International Journal of Non-Linear Mechanics 42 (2007) 1010–1017www.elsevier.com/locate/nlm

The effects of transpiration on the boundary layer flow andheat transfer over a vertical slender cylinder

Anuar Ishaka, Roslinda Nazara,∗, Ioan Popb

aSchool of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, MalaysiabFaculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

Received 14 March 2006; received in revised form 18 May 2007; accepted 30 May 2007

Abstract

This paper deals with a theoretical (numerical) analysis of the effects that blowing/injection and suction have on the steady mixed convectionor combined forced and free convection boundary layer flows over a vertical slender cylinder with a mainstream velocity and a wall surfacetemperature proportional to the axial distance along the surface of the cylinder. Both cases of buoyancy forces aid and oppose the developmentof the boundary layer are considered. Similarity equations are derived and their solutions are dependent upon the mixed convection parameter,the non-dimensional transpiration parameter and the curvature parameter, as well as of the Prandtl number. Dual solutions for the previouslystudied mixed convection boundary layer flows over an impermeable surface of the cylinder are shown to exist also in the present problem foraiding and opposing flow situations.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Combined forced and free convection flows; Vertical slender cylinder; Boundary layer; Suction/injection; Numerical similarity solutions

1. Introduction

Mixed convection flows, or combined forced and free con-vection flows arise in many transport processes in natural andengineering devices, such as, for example, atmospheric bound-ary layer flow, heat exchangers, solar collectors, nuclear reac-tors, electronic equipments, etc. Such a process occurs whenthe effect of the buoyancy force in forced convection or theeffect of forced flow in free convection becomes significant. Theeffect is especially pronounced in situations where the forced-flow velocity is low and the temperature difference is large (see[1]). Over the last several decades several analyses of mixedconvection flow of a viscous and incompressible fluid over avertical flat plate have been performed. Analytical and numer-ical solutions for the temperature and the velocity fields havebeen obtained both for prescribed wall temperatures and for pre-scribed wall heat fluxes. However, the problem of mixed con-vection in axisymmetric boundary layer flow has received littleattention so far. Mahmood and Merkin [2] have considered the

∗ Corresponding author. Tel.: +60 3 89213371; fax: +60 3 89254519.E-mail address: rmn72my@yahoo.com (R. Nazar).

0020-7462/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2007.05.004

similarity equations for the axisymmetric mixed convectionboundary layer flow along a vertical cylinder in the case of op-posing flow that is when the buoyancy force and the flow are inopposite directions. Dual solutions were reported only for thisopposing flow. Ridha [3] has later shown that dual solutionsexist also for the aiding flow regime, that is when the buoy-ancy force acts in the same direction as the flow. The interestin similarity solutions stems from the fact that they provide in-termediate asymptotic solutions related to the more complexnon-similar ones (see [4]). However, to our best knowledge, thesimilarity solutions for the mixed convection boundary layerflow along a vertical permeable cylinder have not been studiedbefore. It is worth mentioning that the similarity equations forthe free convection boundary layer flow on a vertical perme-able (with blowing and suction) flat plate with prescribed wallheat flux and prescribed wall temperature proportional to xn,where x is the usual distance measured along the plate and nis a positive constant, have been considered by Chaudhary andMerkin [5], and Merkin [6]. The differences between blowingand suction were found. Such solutions for a vertical perme-able surface embedded in a fluid-saturated porous medium havealso been studied by Chaudhary et al. [7,8].

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010–1017 1011

The fundamental governing equations for fluid mechanics arethe Navier–Stokes equations. This inherently non-linear set ofpartial differential equations has no general solution, and onlya small number of exact solutions have been found (see [9]).Exact solutions are important for the following reasons: (i)the solutions represent fundamental fluid-dynamic flows. Also,owing to the uniform validity of exact solutions, the basicphenomena described by the Navier–Stokes equations can bemore closely studied. (ii) The exact solutions serve as stan-dards for checking the accuracies of the many approximatemethods, whether they are numerical, asymptotic, or empirical.Explicit solutions are used as models for physical or numericalexperiments, and often reflect the asymptotic behavior of morecomplicated solutions. All explicit solutions for the boundarylayer equations are seemingly similarity solutions in the sensethat the longitudinal velocity component displays the sameshape of profile across any transverse section of the layer, seeSchlichting [10]. By an appropriate choice of the indepen-dent non-dimensional similarity variables, the boundary layerequations can therefore be reduced to ordinary differentialequations. In the rare cases when these equations can be solvedin closed form, the explicit solutions are obtained (see [11]).

The aim of this paper is to study the effects of transpiration(suction or injection) on the mixed convection or combinedforced and free convection flows in boundary layer flow alonga vertical permeable cylinder. Injection or withdrawal of fluidthrough a porous bounding heated or cooled wall is of generalinterest in practical problems involving film cooling, controlof boundary layers, etc. This can lead to enhanced heating (orcooling) of the system and can help to delay the transitionfrom laminar flow (see [5]). We mention to this end that sucha study has also been done by Massoudi [12], Weidman et al.[13,14] and Ishak et al. [15,16] for the classical problems of theboundary layers over a permeable wedge, moving flat platesand permeable vertical flat plates.

2. Governing equations

Consider the convective flow and heat transfer along a verti-cal permeable slender cylinder of radius a placed in a viscousand incompressible fluid of uniform ambient temperature T∞and constant density �∞. The equations of motion for such afluid, see Gebhart et al. [17] or Rajagopal et al. [18], are

∇ · v = 0, (1)

(v · ∇)v = − 1

�∞∇p + �∇2v + � − �∞

�∞g, (2)

(v · ∇)T = �∇2T , (3)

where v is the velocity vector, p is the pressure, T is the tem-perature of the fluid, g is the gravitation acceleration vector,� is the kinematic viscosity, � is the density, � is the ther-mal diffusivity and ∇2 is the Laplacian operator. We assumethat the density difference (� − �∞) in the buoyancy term ofthe momentum equation is given by the Oberbeck–Boussinesq

approximation, see Gebhart et al. [17] or Pop and Ingham [19],

� = �∞[1 − �(T − T∞)], (4)

where � is the thermal expansion coefficient.Further, we shall write Eqs. (1)–(3) in cylindrical coordinates

(x, r) with the x-axis is measured along the surface of thecylinder in the vertical direction and the r-axis is measured inthe radial direction. We will then apply the following boundarylayer approximations:

x = x/a, r = Re1/2(r/a), u = u/U∞,

w = Re1/2(w/U∞),

T = (T − T∞)/�T ,

p = (p − p∞)/�U2∞, (5)

where u and w are the velocity components along the x and yaxes, respectively, U∞ is the characteristic velocity, �T is thecharacteristic temperature and Re = U∞a/� is the Reynoldsnumber with � being the kinematic viscosity. We will considerthat the flow is symmetric relative to the transversal coordinate.Substituting variables (5) into Eqs. (1)–(3), using the bound-ary layer approximation that Re → ∞ and returning back tothe dimensional (without bar) physical variables, we obtain thefollowing boundary layer equations for the problem under con-sideration, see Mahmood and Merkin [2],

�

�x(ru) + �

�r(rw) = 0, (6)

u�u

�x+w

�u

�r= U

dU

dx+�

(�2u

�r2 + 1

r

�u

�r

)+g�(T −T∞), (7)

u�T

�x+ w

�T

�r= �

(�2T

�r2 + 1

r

�T

�r

), (8)

where U(x) is the mainstream velocity. We assume that theappropriate boundary conditions are

u = 0, w = V, T = Tw(x) at r = a,

u → U(x), T → T∞ as r → ∞, (9)

where V is the constant velocity of injection (V >) or suction(V < 0). Further, we assume that the mainstream velocity U(x)

and the temperature of the cylinder surface Tw(x) have the form

U(x) = U∞(x

�

), Tw(x) = T∞ + �T

(x

�

), (10)

where � is a characteristic length, U∞ is the characteristic ve-locity and �T is the characteristic temperature with �T > 0 fora heated surface and �T < 0 for a cooled surface.

3. The solution

It is worth mentioning that the partial differential equations(6)–(8) can be solved numerically using a finite-differencemethod or any other numerical methods, but in this paper wewill solve these equations using a similarity transformation,

1012 A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010–1017

i.e. by reducing these equations to ordinary differential equa-tions. Thus, following Mahmood and Merkin [2], we introducethe similarity variables

� = r2 − a2

2a

(U

�x

)1/2

, � = (U�x)1/2af (�),

�(�) = T − T∞Tw − T∞

, (11)

where � is the stream function defined as u = r−1��/�r andw = −r−1��/�x, which identically satisfy Eq. (6). By usingthis definition, we obtain

u = Uf ′(�), w = −a

r

(�U∞

�

)1/2

f (�), (12)

where primes denote differentiation with respect to �. In orderthat similarity solutions exist, V has to be of the form

V = −a

r

(�U∞

�

)1/2

f0, (13)

where f0 =f (0) and f0 < 0 is for mass injection and f0 > 0 isfor mass suction.

Substituting (11) into Eqs. (7) and (8), we get the followingordinary differential equations:

(1 + 2�)f ′′′ + 2f ′′ + ff ′′ + 1 − f ′2 + � = 0, (14)

(1 + 2�)�′′ + 2�′ + Pr(f �′ − f ′�) = 0, (15)

subject to the boundary conditions (9) which become

f (0) = f0, f ′(0) = 0, �(0) = 1,

f ′ → 1, � → 0 as � → ∞, (16)

where is the curvature parameter and is the buoyancy ormixed convection parameter defined as

=(

��

U∞a2

)1/2

, = g���T

U2∞, (17)

respectively. In Eq. (17), > 0 and < 0 correspond to the aid-ing flow (heated cylinder) and to the opposing flow (cooledcylinder), respectively, while = 0 represents the pure forcedconvection flow (buoyancy force is absent). It is worth men-tioning that, the similarity solution of Eqs. (14) and (15) isnot necessarily the only solution to the problem as the govern-ing equations are non-linear. We notice that when = 0 (i.e.a → ∞), the problem under consideration reduces to the flatplate case considered by Ishak et al. [16], while when f0 = 0 itreduces to the impermeable cylinder considered by Mahmoodand Merkin [2]. Furthermore, when both and f0 are zero,the present problem reduces to the problem considered byRamachandran et al. [20] for the case of an arbitrary surfacetemperature with n = 1 in their paper.

The physical quantities of interest are the skin friction co-efficient Cf and the local Nusselt number Nux , which aredefined by

Cf = �w

�U2/2, Nux = xqw

k(Tw − T∞), (18)

where the skin friction �w and the heat transfer from the plateqw are given by

�w = �

(�u

�r

)r=a

, qw = −k

(�T

�r

)r=a

, (19)

with � and k being the dynamic viscosity and thermal conduc-tivity, respectively. Using the similarity variables (11), we get

1

2Cf Re

1/2x = f ′′(0), Nux/Re

1/2x = −�′(0), (20)

where Rex = Ux/� is the local Reynolds number.

4. Results and discussion

Eqs. (14) and (15) subject to the boundary conditions (16)have been solved numerically for some values of the governingparameters , f0 and using a very efficient finite-differencescheme known as the Keller-box method, which is described inthe book by Cebeci and Bradshaw [21]. The solution is obtainedin the following four steps:

• Reduce Eqs. (14) and (15) to a first-order system.• Write the difference equations using central differences.• Linearize the resulting algebraic equations by Newton’s

method, and write them in matrix–vector form.• Solve the linear system by the block-tridiagonal-elimination

technique.

To conserve space, the details of the solution procedure arenot presented here. Following Mahmood and Merkin [2], weconsidered only Prandtl number unity throughout the paper,except for comparison with previously reported cases, by Ra-machandran et al. [20], Hassanien and Gorla [22] and Lok etal. [23]. This comparison is shown in Tables 1 and 2, and isfound to be in a very good agreement.

The variations of the skin friction coefficient f ′′(0) with together with their velocity profiles are shown in Figs. 1–7 for = 1, f0 = 0.5 and f0 = −0.5, respectively, while the respec-tive local Nusselt number �′(0) together with their temperatureprofiles are shown in Figs. 8–14, to support the validity of thenumerical results obtained. It is worth mentioning that all thevelocity and temperature profiles satisfy the boundary condi-tions (16). The results for the skin friction coefficient f ′′(0)

and the local Nusselt number �′(0) as a function of show thatit is possible to get dual solutions of the similarity equations(14) and (15) subject to the boundary conditions (16) for theaiding flow ( > 0) as well, besides that usually reported in theliterature for the opposing flow ( < 0). Also for > 0, thereis a favorable pressure gradient due to the buoyancy effects,which results in the flow being accelerated in a larger skin fric-tion coefficient than in the non-buoyant case ( = 0). For neg-ative values of , dual solutions (c < < 0), unique solution( = c) or no solution ( < c) is obtained, where c is thecritical value of for which the solution exists. At = c, bothsolution branches are connected, thus a unique solution is ob-tained. For the aiding flow, dual solutions exist for all values of considered in this study, whereas for the opposing flow, the

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010–1017 1013

Table 1Values of f ′′(0) for different values of Pr when = 1, f0 = 0 and = 0 (flat plate)

Pr Ramachandran Hassanien and Lok et al. [23] Present resultset al. [20] Gorla [22]

Upper branch Lower branch

0.7 1.7063 1.70632 1.7064 1.7063 1.23871 – – – 1.6754 1.13327 1.5179 – 1.5180 1.5179 0.582410 – 1.49284 – 1.4928 0.495820 1.4485 – 1.4486 1.4485 0.343640 1.4101 – 1.4102 1.4101 0.211150 – 1.40686 – 1.3989 0.172060 1.3903 – 1.3903 1.3903 0.141380 1.3774 – 1.3773 1.3774 0.0947100 1.3680 1.38471 1.3677 1.3680 0.0601

Table 2Values of −�′

(0) for different values of Pr when = 1, f0 = 0 and = 0 (flat plate)

Pr Ramachandran Hassanien and Lok et al. [23] Present resultset al. [20] Gorla [22]

Upper branch Lower branch

0.7 0.7641 0.76406 0.7641 0.7641 1.02261 – – – 0.8708 1.16917 1.7224 – 1.7226 1.7224 2.219210 – 1.94461 – 1.9446 2.494020 2.4576 – 2.4577 2.4576 3.164640 3.1011 – 3.1023 3.1011 4.108050 – 3.34882 – 3.3415 4.497660 3.5514 – 3.5560 3.5514 4.857280 3.9095 – 3.9195 3.9095 5.5166100 4.2116 4.23372 4.2289 4.2116 6.1230

-5 -4 -3 -2 -1 0 1 2 3 4

-2

-1

0

1

2

3

4

λ

f" (0)

Pr = 1 γ = 1

f0 = -0.5, 0, 0.5

Fig. 1. Skin friction coefficient f ′′(0) as a function of for various valuesof f0 when Pr = 1 and = 1.

solutions exist up to certain values of , i.e. c. Beyond thesecritical values, the boundary layer separates from the surface,thus no solution is obtained using the boundary layer approxi-mations. Moreover, from Figs. 1 and 8, we found that the val-ues of || for which the solution exists increase as f0 increases.

0 2 4 6 8 10 12 14 16 18−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

η

f ′ (η

)

γ = 1

Pr = 1

λ = −2

f0 = −0.5, 0, 0.5

upper branch

lower branch

Fig. 2. Velocity profiles f ′(�) for various values of f0 when Pr = 1, = 1and = −2.

Hence, suction delays the boundary layer separation. It is no-ticed that, for the opposing flow ( < 0), the lower branch so-lutions in Figs. 1–7 correspond to the upper branch solutionsin Figs. 8–14, and vice versa. Numerical results for the localNusselt number as presented in Figs. 8, 11 and 14 show that�′(0) approaches +∞ as → 0−, and −∞ as → 0+.

1014 A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010–1017

0 5 10 15 20 25 30 35 40

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

η

f0 = -0.5, 0, 0.5

upper branch

lower branch

f0 = -0.5, 0, 0.5

f' (η)

λ = 1

γ = 1

Pr = 1

Fig. 3. Velocity profiles f ′(�) for various values of f0 when Pr = 1, = 1and = 1.

-6 -4 -2 0 2 4

-2

-3

-1

0

1

2

3

4

5

λ

f" (0)

Pr = 1

f0 = 0.5

γ = 0, 1, 2

Fig. 4. Skin friction coefficient f ′′(0) as a function of for various valuesof when Pr = 1 and f0 = 0.5.

0 5 10 15 20 25 30−0.6

−0.4

− 0.2

0

0.2

0.4

0.6

0.8

1

η

f ′ (η

)

γ = 0, 1, 2

Pr = 1

λ = −2

f0 = 0.5

upper branch

lower branch

γ = 0, 1, 2

Fig. 5. Velocity profiles f ′(�) for various values of when Pr = 1, f0 = 0.5and = −2.

0 10 20 30 40 50 60 70

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

η

upper branch

lower branch

γ = 0, 1, 2

f' (η)

γ = 0, 1, 2

Pr = 1

f0 = 0.5

λ = 2

Fig. 6. Velocity profiles f ′(�) for various values of when Pr = 1, f0 = 0.5and = 2.

-4 -3 -2 -1 0 1 2 3 4

-2

-1

0

1

2

3

4

λ

f0 = -0.5

f" (0)

γ = 0, 1, 2

Pr = 1

Fig. 7. Skin friction coefficient f ′′(0) as a function of for various valuesof when Pr = 1 and f0 = −0.5.

In Figs. 1, 4 and 7, following the upper branch solution fora particular value of f0 or , one may expect that the solutionsuddenly disappears at the separation point = c, but thisis not the case. The solution makes an U−turn at this pointand form the lower branch solution. It is worth mentioningthat the separation occurs here at a point where f ′′(0) �= 0,that is a different feature from the classical boundary layertheory, where the separation takes place at f ′′(0) = 0. Wilksand Bramley [24] stopped the lower branch solutions whenthe wall heat transfer goes to zero. Although physically it is arealistic thing to do, it was shown in [2] that the lower branchsolutions could be continued further to the point where thebuoyancy parameter goes to zero and terminated at this point.It seems that Ridha [3,4] was the first to show the existence ofdual (non-uniqueness) solutions for both aiding and opposingflow situations. In the present paper, we show that the lowerbranch solutions exist in the opposing flow regime ( < 0) andthey continue to the aiding flow regime ( > 0), which is in

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010–1017 1015

-5 -4 -3 -2 -1 0 1 2 3 4

6

-4

-2

0

2

4

6

λ

Pr = 1

θ' (0)

γ = 1

f0 = -0.5, 0, 0.5

f0 = -0.5, 0, 0.5

Fig. 8. Variation of �′(0) with for various values of f0 when Pr = 1 and = 1.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

η

θ (η

)

γ = 1

Pr = 1

λ = −2

f0 = −0.5, 0, 0.5

upper branch

lower branch

Fig. 9. Temperature profiles �(�) for various values of f0 when Pr=1, =1and = −2.

0 5 10 15 20 25 30 35 40

-1.5

-1

-0.5

0

0.5

1

η

θ (η

)

γ = 1

Pr = 1

λ = 1f0 = -0.5, 0, 0.5

upper branch

lower branch

f0 = -0.5, 0, 0.5

Fig. 10. Temperature profiles �(�) for various values of f0 when Pr = 1, = 1 and = 1.

-6 -4 -2 0 2 4

-3

-2

-1

0

1

2

λ

Pr = 1

f0 = 0.5

γ = 0, 1, 2

θ' (0)

γ = 0, 1, 2

γ = 0

γ = 1

γ = 2

Fig. 11. Variation of �′(0) with for various values of when Pr = 1 andf0 = 0.5.

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

η

θ (η

)

γ = 0, 1, 2

Pr = 1

λ = −2

f0 = 0.5

upper branch

lower branch

γ = 0, 1, 2

Fig. 12. Temperature profiles �(�) for various values of when Pr = 1,f0 = 0.5 and = −2.

0 10 20 30 40 50 60 70

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

η

θ (η

)

Pr = 1

f0 = 0.5

upper branch

lower branch

γ = 0, 1, 2

γ = 0, 1, 2

λ = 2

Fig. 13. Temperature profiles �(�) for various values of when Pr = 1,f0 = 0.5 and = 2.

1016 A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010–1017

-4 -3 -2 -1 0 1 2 3 4

-2

-1

0

1

2

λ

Pr = 1

f0 = -0.5

γ = 0

γ = 2

γ = 1

θ' (0)

γ = 0, 1, 2

γ = 0, 1, 2

Fig. 14. Variation of �′(0) with for various values of when Pr = 1 andf0 = −0.5.

agreement with Ridha [3,4]. However, as discussed by Ridha[3,4] and Ishak et al. [16], the lower branch solutions haveno physical meaning. Although such solutions are deprived ofphysical significance, they are nevertheless of mathematicalinterest as well as of physical terms in so far as the differentialequations are concerned. Besides, similar equations may arisein other situations where the corresponding solutions could havemore realistic meaning.

The effects of the curvature parameter on the skin frictioncoefficient f ′′(0) are shown in Figs. 4 and 7, for f0 =0.5 (suc-tion) and f0 = −0.5 (injection), respectively, while the respec-tive local Nusselt number �′(0) are presented in Figs. 11 and14. The results for f0 = 0 (impermeable plate) were reportedby Mahmood and Merkin [2], but the lower branch solutionsterminate as → 0−, i.e. dual solutions were only reported forsituations of opposing flow. Our further investigation showedthat these dual solutions extend into the aiding flow regime thatis when the buoyancy force and the flow are in the same direc-tion. As mentioned in [3], the reason of this omission is perhapsdue to the misleading behavior of the non-dimensional temper-ature function � used in the similarity formulation, which showsthe existence of singularity at = 0 (i.e. �T = 0). However, itdoes not reflect the solution of the similarity stream functionf terminates in a singularity as → 0−. As can be seen fromFigs. 4 and 7, f ′′(0) remains regular and finite in the neigh-borhood of =0, even �′(0) undergoes a discontinuity at =0(see Figs. 11 and 14). The results for the case of suction andinjection showed similar features as the case of impermeablesurface. From the numerical results shown in Figs. 4 and 7, aswell as Figs. 11 and 14, it can be concluded that larger values of (smaller values of the cylinder diameter) delay the boundarylayer separation, which is in agreement with the results reportedby Mahmood and Merkin [2] for an impermeable surface.

5. Conclusions

We have theoretically studied the existence of dual similaritysolutions in mixed convection boundary layer flow about a

vertical slender cylinder in an incompressible viscous fluid. Thegoverning boundary layer equations were solved numericallyfor both aiding and opposing flow regimes using the Keller-boxmethod. Discussions for the effects of the curvature parameter, suction or blowing parameter f0 and the buoyancy parame-ter on the skin friction coefficient f ′′(0) and the local Nus-selt number �′(0) for Pr = 1 have been done. From the presentinvestigation, it may be concluded that:

• Dual solutions exist also for the aiding/assisting flow ( > 0).• For the opposing flow ( < 0), there are dual solutions, unique

solution or no solution. The solution curves bifurcate at thecritical point c (< 0).

• Suction (f0 > 0) delays the boundary layer separation, whileinjection (f0 < 0) accelerates it.

• Larger values of (smaller values of the cylinder diameter)delay the boundary layer separation.

Acknowledgments

The authors wish to express their very sincere thanks to thereviewers for their valuable comments and suggestions. Thiswork is supported by a research grant (IRPA project code: 09-02-02-10038-EAR) from the Ministry of Science, Technologyand Innovation (MOSTI), Malaysia.

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