u@T@x

+ v@T@y

= �@2T@y2 ; (4)

where u and v are the velocity elements in the xand y directions, respectively, g is the accelerationdue to gravity, � is thermal expansion coe�cient,� is electrical conductivity (assumed constant), � isthermal di�usivity, and T is the uid temperature. InEq. (3), we choose the magnetic �eld B(x) = B0e(x=2L),where B0 is the constant magnetic �eld. Therefore, therelated boundary conditions are:

u = uw(x) = �Uwe(x=L); v = vw(x);

Tw(x) = T1 + T0e(2x=L) at y = 0;

u! 0 T ! T1 as y !1; (5)

where Uw > 0 is the shrinking velocity and vw(x) < 0is for suction.

Non-dimensional variables used in this problemare:

� = y�Uw2�L

�1=2

e(x=2L);

= (2�LUw)1=2f(�)e(x=2L);

�(�) =T � T1Tw � T1 ; (6)

where stream function, , is de�ned as u = @=@y andv = �@=@x.

Substituting Eq. (6) into Eqs. (2)-(4), the gov-erning equations are obtained using the dimensionlessfunctions f(�) and �(�) as follows:�

1 +1

�f 000 + ff 00 � 2 (f 0)2 �M2f 0 + 2�� = 0; (7)

1Pr�00 + f�0 � 4f 0� = 0: (8)

The boundary conditions are reduced to:

f 0(�)=�1; f(�)=s; �(�)=1 at �=0;

f 0(�)! 0; �(�)! 0; as � !1; (9)

where primes denote di�erentiation with respect to�, M2 = 2�B2

0L=�Uw is the Hartman number (alsode�ned as magnetic �eld parameter), Pr = �=� is thePrandtl number, s = �(2L=�Uw)1=2(1=ex=2L)vw(x) >0 is the suction parameter, and � = Gr=Re2 is theconstant mixed convection parameter, where Gr =g�TT0L3=�2 is the Grashof number and Re = UwL=�is the Reynolds number. It is mainly noticeable that� > 0, � < 0, and � = 0 refer to the assisting ow,opposing ow, and forced convection ow, respectively.

This paper will focus on the following physicalquantities: the skin friction coe�cient Cf as well aslocal Nusselt number Nux:

Cf =�w

�u2w(x)

; Nux=L

Tw�T1��@T@y

�y=0

: (10)

Substituting Eq. (6) into Eq. (10), we get:

Cf (2Rex)1=2e(�3x=2L) =�

1 +1

�f 00(0);

Nux�

2Rex

�1=2

e(�x=2L) = ��0(0); (11)

where Rex = xuw(x)=� is a local Reynolds number. Itis meaningful to emphasize that the problem of non-Newtonian Casson uid can be reduced to the case ofNewtonian uid when !1.

640 S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647

2.1. Numerical method for solutionThe shooting technique gives the solutions for theboundary value problems in Eqs. (7), (8), and (9) bychanging them into an initial value problem. Thefourth-order Runge-Kutta integration scheme is ap-plied to solve the initial value problem. We set:

f 0 = fp; fp0 = fpp;

fpp0 =�f(fpp) + 2(fp)2 +M2fp� 2��

(1 + 1= ); (12)

�0 = �p; �p0 = �Pr(f�p) + 4Pr(fp�); (13)

with the boundary conditions:

fp(0) = �1; f(0) = s; �(0) = 1;

f 0(�)! 0 �(�)! 0 as � !1: (14)

Firstly, the values of the physical parameters includedin Eqs. (12)-(14) are �xed. In order to carry out theintegration in Eqs. (12) and (13) as an initial valueproblem, we assume initial values for fpp(0), i.e. f 00(0)and �p(0), i.e. �0(0). We also assume a suitable �nitevalue for � !1 say, �1. Then, we calculate the valuesof f 0(�1) and �(�1) by adjusting the assumed values off 00(0) and �0(0) to give a better approximation of thesolution. Basically, the assumed values of �1, f 00(0)and �0(0) satisfy the boundary conditions in Eq. (14)(f 0(�1) = 0 and �(�1) = 0). This steps is repeatedwith higher values of �1 until 2 consecutive values off 00(0) and �0(0) vary signi�cantly by a speci�ed value.Finally, the most appropriate value of �1 is selected,which is the largest �1 that satis�es the solution.

3. Results and discussion

A shooting method is used to solve the nonlinearordinary di�erential equations (7) and (8), subject toboundary condition (9). Since the problem formulationcan be transformed to the case of Newtonian uid bysetting ! 1, the accuracy of the applied numericalmethod is tested by a direct comparison of the valuesof the skin friction coe�cient when = 100000, withthose reported by Isa et al. [32] for = 0. Theresults are presented in Table 1. It is evident that thevalues obtained for the skin friction coe�cient agreewith values attained by Isa et al. [32]. In conclusion,our numerical method presented here could be usedwith great con�dence for further investigation. Toillustrate the ow and heat transfer characteristicsof the opposing ow case, the variations of the skinfriction coe�cient, Cf (2Rex)1=2e(�3x=2L), local Nusseltnumber, Nux(2=Rex)1=2e(�x=2L), velocity pro�le, f 0(�),and temperature pro�le, �(�), are depicted. Figures 2and 3 show the variations of skin friction coe�cient

Table 1. Comparison of the values of skin frictioncoe�cient Cf (2Rex)1=2e(�3x=2L) with those of Isa etal. [26] for Newtonian uid and present results for a verylarge Casson parameter.

Cf(2Rex)1=2e(�3x=2L)

� Solutionpro�les

0

Isa et al. [26]1

present

-0.5

First 3.3616 3.3615Second -3.2002 -3.2001Third -6.7088 -6.7088Fourth -0.4698 -0.4699

1

First 4.2806 4.2805Second -2.7500 -2.7499Third -3.1944 -3.1943Fourth 0.6641 0.6641

Figure 2. Variations of skin friction coe�cientCf (2Rex)1=2e(�3x=2L) with the suction parameter s fordi�erent values of M2 when Pr = 1, = 2, and � = �1.

and local Nusselt number, respectively, with changesin the rate of suction s for di�erent values of M2. Ithas been evidenced that M2 increases the skin frictioncoe�cient as well as the local Nusselt number accordingto the �rst solution. On the other hand, other pro�lesshow opposite trends. The �rst and second solutionsare connected at the point s = sc, and these two pro�lesexist at s > sc. The third and fourth solutions occurfor all values of the suction parameter s. The e�ectsof the magnetic �eld on the skin friction coe�cientand the local Nusselt number were found to be morepronounced for a larger s. For the third and fourthsolution pro�les, the values of skin friction coe�cientand local Nusselt number were close to each other whenthe rate of suction was reduced. As it is clear, when werefer to Figure 2 for the �rst solution, a large value ofs produced a highly positive value of the skin friction

S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647 641

coe�cient. On the other hand, it was found that thethird solution gave large negative values of the skinfriction coe�cient when s was very large. It is evidentfrom Figure 3 that a high rate of suction led to anincrement of the local Nusselt number for the �rst,third, and fourth solution pro�les. We can concludethat the rate of heat transfer in the third and fourthsolutions decreased with a decrease in s until the rateof suction was close to zero. These two pro�les (thirdand fourth solutions) showed a very slight di�erence inthe values of the local Nusselt number when the walltended to become impermeable.

The dimensionless velocity pro�le when � = �1is depicted in Figure 4, which shows that the third

Figure 3. Variations of local Nusselt numberNux(2=Rex)1=2e(�x=2L) with the suction parameter s fordi�erent values of M2 when Pr = 1, = 2, and � = �1.

Figure 4. Velocity pro�les when � = �1.

Figure 5. Temperature pro�les when � = �1.

solution had the largest magnitude of velocity whenthe point of the uid was close to the shrieked wall.After this, negative f 0(�) in all pro�les tended to attaina constant zero velocity. It implies that there wasreduction in the magnitude of velocity that was inthe reverse direction, until the ow ended. Figure 5shows the temperature distribution �(�) for � = �1. Inthe third and fourth solution pro�les, the temperature�(�) at the point initially decreased and then becamenegative. Subsequently, the temperature started toincrease when the distance from the shrieked wall andthe body of the uid became larger. The secondsolution showed the presence of a peak for a small valueof �, indicating that the maximum temperature occursin the uid at a point near the shrieked wall. Theenhancement of the temperature di�erence between theshrieked wall and the adjacent uid was indicated bythe increment of temperature peak value. As a result,the heat transfer process from the shrieked wall to theambient uid was enhanced.

The skin friction coe�cient and the local Nusseltnumber for several values of M2, for Pr = 1, = 2, and� = 1, are shown in Figures 6 and 7, respectively. FromFigure 6, it is clear that the impact of the magnetic�eld was to enhance the rate of skin friction coe�cientin the �rst and third solutions. On the other hand,the opposite trend occurred in the other 2 pro�les.Moreover, the local Nusselt number increased in the�rst, second, and fourth solutions with the e�ect of themagnetic �eld. When M2 = 0:25, the critical suctionparameter sc connected the third and second pro�les,but the third solution pro�le was connected with thefourth solution at the point sc when M2 = 0:3. Theoccurrence of the �rst solution pro�le was observedfor various values of M2 and s. With regard to thevariations of skin friction coe�cient and local Nusseltnumber, the second solution pro�le continued until

642 S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647

Figure 6. Variations of skin friction coe�cientCf (2Rex)1=2e(�3x=2L) with the suction parameter s fordi�erent values of M2 when Pr = 1, = 2, and � = 1.

Figure 7. Variations of local Nusselt numberNux(2=Rex)1=2e(�x=2L) with the suction parameter s fordi�erent values of M2 when Pr = 1, = 2, and � = 1.

s = 0 when M was large. Meanwhile, the fourthsolution pro�le occurred for all values of s for a smallvalue of M2.

The velocity pro�le for � = 1 is depicted inFigure 8. The magnitude of the uid velocity at a pointnear the stretching sheet shows that the �rst solutionhad the lowest value whereas the third solution hadthe highest one. The graph of the temperature pro�le�(�) when � = 1 is shown in Figure 9. In this set ofresults, the fourth solution shows the highest thermalboundary layer whereas the boundary layer thicknessof the second solution is the lowest.

Figure 8. Velocity pro�les when � = 1.

Figure 9. Temperature pro�les when � = 1.

The skin friction coe�cient and the local Nusseltnumber, with a mixed convection parameter � anddi�erent values of M2, are shown in Figures 10 and11 for Pr = 1, = 2, and s = 4. With thepresence of suction, the �rst solution always had anincrement in the skin friction coe�cient and localNusselt number caused by the e�ect of increasing M2

and �. A decrement in the value of skin frictioncoe�cient was observed for the other pro�les, due toan increment in the magnetic �eld, when the mixedconvection parameter decreased. When examining thesecond and fourth solutions, the local Nusselt numberbecame very large or very small when � was close to 0.The critical value of the mixed convection parameter,�c, is the point which connects the third and fourth

S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647 643

Figure 10. Variations of skin friction coe�cientCf (2Rex)1=2e(�3x=2L) with mixed convection parameter �for di�erent values of M2 when Pr = 1, = 2, and s = 4.

Figure 11. Variations of local Nusselt numberNux(2=Rex)1=2e(�x=2L) with mixed convection parameter� for di�erent values of M2 when Pr = 1, = 2, and s = 4.

solutions, and depends on M2. The point �c moves tothe left for an increasing M2. For the assisting owsituation (� > 0), all solution pro�les occurred in therange 0 < � < �c, whereas only dual solutions (�rstand second solution pro�les) arose for � > �c. Whenwe observed the case of opposing ow (� < 0), all thepro�les showed their existence for all values of �. Inthe case of forced convection (� = 0), only the �rstand third solutions appeared in the variation of heattransfer. However, the occurrence of all pro�les couldbe seen for the variations of the skin friction coe�cientwhen � = 0.

Figure 12. Variations of skin friction coe�cientCf (2Rex)1=2e(�3x=2L) with mixed convection parameter �for di�erent values of when Pr = 1, M2 = 0:3, and s = 4.

Figure 12 shows illustrations of the skin frictioncoe�cient, which depend on the change of mixed con-vection parameter, �, for di�erent rates of the Cassonparameter, ( = 0:5; 2). The points of this �gurewere calculated numerically for s = 4 and M2 = 0:3.Foremost, all solution pro�les occurred when the valueof the Casson parameter was small ( = 0:5), and theirregions were in the range of �1:38807 < � < 0:42200.In addition, dual solutions in = 0:5 for opposing owoccurred when � < �1:38807; this region involved thethird and fourth solution pro�les. Besides, anotherregion of dual solutions in = 0:5 for the assisting ow case occurred when � > 0:42200. This region ofdual solutions involve �rst and fourth solutions. Thefourth solution appeared for all values of � for = 0:5.Next, when the rate of the Casson parameter increasedto 2, all solutions existed in the region of � < 1:14600.Moreover, dual solutions (�rst and second solutions)occurred when the assisting ow parameter was large(� > 1:14600). In conclusion, the rate of skin frictioncoe�cient attempted to increase for very large � and , and it was demonstrated that all the 4 solutionsexpanded because of increase in Casson parameter.

The local Nusselt numbers are plotted versus themixed convection parameter as shown in Figure 13for di�erent values of Casson parameter, . When = 0:5, all solutions came about when the values of themixed convection parameter were �1:38807 < � < 0and 0 < � < 0:42200. Besides, when the values ofthe mixed convection parameter were � < �1:38807,� = 0, and � > 0:42200, there existed two solutionpro�les at = 0:5. These dual solutions were relatedto the third and fourth solutions in the opposing owcase, the �rst and third solutions for forced convection� = 0, and the �rst and fourth solutions when the

644 S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647

Figure 13. (a) Variations of local Nusselt numberNux(2=Rex)1=2e(�x=2L) with mixed convection parameter� for di�erent values of when Pr = 1, M2 = 0:3, ands = 4. (b) Enlargement of (a).

uid ow was assisted by a buoyancy force. Whenthe Casson parameter increased from 0.5 to 2, allsolutions became established when the values of mixedconvection parameter were � < 0 and 0 < � < 1:14600.Furthermore, dual solutions were presented at = 2with the involvement of the �rst and second solutionswhen � > 1:14600. In addition, for = 2, forceconvection also showed the dual solutions, which relate�rst and third solutions. The variation of local Nusseltnumber for = 2 proved that �rst solution occurred forall values of �. Conclusively, the e�ect of for variousvalues of � was to enhance the rate of the local Nusseltnumber for the �rst solution, whereas a reduction wasobserved with the e�ect of Casson rate in the thirdsolution. However, for the assisting ow case � > 0, thesecond solution increased and fourth solution decreaseddue to the increment in the Casson parameter. In

opposing ow � < 0, the two solutions (second andfourth) showed an opposite trend compared to the casewhen � > 0. Figure 13 also shows similar results toFigure 12 on the in uence of the Casson parameter onthe region of multiple solutions, which became widerwhen the Casson parameter became higher.

4. Conclusions

A detailed numerical study is performed for mixedconvection MHD ow of a Casson uid over an ex-ponentially permeable shrinking sheet. E�ects of thenon-Newtonian (Casson) parameter, suction parame-ter, mixed convection parameter, and magnetic �eldparameter on the ow and heat transfer behavior arethoroughly analyzed. As already proved by Isa etal. [32], 4 solutions are obtained by containing theimpact of magnetic �eld in mixed convection boundarylayer ow of viscous Newtonian uid over a permeableshrinking sheet. Therefore, the presence of buoyancyforce, magnetic �eld, and suction in Casson uid alsocontributes to the occurrence of multiple solutions (4solutions). These 4 solutions exist for both the skinfriction coe�cient and the local Nusselt number whenthe ow is opposing and assisting. However, only twopro�les exist for the case of forced convection, for thegraph of local Nusselt number. The enlargement inthe region of 4 solutions is due to the enhancement inthe rate of Casson parameter. The study discloses thatthe quad solutions of velocity and temperature existin certain ranges of mass suction parameter, mixedconvection parameter, magnetic �eld parameter, andCasson parameter. Nevertheless, the e�ect of Cassonparameter in uences the rates of skin friction andlocal Nusselt number and changes the characteristicsof Newtonian Casson uid into those of the non-Newtonian one.

Nomenclature

B(x) Magnetic �eldCf Skin friction coe�cientg Acceleration of gravityGr Grashof numberL Characteristic length of the sheetM2 Hartman number (magnetic �eld

parameter)Nux Local Nusselt numberpy Yield stress of uidRe Reynolds numberRex Local Reynolds numberT Fluid temperatureTw Shrinking sheet temperature

S.S.P.M. Isa et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 637{647 645

T1 Ambient uid temperatureUw Velocity of the shrinking sheetu; v Velocity components in the x and y

directions respectivelyvw(x) Velocity of the suction/mass transferx; y Cartesian coordinates along the

surface of the sheet and normal to it,respectively

Greek letters

� Thermal di�usivity� Thermal expansion coe�cient Casson parameter� Similarity variable� Dimensionless temperature� Mixed convection parameter�g Plastic dynamic viscosity of the

non-Newtonian uid� Kinematic viscosity� Square of the component of the

deformation rate�c Critical value of product� Fluid density� Electrical conductivity� Shear rate Stream function

Subscripts

c A manifestation of the critical pointw Shrinking sheet's state at the surface1 Free-stream condition

Superscripts0 Di�erentiation with respect to �

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Biographies

Siti Suzilliana Putri Mohamed Isa has been atutor of Mathematics since July 2012 at the Centreof Foundation Studies for Agricultural Science, UPM,Malaysia. She received her BSc in Physics in 2007and her MSc degree in 2010 in the �eld of Mathe-matical Sciences and Applications (Fluid Dynamics).Currently, she is a PhD Candidate in the �eld of FluidDynamics at the Institute for Mathematical Researchin UPM, Malaysia. During her PhD, she is working onsteady- and unsteady-convection boundary layer owon a stretching/shrinking sheet in viscous and Casson uids.

Norihan Md. Ari�n is Associate Professor of Math-ematics in the Department of Mathematics, UPM,Malaysia. She received her MSc in Industrial Math-ematics from University of Strathclyde-Glasgow, UK,in 1996 and her PhD in Mathematics from UKM,Malaysia. Her research interests include uid me-chanics and heat transfer with application to thethermal convection and boundary-layer theory, andheat transfer in Newtonian and non-Newtonian aswell as in uid saturated porous media and nano u-ids.

Roslinda Nazar has been a Professor of Mathematics,since January 2012, in the School of MathematicalSciences, Faculty of Science and Technology, UKM,Malaysia. She obtained the BSc in Mathematics

from the University of Illinois at Urbana-Champaign,Illinois, USA, in January 1995. Later, in October1996, she earned the MSc in Mathematics of NonlinearModels at the Heriot-Watt University & University ofEdinburgh, Scotland, UK. She received the PhD inApplied Mathematics in January 2004 in the �eld ofFluid Dynamics and Heat Transfer from the UniversitiTeknologi Malaysia, Skudai, Johor, Malaysia. Her areaof research covers uid dynamics, heat transfer, andmathematical modelling.

Nor�fah Bachok is currently Associate Professorin the Department of Mathematics, UPM, Malaysia.She is the head of program: Mathematical Physicsand Engineering at the Institute for MathematicalResearch, UPM. Her expertise covers uid ow, heattransfer, and boundary layer theory.

Fadzilah Md Ali is a Senior Lecturer in the De-partment of Mathematics, UPM, Malaysia. Her majorresearch areas include uid mechanics and heat transferin boundary layer.

Ioan Pop is a Professor of Applied Mathematics inthe Faculty of Mathematics and Computer Science,Babe�s-Bolyai University, Romania. His �elds of in-terest include uid mechanics and heat transfer withapplication to boundary-layer theory, heat transfer inNewtonian and non-Newtonian uids, convective owin uid-saturated porous media, and magnetohydrody-namics.