Abstract

Purpose - This paper considers the problem of the steady plane obliquestagnation point flow of an electrically conducting Newtonian fluid im-pinging on a heated vertical sheet. The temperature of the plate varieslinearly with the distance from the stagnation point.

Design/methodology/approach - The governing boundary layer equa-tions are transformed into a system of ordinary differential equations usingthe similarity transformations. The system is then solved numerically us-ing the ”bvp4c” function in MATLAB.

Findings - An exact similarity solution of the MHD Navier-Stokes equa-tions under the Boussinesq approximation is obtained. Numerical solu-tions of the relevant functions as well as the structure of the flow fieldare presented and discussed for several values of the parameters whichinfluence the motion: M the Hartmann number, β describing the obliquepart of the motion, the Prandtl number Pr and the Richardson numbersλ and Ri. Dual solutions exist for several values of the parameters.

Originality/value - The present results are original and new for theproblem of MHD mixed convection oblique stagnation-point flow of a New-tonian fluid over a vertical flat plate, with the effect of induced magneticfield and temperature.

Keywords - Newtonian fluids; oblique stagnation-point flow; heat trans-fer; Boussinesq approximation; MHD.

Paper type - Research paper.

1

MHD mixed convection oblique stagnation-point

flow on a vertical plate

G. Giantesioa, A. Vernab, N.C. Roscac, A.V. Roscad, I. Pope

agiulia.giantesio@unicatt.it, Dipartimento di Matematica e Fisica, UniversitaCattolica del Sacro Cuore, via Musei 41, 25121 Brescia, Italy;

bannaverna@hotmail.it, Dipartimento di Matematica e Informatica, Universita diFerrara, via Machiavelli 35, 44121 Ferrara, Italy;

cnatalia@math.ubbcluj.ro, Department of Mathematics, Faculty of Mathematics andComputer Science, Babes-Bolyai University, str Mihail Kogalniceanu 1, Cluj-Napoca,

Romania;dalin.rosca@econ.ubbcluj.ro, Department of Statistics-Forecasts-Mathematics,

Faculty of Economics and Business Administration, Babes-Bolyai University, strMihail Kogalniceanu 1, Cluj-Napoca, Romania;

epopm.ioan@yahoo.co.uk, Department of Mathematics, Faculty of Mathematics and

Computer Science, Babes-Bolyai University, str Mihail Kogalniceanu 1, Cluj-Napoca,

Romania

1 Introduction

During the last several decades the flow and heat transfer characteristic of a vis-cous and incompressible fluid near the orthogonal or oblique stagnation-pointhas received a great attention. This problem has been first studied by Hiemenz(1911) for a two-dimensional flow and by Homann (1936) for an axisymmetricflow. Hiemenz has demonstrated that the Navier-Stokes equations governingthe flow can be reduced to an ordinary differential (similarity) equation of thirdorder by means of a similarity transformation. In the absence of an analyticalsolution, the reduced ordinary differential equation is usually solved numericallysubject to two-point boundary conditions, one of which is prescribed at infin-ity. The study of magnetohydrodynamic (MHD) flow has been actively doneby many researchers until recently due to its many practical applications in themodern industry, such as, for example, in metallurgy, solar physics, geophysics,cosmic fluid dynamics, polymer industry, in the motion of earth’s core, etc. Re-cently, the researchers became interested in the influence of magnetic fields onblood flows primarily with a view to utilizing MHD in controlling blood flow ve-locities in surgical procedures and also establishing the effects of magnetic fieldson blood flows in astronauts, etc. In the bioengineering and medical technology,one of the promising methods to accomplish precise targeting is magnetic drugdelivery (Voltairas et al. (2002)) and cell separation (Haik et al. (1999)).Since the blood is an electrically conducting fluid, the magnetohydrodynamic

2

(MHD) principle has been used to decelerate the flow of blood in human arte-rial system and treat of certain cardiovascular disorders. A seminal paper onhydromagnetic blood flow via a rigid tube network was communicated by Sudand Sekhon (1989). Additionally, Bali and Awasthi (2007) analyzed effectsof the imposed magnetic field on the resistance to blood flow velocities in anidealised stenotic artery. An extensive research work Kenjeres (2008), Midyaet al. (2003), Tzirtzilakis (2005) has been reported on the flow dynamics inthe presence of magnetic field.

Fluid impinging obliquely to the surface is known as oblique stagnation pointflow. Such flows happens when orthogonal stagnation point flow is combinedwith a shear flow parallel to the wall. Investigations about oblique stagnation-point flow of a viscous and incompressible fluid in the absence or in the presenceof a magnetic field have been done by many researchers. It seems that Stuart(1959) did the pioneer work on this stagnation flow without an applied mag-netic field. The oblique solution was later studied by Tamada (1979), Dorrepaal(Dorrepaal (2000, 1986)), Wang (2003); Drazin and Riley (2007), and TookeTooke and Blyth (2012) reviewed the problem and included a free parameterassociated with the shear flow component, which is related to the pressure gra-dient. Notable work on this oblique stagnation point flows was also done by Liu(1992), Tittley and Weidman (1998), Weidman and Putkaradze (2003), Blythand Pozrikidis (2005), Reza and Gupta (2005), Lok et al. (2010), Mahapatraet al. (2007), Husain et al. (2011), Mahapatra et al. (2012), Lok et al. (2010).Labropulu et al. (2010) studied the flow of non-Newtonian second grade fluidin the region of oblique stagnation point flow toward a stretching surface withheat transfer and they found that with the increase of Weissenberg number thevelocity, the skin friction and thermal boundary layer near the wall increases.On the other hand, it should be mentioned the several existing papers. Thus,Grosan et al. (2009) have studied the laminar two-dimensional stagnation flowof a viscous and electrically conducting fluid obliquely impinging on a flat platein the presence of a uniform applied magnetic field as a similarity solution of theNavier-Stokes equations. The relative importance of this flow is measured bythe dimensionless strain rate and magnetohydrodynamic parameters. Further,we mention the papers by Borrelli et al. (2012) on MHD oblique stagnation-point flow of a Newtonian fluid and on a micropolar fluids.Since then many investigators have considered various aspects of such flow andobtained similarity solutions. Actually, a great deal of interest has been gener-ated in understanding the influence of the temperature on the stagnation-point(Yajun and Liancun (2013), Ishak et al. (2008), Siddiqa and Hossain (2012),Lok et al. (2007), Mahapatra et al. (2012)).

To the best of our knowledge, no investigation is made for oblique stagnationpoint flow when the obstacle is not uniformed heated. In this paper, we studythe steady MHD two-dimensional oblique stagnation-point flow and heat trans-fer of an incompressible viscous fluid over a vertical sheet. The temperaturedistribution in the flow is determined when the surface is held at a temperaturewhich varies linearly with the distance x from the stagnation point. The in-fluence of the Hartmann number M , the Prandtl number (Pr), the Richardson

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numbers (λ and Ri) and the free oblique parameter (β) on the flow and on thetemperature distribution of the fluid is investigated and analyzed with the helpof their graphical representations.

Nomenclature

a, b constants related to the outer flow (a > 0)A orthogonal displacement thicknessB shear displacement thicknessCfϕ, Cfγ skin friction coefficients defined in (12)dT thermal diffusivity(ex, ey) canonical baseE0 external electric fieldf(y), g(y), ϑ1(y), ϑ2(y) similarity functionsg0 = −g0ex gravity accelerationH0 external uniform magnetic fieldH0 constant related to the strength of the magnetic fieldk thermal conductivityL characteristic length given in (10)M Hartmann numberms slope of the dividing streamline at the wallm∞ slope of the dividing streamline at infinityNuxθ1 , Nuxθ2 local Nusselt numbers defined in (12)p pressurep∗ pressure at the stagnation-pointp∞ pressure of the inviscid fluidPr Prandtl numberqwθ1 , qwθ2 heat fluxes from the wall

Rex = ax2

ν local Reynolds numberRi Richardson number for the oblique part of the motionS region of motionT temperatureT1 arbitrary constantTw(x) temperature of the plateT∞ uniform temperature of the ambient fluidv = (u, v) velocity fieldV∞ = (u∞, v∞) oblique velocity of the inviscid fluid(x, y) spatial coordinates

Greek symbols

4

α, β dimensionless quantities related to a, bαT thermal expansion coefficientδ thickness of the boundary layerηγ boundary layer for γηϕ boundary layer for ϕλ Richardson numberµ viscosity coefficientµe magnetic permeability (µe > 0)ρ∞ mass density at the reference temperature T∞σe electrical conductivity (σe > 0)τw skin friction on the flat plateτwϕ, τwγ skin frictions along the wallϕ(η), γ(η), θ1(η), θ2(η) similarity dimensionless functions(ξ, η) dimensionless variablesξp point of maximum pressure on the wallξs point of zero wall shear stress

2 Mathematical Formulation

Consider a two-dimensional stagnation-point flow of a Newtonian fluid imping-ing on a vertical heated fix plate, as shown in Figure 1.

The oblique velocity of the inviscid fluid is V∞ = (u∞, v∞), which consistsof irrotational stagnation point flow and a uniform shear flow parallel to thewall.The coordinate system are fixed in order to have that the stagnation-pointcoincides with the origin, the wall has equation y = 0 and x is vertical up-ward.It is assumed that the temperature of the plate Tw(x) varies linearly with thedistance x from the stagnation point located at the origin, actually

Tw(x) = T1x+ T∞. (1)

According to the results proved in Borrelli et al. (2012), we further assumethat an external uniform electromagnetic field is impressed of the form

H0 =H0√

4a2 + b2(−bex + 2aey), E0 = 0.

Under these assumptions, along with the Boussinesq approximation, if theinduced magnetic field is neglected, then the steady equations governing theflow are Borrelli et al. (2017)

ρ∞v · ∇v = −∇p+ µ4v + µeσe(v ×H0)×H0 + ρ∞[1− αT (T − T∞)]g0,

∇ · v = 0,

∇T · v = dT4T,∇×H = σe(E + µev ×H),

∇×E = 0, ∇ ·E = 0, ∇ ·H = 0, in S. (2)

5

We underline that in equations (2) the effects of the induced magnetic field areneglected as it is customary in the literature.

The boundary conditions for equations (2) are

u = v =0, T = Tw(x), at y = 0,

u = u∞ =ax+ b(y −B), v = v∞ = −a(y −A), T = T∞, as y → +∞,

p = p∞ =− ρ∞a2

2

{[x− b

a(B −A)

]2+ (y −A)2

}− ρ∞g0

[x− b

a(B −A)

]

− σea

2

B20

4a2 + b2

{2a

[x− b

a(B −A)

]+ b(y −A)

}2

+ p∗, as y → +∞.

(3)

In particular we recall that the above boundary conditions at infinity mean thatwe are asking that the oblique stagnation-point flow of a Newtonian fluid ap-proaches at infinity the flow of an inviscid fluid whose stagnation point is locatedat(ba (B −A), A

), mass density is ρ∞, pressure is p∞ and the temperature is

T∞. This condition is deduced from the physical experience because in such aproblem the effects of the viscosity appear only in a layer near the boundary andfar from it there is no trace of the viscous nature of the fluid. The parameter bdescribes the shear flow directed parallel to the wall (when b = 0 the orthogonalstagnation-point flow is obtained).

As usual in the oblique stagnation-point flow (Dorrepaal (1986), Dorrepaal(2000), Drazin and Riley (2007)), we use some similarity unknown functions inorder to describe the velocity and the temperature:

u = axf ′(y) + bg(y), v = −af(y),

T = T1xϑ1(y) + T2ϑ2(y) + T∞. (4)

We underline that relations (4)1,2 assure that v is divergence free.The unknown functions f(y), g(y), ϑ1(y), ϑ2(y) must satisfy boundary condi-tions (3), which can be written as

f(0) = 0, f ′(0) = 0, g(0) = 0, ϑ1(0) = 1, ϑ2(0) = 0,

limy→+∞

f ′(y) = 1, limy→+∞

g′(y) = 1,

limy→+∞

ϑ1(y) = 0, limy→+∞

ϑ2(y) = 0. (5)

More precisely (Tamada (1979), Stuart (1959), Tooke and Blyth (2012)), theasymptotic behavior of f and g at infinity is related to the constants A,B in(3) in the following way:

limy→+∞

[f(y)− y] = −A, limy→+∞

[g(y)− y] = −B. (6)

6

Substituting (4) into equations (2)1,3 we get

a2xf ′2 − a2xff ′′ + ab (f ′g − fg′) = −g0 [1− αT (T − T∞)] + νaxf ′′′ + νbg′′

− 1

ρ∞

∂p

∂x− σeB

20

ρ∞(4a2 + b2)[4a3xf ′ + 2a2b(2g − f)],

a2f ′f = −aνf ′′ − 1

ρ∞

∂p

∂y− σeB

20

ρ∞(4a2 + b2)[2a2bxf ′ + ab2(2g − f)],

T1ϑ1 (axf ′ + bg)− af (T1xϑ′1 + T2ϑ

′2) = dT (T1xϑ

′′1 + T2ϑ

′′2) . (7)

By integrating (7)2 and thanks to conditions (3)6, (5)6,7, (6), we can computethe pressure:

p =− ρ∞a2

2

[x2 − 2

b

a(B −A)x+ f2

]− ρ∞ (aνf ′ + g0x)

− σeB20

4a2 + b2a

2

{4abxf + 2b2

∫(2g − f)dy + 4a2x

[x− 2

b

a(B −A)

]}+ p0,

(8)

where p0 is the pressure at the stagnation point. We remark that ∇p hasa constant component in the x direction proportional to B − A, B2

0 and g0,which does not appear in the orthogonal stagnation-point flow. This componentdetermines the displacement of the uniform shear flow parallel to the wall y = 0.In consideration of (8) and (7)1,3, we obtain that the flow is described by thefollowing ordinary differential equations

ν

af ′′′ − f ′2 + ff ′′ +

g0αTT1a2

ϑ1 +σeB

204a

ρ∞(4a2 + b2)(1− f ′) + 1 = 0,

ν

ag′′ − f ′g + fg′ +

g0αTT2ab

ϑ2 +σeB

204a

ρ∞(4a2 + b2)(f − g) = (B −A)

[1 +

σeB204a

ρ∞(4a2 + b2)

],

ϑ′′1 +a

dTfϑ′1 −

a

dTf ′ϑ1 = 0,

ϑ′′2 +a

dTfϑ′2 −

b

dT

T1T2gϑ1 = 0, (9)

along with the boundary conditions (5).In order to obtain the governing problem in simpler form, the following

similarity variables are introduced:

ξ =x

L, η =

y

L, L =

√ν

a,

ϕ(η) =f(Lη)

L, γ(η) =

g(Lη)

L, θ1(η) = ϑ1(Lη), θ2(η) = ϑ2(Lη). (10)

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Problem (9), (5) in dimensionless form becomes

ϕ′′′ + ϕϕ′′ − ϕ′2 + λθ1 +M2(1− ϕ′) + 1 = 0,

γ′′ + ϕγ ′ − ϕ′γ + Riθ2 +M2(ϕ− γ) = (1 +M2)(β − α),

θ′′1 + Pr (θ′1ϕ− θ1ϕ′) = 0,

θ′′2 + Pr

(θ ′2ϕ−

λ

Riθ1γ

)= 0,

ϕ(0) = 0, ϕ′(0) = 0, γ(0) = 0,

θ1(0) = 1, θ2(0) = 0,

limη→+∞

ϕ′(η) = 1, limη→+∞

γ′(η) = 1,

limη→+∞

θ1(η) = 0, limη→+∞

θ2(η) = 0, (11)

where α =A

L, β =

B

L, Pr =

ν

dTPrandtl number, M =

√σeB2

04a

ρ∞(4a2 + b2)Hart-

mann number, Ri =g0αTT2ab

√a

νand λ =

g0αTT1a2

Richardson numbers.

The system depends on the values of the parameters λ, Ri, Pr, α, β, M, how-ever α is not free but its value is determined by solving numerically problem(11)1,3,5,6,8,10,12 which governs the orthogonal stagnation-point flow (Ishak etal. (2008), Ishak et al. (2008)).

Remark 1 It is important to compute some quantities of physical interest inthis problem such as the skin friction coefficients Cfϕ, Cfγ and the local Nusseltnumbers Nuxθ1 , Nuxθ2 , which are defined as

Cfϕ =τwϕρ(ax)2

, Cfγ =τwγρ(by)2

,

Nuxθ1 =qwθ1kT1

, Nuxθ2 =xqwθ2kT2

. (12)

Using (4) and (12), we get√RexCfϕ = ϕ′′(0),

√RexCfγ = γ′(0),

1√Rex

Nuxθ1 = −θ′1(0),1√Rex

Nuxθ2 = −θ′2(0). (13)

Remark 2 Thanks to the previous remark, the skin friction on the flat plate isgiven by

τw = aξϕ′′(0) + bγ′(0). (14)

8

From (14), we have that the dividing stream lines

aξϕ(η) + b

∫ η

0

γ(s) ds = 0

meets the boundary at

ξs = −b√

ν

a3γ′(0)

ϕ′′(0)(point of zero wall shear stress). (15)

Further, the point of maximum pressure on the wall is

ξp = b

√ν

a3(β − α)− g0

a2(1 +M2). (16)

Studying the small-η behavior of

∫ η0γ(s)ds

φ(η)(Dorrepaal (2000)) the slope of

the dividing streamline at the wall is given by

ms = − 3aϕ′′2(0)

b [(1 +M2)(β − α)ϕ′′(0) + (1 +M2 + λ)γ′(0)]

and does not depend on the kinematic viscosity. Thus, the ratio of this slope

to that of the dividing streamline at infinity

(m∞ = −2a

b

)is the same for all

oblique stagnation-point flows and it is given by

ms

m∞=

3ϕ′′2(0)

2 [(1 +M2)(β − α)ϕ′′(0) + (1 +M2 + λ)γ ′(0)]. (17)

3 Numerical solution

In the absence of an analytic solution, a numerical solution is indeed an obviousand natural choice. Hence the governing problem (11) is solved numerically us-ing the bvp4c MATLAB routine. The “infinity” in the boundary value problem(11) is replaced in practice with a finite number. We have chosen a suitablefinite value of η → ∞, namely η = η∞ = 40 for the upper branch solution andη = η∞ in the range 60-90 for the lower branch solution. Mesh selection anderror control are based on the residual of the continuous solution. For this prob-lem we set the relative and the absolute tolerance equal to 10−7. The methodwas used and described in Shampine et al. (2003).

As we can see from problem (11), the flow depends on the choice of severalparameters: β − α describing the oblique part of the motion, M related to themagnetic properties, Pr proportional to the thermal diffusivity, the Richardsonnumbers λ and Ri which are related to the temperature. The values of Pr,λ and Ri are chosen according to Ishak et al. (2008), Siddiqa and Hossain(2012), Lok et al. (2007), Ishak et al. (2008). As we have already said,

9

α is not free but its value is computed by solving the corresponding problemfor the orthogonal stagnation-point flow. We take the values of β according toTooke and Blyth (2012), Stuart (1959) ,Tamada (1979), Dorrepaal (2000)and Dorrepaal (1986).

Figure 2 shows the profiles of the functions describing the flow.The profiles of ϕ,ϕ′, ϕ′′, θ1, θ

′1, γ, γ

′, θ2, θ′2 reported in this Figure corre-

sponds to the solution which is called upper branch. Actually, as it is reportedin the literature, this kind of boundary value problem may have more than one(dual) solution, upper and lower branch solutions, when the parameters change,as we can see in Figure 3 for the oblique part of the motion.

Figures 4 and 5 show that multiple (dual) solutions exist for some values ofthe parameters M and the mixed convection parameter λ.

There are two solutions when λ > λc in a certain range of λ < 0, one solutionwhen λ = λc and no solution when λ < λc, where λc < 0 is the critical value ofλ for which the solution exists.The effect of λc on the reduced skin friction coefficients ϕ′′(0), γ′(0) and reducedheat fluxes from the wall θ′1(0), θ′2(0) when the parameter M varies is shown inFigure 6.

Further, it should be mentioned that the existence of the multiple solutionsresults in from a linear stability analysis. However, we do not present such ananalysis in this paper. From the stability analysis one can see that the upperbranch solutions are stable and, thus, physically realizable, while the lowerbranch solutions are unstable and, therefore, not physically realizable. Theselower branch solutions appear only from numerical reasons. The existence ofdual solutions for the similar problems was also reported by Weidman et al.(Weidman et al. (2006)), Ishak et al. (Ishak et al. (2007, 2008, 2006, 2011,2008)), Bhattacharyya (Bhattacharyya (2011)), Bhattacharyya and Vajravelu(Bhattacharyya and Vajravelu (2012)), Bachok et al. (Bachok et al. (2012a,b,2013)) and Rosca and Pop (Rosca and Pop (2013b,a)) among others.

Tables 2 and 3 elucidate the values of the descriptive quantities of the or-thogonal stagnation-point for several values of M , Pr and λ. For the obliquequantities we refer to Table 4.

Since the lower branch solutions appear only from numerical reasons, in thefollowing we analyze the behavior of the upper branch solution. As one can seefrom Figure 2, the numerical solution (ϕ, γ, θ1, θ2) of problem (11) satisfies

limη→+∞

[ϕ(η)− η] = −α, limη→+∞

[γ(η)− η] = −β,

limη→+∞

θ1(η) = 0, limη→+∞

θ2(η) = 0. (18)

We recall that the parameter α is proportional to the displacement thicknesswhich represents the quote of the plane towards which the inviscid fluid, whoseflow is approached at infinity by the Newtonian fluid, is pointed. We denote by

• ηϕ the value of η such that if η > ηϕ then ϕ ∼= η − α;

• ηγ the value of η such that if η > ηγ then γ ∼= η − β.

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The values of ηϕ and ηγ are computed numerically by requiring an accuracy of99% and they are furnished in Table 1 when the parameters change.Thanks to these notations, the influence of the viscosity on the velocity appearsonly in a layer lining the boundary whose thickness is defined as

δ := max(ηϕ, ηγ).

As it can be expected from the physical point of view, beyond the boundarylayer the fluid behaves as an inviscid one (Borrelli et al. (2013), Borrelli et al.(2015)).

These values are in agreement with the results of Ishak et al. (2008) andIshak et al. (2008) and they show that

• if M , Pr and λ increase, then ϕ′′(0) and |θ′1(0)| increase;

• α and ηϕ increase when λ decreases, while if λ < 0(> 0), then α and ηϕdecrease (increase) when Pr increases.

These results underline that the thickness of the boundary layer related tothe orthogonal flow decreases as M increases (Fig. 71), as it is customary in theliterature (Borrelli et al. (2012)). Moreover it decreases as λ increases (Fig. 72)and as Pr increases if λ < 0 (Fig. 73) or as Pr decreases if λ > 0 (Fig. 74).

The behavior of θ1 when one parameters varies is shown in Figure 8.From the Tables we see that γ′(0) and ηγ are not influenced directly by Ri,

while θ′2(0) is positive (negative) when Ri is positive (negative), as can be easilyseen in Fig. 9.

Further we have that |γ′(0)| and ηγ decrease as λ increases.The most important thing is that for any value of β, we have that ηγ is biggerthan ηφ so that the thickness δ of the boundary layer is determined by theoblique part of the flow and

• δ decreases as M , Pr and λ increase (Fig. 10);

• δ is not influenced by Ri.

Finally, Fig. 11 shows the streamline patterns for the flow and the location ofξs (i.e. the point of zero wall shear stress).

4 Conclusions

We have obtained an exact similarity solution of the Navier-Stokes equationsunder the Boussinesq approximation which represent the steady plane obliquestagnation point flow of a viscous fluid impinging on a heated sheet. The tem-perature of the plate varies linearly with the distance from the stagnation pointlocated at the origin.Numerical solutions of the similarity equations are obtained and the behaviorof the flow is discussed for several values of the parameters which influence themotion: β describing the oblique part of the motion, the Hartmann number M ,

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the Prandtl number Pr and the Richardson numbers λ and Ri. In particular,it is found that the thickness of the boundary layer decreases as M , Pr and λincrease. It is shown that the position of the point ξs of zero skin friction (shearstress on the wall) is shifted to the left or to the right of the origin dependingon the values of β − α, on the influence of temperature and on the buoyancyeffects. Moreover, we have that dual solutions may arise for suitable values ofλ.

Acknowledgements The authors are thankful to the competent reviewersfor their suggestions.

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13

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14

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15

x

yO

T (x)w

T∞

g0

V

H0

∞

Figure 1: Physical model and coordinate system.

Yajun, L.V. and Liancun, Z. (2013), “MHD oblique stagnation-point flow andheat transfer of a micropolar fluid towards to a moving plate with radiation”,International Journal of Engineering Science and Innovative Technology, Vol.2 No. 2, pp. 200-209.

16

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

0

1

2

3

4

5

η

M =1, P r =0.7, Ri =0.5, λ =1, β − α =0.

ϕ

ϕ′

ϕ′′

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

η

M =1, P r =0.7, Ri =0.5, λ =1, β − α =0.

θ1θ′1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−2

0

2

4

6

8

10

η

γ

M =1, P r =0.7, Ri =0.5, λ =1.

β−α =-5.3933

β−α =-0.3933

β−α =0

β−α =0.3933

β−α =4.6067

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−8

−6

−4

−2

0

2

4

6

8

10

η

γ ′

M =1, P r =0.7, Ri =0.5, λ =1.

β−α =-5.3933

β−α =-0.3933

β−α =0

β−α =0.3933

β−α =4.6067

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

η

θ2

M =1, P r =0.7, Ri =0.5, λ =1.

β−α =-5.3933

β−α =-0.3933

β−α =0

β−α =0.3933

β−α =4.6067

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

η

θ ′2

M =1, P r =0.7, Ri =0.5, λ =1.

β−α =-5.3933

β−α =-0.3933

β−α =0

β−α =0.3933

β−α =4.6067

Figure 2: The first two figures show the velocity (ϕ,ϕ′, ϕ′′) and the tempera-ture (θ1, θ

′1) in the orthogonal stagnation-point flow, while the other pictures

represent the oblique part of the velocity (γ, γ′) and of the temperature (θ2, θ′2).

17

0 2 4 6 8 10 12 14 16−5

0

5

10

15

20

25

η

γ

β = − 5, − α, 0, α, 5

0 2 4 6 8 10 12 14 16−25

−20

−15

−10

−5

0

5

10

15

20

25

η

γ

β = − 5, − α, 0, α, 5

β = − 5, − α, 0, α, 5

0 1 2 3 4 5 6−10

−5

0

5

10

15

η

γ’

β = − 5, − α, 0, α, 5

β = − 5, − α, 0, α, 5

0 2 4 6 8 10−20

−15

−10

−5

0

5

10

15

η

γ’

β = − 5, − α, 0, α, 5

β = − 5, − α, 0, α, 5

0 2 4 6 8 10

−14

−12

−10

−8

−6

−4

−2

0

2

4

6

η

θ2

β = − 5, − α, 0, α, 5

0 2 4 6 8 10

−20

0

20

40

60

80

100

120

η

θ2

β = − 5, − α, 0, α, 5

Figure 3: Oblique part of the flow (left upper branch, right lower branch) forseveral values of β when M = 1, λ = −3, Pr = 0.7 and Ri = −0.5.

18

0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

η

ϕ’

λ = − 4, − 3, − 2

λ = − 4, − 3, − 2

Upper solution branch

Lower solution branch

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

η

θ1

λ = − 4, − 3, − 2

Upper solution branch

Lower solution branch

0 5 10 15 20−5

0

5

10

15

20

η

γ

λ = − 4, − 3, − 2

λ = − 4, − 3, − 2

Upper solution branch

Lower solution branch

0 2 4 6 8 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

η

γ’

λ = − 4, − 3, − 2

Upper solution branch

Lower solution branch

0 2 4 6 8 10−15

−10

−5

0

5

10

15

η

θ2

Upper solution branch

Lower solution branch

λ = − 4, − 3, − 2

0 2 4 6 8 10

−10

−5

0

5

10

15

η

θ2’

Upper solution branch

Lower solution branch

λ = − 4, − 3, − 2

λ = − 4, − 3, − 2

Figure 4: Comparison between the upper and lower branch solutions for severalvalues of λ < 0 (opposing flow) when β = α, M = 2, Pr = 0.7 and Ri = −0.5.

19

0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

η

ϕ’

Upper solution branch

Lower solution branch

M = 0, 1, 2

M = 0, 1, 2

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

η

θ1

Upper solution branch

Lower solution branch

M = 0, 1, 2

M = 0, 1, 2

0 2 4 6 8 10−10

−5

0

5

10

15

η

γ

M = 0, 1, 2

M = 0, 1, 2

M = 0, 1, 2

M = 0, 1, 2

Lower solution branch

Upper solution branch

0 1 2 3 4 5 6 7 8 9 10 11−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

η

γ’

M = 0, 1, 2

Upper solution branch

Lower solution branch

M = 0, 1, 2

0 2 4 6 8 10

−10

−5

0

5

10

15

η

θ2

Upper solution branch

Lower solution branch

M = 0, 1, 2

M = 0, 1, 2

0 2 4 6 8 10

−10

−5

0

5

10

η

θ2’

M = 0, 1, 2

Upper solution branch

Lower solution branch

M = 0, 1, 2

M = 0, 1, 2

M = 0, 1, 2

Figure 5: Comparison between the upper and lower branch solutions for severalvalues of M when β = α, λ = −2, Pr = 0.7 and Ri = −0.5.

20

−16 −14 −12 −10 −8 −6 −4 −2 0−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

λ

ϕ′′(0)

upper solution branch

lower solution branch

M = 3

M = 0M = 2 M = 1

λc1

= − 2.2025 λ

c3 = − 7.9857

λc2

= − 3.6445

λc4

= − 15.2028

−16 −14 −12 −10 −8 −6 −4 −2 0−15

−10

−5

0

5

10

15

λ

γ’(0)

upper solution branch

lower solution branch

M = 3

λc2

= − 3.6445 λc3

= − 7.9857

λc4

= − 15.2028

M = 2

λc1

= − 2.2025

M = 0M = 1

−16 −14 −12 −10 −8 −6 −4 −2 0−15

−10

−5

0

5

λ

−θ1’(0)

M = 3 M = 1 M = 0

upper solution branch

lower solution branch

λc2

= − 3.6445 λ

c3 = − 7.9857

λc4

= − 15.2028

λc1

= − 2.2025

M = 2

−16 −14 −12 −10 −8 −6 −4 −2 0−30

−20

−10

0

10

20

30

40

50

60

λ

−θ2’(0)

M = 3

M = 2

M = 0

M = 3

λc2

= − 3.6445

λc3

= − 7.9857

λc4

= − 15.2028

M = 0M = 1

λc1

= − 2.2025

upper solution branch

lower solution branch

M = 2

M = 1

Figure 6: Variation of ϕ′′(0), γ′(0), θ′1(0) and θ′2(0) with λ for several values ofM when β = α, Pr = 0.7 and Ri = −0.5.

21

Table 1: Thickness of the boundary layer.M λ Pr Ri α β − α ηϕ ηγ0 -1.0 0.7 -0.5 0.9332 -0.9332 3.0386 4.1030

0 3.95050.9332 3.6873

0.5 -0.9332 4.10300 3.9505

0.9332 3.687310.0 -0.5 0.7049 -0.7049 2.4481 3.2239

0 3.27230.7049 3.3123

0.5 -0.7049 3.22390 3.2723

0.7049 3.31231.0 0.7 -0.5 0.4455 -0.4455 1.4692 3.4213

0 3.18040.4455 2.4911

0.5 -0.4455 3.42130 3.1804

0.4455 2.491110.0 -0.5 0.5999 -0.5999 2.3179 3.0066

0 3.11910.5999 3.1973

0.5 -0.5999 3.00660 3.1191

0.5999 3.19731 -1.0 0.7 -0.5 0.7222 -0.7222 2.7138 3.7085

0 3.56330.7222 3.3498

0.5 -0.7222 3.70850 3.5633

0.7222 3.349810.0 -0.5 0.5780 -0.5780 2.1547 2.7958

0 2.86310.5780 2.9154

0.5 -0.5780 2.79580 2.8631

0.5780 2.91541.0 0.7 -0.5 0.3933 -0.3933 1.9396 3.2296

0 3.02990.3933 2.1961

0.5 -0.3933 3.22960 3.0299

0.3933 2.196110.0 -0.5 0.5079 -0.5079 2.0594 2.6289

0 2.75160.5079 2.8338

0.5 -0.5079 2.62890 2.7516

0.5079 2.8338

22

Table 2: Comparisons with A. Ishak, R. Nazar, I. Pop Ishak et al. (2008) forthe orthogonal stagnation-point flow and some values of α for the upper branchsolution.M Pr λ α Present Results (upper branch solution) A. Ishak, R. Nazar, I. Pop Ishak et al. (2008)

ϕ′′(0) −θ′1(0) ϕ′′(0) −θ′1(0)

0 0.7 -1 0.9332 0.6917 0.6333 0.6917 0.63327 0.7190 0.9235 1.5460 -1.5460 1.540320 0.6847 1.0031 2.2683 1.0031 2.268340 0.6716 1.0459 2.9054 1.0459 2.9054100 0.6610 1.0918 4.0097 1.0918 4.00970.7 1 0.4455 1.7063 0.7641 1.7063 0.76417 0.5891 1.5179 1.7224 1.5179 1.722420 0.6158 1.4485 2.4576 1.4485 2.457640 0.6267 1.4101 3.1011 1.4101 3.1011100 0.6358 1.3680 4.2116 1.3680 4.2116

Table 3: Comparisons with A. Ishak, R. Nazar, N. M. Arifin, I. Pop Ishak etal. (2008) for the orthogonal stagnation-point flow and some values of α forthe lower branch solution.M Pr λ α Present Results (lower branch solution) A. Ishak, R. Nazar, N. M. Arifin, I. Pop Ishak et al. (2008)

ϕ′′(0) −θ′1(0) ϕ′′(0) −θ′1(0)

0 0.7 1 6.5688 1.2387 1.0226 1.2387 1.02267 3.4456 0.5824 2.2192 0.5824 2.219220 2.6198 0.3436 3.1646 0.3436 3.164640 2.2034 0.2111 4.1080 0.2111 4.1080100 1.7717 0.0600 6.1230 0.0601 6.1230

23

Table 4: Descriptive quantities of the oblique stagnation-point flow.M Pr Ri λ upper branch solution lower branch solution

α β γ′(0) θ′2(0) α β γ′(0) θ′2(0)

1 0.7 -0.5 -3 1.3424 -5 14.2205 -18.8787 3.4971 -5 -10.0430 82.5765-α 6.3387 -8.7172 -α -8.2809 67.98830 3.4236 -4.9588 0 -4.1810 34.0465α 0.5300 -1.2283 α -0.0693 0.00755 -7.3732 8.9610 5 1.6927 -14.5806

-1 0.7222 -5 10.1614 -3.3438 6.5260 -5 24.5722 -22.3844-α 2.9900 -1.1017 -α 27.7318 -25.07420 1.7640 -0.7184 0 14.1788 -13.5366α 0.5547 -0.3404 α 0.6259 -1.99905 -6.6334 1.9068 5 3.8063 -4.7065

10 -0.5 -3 0.6698 -5 10.8594 -28.0861 1.8825 -5 -2.6144 96.8083-α 3.0253 -8.0938 -α -1.5206 54.15860 1.8131 -5.0003 0 -0.8594 28.3768α 0.6009 -1.9068 α -0.1982 2.59515 -7.2330 18.0855 5 0.8990 -40.1916

-1 0.5780 -5 9.7320 -7.4096 2.4649 -5 1.7474 49.3895-α 2.4668 -1.9615 -α 1.1010 33.28440 1.5320 -1.2606 0 0.4674 17.4989α 0.5808 -0.5473 α -0.1635 1.77735 -6.6678 4.8884 5 -0.8150 -14.4555

3 0.7 -0.5 -3 0.4633 -5 12.3418 -16.4566 6.3218 -5 35.5168 24.8883-α 2.5382 -3.8174 -α 39.5246 30.55060 1.5449 -2.5368 0 20.4800 3.6438α 0.5300 -1.2283 α 1.4353 -23.26305 -9.2519 11.3830 5 5.4130 -17.6432

-1 0.3483 -5 9.5400 -3.1495 6.2384 -5 32.2169 86.0758-α 1.7304 -0.7079 -α 35.6157 98.01030 1.1426 -0.5242 0 18.5399 38.0487α 0.5547 -0.3404 α 1.4366 -22.00925 -7.2548 2.1011 5 4.8354 -10.0746

10 -0.5 -3 0.3293 -5 18.2358 -25.5432 2.1923 -5 16.7101 79.0906-α 2.7080 -4.1685 -α 10.2890 46.55390 1.6108 -2.6581 0 5.2256 20.8977α 0.5135 -1.1477 α 0.1854 -4.64175 -15.0141 20.2270 5 -6.2818 -37.4117

-1 0.3062 -5 17.8685 -8.0640 2.2161 -5 13.9826 174.5263-α 2.5133 -1.2581 -α 8.6926 104.92090 1.5331 -0.8237 0 4.4873 49.5872α 0.5203 -0.3748 α 0.2630 -5.99695 -14.8021 6.4164 5 -5.0268 -75.6023

24

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

η

ϕ′

P r =0.7, λ =1.

M=0, 1, 2, 5, 10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

η

ϕ′

M =1, P r =0.7.

λ = −1,−0.5, 0.5, 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ϕ′

Pr=0.7, 10, 40, 60, 100

M =1, λ =-1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

η

ϕ′

Pr=0.7, 10, 40, 60, 100

M =1, λ =1.

Figure 7: Boundary layer in the orthogonal stagnation-point flow when M , Prand λ change.

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ1

Pr=0.7, 10, 40, 60, 100

M =1, λ =1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ1

P r =0.7, λ =1.

M=0, 1, 2, 5, 10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ1

M =1, P r =0.7.

λ = −1,−0.5, 0.5, 1

Figure 8: Effect of Pr, M and λ on θ1.

26

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

η

θ2

β − α = α, M =1, P r =0.7, Ri =1.

λ = 1, 0.5,−0.5,−1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

η

θ2

β − α = α, M =1, P r =0.7, λ =1.

Ri = 0.5, 1,−1,−0.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

η

θ2

β − α = α, M =1, λ =1, Ri =1.

Pr=0.7, 10, 40, 60, 100

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

η

θ2

β − α = α, P r =0.7, λ =1, Ri =1.

M=0, 1, 2, 5, 10

Figure 9: Effect of the parameters on θ2.

27

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

η

γ ′

β − α = α, M =1, P r =0.7, Ri =1.

λ = −1,−0.5, 0.5, 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

η

γ ′

Pr=0.7, 10, 40, 60, 100

β − α = α, M =1, λ =1, Ri =1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

η

γ ′

β − α = α, P r =0.7, λ =1, Ri =1.

M=0, 1, 2, 5, 10

Figure 10: Boundary layer in the oblique stagnation-point flow when M , λ andPr change.

28

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

−4

−3

−2

−1

0

1

2

3

4

5

ξ

η

ξs=−1.5316

b

a=1, M =1, β−α =-0.72219, Pr =0.7, Ri =0.5, λ =-1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

−4

−3

−2

−1

0

1

2

3

4

5

ξ

η

ξs=−0.59221

b

a=1, M =1, β−α =-0.39331, Pr =0.7, Ri =0.5, λ =1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

−4

−3

−2

−1

0

1

2

3

4

5

ξ

η

ξs=−0.47821

b

a=1, M =1, β−α =0, Pr =0.7, Ri =0.5, λ =-1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

−4

−3

−2

−1

0

1

2

3

4

5

ξ

η

ξs=−0.2921

b

a=1, M =1, β−α =0, Pr =0.7, Ri =0.5, λ =1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

−4

−3

−2

−1

0

1

2

3

4

5

ξ

η

ξs=0.57516

b

a=1, M =1, β−α =0.72219, Pr =0.7, Ri =0.5, λ =-1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

−4

−3

−2

−1

0

1

2

3

4

5

ξ

η

ξs=0.0080001

b

a=1, M =1, β−α =0.39331, Pr =0.7, Ri =0.5, λ =1.

Figure 11: Streamlines and point of zero wall shear stress of the flow.

29