The boundary layers of an unsteady stagnation-point flow in a nanofluid

The boundary layers of an unsteady stagnation-point flow in a nanofluid

International Journal of Heat and Mass Transfer 55 (2012) 6499–6505 Contents lists available at SciVerse ScienceDirect International Journal of Heat...

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International Journal of Heat and Mass Transfer 55 (2012) 6499–6505

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

The boundary layers of an unsteady stagnation-point flow in a nanofluid Norfifah Bachok a, Anuar Ishak b,⇑, Ioan Pop c a

Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia c Department of Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania b

a r t i c l e

i n f o

Article history: Received 18 February 2012 Received in revised form 22 May 2012 Accepted 17 June 2012 Available online 13 July 2012 Keywords: Nanofluids Stagnation-point flow Unsteady boundary layer Dual solutions

a b s t r a c t The boundary layer of an unsteady two-dimensional stagnation-point flow of a nanofluid is further investigated. The similarity equations are solved numerically for three types of nanoparticles, namely copper (Cu), alumina (Al2O3), and titania (TiO2) in the water based fluid with Prandtl number Pr = 6.2. The skin friction coefficient, the local Nusselt number, and the velocity and temperature profiles are presented and discussed. Effects of the solid volume fraction parameter u on the fluid flow and heat transfer characteristics are thoroughly examined. Interesting observation is that there are dual solutions seen for negative values of the unsteadiness parameter A (decelerating flow with A < 0). Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Unsteady boundary layer plays important roles in many engineering problems like start-up process and periodic fluid motion. It has different behavior due to extra time-dependent terms, which will influence the fluid motion pattern and the boundary layer separation [1–3]. Some typical examples of unsteady boundary layers in the history of fluid mechanics are the Rayleigh and the Stokes oscillating plate problems [2,3]. Yang [4] investigated the unsteady boundary layer for a stagnation flow involving the starting up of a cylinder. Following the pioneer work by Yang [4], the problem was extended to unsteady axis-symmetric stagnation-point flow by William III [5] and to general three dimensional stagnation-point flow by Jankowski and Gersting [6]. The heat transfer behavior for the three dimensional unsteady stagnation-point flow was studied by Teipel [7]. The effect of the unsteadiness parameter was discussed and it has been found that the heat transfer was reduced with increasing unsteadiness parameter. The unsteady oblique stagnation-point flow was also investigated by Wang [8]. Recently, the boundary layers of an unsteady incompressible stagnation-point flow with mass transfer was considered by Fang et al. [9] and found multiple solutions for negative values of the unsteadiness parameters. The unsteady stagnation-point boundary layer problem is still a quite active area with many papers published in various journals. In a series of papers it has been discussed the axisymmetric stagnation flow over a cylinder [10], mixed convection flow near the ⇑ Corresponding author. E-mail address: [email protected] (A. Ishak). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.050

stagnation-point [11], and stagnation-point flow for hydromagnetic fluid [12], stagnation-point flow over a stretching sheet [13], and stagnation-point flow over a shrinking sheet [14]. Meanwhile, the unsteady stagnation-point flows for non-Newtonian fluids were investigated by Seshadri et al. [15], Xu et al. [16] and Baris and Dokuz [17], among others. Many problems on the unsteady viscous fluids can be found in the excellent book by Telionis [18]. Different from all studies mentioned above, the present paper deals with the problem of an unsteady two-dimensional stagnation-point flow of a nanofluid, with water as the based fluid. Most conventional heat transfer fluids, such as water, ethylene glycol, and engine oil, have limited capabilities in term of thermal properties, which, in turn, may impose serve restrictions in many thermal applications. On the other hand, most solids, in particular, metals, have thermal conductivities much higher, say, by one to three orders of magnitude, compared with that of liquids. Hence, one can then expect that fluid containing solid particles may significantly increase its conductivity. Many of the publications on nanofluids are about understanding their behavior so that they can be utilized where straight heat transfer enhancement is paramount as in many industrial applications, nuclear reactors, transportation, electronics as well as biomedicine and food. The broad range of current and future applications involving nanofluids have been given by Wong and Leon [19]. Nanofluid as a smart fluid, where heat transfer can be reduced or enhanced at will, has also been reported. These fluids enhance thermal conductivity of the base fluid enormously, which is beyond the explanation of any existing theory. They are also very stable and have no additional problems, such as sedimentation, erosion, additional pressure drop and non-Newtonian

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Nomenclature A Cf f Cp k Nux Pr qw Rex t T Tw T1 u,v U1 x, y

unsteadiness parameter skin friction coefficient dimensionless stream function specific heat at constant pressure thermal conductivity local Nusselt number Prandtl number surface heat flux local Reynolds number time fluid temperature plate temperature ambient temperature velocity components along the x - and y - directions, respectively free stream velocity Cartesian coordinates along the surface and normal to it, respectively

dimensionless temperature velocity ratio parameter kinematic viscosity dynamic viscosity fluid density surface shear stress stream function similarity variable

h k

m l q sw w g

Subscripts w condition at the surface of the plate 1 ambient condition nf nanofluid f fluid s solid Superscript 0 differentiation with respect to g

Greek symbols a thermal diffusivity u nanoparticle volume fraction parameter

behavior, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement. These suspended nanoparticles can change the transport and thermal properties of the base fluid. The comprehensive references on nanofluids can be found in the recent book by Das et al. [20] and in the review papers by Buongiorno [21], Daungthongsuk and Wongwises [22], Trisaksri and Wongwises [23], Ding et al. [24], Kakaç and Pramuanjaroenkij [25], Lee et al. [26], Eagen et al. [27], and Fan and Wang [28]. It is worth mentioning that the nanofluid model proposed by Buongiorno [21] was very recently used by Nield and Kuznetsov [29–31], Neild and Kuznetsov [32–34], Khan and Pop [35] and Bachok et al. [36,37] in their papers. Different from the above model, the present paper considers a problem using the nanofluid model proposed by Tiwari and Das [38], which was also been used by several authors (cf. Abu-Nada [39], Muthtamilselvan et al. [40], Abu-Nada and Oztop [41], Talebi et al. [42], Bachok et al. [43–46]). We investigate theoretically here the boundary layer of an unsteady two-dimensional stagnation-point flow of a nanofluid. To the best of our knowledge, the results of this paper are new and they have not been published before.

2. Mathematical formulation Consider a two-dimensional laminar viscous and incompressible stagnation-point flow of an unsteady nanofluid with a velocity of the outer or inviscid flow of the form U1(x, t) = ax(1  ct)1, where a and c are positive constants. We assume that the surface and ambient temperatures are constants and are Tw and T1, respectively. The x-axis runs along the free stream direction and the y-axis is perpendicular to it. The simplified two-dimensional equations governing the forced convection flow in the boundary layer of this problem are

@u @ v þ ¼0 @x @y

@T @T @T @2T þu þv ¼ anf 2 @t @x @y @y with the initial and boundary conditions

t<0:

u ¼ v ¼ 0;

tP0:

u ¼ 0;

u ! U 1 ðx; tÞ;

ð2Þ

T ! T1

for any x; y

T ¼ Tw

at y ¼ 0

ð4Þ

as y ! 1

lf ; ð1  uÞ2:5 knf ðks þ 2kf Þ  2uðkf  ks Þ ¼ ðqC p Þnf ¼ ð1  uÞðqC p Þf þ uðqC p Þs ; : kf ðks þ 2kf Þ þ uðkf  ks Þ

anf ¼

knf ; qnf ¼ ð1  uÞqf þ uqs ; ðqC p Þnf

lnf ¼

ð5Þ Here, u is the nanoparticle volume fraction, (qCp)nf is the heat capacity of the nanofluid, knf is the thermal conductivity of the nanofluid, kf and ks are the thermal conductivities of the fluid and of the solid fractions, respectively, and qf and qs are the densities of the fluid and of the solid fractions, respectively. It should be mentioned that the use of the above expression for knf/kf is restricted to spherical nanoparticles where it does not account for other shapes of nanoparticles (Abu-Nada [39]). Also, the viscosity of the nanofluid lnf has been approximated by Brinkman [48] as viscosity of a base fluid lf containing dilute suspension of fine spherical particles. The governing Eqs. (1)–(3) subject to the initial and boundary conditions (4) can be expressed in a simpler form if we assume that U1(x, t) = ax/(1  ct), where both a (>0) and c are constants of the dimension t1 with c showing the unsteadiness of the problem. Further, we introduce the following similarity variables



@u @u @u @U 1 @U 1 lnf @ u þu þv ¼ þ U1 þ @t @x @y @t @x qnf @y2

T ¼ T1

v ¼ 0;

where u and v are the velocity components along the x- and y- axes, respectively, T is the temperature of the nanofluid, lnf is the viscosity of the nanofluid, anf is the thermal diffusivity of the nanofluid and qnf is the density of the nanofluid, which are given by (Oztop and Abu-Nada [47])

ð1Þ 2

ð3Þ



a mf ð1  ctÞ

1=2

y;





mf a 1=2 xf ðgÞ; 1  ct

hðgÞ ¼

T  T1 Tw  T1

ð6Þ

where w is the stream function defined as u = ow/oy and v = ow/ox, which identically satisfies Eq. (1). Employing the similarity

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variables (6), Eqs. (2) and (3) reduce to the following nonlinear ordinary differential equations:   1 00 f 000 þ ff  f 02 þ 1  A f 0 þ gf 00  1 ¼ 0 2 ð1  uÞ ð1  u þ uqs =qf Þ

ð7Þ

  knf =kf 1 A h i h00 þ f  g h0 ¼ 0 Pr 1  u þ uðqC p Þ =ðqC p Þ 2 s f

ð8Þ

1

2:5

where primes denote differentiation with respect to g, Pr (=mf/af) is the Prandtl number and A = c/a is the parameter that measures the unsteadiness (Fang et al. [49]). The initial and boundary conditions (4) now become

f ð0Þ ¼ 0; f 0 ðgÞ ! 1;

f 0 ð0Þ ¼ 0;

hð0Þ ¼ 1

hðgÞ ! 0 as g ! 1:

ð9Þ

It is worth mentioning that for A = 0 (steady-state flow) and u = 0 (regular fluid), Eq. (7) reduces to the classical equation of the steady two-dimensional flow near the stagnation point first studied by Hiemenz [50]. The physical quantities of interest are the skin friction coefficient Cf and the local Nusselt number Nux, which are defined as

Cf ¼

sw xqw ; Nux ¼ ; kf ðT w  T 1 Þ qf U 21

ð10Þ

where the surface shear stress sw and the surface heat flux qw are given by

sw ¼ lnf

  @u ; @y y¼0

qw ¼ knf



@T @y

 ;

ð11Þ

y¼0

with lnf and knf being the dynamic viscosity and thermal conductivity of the nanofluids, respectively. Using the similarity variables (6), we obtain

C f Re1=2 ¼ x

1 ð1  uÞ2:5

¼ Nux =Re1=2 x

f 00 ð0Þ;

knf 0 h ð0Þ; kf

ð12Þ

ð13Þ

where Rex = U1x/mf is the local Reynolds number. 3. Results and discussion Numerical solutions to the governing ordinary differential Eqs. (7) and (8) with the boundary conditions (9) were obtained using a shooting method. This method has been successfully used by Bhattacharyya [14] and Bachok et al. [44–46] to solve various problems related to boundary layer flow and heat transfer. The details description of this method also can be found in Refs. [51–53]. Further, it is noticed that the second solution was obtained by setting different initial guesses for the missing values of f 00 ð0Þ and h0 (0), where both profiles (first and second solutions) satisfy the boundary conditions (9) asymptotically but with different shapes. The effects of the solid volume fraction of the nanofluid u and the parameter A that measures the unsteadiness are analyzed for three different nanofluids, namely Cu-water, Al2O3-water, and TiO2water, as the working fluids. Following Oztop and Abu-Nada [47] or Khanafer et al. [54], the value of the Prandtl number Pr is taken as 6.2 (water) and the volume fraction of nanoparticles is from 0 to 0.2 (0 6 u 6 0.2) in which u = 0 corresponds to the regular fluid. Thus, for A = 0 and u = 0 it is found that f 00 ð0Þ ¼ 1:232587669, while Hiemenz [50] has reported the value f 00 ð0Þ ¼ 1:232588, which shows an excellent agreement. Table 2 present the values of f 00 ð0Þ and –h0 (0) for selected values of A and different nanoparticles

Table 1 Thermophysical properties of fluid and nanoparticles (Oztop and Abu-Nada, [47]). Physical properties

Fluid phase (water)

Cu

Al2O3

TiO2

Cp (J/kgK) q (kg/m3) k (W/mK)

4179 997.1 0.613

385 8933 400

765 3970 40

686.2 4250 8.9538

when u = 0.1. It is seen from this table that the values of jf 00 ð0Þj and –h0 (0) have greater values for Cu than for Al2O3 and TiO2. This is due to the physical properties of fluid and nanoparticles (i.e., thermal conductivity of Cu is much higher than that of Al2O3 and TiO2), see Table 1. The variation with A of the reduced skin friction f 00 ð0Þ and reduced surface heat flux –h0 (0) are shown in Figs. 1–4 for some values of parameter A that measures the unsteadiness and the nanoparticle volume fraction u. It is seen that there is one solution for (A > 0) [ (A = Ac), two solutions, an upper branch and a lower branch solutions, respectively, for Ac < A < 0, where Ac is a critical value of A for which the solution exists, and no solution for A < Ac < 0. Based on our computation, Ac = 4.5066. The values of f 00 ð0Þ decrease with decreasing A for the upper branch solution from positive to negative values, while for the lower branch, f 00 ð0Þ is also changing from a positive to a negative value. The upper branch (first) solution makes a U-turn at this point and continues to the lower branch (second) solution. The lower branch solutions continue further and terminate at certain values of A. It should be remarked that the computations have been performed until the point where the solution does not converge, and the calculations were terminated at that point. The solution domain of the temperature gradient at the surface –h0 (0), which is proportional to the local Nusselt number, are plotted in Figs. 2 and 4. The first (upper branch) solution and second (lower branch) solution refer to the curves shown in Figs. 1–4, where the first solution has larger values of f 00 ð0Þ and –h0 (0) compared to the second solution. It is seen that h0 (0) decreases with increasing values of the unsteadiness parameter A. It is notice that the first solutions of f 00 ð0Þ and h0 (0) are stable and physically realizable, while the second solutions are not. The procedure for showing this has been described by Weidman et al. [55], Merkin [56] and very recently by Postelnicu and Pop [57], so that we will not repeat it here. For the second solution, the behavior differs greatly from the first solution. When A is less than certain value, A0, the quantity h0 (0) decrease with the increase of A and when A > A0 an opposite trend is observed. This value of A0 does not depend on solid volume fraction of nanofluid and types of the nanoparticles. Figs. 5–8 illustrate the variations of the skin friction coefficient C f Re1=2 and the local Nusselt number Nux Re1=2 , given by Eqs. (12) x x and (13) with the nanoparticle volume fraction parameter u for

Table 2 Values of f 00 ð0Þ and h0 (0) for some values of A and different nanoparticles when u = 0.1 and Pr = 6.2. A

1 1 2 3 4

Cu

Al2O3

TiO2

f 00 ð0Þ

h0 (0)

f 00 ð0Þ

h0 (0)

f 00 ð0Þ

h0 (0)

1.7604 1.0845 [-1.1573] 0.6499 [1.9885] 0.1045 [2.3251] 0.6757 [2.2646]

0.4681 1.4957 [0.4638] 1.8532 [1.3327] 2.1550 [1.8455] 2.4089 [2.2590]

1.4967 0.9221 [0.9839] 0.5525 [1.6906] 0.0888 [1.9768] 0.5745 [1.9254]

0.4032 1.4663 [0.4420] 1.8335 [1.3565] 2.1428 [1.8625] 2.4039 [2.2688]

1.5128 0.9320 [-0.9945] 0.5585 [1.7088] 0.0898 [1.9981] 0.5806 [1.9461]

0.4076 1.4910 [0.4491] 1.8651 [1.3817] 2.1801 [1.8962] 2.4461 [2.3094]

[ ] Second solution.

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Cu

1.5

3

ϕ = 0, 0.1, 0.2

1

A = -4.5066 c

first solution second solution

2.5

0.5

2

f ′′ (0)

0

- θ′(0)

-0.5 -1

2

-2

ϕ = 0, 0.1, 0.2 c

0 -4

-3

-2 A

-1

0

1

3

-5

-3

2

-2

-1

0

3

1

Fig. 4. Variation of –h0 (0) with A for different nanoparticles with u = 0.1 and Pr = 6.2.

A = -4.5066

ϕ=0

-4

Cu, TiO , 2 Al O

A

Fig. 1. Variation of f 00 ð0Þ with A for some values of u(0 6 u 6 0.2) for Cu-water working fluid.

3

c

A = 0.5 Cu

2.5 ϕ = 0.1

Al O

ϕ = 0.2

2

2.5

3

TiO

2

1/2 x

1.5

C Re

- θ′(0)

2

ϕ = 0.1 first solution second solution

0.5

A = -4.5066

-3

2

3

1

-1.5

-2.5

Cu, Al O , TiO

1.5

f

1

2

Cu

0.5

first solution second solution

0 -5

-4

-3

1.5

-2 A

-1

0

1 1

Fig. 2. Variation of –h0 (0) with A for some values of u(0 6 u 6 0.2) for Cu-water working fluid and Pr = 6.2.

0

0.05

0.1 ϕ

0.15

0.2

Fig. 5. Variation of the skin friction coefficient with u for different nanoparticles with A = 0.5.

ϕ = 0.1 Cu, TiO , Al O 2 2 3 first solution second solution

1

A = 0.5 Cu Al O

1.2

0

f ′′(0)

2

3

TiO

x

Nu Re

-1/2 x

-1

-2

Cu, TiO , Al O 2

A = -4.5066

2

2

1.1

1

0.9

3

c

-3

-4

-3

-2

-1

0

1

A Fig. 3. Variation of f 00 ð0Þ with A for different nanoparticles with u = 0.1.

three different nanoparticles: copper Cu, alumina Al2O3, and titania TiO2 with A = 0.5 and A = 2.0, respectively. One can see that these

0.8

0.7

0

0.05

0.1 ϕ

0.15

0.2

Fig. 6. Variation of the Nusselt number with u for different nanoparticles with A = 0.5 and Pr = 6.2.

N. Bachok et al. / International Journal of Heat and Mass Transfer 55 (2012) 6499–6505

6503

1 0.5 A = - 2.0 Cu

-0.5

Al O 2

C Re

1/2 x

0

f

3

TiO

-1

2

-1.5 -2 -2.5 -3 -3.5 0

0.05

0.1

ϕ

0.15

0.2

Fig. 7. Variation of the skin friction coefficient with u for different nanoparticles with A =  2.0. Fig. 10. Temperature profiles for some values of u(0 6 u 6 0.2) for Cu-water working fluid with A =  0.5 and Pr = 6.2.

3

A = - 2.0 Cu Al O 2

TiO

2

x

Nu Re

-1/2 x

2.5

3

2

1.5

0

0.05

0.1

ϕ

0.15

0.2

Fig. 8. Variation of the Nusselt number with u for different nanoparticles with A =  2.0 and Pr = 6.2.

Fig. 11. Velocity profiles for different nanoparticles with u = 0.1 and A =  3.0.

Fig. 9. Velocity profiles for some values of u(0 6 u 6 0.2) for Cu-water working fluid with A =  0.5.

quantities increase almost linearly with u. In addition, it is noted that the lowest heat transfer rate is obtained for the TiO2 nanoparticles due to domination of conduction mode of heat transfer. This is because TiO2 has the lowest thermal conductivity compared to Cu and Al2O3, as can be seen from Table 1. This behavior of the local Nusselt number is similar with that reported by Oztop and AbuNada [47], but for the convective flow in a cavity. The thermal conductivity of Al2O3 is approximately one tenth of Cu, as given in Table 1. However, a unique property of Al2O3 is its low thermal diffusivity. The reduced value of thermal diffusivity leads to higher temperature gradients and, therefore, higher enhancement in heat transfers. The Cu nanoparticles have high values of thermal diffusivity and, therefore, this reduces the temperature gradients which affect the performance of the Cu-water working fluid. The velocity and temperature profiles for some values of the governing parameters are presented in Figs. 9–14. These profiles have essentially the same form as in the case of regular fluid (u = 0). Figs. 9–14 show that the far field boundary conditions (9) are satisfied asymptotically, thus support the validity of the

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Fig. 12. Temperature profiles for different nanoparticles with u = 0.1, A =  3.0 and Pr = 6.2.

Fig. 14. Temperature profiles for some values of A for Cu-water working fluid with u = 0.1 and Pr = 6.2.

differential equations using a similarity transformation. The resulting system of nonlinear ordinary differential equations is solved numerically for three types of nanoparticles, namely copper (Cu), alumina (Al2O3), and titania (TiO2) in the water based fluid with Prandtl number Pr = 6.2, to investigate the effect of the solid volume fraction parameter u and the parameter A that measures the unsteadiness on the fluid flow and heat transfer characteristics. Dual solutions were found for the boundary layer equations for negative values of the parameter A (decelerated flow). It is found that the inclusion of nanoparticles into the water base fluid has produced an increase in the skin friction and heat transfer coefficients, which increases appreciably with an increase of the nanoparticle volume fraction parameter u. Nanofluids are capable to change the velocity and temperature profiles in the boundary layer. The type of nanofluids is a key factor for heat transfer enhancement. The highest values of the skin friction coefficient and the local Nusselt number were obtained for the Cu nanoparticles compared with the others. In the future, this study can be extended to different models of nanofluid (see Ref. [21]). Acknowledgement Fig. 13. Velocity profiles for some values of A for Cu-water working fluid with u = 0.1.

numerical results, besides supporting the existence of the dual solutions shown in Table 2 as well as in Figs. 1–4. It can be further noticed that the boundary layer thicknesses for the lower branch (second) solutions are higher than those of the upper branch (first) solutions, which indicates that the lower branch solutions are unstable. For the stable solution shown in Fig. 10, the magnitude of the temperature gradient at the surface decreases as the solid volume fraction of the nanofluid u increases, and consequently decreases the reduced surface heat flux. Opposite behaviors are observed for the temperature profiles shown in Figs. 12 and 14, for different values of A. This observation is consistent the variation of h0 (0) presented in Fig. 2.

4. Conclusions We have presented an analysis for the flow and heat transfer characteristics of an unsteady stagnation-point flow in a nanofluid. The governing partial differential equations are reduced to ordinary

The authors wish to express their thanks to the Reviewers for the valuable comments and suggestions. This work was supported by research Grants from the Ministry of Higher Education, Malaysia (Project Code: FRGS/1/2012/SG04/UKM/01/1) and the Universiti Kebangsaan Malaysia (Project Code: DIP-2012-31). References [1] F.T. Smith, Steady and unsteady boundary layer separation, Ann. Rev. Fluid Mech. 18 (1986) 197–220. [2] F.M. White, Viscous Fluid Flow, second ed., McGraw-Hill, New York, 1991. [3] H. Schlichting, K. Gersten, Boundary Layer Theory, 8th revised and enlarged ed. (English)., Springer, New York, 2000. [4] K.T. Yang, Unsteady laminar boundary layers in an incompressible stagnation flow, Trans. ASME: J. Appl. Mech. 25 (1958) 421–427. [5] J.C. Williams III, Nonsteady stagnation-point flow, Amer. Inst. Aeronaut. Astronaut. J. 6 (1968) 2417–2419. [6] D.F. Jankowski, J.M. Gersting, Unsteady three-dimensional stagnation-point flow, Amer. Inst. Aeronaut. Astronaut. J. 8 (1970) 187–188. [7] I. Teipel, Heat transfer in unsteady laminar boundary layers at an incompressible three-dimensional stagnation flow, Mech. Res. Commun. 6 (1979) 27–32. [8] C.Y. Wang, The unsteady oblique stagnation point flow, Phys. Fluids 28 (1985) 2046–2049.

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