Afr Mar D0I10.1007k13370-015-0323-x Boundary layer flow past an inclined stationary/moving flat plate with convective boundary condition G.K.Ramesh·A.J.Chamkha·B.J.Gireesha Abstract In this study,the mathematical modeling for boundary layer flow and heat transfe ered.Uncline p statio nary/m oving flat te with a convective b ry co tion is consid y ua coupl third 30 ng t method.T ns are compa in the and b agre the angl ber,local Grashe umber and the Biot num are analyzed and disc t is foun that the temperature of the stationary flat plate is higher than the temperature of the moving fat plate Keywords Boundary layer flow.Inclined plateGrashof numberConvective boundary condition·Numerical Mathematics Subject Classification 76T15.80A20 1 Introduction Investigations of laminar boundary layer flow about a flat plate in a uniform stream of fluid continues to receive considerable attention because of its importance in many practical applications in a broad spectrum of engineering systems such as geothermal reservoirs, cooling of nuclear reactors.thermal insulation.combustion chamber.rocket engine.etc. 4S物a品 es and Re rch in Mathematics,Kuvempu University. &Chamkha☒ e-mail:achamkha@pmu.edu.sa Published online:11 March 2015 么Springer
Afr. Mat. DOI 10.1007/s13370-015-0323-x Boundary layer flow past an inclined stationary/moving flat plate with convective boundary condition G. K. Ramesh · A. J. Chamkha · B. J. Gireesha Received: 24 May 2014 / Accepted: 3 March 2015 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2015 Abstract In this study, the mathematical modeling for boundary layer flow and heat transfer past an inclined stationary/moving flat plate with a convective boundary condition is considered. Using a similarity transformation, the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations and are solved numerically using the shooting method. The obtained numerical solutions are compared with the available results in the literature and are found to be in excellent agreement. The features of the flow and heat transfer characteristics for various values of the angle of inclination, Prandtl number, local Grashof number and the Biot number are analyzed and discussed. It is found that the temperature of the stationary flat plate is higher than the temperature of the moving flat plate. Keywords Boundary layer flow · Inclined plate · Grashof number · Convective boundary condition · Numerical solution Mathematics Subject Classification 76T15 · 80A20 1 Introduction Investigations of laminar boundary layer flow about a flat plate in a uniform stream of fluid continues to receive considerable attention because of its importance in many practical applications in a broad spectrum of engineering systems such as geothermal reservoirs, cooling of nuclear reactors, thermal insulation, combustion chamber, rocket engine, etc. G. K. Ramesh · B. J. Gireesha Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, Shimoga 577 451, Karnataka, India A. J. Chamkha (B) Mechanical Engineering Department, Prince Mohammad Bin Fahd University (PMU), P.O. Box 1664, Al-Khobar 31952, Kingdom of Saudi Arabia e-mail: achamkha@pmu.edu.sa 123
G.K.Ramesh et al. layer flo over a flat t plate in m.The beh fl retically by a te met aler [3 ed t ation on the rder oundary layer about a flat plate via Kutta r with s ing me e work by many res archers (see [4-8)). deling the oundary layer flow and heat transfer about aflat plate.the boundar conditions that are usually applied are either a specified surface temperature or a specifie surface heat flux.However,there are boundary layer flow and heat transfer problems in which the surface heat transfer depends on the surface temperature.This situation arises in conjugate heat transfer problems and when there is Newtonian heating of the convective fluid from the surface.Newtonian heating occurs in many important engineering devices,for example. in heat exchangers,where the conduction in a solid tube wall is greatly influenced by the convection in the fluid flowing over it.On the basis of above discussions and applications, Bataller[9]analyzed the effects of thermal radiation on the laminar boundary layer about a flat plate in a uniform stream of fluid,and about a moving plate in a quiescent ambient fluid both under a convective surface boundary condition.Later.Aziz [10]investigated the heat transfer problems for boundary layer flow concerning with a convective boundary condition. Ishak et al.[11]studied the steady laminar boundary layer flow over a moving plate in a moving fluid with convective surface boundary condition and in the presence of thermal radiation.In this problem they combine two problems i.e.,Blasius flow and Sakiadis flow using the composite velocity (U=U+)which was introduced by Afzal et al.[12]. Makinde [13,14]studied the hydromagnetic flow over a vertical flat plate with a convective boundary condition,in this analysis he studied both heat and mass transfer analysis.Further. they extended their work and investigate the MHD mixed convection flow of a vertical plate embedded in a porous medium with a convective boundary condition.Recently,Ramesh et al.[15]obtained a numerical solution for MHD mixed convection flow of a viscoelastic fluid over an inclined surface with a non-uniform heat source/sink.Rajesh and Chamkha [16]studied the effects of ramped wall temperature on unsteady two-dimensional flow past a vertical plate with thermal radiation and chemical reaction.Chamkha et al.[17]investigated the coupled heat and mass transfer by MHD free convection flow along a vertical plate with stream-wise temperature and species concentration variations The aim of this paper is to extend the work by Ishak et al.[in the absence of radiation effect and by considering the angle of inclination.Appropriate similarity transformations reduce the s veming partial differential equations into a set of nonlinear ordinary differen- tial equations.The resulting equations are solved numerically using the shooting method. 2 Problem formulation We consider a steady two-dimensional flow of a stream of cold incompressible fluid about a vertical plate which is inclined with an acute angle a.and the temperatu re T over the surface of the flat plate with ac ostant fre velocity and mov ing flat plate ith locity U.. while the l ted by fluid at temper ure T which vides a heat transfer.it is assumed 么Springer
G. K. Ramesh et al. Blasius [1] was the first to investigate and presented a theoretical result for the boundary layer flow over a flat plate in a uniform stream. The behavior of boundary layer flow due to a moving flat surface immersed in an otherwise quiescent fluid was first studied by Sakiadis [2], who investigated it theoretically by both exact and approximate methods. Bataller [3] studied the effects of thermal radiation on the laminar boundary layer about a flat plate via fourth-order Runge-Kutta algorithm together with shooting method. Apart from these works, various aspects of flow and heat transfer of viscous fluid over a flat plate were investigated by many researchers (see [4–8]). When modeling the boundary layer flow and heat transfer about a flat plate, the boundary conditions that are usually applied are either a specified surface temperature or a specified surface heat flux. However, there are boundary layer flow and heat transfer problems in which the surface heat transfer depends on the surface temperature. This situation arises in conjugate heat transfer problems and when there is Newtonian heating of the convective fluid from the surface. Newtonian heating occurs in many important engineering devices, for example, in heat exchangers, where the conduction in a solid tube wall is greatly influenced by the convection in the fluid flowing over it. On the basis of above discussions and applications, Bataller [9] analyzed the effects of thermal radiation on the laminar boundary layer about a flat plate in a uniform stream of fluid, and about a moving plate in a quiescent ambient fluid both under a convective surface boundary condition. Later, Aziz [10] investigated the heat transfer problems for boundary layer flow concerning with a convective boundary condition. Ishak et al. [11] studied the steady laminar boundary layer flow over a moving plate in a moving fluid with convective surface boundary condition and in the presence of thermal radiation. In this problem they combine two problems i.e., Blasius flow and Sakiadis flow using the composite velocity (U = Uw + U∞) which was introduced by Afzal et al. [12]. Makinde [13,14] studied the hydromagnetic flow over a vertical flat plate with a convective boundary condition, in this analysis he studied both heat and mass transfer analysis. Further, they extended their work and investigate the MHD mixed convection flow of a vertical plate embedded in a porous medium with a convective boundary condition. Recently, Ramesh et al. [15] obtained a numerical solution for MHD mixed convection flow of a viscoelastic fluid over an inclined surface with a non-uniform heat source/sink. Rajesh and Chamkha [16] studied the effects of ramped wall temperature on unsteady two-dimensional flow past a vertical plate with thermal radiation and chemical reaction. Chamkha et al. [17] investigated the coupled heat and mass transfer by MHD free convection flow along a vertical plate with stream-wise temperature and species concentration variations. The aim of this paper is to extend the work by Ishak et al. [11] in the absence of radiation effect and by considering the angle of inclination. Appropriate similarity transformations reduce the governing partial differential equations into a set of nonlinear ordinary differential equations. The resulting equations are solved numerically using the shooting method. Variations of several pertinent emerging parameters are analyzed in detail. To the authors’ knowledge, no previous attempts have been made to analyze this problem. 2 Problem formulation We consider a steady two-dimensional flow of a stream of cold incompressible fluid about a vertical plate which is inclined with an acute angle α, and the temperature T∞ over the upper surface of the flat plate with a constant free stream velocity U∞ and moving flat plate with constant velocity Uw, while the lower surface of the plate is heated by convection from a hot fluid at temperature Tf which provides a heat transfer coefficient h f . Further, it is assumed 123
Boundary layer flow past an inclined stationary that the viscous dissipation and radiation effects are neglected.The velocity and temperature profiles in the fluid flow must obey the usual boundary layer equations are given by Ishak et al.[1 au ax a y (+)）= +gB (T-Too)cosa (2) 82T (3) ←remh ud longand y f the fluid. etric 二 s the the T =U:v=0.-kay=hy(Ty-T)aty =0 →,T→Te,asy→o (④ where T is the hot fluid temperature andhis the heat transfer coefficient In order to reduce the number of independent variables and to get the dimensionless equations,we define the new vanables as, T-Too =xf.=)= Tf-Too (5) and the stream function is defined by M= (⑤ ransfo ns.the quation of(1)isidentically satisfied and Eqs. following forms as 2fT"+ff"+2Gr0 cosa =0. (7) 20+Prf0'=0 (8 when a =90 our problem reduces to the horizontal flat plate case,while when =0. it reduces to the vertical flat plate.To exit Eqs.(1-4),here we takehf=where c is a constant. The boundary conditions defined as in(4)will become, f=0,f'=入，6=-Bi(1-)atn=0 f'→1-入，0→0asn→0∞. (9) whereis the Biot number andis the velocity ratio parameter.Here.o can observe that when A=0,the problem reduces to the Blasius flow (stationary flat plate) and when A=1.the problem reduces to the Sakiadis flow(moving flat plate),respectively. Springer
Boundary layer flow past an inclined stationary that the viscous dissipation and radiation effects are neglected. The velocity and temperature profiles in the fluid flow must obey the usual boundary layer equations are given by Ishak et al. [11] ∂ u ∂ x + ∂ v ∂ y = 0, (1) ρ u ∂ u ∂ x + v ∂ u ∂ y = μ ∂2 u ∂ y2 + gβ (T − T∞) cos α, (2) ρ cp u ∂ T ∂ x + v ∂ T ∂ y = k ∂2 T ∂ y2 , (3) where u and v are the velocity components of the fluid along x and y directions respectively. μ, ρ and cpare the co-efficient of viscosity of the fluid, density of the fluid, and specific heat of fluid, respectively. g is the acceleration due to gravity, β is the volumetric coefficient of thermal expansion, T is the temperature of the fluid, k is the thermal conductivity. The appropriate boundary conditions for the flow problem are given by [11] u = Uw, v = 0, −k ∂T ∂y = h f (Tf − T ) at y = 0 u → U∞, T → T∞, as y → ∞. (4) where Tf is the hot fluid temperature and h f is the heat transfer coefficient. In order to reduce the number of independent variables and to get the dimensionless equations, we define the new variables as, ψ = √ U xν f (η), η = U ν x y, θ (η) = T − T∞ Tf − T∞ , (5) and the stream function is defined by u = ∂ψ ∂y and v = −∂ψ ∂x , (6) with the above transformations, the equation of continuity (1) is identically satisfied and Eqs. (2) and (3) reduce to the following forms as: 2 f + f f + 2Gr θ cos α = 0, (7) 2θ + Pr f θ = 0 (8) where a prime denotes differentiation with respect to η and Gr = gβ(Tf −T ∞)x U2 is the local Grashof number (Kierkus [18]), Pr = ν/α is the Prandtl number. From Eq. (7) we note that, when α = 90◦, our problem reduces to the horizontal flat plate case, while when α = 0◦, it reduces to the vertical flat plate. To exit Eqs. (1–4), here we take h f = √c x , where c is a constant. The boundary conditions defined as in (4) will become, f = 0, f = λ, θ = −Bi(1 − θ ) at η = 0, f → 1 − λ, θ → 0 as η → ∞. (9) where Bi = c k ν U is the Biot number and λ = Uw U is the velocity ratio parameter. Here, one can observe that when λ = 0, the problem reduces to the Blasius flow (stationary flat plate) and when λ = 1, the problem reduces to the Sakiadis flow (moving flat plate), respectively. 123
G.K.Ramesh et al. Bi Bataller [9] Aziz [10] Ishak et al.[11] Present result a=90° a=30° a=00 0.05 0.1446 0.1447 0.1446 0.1446 0.1394 0.1388 0 0.2528 0.2527 02527 0.2401 0.238 02 04035 0403 0403 0.4035 0380 0.3774 0 0.5750 0.575 0.5750 0.543 0.6 0.6699 0.669 0.669 0.669 0.637 0.633 0.8 0.730 0.730 0.7301 0.698 0.6954 1.0 0.7718 0.7718 0.7718 0.7718 0.7422 0.739 50 0044 00441 00441 00324 00323 10 09712 0.9713 0.9712 09712 0.9654 0.9648 Table 2 Computations values of(0)for different t values of Biot number (Bi)when Pr =0.72.Gr =0.5 and=1(moving fat plate) a0) a=900 =30 -0 0.05 0.1227 0.119 0.1190 0.1 0.2185 0.2102 0.209 0.2 03587 0.3420 0.340 0.4 0.5280 0.5035 0.5010 0.6266 0.600 0.5976 0.6651 0.71 8c 5.0 0.9332 0.9234 0.922 10 0.9654 0.9600 0.9595 3 Results and discussion The nonlinear coupled differential Eqs.(7)and(8)along with the boundary conditions(9) are solved numerically using Runge-Kutta method along with the shooting technique.The accuracy of the e employed numerical method is tested by direct comparisons with the values of(0)(at=0)with those reported by [11]in Table 1.for the special case of the t problem and ults fo 9(0)when of fre strea ented in Table 2 The mputations uted for ral values of the ameters involved in the ations viz.the mber (r)local Grashof mber (Gr Ithe biot er(Bi).So me figures are plotted the nd als phys The and t CSpoiesfordfcerentvalues of the angles of ation(a=0，30°，90°)are presented in Figs.1and2.respectively. ②Springer
G. K. Ramesh et al. Table 1 Comparison results of θ (0) for different values of Biot number (Bi) when Pr = 0.72, Gr = 0.5 and λ = 0 (stationary flat plate) Bi Bataller [9] Aziz [10] Ishak et al. [11] Present result α = 90◦ α = 30◦ α = 0◦ 0.05 0.1446 0.1447 0.1446 0.1446 0.1394 0.1388 0.1 – 0.2528 0.2527 0.2527 0.2401 0.2386 0.2 0.4035 0.4035 0.4035 0.4035 0.3800 0.3774 0.4 – 0.5750 0.5750 0.5750 0.5431 0.5398 0.6 0.6699 0.6699 0.6699 0.6699 0.6371 0.6337 0.8 – 0.7302 0.7301 0.7301 0.6986 0.6954 1.0 0.7718 0.7718 0.7718 0.7718 0.7422 0.7392 5.0 – 0.9441 0.9441 0.9441 0.9334 0.9323 10 0.9712 0.9713 0.9712 0.9712 0.9654 0.9648 Table 2 Computations values of θ (0) for different values of Biot number (Bi) when Pr = 0.72, Gr = 0.5 and λ = 1 (moving flat plate) Bi θ (0) α = 90◦ α = 30◦ α = 0◦ 0.05 0.1227 0.1194 0.1190 0.1 0.2185 0.2102 0.2092 0.2 0.3587 0.3420 0.3402 0.4 0.5280 0.5035 0.5010 0.6 0.6266 0.6003 0.5976 0.8 0.6911 0.6651 0.6625 1.0 0.7366 0.7117 0.7092 5.0 0.9332 0.9234 0.9224 10 0.9654 0.9600 0.9595 3 Results and discussion The nonlinear coupled differential Eqs. (7) and (8) along with the boundary conditions (9) are solved numerically using Runge-Kutta method along with the shooting technique. The accuracy of the employed numerical method is tested by direct comparisons with the values of θ (0) (at λ = 0) with those reported by [9–11] in Table 1, for the special case of the present problem and an excellent agreement between the results is found. Also, it provides a sample of our results for θ (0) when the direction of free stream is fixed (i.e., λ = 1) which is presented in Table 2. The numerical computations are executed for several values of the dimensionless parameters involved in the equations viz. the angle of inclination (α), Prandtl number (Pr), local Grashof number (Gr) and the Biot number (Bi). Some figures are plotted to illustrate the computed results and also to give the physical explanations. The variations of the dimensionless velocity and temperature profiles for different values of the angles of inclination (α = 0◦, 30◦, 90◦) are presented in Figs. 1 and 2, respectively. 123
Boundary layer flow past an inclined stationary 1.0 0.8 a=0,30,90 Gr=0.5,Pm-0.72.Bl 0.2 0 8 n Fig1 Effect of a on velocity profiles 0.8 0.7- 0.6 Gr0.5,Pr0.72,B-1 0.5 0.4 0.3 a=0,30,90 0.1 0.0 0 Fig2 Effect of on temperature profiles s with the increase is due e ang buoyancy force aue to ther ases by a r we buoya (WD maximum for ream ve the similar effect c Furth we observe that the temperature increases s the angle of inclination incr eases as shown in Fig. e can note that if o=90,the problem reduces to the horizont al flat plate he vertical flat plate (at=0)and the vertic moving nat plate (at =1)an when o =30,the problem reduces to the inclined flat plate (at A=0)and the inclined moving flat plate (at=1). Figure 3 depicts the variation in the velocity profiles for different values of the Grasho number.It is found that for a fixed value of a(=30),both the stationary flat plate(=0) Springer
Boundary layer flow past an inclined stationary Fig. 1 Effect of α on velocity profiles Fig. 2 Effect of α on temperature profiles For λ = 0, it is observed that boundary layer flow for the velocity decreases with the increase of the angle of inclination. This is due to the fact that as the angle of inclination increases, the effect of the buoyancy force due to thermal variations decreases by a factor of cos α. Also, we notice that the effect of the buoyancy force (which is maximum for α = 0) overshoots the main stream velocity significantly. At λ = 1, the similar effect can be found as shown in Fig. 1. Further, we observe that the temperature increases as the angle of inclination increases as shown in Fig. 2. One can note that if α = 90◦, the problem reduces to the horizontal flat plate (at λ = 0) and the horizontal moving flat plate (at λ = 1), while when α = 0◦ the problem reduces to the vertical flat plate (at λ = 0) and the vertical moving flat plate (at λ = 1) and when α = 30◦, the problem reduces to the inclined flat plate (at λ = 0) and the inclined moving flat plate (at λ = 1). Figure 3 depicts the variation in the velocity profiles for different values of the Grashof number. It is found that for a fixed value of α(α = 30◦), both the stationary flat plate (λ = 0) 123