Explicit solutions of wall jet flow subject to a convective boundary condition

Abstract

In this paper, an analysis is made on the laminar jet flow and heat transfer of a copper-water nanofluid over an impermeable resting wall. With the homogeneous model (Maïga et al. in Int. J. Heat Fluid Flow 26(4): 530-546, 2005), the Navier-Stokes equations describing this heat fluid flows are reduced to a set of differential equations via similarity transformations. An implicitly analytical solution overlooked in previous publications is discovered for the velocity distribution. We further present the explicit solutions with high precision for both the velocity and the temperature distributions. A mathematical analysis shows that those explicit solutions have exponential behaviors at far field. Besides, the effects of the volumetric fraction parameter and the dimensionless heat transfer parameter γ on the velocity and temperature profiles, as well as on the reduced local skin friction coefficient and the reduced Nusselt number, are examined in detail.

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R E S E A R C H Open Access

Explicit solutions of wall jet ow subject to a convective boundary condition

Ammarah Raees1*, Hang Xu1 and Muhammad Raees-ul-Haq2

*Correspondence: mailto:[email protected]

Web End [email protected]

1State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China Full list of author information is available at the end of the article

Abstract

In this paper, an analysis is made on the laminar jet ow and heat transfer of a copper-water nanouid over an impermeable resting wall. With the homogeneous model (Maga et al. in Int. J. Heat Fluid Flow 26(4): 530-546, 2005), the Navier-Stokes equations describing this heat uid ows are reduced to a set of dierential equations via similarity transformations. An implicitly analytical solution overlooked in previous publications is discovered for the velocity distribution. We further present the explicit solutions with high precision for both the velocity and the temperature distributions. A mathematical analysis shows that those explicit solutions have exponential behaviors at far eld. Besides, the eects of the volumetric fraction parameter and the dimensionless heat transfer parameter on the velocity and temperature proles, as well as on the reduced local skin friction coecient and the reduced Nusselt number, are examined in detail.

Keywords: jet ow; convective heat transfer; exponential behavior; nanouid ow; convective boundary condition; explicit solutions

1 Introduction

In an actual environment, the turbulent wall jets are dominant. While the laminar ones still attract many researchers owing to their numerous practical and potential applications including the cooling systems for the central processing units of laptops, spray-paint processing for vehicles or buildings, downwards-directed jets from a vertical-take-o aircraft spreading out over the ground, cooling jets over turbo-machinery components, sluice gate ows, and so on. The boundary layer approximation is used commonly as an eective approach for simplication of the laminar wall jet problems. The corresponding similarity (or non-similarity) solutions were found to be adequate for the prediction of their behaviors. Among those studies of the laminar wall jet, Glauert [] was the rst to use the name of wall jet for the description of such ows. By means of the similarity method, he reduced the two-dimensional Navier-Stokes equations to an ordinary dierential equation and then obtained a well-known implicit analytical solution. Merkin and Needham [] and Needham and Merkin [] extended Glauerts problem [] to the cases that both the wall moving and the wall blowing/suction are allowed. They then concluded that if wall moving is introduced, the similarity solution could be only possible when an appropriately lateral suction is applied through the moving surface. Magyari and Keller [] conrmed Merkin and Needhams conclusion [] that no similarity solution of Glauerts type exists for the wall jet owing over a resting surface in the presence of suction and/or injection.

2014 Raees et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Instead, they found that a particular kind of solution with algebraically decaying behavior is in existence for the wall suction case. While Cohen et al. [] argued that the solutions of Glauerts type could be possible for the resting surface case when the suction/injection velocity is proportional to xb for < b < . Xu et al. [] further found that except for the solutions given by Cohen et al. [], an innite number of solutions of an algebraical nature exist for the considered case. For heat transfer problems, Magyari et al. [] made an analysis on heat transfer characteristics of a boundary layer ow driven by a power-law shear at far eld over an impermeable at surface. Mathematically, their problem is equivalent to a wall jet owing over a heated at surface. They then presented both the analytical and the numerical solutions for the special cases of the isothermal and of the adiabatic at plate. Cossali [] considered a forced convection thermal boundary layer over an impermeable at surface driven by an outer power-law shear. He obtained a family of similarity solutions for various values of the exponent of the decaying exterior velocity prole and the exponent of the power-law prescribing the thermal condition on the wall. Very recently, Fan and Xu [] extended Magyari et al.s [] problem to the case that the plate is permeable. They found that both exponentially and algebraically decaying solutions could be possible when the suction is applied through the wall. They then presented a family of solutions covered for various power-law distributions of the outer velocity and the wall temperature both analytically and numerically.

As a new generation heat transfer uids, nanouids have received more and more attention for the reason that they possess a better thermal conductivity than that of the traditional heat transfer uids. Many researchers have made great eorts to investigate the heat uid ow in nanouids under various circumstances since the heat transfer intensication due to dilute nanouids could provide opportunities for a number of innovative applications in industrial sectors such as transportation, power generation, micro-manufacturing, thermal therapy for cancer treatment, chemical and metallurgical sectors, as well as heating, cooling, ventilation, and air-conditioning. Several approaches have been suggested for modeling nanouids. Typical models include the homogeneous model [, ], the dispersion model [], the Buongiorno model [] and so on. Note that there are controversies about the validity and applicability of those nanouids models for the prediction of the behavior of nanouids. Readers are referred to [] for more details. Among those models, the homogeneous model is the most popular one since it is very convenient to extend the conventional conservation equations for pure uids to nanouids. As a result, all traditional heat transfer correlations regarding the computation of thermophysical properties could be suitable for nanouids as well. This model is only valid for dilute nanouids with a similar behavior to Newtonian uids. When the concentration of the nanouids grows high, the nanouids no longer have the nature of Newtonian uids, but exhibit behaviors like non-Newtonian uids. In such a situation, this model cannot be used for modeling nanouids. Fortunately, it has been found that, even for dilute nanouids, the heat transfer enhancement can still be improved signicantly as compared with the traditional heat transfer uids. With the homogeneous model, some investigations have been done towards understandings of the heat transfer characteristics of nanouids. For examples, Bachok et al. [] investigated a boundary layer ow and heat transfer due to a rotating disk immersed in a nanouid. Their theoretical results show that one can distinguish how the addition of nanoparticles into pure uids can improve the heat transfer capability of the uids even if the amount of added nanoparticles is small. Rohni []

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made an analysis for a viscous nanouid ow and heat transfer over an unsteady shrinking at surface. Considering three kinds of nanouids, including Cu-water, AlO-water and TiO-water, they concluded that the nanoparticle volume fraction parameter (which is associated with the concentration of nanouids) and types of nanouid play a key role for determination of the ow behavior. Similar conclusions were, respectively, given by Yacob et al. [] and Vajravelu et al. [] via their investigations about the ow and heat transfer of nanouids past a wedge and over a at surface.

In previous studies about the heat transfer problems in the boundary layer, the isothermal wall condition and the constant wall heat ux condition were frequently used since they possess a simple mathematical structure and often admit similarity solutions. Recently, Aziz [] made an analysis on a thermal boundary layer over a convectively heated at surface in a uniform stream of uid. He found that the similarity solutions could also be possible if the convective heat transfer associated with the hot uid on the lower surface of the plate is inversely proportional to the square root of the distance along the wall. Ishak [] extended Azizs work [] by introducing the eects of suction and injection through the at surface and then presented similarity solutions with the same heat transfer coecient. Aziz and Khan [] discovered similarity solutions for a free convective ow of a nanouid about a vertical surface with a convective boundary condition by assuming that the convective heat transfer coecient for the hot uid varies inversely with the fourth root of the distance along the vertical wall. Makinde and Aziz [] derived a similarity solution for a hydromagnetic mixed convection ow past a convectively heated vertical plate embedded in a porous medium with the convective heat transfer coecient for the hot uid being a constant. Hayat et al. [] noticed that the convective boundary condition could also be applied to derive similarity solutions for non-Newtonian uids. They then obtained the similarity solutions for the problem of the ow and heat transfer of an Eyring Powell uid over a continuously moving surface in the presence of a free stream velocity. It should be noted to this end that investigations of ow and heat transfer problems with convective boundary conditions are very attractive and unique, since they are more realistic and practically useful than those with commonly used conditions of a constant surface temperature or constant heat ux.

The aim of this paper is to investigate the laminar nanouid ow and heat transfer due to a jet spreading out over a convectively heated at surface. The homogeneous model will be introduced to the boundary layer equations for modeling the nanouid. Similarity solutions of the boundary layer equations will be sought, according to which the forms of the velocity distribution across the jet and the heat transfer coecient associated with the hot uid on the lower surface of the plate are assumed, respectively, to vary inversely proportional to the square root and the three-quarter root of the distance along the at surface from the leading edge. Explicit analytical approximations with high precision will be given for both the velocity and the temperature distributions. Besides, the important quantities of practical interests including the boundary layer thickness, the skin friction coecient, the Nusselt number, as well as the overall surface heat transfer rate are computed and discussed. To the best of our knowledge this problem has not been considered before and the results are original and new.

2 Mathematical description

Consider a two-dimensional laminar jet owing over a at plate in a viscous nanouid of temperature T. As shown schematically in Figure , a jet of uid spreads out from a

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Figure 1 The physical sketch.

narrow slit to the upper surface of the plate, while the lower surface of the plate is heated or cooled by convection from a uid of temperature Tf . With the assumptions that () the uid is incompressible and the ow is laminar, () the nanouid is dilute, () the shape of the metal nanoparticles suspended in the uid is spherical, and () the homogeneous model developed by Maga et al. [] is employed, the velocity and temperature within the momentum and thermal boundary layers, which develop along the surface, can be written as

u

x +

y = , ()

u

u

x +

u

u

y =

nf

nf

y , ()

u

T

x +

y , ()

subject to boundary conditions

u = , = , knf

T

T

y =

nf

T

y = hf (x)(Tf T) at y = ,

u , T T, as y .

()

Here the Cartesian coordinates system (x, y) is chosen with the x-axis being measured along the plate and the y-axis being normal to it, respectively. u and v are the velocity components along the x- and y-axes, T is the temperature, hf (x) is the heat transfer coefcient due to Tf ; nf , nf , nf , and knf are, respectively, the viscosity, the density, and the diusivity and thermal conductivity of the nanouid, which are given by

nf =

f( ). ,

nf = ( )f + s, nf = knf

(Cp)nf ,

()

where is the solid volume fraction of the nanouid, is the viscosity, is the density, k is the thermal conductivity, Cp is the specic heat capacitance, the subscript nf represents the nanouid, f represents the base uid, and s represents the nanoparticles, respectively. It should be noted here that the expression for nf is estimated by Brinkman [] using the existing relations for two-phase mixtures, the expressions for nf and (Cp)nf are given by Xuan and Li [], and the expression for knf is approximated by the Maxwell-

(Cp)nf = ( )(Cp)f + (Cp)s, knf

kf =

(ks + kf ) (kf ks)

(ks + kf ) + (kf ks) ,

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Garnetts model [] with the assumption that the shape of the nanoparticles is restricted to spherical ones.

We now introduce the following similarity variables:

= f x

/ f (), =

/ y, () =

T T

Tf T, ()

where is the stream function dened by u = /y and v = /x, and f is the kinematic viscosity of the base uid.

Substituting Eq. () into Eqs. ()-(), the continuity equation () is automatically satised, and the momentum equation () and the energy equation () are reduced to

f + + f = , ()

Pr

+ f = , ()

subject to the boundary conditions

f () = , f () = , () =

()

f x

, f () = , () = , ()

where Pr = f /f is the Prandtl number, is the reduced heat transfer parameter, and and are two constants related to the properties of nanouids and are given as

= ( ).[( ) + s/f ],

= knf /kf( ) + (Cp)s/(Cp)f . ()

The physical quantities of practical interest are the local skin friction coecient Cfx and

the local Nusselt number Nux, which are given by

Cfx =

wf ur , Nux =

xqwkf (Tw T)

, ()

where w = nf (u/y)y= is the wall shear stress, qw = knf (T/y)y= is the wall heat ux, and ur = (/)x/ is the reference velocity.

In terms of similarity variables (), we are able to obtain the non-dimensional expressions for Cfx and Nux via Eq. () as

CfxRe/x = f ()

( ). , NuxRex/ =

knf

kf

()

() , ()

where Rex = (urx)/f is the local Reynolds number.

3 Analytical solutions3.1 Asymptotic analysis

Here we check the asymptotic behaviors of the velocity prole f () and the temperature prole () in the limit case . Due to the boundary conditions (), for considerably

large , we have

lim

f (

) , lim

() .

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Integrating f () over one time, we obtain

lim

f (

+ = , ()

which have the explicit solutions

F () = C exp

[parenrightbigg]+ C, ()

() = C exp

Pr

. ()

It can be seen from Eqs. () and () that F () and () (hence f () and ()) decay exponentially as .

3.2 The implicit solution for f()

We multiply Eq. () by f () and integrate it from to , obtaining

f + f f = . ()

Multiplying Eq. () by f / and integrating again, we obtain

f /f +

f / = constant. ()

) , ()

where is an integral constant. For the physical constraint f , it is readily seen that

.

For , we suppose f () and () can be expressed by

f () = + F(), () = (), ()

where F() and () are negligibly small.Substituting Eq. () into Eqs. () and () and then linearizing them, we obtain

F + F = , ()

Pr

, ()

where C, C, and C are integral constants.

Due to Eqs. () and (), it is known that

F = = as . ()

With substitution of the boundary condition () into Eq. (), we nd C = , which leads to

F () = C exp

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It is found here that if f() is the solution of Eq. (), so is f() = Af(A), since for any constant A, f() always satises the boundary conditions (). When A varies, the expressions for and change accordingly, and u is precisely the same as that of changing the value of the arbitrary constant velocity U to U/A. Without loss of generality, we choose f () = so that the constant in Eq. () is equal to /. Write f = g; we then have f = gg .

Therefore Eq. () takes the form

g =

, ()

which has the implicit solution

=

log[parenleftbigg][radicalbig] +g + g g

g

[parenrightbigg]

+ arctan

[parenleftbigg]

g + g

[parenrightbigg][bracketrightbigg]

. ()

By replacing g with f /, Eq. () can further be written as

=

log[parenleftbigg][radicalbig] +f / + f f /

[parenrightbigg]+ arctan

[parenleftbigg]

f /

+ f /

[parenrightbigg][bracketrightbigg]

. ()

3.3 The explicit solutions for f() and ()

Based on the above mentioned asymptotic analysis for f () and () and the homotopy analysis method (HAM), it is assumed that they can be expressed by a set of real functions in the following forms:

f () =

k= fk(),

() =

()

k= k(),

where fk() and k() are the high order deformation derivatives and are given by

fk() =k +

k+

j=jkjk exp(j), ()

k() =

k+

j=jk Bjk exp(j), ()

where is a given positive constant, and jk and jk are the coecients dened as

jk =

, j k + ,

, other cases,

, j k ,

, other cases. ()

The recursive coecientsjk and Bjk for k are determined by

k = , ()

k = kkk + Ck, ()

jk =

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k = kkk + Ck, ()

k =

[planckover2pi1]f

Gk + Ck

, ()

jk =

[planckover2pi1]f(j )(j )j

, ()

Bk = kk Bk + Ck, ()

Bk = kk Bk +

Gjk +

Cjk +

Bjk + Ajk

[parenleftbigg]

Pr Fk +

Ek[parenrightbigg]

, ()

Bjk = kjk Bjk +

j(j + )

[parenleftbigg]

Pr Fjk +

Ejk +

Djk[parenrightbigg]

, ()

where [planckover2pi1]f and [planckover2pi1] are the HAM auxiliary parameters; Ck, Ck, and Ck are integral constants given by

Ck =

[planckover2pi1]f

Gk +

Ck[parenrightbig]

k+

j=

[planckover2pi1]f(j + )(j + )

Gjk +

Cjk +

Bjk + Ajk

,

Ck =

[planckover2pi1]f

Gk +

Ck[parenrightbig]+

k+

j=

[planckover2pi1]f j(j + )

Gjk +

Cjk +

Bjk + Ajk

,

()

Ck =

+

k+

j=

[bracketleftbigg]

j +

j(j + )

[bracketrightbigg][parenleftbigg]

Pr Fjk +

Ejk +

Djk[parenrightbigg]

+

[parenleftbigg]

+

[parenrightbigg][parenleftbigg]

Pr Fk +

Ek[parenrightbigg]

,

and

, k = ,

, k > . ()

Other coecients appearing in the above recursive formulas are dened as

Cjk = (j)jkjk,jk = (j)jkjk, Gjk = (j)jkjk,

Djk = (j)jk Bjk, Fjk = (j)jk Bjk,

()

k =

and

Ajk =

k

n=

max{kn,j}

s=max{,jn}Cskn Cjsn,

Bjk =

k

n=

max{kn,j}

s=max{,jn}sknjsn,

Cjk =

k

n=[ j

]

knjn, ()

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Djk =

k

n=

max{kn,j}

s=max{,jn}skn Djsn,

Ejk =

k

n=[ j

]

kn Djn,

where [x] gives the greatest integer less than or equal to x.

When we set = , = , = , B = , B = /( + ), the purely explicit solutions for fk() and k() (f () and ()) can be determined successively for k = , , , . . . .

Note that here the homotopy auxiliary linear operators are selected as

Lf =

+

, ()

and the homotopy auxiliary functions are chosen by

f () = exp(), () = exp(). ()

4 Results and discussion

We rst examine the reliability and consistency of the explicit solution f () denoted in Eq. (). As shown in Figure , for various values of , our explicit solution agrees well with the implicit one given in () in the whole range < . Further, to check the

+

, L =

+

Figure 2 Comparison of the implicit solutions (25) with the explicit solutions at 100th order truncations for various values of in the case of [planckover2pi1]f = 3/2. Circle: the implicit solutions; line: the explicit solutions.

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Table 1 Computational errors for Err with [planckover2pi1]f = 3/2 and [planckover2pi1] = 1 in the case of Pr = 1

kth order = 20/100, = 1 = 8/100, = 1/2

0 8.18646 103 4.04103 103

30 1.48150 104 1.44346 105

60 6.73996 105 4.55130 106

90 3.01452 105 2.26475 106

120 2.74295 105 1.37636 106

150 2.31459 105 9.35413 107

180 6.82566 107

The nanouid is Cu-water.

Figure 3 Velocity prole f[prime]() for various values of .

accuracy of the explicit solution (), we dene the following error estimation function:

Err = lim

s

[integraldisplay]

Substituting various orders of explicit solutions f () and () into Eq. (), the corresponding errors can be obtained, as listed in Table . It is clearly seen from this table that the errors decrease monotonously with the increase of the computational order k for both considered cases. The accuracies of those explicit solutions can be further improved when more and more orders of HAM truncations are involved.

The variation of the non-dimensional velocity proles f () with for various values of are plotted in Figure . It is found from this gure that for each selected value of , there is a critical value of c corresponding to the maximum velocity, when < c, the velocity prole f () increases dramatically as enlarges, while, when c, f () decreases grad

ually as evolves. This is due to the uid ow near the at surface being restrained by the resting wall, while the ow at far eld is retarded by the ambient quiescent uid. On the

[parenleftbigg]

s

Pr

+ f

d. ()

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Figure 4 Temperature proles () for various values of with Pr = 1 and = 1.

other hand, we notice that the solid volumetric fraction parameter plays an important role in the variation of f (); the larger is the value of , the greater is the maximum velocity f (). We further notice that the peak velocity of the ow increases with enlarging, but the critical value of c varies contrarily, it decreases as increases. This can be explained by two reasons. One is that the higher concentration nanouid ow possesses more kinetic energy than the lower concentration nanouid or Newtonian uid ows since they have the same incidence velocities. The other reason is that, unlike Newtonian uid ows that are usually subject to a solid boundary, the nanouid ows past a geometry are often restricted by a slip boundary, which is of help to reduce the ow drag near the resting wall.

We then consider the variation of the non-dimensional temperature proles () with for various values of in the case of Pr = and = . As illustrated in Figure , () decreases continuously as evolves for any prescribed value of . On the other hand, () enhances smoothly with increasing. This shows that the heat transfer characteristics of the considered uid can be gradually improved as appropriate nanoparticles are continually added to it. We also notice that the reduced heat transfer parameter has an obvious eect on (). As shown in Figure , the temperature prole () increases consecutively as enlarges for all . This changing trend becomes more evident for small values of . We further notice that, as tends to , () approaches as well. In this limiting case, there is no heat transfer by convection through the wall at all. As tends to , () ap

proaches . In this limiting case, the wall temperature is equal to the temperature of the uid under the wall. Following that, we discuss the eect of the Prandtl number Pr on the variation of (). As shown in Figure , () diminishes consecutively as Pr increases. This trend is very similar to the case of the Newtonian uid ow, but the variation range for () is apparently smaller as compared with that of the Newtonian uid ow. When Pr is suciently large, its eect on () becomes negligibly small.

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Figure 5 Temperature proles () for different values of with = 0.1 and Pr = 1.

Figure 6 Temperature proles () for different values of Pr with = 0.1 and = 1.

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Figure 7 Variation of the reduced local skin friction coefcient Cfx with .

The local skin friction coecient Cfx and the local Nusselt number Nux are practically important in various applications regarding the ow and heat transfer in the boundary layer region. We therefore discuss them successively. As shown in Figure , Cfx enlarges almost linearly as evolves from to .. Using the explicit solution for f (), one readily gives the explicit expression for CfxRe/x, due to Eq. (),

CfxRe/x = ( ).

k+

+

k=

k+

j=(j)jkjk. ()

Figure gives the variation of the reduced Nusselt number NuxRe/x with for Pr = and = /. It is found that NuxRe/x increases gradually with increasing, but its changing range is slower than that of CfxRe/x. Though it is not shown in this paper, it is found that

has little eect on the variation of NuxRe/x, it is almost unchanged for all ranges <

< . Similarly, we are able to give the explicit expression for NuxRe/x in the following form:

NuxRex/ = knf kf

+

k=

j=(j)jk Bjk[parenrightBigg][slashBig][parenleftBigg]

+

k=

j= Bjk[parenrightBigg]. ()

The boundary layer thicknesses are of physical importance for this problem, dened by

f /x = f/Re/x, /x = /Re/x, ()

where f is the thickness of the velocity boundary layer and is the thickness of the thermal boundary layer. Here it is assumed that the edges of the velocity and the thermal

k+

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Figure 8 Variation of the reduced Nusselt number Nux with for = 1/10 and Pr = 1.

boundary layers are located at the points f and with f (f) and () having the same value of .. The variations of the boundary layer thicknesses f and with are plotted in Figure . It is seen from the gure that the f increases continuously as enlarges from to .; beyond this value, its value decreases as evolves. exhibits a totally different trend, its values increase gradually as increases from to .. But its increasing rate gradually descends as evolves, particularly, when ., its eect on becomes

dramatically small. It is worth mentioning that the homogeneous model is only valid for small values of , i.e. .. When is considerably large, the nanouid shows the

characteristics of non-Newtonian uids and this model cannot predict the behaviors of the nanouid accurately.

5 Conclusions

In this paper, the laminar jet ow and heat transfer of a copper-water nanouid over a resting wall has been examined in detail. By means of the homogeneous model, the original Navier-Stokes equations describing these jet ows have been reduced to a set of dierential equations. An analytical solution for the velocity distribution has been obtained. The explicit solutions for both the velocity and the temperature distributions have been given by means of the HAM technique. The eects of the volumetric fraction parameter and the dimensionless heat transfer coecient on the velocity and temperature proles, as well as on the reduced local skin friction coecient and the reduced Nusselt number have been investigated. Some novel results and ndings of this study have been presented:

. An implicitly analytical solution for the velocity distribution is given, which has not been reported before.

. A mathematical analysis for solution behavior at far eld is presented. It is found that both the velocity and the temperature proles decay exponentially at innity.

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Figure 9 Variation of the boundary layer thicknesses of f and with for = 1/10 and Pr = 1.

. Purely explicit solutions with high precision for both the velocity and the temperature distributions are obtained.

. The volumetric fraction parameter has an important eect on the velocity and temperature distribution. The maximum velocity increases as enlarges. The temperature proles increase as evolves, too. This means that the addition of nanoparticles into pure uids is of help to reduce the ow drag near the wall and to improve the heat transfer capability of the base uids.

. The dimensionless heat transfer parameter plays a key role on the variation of the temperature proles. The temperature proles enhances as increases.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Author details

1State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China. 2Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China.

Acknowledgements

We extend our sincere appreciations to the Program for New Century Excellent Talents in University (Grant No. NCET-12-0347), and to the Program of Innovative Fundings for Youth of the State Key Laboratory of Ocean Engineering (Shanghai Jiao Tong University) (Grant No. GKZD010059-17) for their nancial supports.

Received: 13 December 2013 Accepted: 15 April 2014 Published: 1 September 2014

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doi:10.1186/1687-2770-2014-163Cite this article as: Raees et al.: Explicit solutions of wall jet ow subject to a convective boundary condition. Boundary Value Problems 2014 2014:163.