Explicit solutions of wall jet flow subject to a convective boundary condition
© Raees et al.; licensee Springer 2014
Received: 13 December 2013
Accepted: 15 April 2014
Published: 1 September 2014
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 ϕ 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.
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-off aircraft spreading out over the ground, cooling jets over turbo-machinery components, sluice gate flows, and so on. The boundary layer approximation is used commonly as an effective approach for simplification 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 first to use the name of wall jet for the description of such flows. By means of the similarity method, he reduced the two-dimensional Navier-Stokes equations to an ordinary differential equation and then obtained a well-known implicit analytical solution. Merkin and Needham  and Needham and Merkin  extended Glauert’s 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  confirmed Merkin and Needham’s conclusion  that no similarity solution of Glauert’s type exists for the wall jet flowing over a resting surface in the presence of suction and/or injection. 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 Glauert’s type could be possible for the resting surface case when the suction/injection velocity is proportional to for . Xu et al.  further found that except for the solutions given by Cohen et al. , an infinite 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 flow driven by a power-law shear at far field over an impermeable flat surface. Mathematically, their problem is equivalent to a wall jet flowing over a heated flat surface. They then presented both the analytical and the numerical solutions for the special cases of the isothermal and of the adiabatic flat plate. Cossali  considered a forced convection thermal boundary layer over an impermeable flat 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 profile 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 fluids, nanofluids have received more and more attention for the reason that they possess a better thermal conductivity than that of the traditional heat transfer fluids. Many researchers have made great efforts to investigate the heat fluid flow in nanofluids under various circumstances since the heat transfer intensification due to dilute nanofluids 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 nanofluids. Typical models include the homogeneous model [10, 11], the dispersion model , the Buongiorno model  and so on. Note that there are controversies about the validity and applicability of those nanofluids models for the prediction of the behavior of nanofluids. Readers are referred to [14–18] 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 fluids to nanofluids. As a result, all traditional heat transfer correlations regarding the computation of thermophysical properties could be suitable for nanofluids as well. This model is only valid for dilute nanofluids with a similar behavior to Newtonian fluids. When the concentration of the nanofluids grows high, the nanofluids no longer have the nature of Newtonian fluids, but exhibit behaviors like non-Newtonian fluids. In such a situation, this model cannot be used for modeling nanofluids. Fortunately, it has been found that, even for dilute nanofluids, the heat transfer enhancement can still be improved significantly as compared with the traditional heat transfer fluids. With the homogeneous model, some investigations have been done towards understandings of the heat transfer characteristics of nanofluids. For examples, Bachok et al.  investigated a boundary layer flow and heat transfer due to a rotating disk immersed in a nanofluid. Their theoretical results show that one can distinguish how the addition of nanoparticles into pure fluids can improve the heat transfer capability of the fluids even if the amount of added nanoparticles is small. Rohni  made an analysis for a viscous nanofluid flow and heat transfer over an unsteady shrinking flat surface. Considering three kinds of nanofluids, including Cu-water, -water and -water, they concluded that the nanoparticle volume fraction parameter (which is associated with the concentration of nanofluids) and types of nanofluid play a key role for determination of the flow behavior. Similar conclusions were, respectively, given by Yacob et al.  and Vajravelu et al.  via their investigations about the flow and heat transfer of nanofluids past a wedge and over a flat surface.
In previous studies about the heat transfer problems in the boundary layer, the isothermal wall condition and the constant wall heat flux 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 flat surface in a uniform stream of fluid. He found that the similarity solutions could also be possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is inversely proportional to the square root of the distance along the wall. Ishak  extended Aziz’s work  by introducing the effects of suction and injection through the flat surface and then presented similarity solutions with the same heat transfer coefficient. Aziz and Khan  discovered similarity solutions for a free convective flow of a nanofluid about a vertical surface with a convective boundary condition by assuming that the convective heat transfer coefficient for the hot fluid 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 flow past a convectively heated vertical plate embedded in a porous medium with the convective heat transfer coefficient for the hot fluid being a constant. Hayat et al.  noticed that the convective boundary condition could also be applied to derive similarity solutions for non-Newtonian fluids. They then obtained the similarity solutions for the problem of the flow and heat transfer of an Eyring Powell fluid over a continuously moving surface in the presence of a free stream velocity. It should be noted to this end that investigations of flow 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 flux.
The aim of this paper is to investigate the laminar nanofluid flow and heat transfer due to a jet spreading out over a convectively heated flat surface. The homogeneous model will be introduced to the boundary layer equations for modeling the nanofluid. 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 coefficient associated with the hot fluid 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 flat 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 coefficient, 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
where ψ is the stream function defined by and , and is the kinematic viscosity of the base fluid.
where is the wall shear stress, is the wall heat flux, and is the reference velocity.
where is the local Reynolds number.
3 Analytical solutions
3.1 Asymptotic analysis
where λ is an integral constant. For the physical constraint , it is readily seen that .
where and are negligibly small.
where , , and are integral constants.
3.2 The implicit solution for
3.3 The explicit solutions for and
where gives the greatest integer less than or equal to x.
4 Results and discussion
Computational errors for with and in the case of
k th order
ϕ = 20/100, γ = 1
ϕ = 8/100, γ = 1/2
8.18646 × 10−3
4.04103 × 10−3
1.48150 × 10−4
1.44346 × 10−5
6.73996 × 10−5
4.55130 × 10−6
3.01452 × 10−5
2.26475 × 10−6
2.74295 × 10−5
1.37636 × 10−6
2.31459 × 10−5
9.35413 × 10−7
6.82566 × 10−7
An implicitly analytical solution for the velocity distribution is given, which has not been reported before.
A mathematical analysis for solution behavior at far field is presented. It is found that both the velocity and the temperature profiles decay exponentially at infinity.
Purely explicit solutions with high precision for both the velocity and the temperature distributions are obtained.
The volumetric fraction parameter ϕ has an important effect on the velocity and temperature distribution. The maximum velocity increases as ϕ enlarges. The temperature profiles increase as ϕ evolves, too. This means that the addition of nanoparticles into pure fluids is of help to reduce the flow drag near the wall and to improve the heat transfer capability of the base fluids.
The dimensionless heat transfer parameter γ plays a key role on the variation of the temperature profiles. The temperature profiles enhances as γ increases.
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 financial supports.
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