Similarity solutions of axisymmetric stagnation-point ow...

Scientia Iranica B (2014) 21(4), 1440{1450

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Similarity solutions of axisymmetric stagnation-point ow and heat transfer of a viscous, Boussinesq-relateddensity uid on a moving at plate

H.R. Mozayyeni and A.B. Rahimi�

Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box 91775-1111, Iran.

Received 8 April 2013; received in revised form 15 October 2013; accepted 13 January 2014

KEYWORDSUnsteady;Axisymmetricthree-dimensional;Similarity solution;Heat transfer;Moving plate.

Abstract. The problem of unsteady three-dimensional axisymmetric stagnation-point ow and heat transfer of a viscous compressible uid on a at plate is solved when theplate can move with any arbitrary time-dependently variable or constant velocity. Anexternal low Mach number potential ow impinges, along z-direction, on the at platewith strain rate a to produce three-dimensional axisymmetric stagnation-point ow wherethe plate moves toward or away from impinging ow, concurrently. An exact solutionof the governing Navier-Stokes and energy equations is obtained by the use of suitably-introduced similarity transformations. The temperature of the plate wall is kept constantwhich is di�erent with that of the main stream. A Boussinesq approximation is used totake into account the density variations of the uid. The results are presented for a widerange of parameters characterizing the problem including volumetric expansion coe�cient(�), wall temperature, Prandtl number and plate velocity at both steady and unsteadycases. According to the results obtained, it is revealed that when the plate moves awayfrom the impinging ow, thermal and velocity boundary layer thicknesses get higher valuescompared to the plate moving upward. Besides, it is captured that the value of � and Prnumber do not have any signi�cant e�ect on shear stress and, also, heat transfer for a platemoving away from the incoming potential ow.© 2014 Sharif University of Technology. All rights reserved.

1. Introduction

The study of impinging jet problems is of consid-erable interest in last decades because of its greattechnical importance in many industrial branches spe-cially cooling applications of electronic components,gas-turbine combustion chambers and mechanical de-vices. Obtaining exact solutions of Navier-Stokes andenergy equations regarding the impinging problemsis one of the most e�cient methods to solve suchproblems. There are some publications available inthe literature which studied stagnation ow and heat

*. Corresponding author.E-mail address: [email protected] (A.B. Rahimi)

transfer based on incompressible or compressible u-ids. The incompressible-based papers were started byHiemenz [1] and Homann [2] who discussed steadytwo-dimensional and axisymmetric three-dimensional,respectively, and stagnation ow towards a circularcylinder. A three-dimensional stagnation-point ow ona plane boundary was considered �rstly by Howarth [3].In the more general context of a three-dimensionalstagnation point, the at plate can be allowed toslide in its own plane with constant velocity [4] and,also, can be assumed to be porous to allow for tran-spiration across it [5]. In another paper, Wang [6]considered axisymmetric case of stagnation ow againsta sliding plate. Axisymmetric and nonaxisymmetricstagnation-point ow and heat transfer of a viscous,

H.R. Mozayyeni and A.B. Rahimi/Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1440{1450 1441

incompressible uid on a moving cylinder in di�erentphysical situations are the main subjects of papersconducted by Saleh and Rahimi [7-9]. The steadythree-dimensional stagnation-point ow of a secondgrade uid against a moving at plate is anotherresearch written by Baris [10]. In another study, exactsolutions of the Navier-Stokes and energy equations ofa viscous obliquely impinging ow on a moving cylinderwere studied by Rahimi and Esmaeilpour [11]. Exactsolutions of the Navier-Stokes and energy equationswere derived by Shokrgozar Abbasi and Rahimi [12,13]to solve the problem of stagnation-point ow and heattransfer of an incompressible uid on a at platewith and without transpiration. Also, Abbasi etal. [14] investigated the unsteady case of this problem.Moreover, it was shown by Weidman et al. [15] that self-similar solution of the Navier-Stokes equation exists ifthe isolated in�nite at plate moves at a constant speednormal to the oncoming stagnation point ow.

Some papers available in the literature studied thecompressible ow in the stagnation region of bodiesusing boundary layer equations. The characteristics ofsuch a ow were scrutinized under di�erent physicalconsiderations in [16-17]. Kumari and Nath [18]studied the theory of the response of the compressiblelaminar boundary layer ow to the variation of the ex-ternal stream velocity with time at a three-dimensionalstagnation point, numerically. They solved such aproblem when the ow is asymmetric with respectto the stagnation point [19]. Subsequently, in an-other paper, Kumari and Nath [20] gained the self-similar solution of the forgoing problem with masstransfer only when the free stream velocity variesinversely as a linear function of time. Vasanthaand Nath [21,22] obtained solutions of the unsteadycompressible second-order boundary layer ow at thestagnation point, analytically. Additionally, Zhenget al. [23] obtained similarity solutions to a secondorder heat equation with convection in an in�nitemedium. They used suitable similarity transformationsin order to reduce the parabolic heat equation to aclass of singular nonlinear boundary value problems.The same authors, in another research [24], solvedthe problem of compressible boundary layer behind athin expansion wave by using the application of thesimilarity transformation and shooting technique. Theobjective of the article presented by Zuccher et al. [25]was to analyze the compressible, non-parallel boundarylayer of the ow passing a at plate and sphere.Furthermore, Turkyilmazoglu [26] was concerned withthe case in which exact solution of the steady laminar ow of a compressible viscous uid over a rotatingdisk was obtained in the presence of uniformly appliedsuction or blowing. The steady stagnation-point owand heat transfer of a viscous, compressible uid on anin�nite stationary cylinder is the subject of the paper

recently written by Mohammadiun and Rahimi [27].The potential ow impinging on the cylinder wasassumed to be low Mach number one. They foundsimilarity parameters of their problem for the �rst time.

In this paper, the general unsteady three-dimensional axisymmetric stagnation-point ow andheat transfer of a viscous, compressible uid of alow Mach number ow impinging on a at plate areintended to be solved for the �rst time. The at plateis moving toward or away from the impinging ow atboth constant and time-dependently variable velocity.The density of the uid, also, changes due to thetemperature di�erence existing between the plate andincoming in�nite uid. New similarity transformationsare introduced in order to reduce the governing Navier-Stokes and energy equations to ordinary di�erentialequations which are much easier to solve. The resultsare presented over a wide range of parameters charac-terizing the problems such as coe�cient of volumetricexpansion, wall temperature and Prandtl number atboth steady and unsteady cases.

2. Problem formulation

The problem of steady and unsteady three-dimensionalaxisymmetric stagnation-point ow and heat transferof a viscous compressible uid on a at plate is aimedto be solved when the plate is moving toward or awayfrom the oncoming low Mach number ow at both time-dependently variable and constant velocity. In orderto solve this problem, the axisymmetric cylindricalcoordinate system (r; z) with corresponding velocitycomponents (u;w) is selected, as illustrated in Figure 1.An external potential ow impinges along z-directionon the moving plate, �rstly centered at z = 0, withstrain rate a. Moreover, the temperature of the platewall is maintained constant which is di�erent with thatof the main stream �xed at 25�C. The Navier-Stokes

Figure 1. Schematic of the problem.

1442 H.R. Mozayyeni and A.B. Rahimi/Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1440{1450

and energy equations governing this problem are:

@�@t

+1r@@r

(�ru) +@@z

(�w) = 0; (1)

�@u@t

+ �u@u@r

+ �w@u@z

= �@P@r

+ ��

1r@@r

(r@u@r

)� ur2

+@2u@z2

�; (2)

�@w@t

+ �u@w@r

+ �w@w@z

= �@P@z

+ ��

1r@@r

(r@w@r

) +@2w@z2

�; (3)

�cp�@T@t

+ u@T@r

+ w@T@z

�=k

�1r@@r

(r@T@r

)+@2T@z2

�;(4)

where, p, �, �, k and T are pressure, density, dynamicviscosity, thermal conductivity and temperature, re-spectively. It is worth noting that dynamic viscosityand thermal conductivity of the uid are assumed tobe constant. Furthermore, the dissipation terms of theenergy equation are negligible at the stagnation region.

3. Similarity solutions

3.1. Fluid ow solutionBy solving the momentum equations in potential re-gion, the velocity components can be gained as:

U = a(t)r; (5)

W = �2a(t)�; (6)

where � = z � S(t) and a(t) = @w@� . Here, S(t) is the

amount of vertical displacement of the plate, positivelyde�ned when the plate moves toward the incoming far�eld ow, and is a function of time. Hence, � andthen ow strain rate a(t) can be expressed as time-dependent functions.

A reduction of Navier-Stokes and energy equa-tions to ordinary di�erential equations is sought by us-ing suitably introduced new similarity transformationsas below:

� =��1

ao�

� 12�Z z

0(��1

)dz � S(t)�; (7)

u = a(t)rf 0(�); (8)

w = �2ca(t)

��

�1a0

� 12

f(�) + _Sln cc; (9)

where � is the similarity variable, the terms involv-ing f(�) comprise the cylindrical similarity form for

stagnation-point ow, prime denotes di�erentiationwith respect to �, ao is the reference potential owstrain rate at the time = 0, the subscript w and1 referto the conditions at the wall and in the free stream,respectively, and _S is the plate velocity. It is interestingto note how the e�ect of the plate velocity shows itselfin w-component of velocity as in Eq. (9). In case ofincompressible uid, c(�) = 1, this part has no role inthe results.

In order to capture the e�ects of variations oftemperature on the density of the uid, a parameter,c(�), named density ratio, is introduced as:

c(�) =�(�)�1

: (10)

From Boussinesq approximation for low Mach number ow:

� ��1�1� �(T � T1)� �2(T � T1)2

2

� �3(T � T1)3

3!� :::

�) �

�1= c(�) � 1� �(T � T1)

� �2(T � T1)2

2� �3(T � T1)3

3!� :::; (11)

in which, � is the volumetric expansion coe�cient.It is clear that for the case of incompressible uid,� = 0. Hence � = �1 and c(�) = 1. Insertingthe Transformations (7)-(10) into Eqs. (1)-(3) causesthe continuity equation to be satis�ed automatically,yeilding an ordinary di�erential equations reduced fromr-momentum, and also an expression for the pressure,obtained by integrating Eq. (3) in z-direction as:

cf 000 +�

2~af + ~_S(1� ln c) + c0�f 00

+��1

~a@~a@�� ~af 0

�f 0 � 1

c~a�@ ~P@�

= 0; (12)

where:

1c~a�

@ ~P@�

= �1c

(~a)� 1c

�1~a@~a@�

�;

~P � ~Po = ��2

2

�@~a@�

+ (~a)2�� 2~a ~_S(

fc

) + ( ~_S)2 ln cc

+Z �

2@~a@�

fc� ~�S

ln cc

+~ac2

(f 0c� c0f)

:(2 ~_S ln c� 4~af) +~_Sc2c0(1� ln c):(2~af � ~_S ln c)


� 2~ac

�f 00c� c00f � c0f 0 + (c0)2 f

c

�+

~_Sc

�c00(1� ln c)� (c0)2

c(2� ln c)

��d�: (13)

There are several dimensionless parameters introducedin Eqs. (12) and (13), which are de�ned as:

~a(�) =a(t)a0

; ~P =P

�1a0�1; � = aot;

~_S = _S(t)=p

(a0�1); ~�S =�S

ao(�1ao) 12;

� =ra0

�1r;

@~a@�

=~�S�0

+( ~_S)2

�20

+~_S ~Wo

�20; (14)

where, ~a(�), ~P , � , ~_S, ~�S and � are dimensionless formsof the quantities strain rate, pressure, time, platevelocity, plate acceleration and r, respectively. Ingeneral, when the plate moves with time-dependentlyvariable velocity, the strain rate, a, can be expressedas a function of time. Hence, @~a

@� represents thestrain variation with respect to time, and is taken intoaccount when the plate moves with time-dependentvelocity and acceleration. The quantity �o used in thisrelation expresses the amount of vertical distance fromthe plate in which the velocity of ow incoming to theplate is a�ected by the movement of the plate and startsdecreasing. The quantity ~Wo is dimensionless velocityof potential ow at �o.

The boundary conditions used for Eq. (12) are:

� = 0 : f =~_S ln cw

2~a; f 0 = 0; (15)

� !1 : f 0 = 1; (16)

where:

cw = 1� �(Tw � T1): (17)

Note that for an incompressible uid impinging on astationary plate at steady state conditions, � = 0:0,_S = 0, �S = 0 and a(t) = ao, Eq. (12) simpli�es to the

case of Homann ow obtained in [2], and this is oneway of validation of the results achieved.

3.2. Heat transfer solutionTo transform the energy equation into a non-dimensional form for the case of de�ned wall tempera-ture, we introduce:

� =T (�)� T1Tw � T1 : (18)

Making use of Transformations (7)-(10) and (18), thisequation may be written as:

c�00 + c0�0 +�

2~af + ~_S(1� ln c)�

Pr:�0 = 0; (19)

in which Pr is the Prandtl number.It is worth noting that the coupled system of

(12), (13) and (19) is the most general form for anyarbitrary at plate movement in vertical direction.The boundary conditions needed to solve Eq. (19) arede�ned as:

� = 0 : � = 1

� !1 : � = 0: (20)

The local heat transfer coe�cient on the at plate iscalculated from:

h =qw

Tw � T1 : (21)

Using Eq. (21) and dimensionless parameters, thedimensionless form of heat transfer coe�cient for thisstudy can be gained as:

H = ��0(� = 0)cw; (22)

where:

H =h

k��1 a

�

� 12: (23)

A �nite di�erence procedure consisting of Tri-DiagonalMatrix Algorithm (TDMA) is used to numerically solvethe governing equations (12), (13) and (19). Thenumerical procedure is repeated until the di�erencebetween the results of two repeated sequences of eachof the equations becomes less than 0.00001.

4. Shear stress

Shear stress at the wall surface is given by:

� = �@u@z z=0

: (24)

By introducing the dimensionless parameters de�nedin Section 3, the dimensionless form of shear stress onthe at plate is obtained as:

~� = �f 00(� = 0)cw; (25)

where:

~� =�

�1ao�1:


Figure 2a. Comparison of f 0 pro�les with [28] whenTw = 200�C and Pr=0.7.

Figure 2b. E�ect of � parameter on f 0 pro�les fordi�erent values of plate velocity when Tw = 100�C andPr=0.7.

5. Presentation of results

In order to validate the results obtained, f 0 distri-butions are compared with those of [28] for the caseof Tw = 200�C and Pr=0.7. As can be seen fromFigure 2(a), there is no signi�cant di�erence betweenthe results achieved and those in [28].

At �rst, the in uences of the plate velocity onf 0 distributions are investigated for selected values of� number, in Figure 2(b), and wall temperature, inFigure 3. As it can be noticed from these two �gures,when the plate is moving away from the incomingpotential ow, i.e. with a negative value of velocity, thethickness of the velocity boundary layer is considerablyhigher compared to that of the plate with zero or

Figure 3. E�ect of wall temperature on f 0 pro�les fordi�erent values of plate velocity when � = 0:003 andPr=0.7.

positive value of velocity. Moreover, it can be foundout that the increase of both � number from 0 to0.004 and wall temperature from 50�C to 150�C hassomehow the same e�ect on f 0 pro�les whether theplate moves toward the impinging ow or away from it.The e�ects of enhancement of these two characterizingparameters can be more noticeably seen when the plateis moving with high negative values of velocity. As thespeed of the plate approaches zero and, then, positiveones the e�ect of the change of � parameter and walltemperature on f 0 distributions decreases, gradually. Itcan be claimed, from these �gures, that f 0 distributionswill be, somehow, independent of � coe�cient or walltemperature in physical situations in which the plateis moving toward the impinging ow with high speeds;~_S = 5 for example.

Next, in Figure 4 e�ects of the plate velocityalong with � parameter on dimensionless distribu-tions of velocity component in z-direction is reported.With the increase of � number, the general ten-dency for the absolute values of w-component is toincrease for a plate receding from the main streamand to decrease for a plate advancing toward the mainstream.

Distributions of the dimensionless temperatureare shown for di�erent values of dimensionless platevelocity and selected � numbers in Figure 5, andPr numbers in Figure 6, when Tw = 100�C andPr=0.7. It is revealed from these two �gures thatthe thermal boundary layer thickness becomes smalleras the negative plate velocity tends to zero and then,afterwards, positive ones. Besides, it is understood thatthe increase of � parameter from 0, incompressible-stated uid, to 0.004 has a signi�cant e�ect on temper-ature pro�les if the plate recedes from the impinging


Figure 4. E�ect of � parameter on dimensionless wcomponent pro�les for di�erent values of plate velocitywhen Tw = 100�C and Pr=0.7.

Figure 5. E�ects of � parameter on � pro�les for di�erentvalues of plate velocity when Tw = 100�C and Pr=0.7.

ow. However, the more the speed of the plateadvancing toward the normal incoming ow enhances,the less important � number becomes. Also, it canbe seen from Figure 6 that the increase of Pr numberat any plate velocity has a considerable in uence ontemperature pro�les and causes the thermal boundarylayer thickness to decrease.

The changes of dimensionless pressure, due to theincrease of � parameter and wall temperature, are,respectively, shown in Figures 7 and 8. As can be foundout, the absolute values of the pressure in the vicinity ofthe plate receding from the incoming potential ow areconsiderably lower than those when the plate is movingtoward the main stream. Moreover, as it is reported,

Figure 6. E�ects of Pr no. on � pro�les for di�erentvalues of plate velocity when Tw = 100�C and Pr=0.003.

Figure 7. E�ects of � parameter on dimensionlesspressure distributions for di�erent values of plate velocitywhen Tw = 100�C and Pr=0.7.

there is no noticeable change in pressure values in theregion close to the plate, up to � = 2, for ~_S = �2:0.

The in uences of � and Pr numbers on dimen-sionless heat transfer coe�cient are investigated in Fig-ures 9 and 10 at di�erent dimensionless plate velocity.For a plate with a negative value of velocity, the en-hancement of these two characterizing parameters doesnot have any signi�cant e�ect on heat transfer betweenthe plate and viscous uid close to the plate. With theincrease of the plate speed toward the impinging ow,the e�ects of the increase of � and Pr numbers on heattransfer become more dominated in such a way that theincrease of � form 0.0 to 0.004 causes the H coe�cientto decrease, and the increase of Pr number from 0.3


Figure 8. E�ect of wall temperature on dimensionlesspressure distributions for di�erent values of plate velocitywhen � = 0:003 and Pr=0.7.

Figure 9. E�ect of � parameter on H values for di�erentamounts of plate velocity when Tw = 100�C and Pr=0.7.

to 1.0 brings about the increase of this coe�cient.Note that with constant uid characteristics, as theplate velocity tends to zero and then positive ones, theamount of heat transfer between the plate and viscous uid increases noticeably. Furthermore, it is shown inFigures 11 and 12 that shear stress on a stationary plateor a plate receding from the incoming far �eld ow isindependent of the value of � and Pr numbers. Notethat if characterizing parameters are kept constant,the change of the plate velocity from negative valuesto positive ones has a signi�cant e�ect on shear stressexisting on the plate wall.

In unsteady cases, the plate can move with anyarbitrary time-dependent velocity function. As themost practical example for time-dependently moving

Figure 10. E�ect of Pr no. on H values for di�erentamounts of plate velocity when Tw = 100�C and Pr=0.7.

Figure 11. E�ect of � parameter on shear stress fordi�erent values of plate velocity when Tw = 100�C andPr=0.7.

plate, the exponential function, which can be used tomodel a three-dimensional axisymmetric solidi�cationproblem, is taken to have the form:

_S = Aexp(�t); (26)

where A is a constant. The results obtained byusing the plate velocity function mentioned above arepresented in Figures 13 through 19.

Velocity component in r-direction is distributedin terms of time for selected values of � number inFigure 13, and wall temperature in Figure 14. Withthe passage of time, the velocity and acceleration ofthe plate approach zero which cause the f 0 value todecrease. This reality is more noticeable when the


Figure 12. E�ect of Pr no. on shear stress for di�erentvalues of plate velocity when Tw = 100�C and Pr=0.7.

Figure 13. E�ect of � parameter on f 0 pro�les atdi�erent times when Tw = 125�C and Pr=0.7.

density variations with respect to temperature arenegligible, which are evident for the cases of � = 0 inFigure 13 and a wall temperature of 50�C in Figure 14.It is worth mentioning that for an incompressible uidwith � = 0, the increase of time, which makes the platevelocity and acceleration close to zero, approaches thef 0 pro�le to that of Homann ow, which is capturedfor � > 5:0 in Figure 13.

In Figure 15, one can see the sample results ofw-component distributions with respect to time andselected wall temperatures. The e�ect of passage oftime on distributions of dimensionless heat transfercoe�cient and shear stress in a wide range of walltemperature is reported in Figures 16 and 17. As itis revealed, the increase of time, �rstly, causes thedecrease in the amounts of both heat transfer coe�cient

Figure 14. E�ect of wall temperature on f 0 pro�les atdi�erent times when � = 0:003 and Pr=0.7.

Figure 15. E�ects of wall temperature on dimensionlessform of w component distributions at di�erent times when� = 0:003 and Pr=0.7.

and shear stress. This trend is continued to reach aconstant value at steady state conditions. It should benoted that the more the wall temperature, the less Hand shear stress will be at any selected time.

Later on, the variations of dimensionless temper-ature and pressure versus time are shown in Figures 18and 19 for selected Pr numbers, 0.3 and 1, respectively.As the velocity and acceleration of the plate vanish, theincrease in temperature and decrease in the absolutevalue of pressure are captured. Besides, the enhance-ment of Pr number causes the thermal boundary layerthickness to decrease (Figure 18), however, it does nothave any considerable e�ect on pressure distributions(Figure 19).


Figure 16. Distributions of dimensionless heat transfercoe�cient at unsteady procedure for di�erent values ofwall temperature when � = 0:003 and Pr=0.7.

Figure 17. Distributions of shear stress on the plate inunsteady procedure for di�erent values of walltemperature when � = 0:003 and Pr=0.7.

6. Conclusions

A similarity solution for the problem of unsteady three-dimensional axisymmetric stagnation-point ow andheat transfer of a viscous, compressible uid on anaccelerated at plate has been obtained in this paperwhen an external low Mach number ow with strainrate a impinges on this plate. Firstly, the intro-duced similarity transformations are used to reducethe unsteady Navier-Stokes and energy equations to acoupled system of nonlinear ordinary di�erential equa-tions. The density of the uid changes because of thetemperature di�erence existing between the plate andincoming far �eld ow. A Boussinesq approximationis used to take the density variations into account.

Figure 18. E�ect of Pr on � pro�les at di�erent timeswhen Tw = 125�C and � = 0:003.

Figure 19. E�ect of Pr on pressure distributions atdi�erent times when Tw = 125�C and � = 0:003.

The results achieved in this paper were presented for awide range of parameters characterizing the problemincluding volumetric expansion coe�cient (�), walltemperature, Prandtl number and plate velocity. Thesolutions obtained show that the thickness of velocityand thermal boundary layer for a plate receding fromthe impinging ow is much more than those whenthe plate moves toward the incoming potential ow.Moreover, it was shown that the increase of � andPr numbers does not have any signi�cant e�ect ondimensionless heat transfer coe�cient and shear stressat steady state conditions. However, the passage oftime causes H and ~� to decrease for an exponentiallymoving plate.


Nomenclature

a(t) Time-dependent ow strain rateao Flow strain rate at time = 0c Density ratiof; g Similarity functionsh Local heat transfer coe�cientH Dimensionless heat transfer coe�cientk Thermal conductivity of the uidp PressureP Dimensionless pressurePr Prandtl numberS; _S; �S Displacement, velocity and acceleration

of the plate, respectively, in z-direction~S; ~_S; ~�S Dimensionless displacement, velocity

and acceleration of the plate,respectively, in z-direction

T Temperatureu;w Velocity components near the plate in

x and z directionsU;W Potential region velocity components

in x and z directionsr; z Cylindrical coordinates

Greeks

� Volumetric expansion coe�cient� Similarity variable� Dynamic viscosity� Dimensionless temperature� Shear stress� Dimensionless time� Variable (z � S(t))� Dimensionless x axis� Density� Kinematic viscosity

Subscripts

o Stagnation pointw Wall1 In�nite

References

1. Hiemenz, K. \Boundary layer for a homogeneous owaround a dropping cylinder", Dinglers Polytechnic J.,326, pp. 321-410 (1911).

2. Homann, F.Z. \E�ect of high speed over cylinder andsphere", Z. Angew. Math. Mech., 16, pp. 153-164(1936).

3. Howarth, L. \The boundary layer in three-dimensional ow. Part II: The ow near a stagnation point", Philos.Mag., 42(7), pp. 1433-1440 (1951).

4. Libby, P.A. \Wall shear at a three-dimensional stag-nation point with a moving wall", AIAA J., 12, pp.408-409 (1974).

5. Libby, P.A. \Laminar ow at a three-dimensionalstagnation point with large rates of injection", AIAAJ., 14, pp. 1273-1279 (1976).

6. Wang, C.Y. \Axisymmetric stagnation ow towards amoving plate", AIChE J., 19, pp. 1080-1081 (1973).

7. Saleh, R. and Rahimi, A.B. \Axisymmetric stagnation-point ow and heat transfer of a viscous uid on a mov-ing cylinder with time-dependent axial velocity anduniform transpiration", Journal of Fluids Engineering,126, pp. 997-1005 (2004).

8. Rahimi, A.B. and Saleh, R. \Axisymmetric stagnation-point ow and heat transfer of a viscous uid on arotating cylinder with time-dependent angular velocityand uniform transpiration", Journal of Fluids Engi-neering, 129, pp. 107-115 (2007).

9. Rahimi, A.B. and Saleh, R. \Similarity solution ofunaxisymmetric heat transfer in stagnation-point owon a cylinder with simultaneous axial and rotationalmovements", Journal of Heat Transfer, TechnicalBrief, 130, pp. 054502-1-054502-5 (2008).

10. Baris, S. \Steady three-dimensional ow of a secondgrade uid towards a stagnation point at a moving atplate", Turkish J. Eng. Env. Sci., 27, pp. 21-29 (2003).

11. Rahimi, A.B. and Esmaeilpour, M. \Axisymmetricstagnation ow obliquely impinging on a moving cir-cular cylinder with uniform transpiration", Int. J.Numer. Meth. Fluids, 65, pp. 1084-1095 (2011).

12. Abbasi, A.S. and Rahimi, A.B. \Non-axisymmetricthree-dimensional stagnation-point ow and heattransfer on a at plate", Journal of Fluids Engineering,131, pp. 074501.1-074501.5 (2009).

13. Abbasi, A.S. and Rahimi, A.B. \Three-dimensionalstagnation ow and heat transfer on a at plate withtranspiration", Journal of Thermophysics and HeatTtransfer, 23, pp. 513-521 (2009).

14. Abbasi, A.S., Rahimi, A.B. and Niazman, H. \Exactsolution of three-dimensional unsteady stagnation owon a heated plate", Journal of Thermodynamics andHeat Transfer, 25, pp. 55-58 (2011).

15. Weidman, P.D. and Sprague, M.A. \Flows inducedby a plate moving normal to stagnation-point ow",Acta Mech., 219(3,4), pp. 219-229 (2011). DOI10.1007/s00707-011-0458-2

16. Libby, P.A. \Heat and mass transfer at a general three-dimensional stagnation point", AIAA J., 5, pp. 507-517 (1967).

17. Nath, G. \Compressible axially symmetric laminarboundary layer ow in the stagnation region of ablunt body in the presence of magnetic �eld", ActaMechanica, 12, pp. 267-273 (1971).

18. Kumari, M. and Nath, G. \Unsteady laminar com-pressible boundary-layer ow at a three-dimensionalstagnation point", Journal of Fluid Mechanics, 87, pp.705-717 (1978).


19. Kumari, M. and Nath, G. \Unsteady compressible 3-dimensional boundary-layer ow near an asymmetricstagnation point with mass transfer", Int. J. Engineer-ing Science., 18, pp. 1285-1300 (1980).

20. Kumari, M. and Nath, G. \Self-similar solution ofunsteady compressible three-dimensional stagnation-point boundary layers", Journal of Applied Mathemat-ics and Physics (ZAMP), 32(3), pp. 267-276 (1981).

21. Vasantha, R. and Nath, G. \Unsteady compressiblesecond-order boundary layers at the stagnation pointof two-dimensional and axisymmetric bodies", Warme-und Sto�ubertragung, 20, pp. 273-281 (1986).

22. Vasantha, R. and Nath, G. \Semi-similar solutionsof the unsteady compressible second-order boundarylayer ow at the stagnation point", Int. J. Heat MossTransfer, 32, pp. 435-444 (1989).

23. Zheng, L., Zhang, X. and He, J. \Similarity solutionto a heat equation with convection in an in�nitemedium", Journal of University of Science and Tech-nology Beijing, 10(4), pp. 29-32 (2003).

24. Zheng, L., Zhang, X. and He, J. \Analytical solutionsto a compressible boundary layer problem with heattransfer", Journal of University of Science and Tech-nology Beijing, 11(2), pp. 120-122 (2004).

25. Zuccher, S., Tumin, A. and Reshotko, E. \Parabolicapproach to optimal perturbations in compressibleboundary layers", Journal of Fluid Mechanics, 556,pp. 189-216 (2006).

26. Turkyilmazoglu, M. \ Purely analytic solutions of thecompressible boundary layer ow due to a porousrotating disk with heat transfer", Journal of Physicsof Fluids, 21(10), pp. 106-104 (2009).

27. Mohammadiun, H. and Rahimi, A.B. \Axisymmetricstagnation-point ow and heat transfer of a viscous,

compressible uid on a cylinder with constant walltemperature", Journal of Thermo Physics and HeatTransfer, 26(3), pp. 494-502 (2012).

28. Mozayyeni, H.R. and Rahimi, A.B. \Three-dimensional stagnation ow and heat transfer of aviscous, compressible uid on a at plate", Journalof Heat Transfer-Transaction of ASME, 135(10),101702-1 - 101702-12 (2013).

Acknowledgement

This research has been supported �nancially by Fer-dowsi University of Mashhad under the contract num-ber 2/28533.

Biographies

Hamid Reza Mozayyeni was born in Torbat, Iran,in 1986. He received his BS degree in Mechanical En-gineering from Shahid Bahonar University of Kermanin 2008 and his M.S. degree in Mechanical Engineeringfrom Ferdowsi University of Mashhad in 2010, wherehe is currently a PhD degree student.

Asghar Baradaran Rahimi was born in Mashhad,Iran, in 1951. He received his B.S. degree in MechanicalEngineering from Tehran polytechnic, in 1974, anda PhD degree in Mechanical Engineering from TheUniversity of Akron, Ohio, USA, in 1986. He has been aprofessor in the Department of Mechanical Engineeringat Ferdowsi University of Mashhad since 2001. His re-search and teaching interests include heat transfer and uid dynamics, gas dynamics, continuum mechanics,applied mathematics and singular perturbation.

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