c 2012 Modern Science Publishers ⃝ www.m-sciences.com
J. of Mod. Meth. in Numer. Math. eISSN: 2090-4770 pISSN: 2090-8296 Vol. 3, No. 1, 2012, 53–63
RESEARCH ARTICLE Effect of Suction/Injection on Natural Convective Boundary-Layer Flow of A Nanofluid Past A Vertical Porous Plate Through A Porous Medium F. Hady, F. Ibrahim, H. El-Hawary and A. Abdelhady
∗
Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt. (Received: 22 August 2011, Accepted: 2 November 2011) In the present work, an analysis has been carried out to study a problem of natural convection past a vertical porous plate, in a porous medium saturated by a nanofluid with the streamwise distance x. The employed mathematical model for the nanofluid takes into account the effects of Brownian motion and thermophoresis. The Darcy model is employed for the porous medium. Non-similar solution has been obtained. This solution depends on a Lewis number Le, a buoyancy-ratio number N r, a Brownian motion number N b and a thermophoresis number N t. The variation of the reduced Nusselt number with N r, N b and N t is expressed by correlation formulas. The dependency of the Nusselt number on these four parameters and the effect of suction and injection are investigated graphicly. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change the flow pattern. Keywords: Natural convection; nanofluid; nonsimilar solution; porous medium; suction; injection. AMS Subject Classification: 35Q35, 76D10, 80A20.
1.
Introduction
Engineers have been working for decades to develop more efficient heat transfer fluids for use in car motors and industrial equipment. Nanofluids may solve a number of problems plaguing the heating, ventilation, air conditioning (HVAC) industry and improving the efficiency of high-heat flux devices like supercomputers circuits, high-power microwave tubes and providing new cancer treatment techniques etc. Nanofluids are a new class of advanced heat-transfer fluids, which are liquids containing a dispersion of submicronic solid particles (nanoparticles). Typical dimension of the nanoparticles is in the range of a few to about 100nm. Thermal conductivity is the characteristic feature of nanofluids. For examples 40 % increase of the effective thermal conductivity in ethylene-glycol nanofluid with 10-nm copper nanoparticles of 0.3 % in volume fraction [1] and the effective thermal conductivity increases 10−30 % in alumina/water nanofluids with 1 − 4 vol % of alumina [2]. In nanoparticles at extremely low volume fractions property enhancements are much higher than those expected from the macroscopic models. The term “nanofluids ” was coined by Choi [3]. The phenomenon of thermal conductivity observed by Masuda et al. [4]. This phenomenon suggests the possibility of using nanofluids in air conditioning systems [5]. Buongiorno [6] made a comprehensive survey of convective transport in nanofluids. Who says that a satisfactory explanation for the abnormal increase of the thermal conductivity and viscosity is yet to be found. He focused on the further heat transfer enhancement observed in convective situations. Buongiorno notes that several authors have suggested that convective heat transfer enhancement could be due to the dispersion of the suspended ∗ Corresponding author Email address:
[email protected] (A. Abdelhady).
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nanoparticles and particle rotation but he argues that these effects are too small to explain the observed enhancement. He also concludes that turbulence is not affected by the presence of the nanoparticles so this cannot explain the observed enhancement. Buongiorno [6] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity ( slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage and gravity settling. After examining each of these in turn, he concluded that in the absence of turbulent effects it is the Brownian diffusion and the thermophoresis that will be important. The problem of natural convection in a porous medium past a vertical plate is a classical problem first studied by Cheng and Minkowycz [7]. In the book by Bejan [8] the problem is presented as a paradigmatic configuration and solution. Bejan and Khair [9] made the extension to the case of heat and mass transfer. Further work on this topic is surveyed in sections 5.1 and 9.2.1 in the book by Nield and Bejan [10]. Mahdy and Hady [11] studied the effect of thermophoresis and Brownian diffusion in non-Newtonian free convection flow over a vertical plate with magnetic field effect. A review of the heat transfer characteristics of nanofluids has been made by Wang and Mujumdar [12]. Nield and Kuznetsov [13] extended the study of the Cheng and Minkowycz problem to the case of a nanofluid using the model of Buongiorno. In the present work the problem of Cheng−Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid [13] is extended to study the effect of the prosity of the plate. A nonsimilar solution is obtained. The present results are useful in cancer treatment and complex biological nanofluids, such as drug delivery systems. The unusual properties of the nanoparticles enable cells to suction these nanoparticles because of their size. Many diseases depend upon processes within the cell and can only be impeded by drugs that make their way into the cell. So, using nanofluid in medicine will lead to development of completely new drugs with more useful behavior and less side effects.
2.
Analysis
We consider a two-dimensional steady flow of a nanofluid through a porous medium past a vertical porous plate. The x−axis is taken along the plate in the vertical upward direction and y−axis is taken normal to the plate. At the plate y = 0, the temperature T and the nanoparticle fraction ϕ take constant values Tw and ϕw , respectively. The ambient values, attained as y tends to infinity, of T and ϕ are denoted by T∞ and ϕ∞ , respectively, as shown in Fig. 1. We employed the Oberbeck-Boussinesq approximation for this problem. We assumed the porous medium, whose porosity is denoted by ε and permeability by K, is homogeneous and in a local ⃗ . The variables of motion are the Darcy thermal equilibrium. The Darcy velocity is denoted by V ⃗ velocity V , the temperature T and the nanoparticle volume fraction ϕ. The boundary layer equations governing the flow and temperature field in the presence of nanoparticles are
⃗ V ⃗ = ⃗0, ∇.
(1)
⃗ ρf ∂ V ⃗ − µV ⃗ + [ϕρp + (1 − ϕ){ρf (1 − β(T − T∞ ))}]⃗g , = −∇p ε ∂t K
(2)
(ρc)m
∂T ⃗ ∇T ⃗ ∇T ⃗ = km ∇2 T + ε(ρc)p [DB ∇ϕ. ⃗ + ( DT )∇T ⃗ .∇T ⃗ ], + (ρc)f V. ∂t T∞
(3)
Effect of Suction/Injection on Natural Convective Boundary-Layer Flow of . . .
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Figure 1.: Physical model and coordinate system.
1.8 1.6
s θ f s’
1.4
s,s’,θ,f
1.2 1 0.8 0.6 0.4 0.2 0
0
2
4
η
6
8
10
Figure 2.: Plots of dimensionless similarity functions s(η, ξ), s′ (η, ξ),θ(η, ξ) and f (η, ξ) for the case Le = 10, ξ = 0andN r = N b = N t = 0.5. ∂ϕ 1 ⃗ ⃗ DT 2 + V .∇ϕ = DB ∇2 ϕ + ( )∇ T. ∂t ε T∞
(4)
Here ρf , µ and β are the density, viscosity, and volumetric volume expansion coefficient of the fluid while ρp is the density of the particles. Further, ⃗g is the gravitational acceleration, (ρc)m is the effective
F. Hady et al.
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heat capacity, km is the effective thermal conductivity, DB is the Brownian diffusion coefficient and DT is the thermophoretic diffusion coefficient. The derivation of Eqs. (3) and (4) are given in the papers by Buongiorno [6], Tzou [14, 15] and Nield and Kuznetsov [16]. The advective term and a Forchheimer quadratic drag term does not appear in the momentum equation because we assumed the flow is very slow. The appropriate boundary conditions for the problem are given by υ = υ0 , T = Tw , ϕ = ϕw at y = 0,
(5)
u = υ = 0, T → T∞ , ϕ → ϕ∞ as y → ∞,
(6)
where u and υ are components of dimensional velocities along x and y directions, respectively. υ0 is a scale of suction and injection velocity. In the spirit of the Oberbeck-Boussinesq approximation, Eq. (2) can be linearized by the neglect of a term proportional to the product of ϕ and T . This assumption is likely to be valid in the case of small temperature gradients in a dilute suspension of nanoparticles, thus we can write Eq. (2) as ⃗ − µV ⃗ + [(ρp − ρf ∞ )(ϕ − ϕ∞ ) + (1 − ϕ∞ )ρf ∞ β(T − T∞ )]⃗g . 0 = −∇p K
(7)
Under the boundary layer approximations, we can write the governing equations as ∂u ∂υ + = 0, ∂x ∂y
(8)
∂p µ = − u + [(1 − ϕ∞ )ρf∞ βg(T − T∞ ) − (ρp − ρf ∞ )g(ϕ − ϕ∞ )], ∂x K
(9)
∂p = 0, ∂y
(10)
∂T ∂T ∂ϕ ∂T DT ∂T 2 +υ = αm ∇2 T + τ [DB +( )( ) ], ∂x ∂y ∂y ∂y T∞ ∂y
(11)
1 ∂ϕ ∂ϕ ∂2ϕ DT ∂ 2 T (u + υ ) = DB 2 + ( ) , ε ∂x ∂y ∂y T∞ ∂y 2
(12)
u
where αm =
ε(ρc)p km ,τ = . (ρc)f (ρc)f
(13)
One can eliminate p from Eqs. (9) and (10) by cross-differentiation. We introduce a stream function ψ defined by u=
∂ψ ∂ψ ,υ = − , ∂y ∂x
(14)
Effect of Suction/Injection on Natural Convective Boundary-Layer Flow of . . .
57
so that Eq. (8) is satisfied identically. We are then left with the following three equations (1 − ϕ∞ )ρf∞ βgK ∂T (ρp − ρf ∞ )gK ∂ϕ ∂2ψ = − , 2 ∂y µ ∂y µ ∂y
(15)
∂ψ ∂T ∂ψ ∂T ∂ϕ ∂T DT ∂T 2 )( − = αm ∇2 T + τ [DB +( ) ], ∂y ∂x ∂x ∂y ∂y ∂y T∞ ∂y
(16)
∂2ϕ 1 ∂ψ ∂ϕ ∂ψ ∂ϕ DT ∂ 2 T ) ( − ) = DB 2 + ( . ε ∂y ∂x ∂x ∂y ∂y T∞ ∂y 2
(17)
We now introduce the local Rayleigh number Rax defined by Rax =
(1 − ϕ∞ )ρf ∞ βgKx(Tw − T∞ ) . µαm
(18)
Using the nonsimilarity transformations η=
y 1/2 2υ0 x Rax , ξ = 1/2 x αm Rax
(19)
and s(η, ξ) =
ψ 1/2 αm Rax
,
θ(η, ξ) =
T − T∞ , Tw − T∞
f (η, ξ) =
ϕ − ϕ∞ . ϕw − ϕ∞
(20)
Then, on substitution in Eqs. (15)-(17), we obtain the differential equations s′′ − θ′ + N r f ′ = 0,
(21)
1 ∂θ ∂s 1 − θ′ ), θ′′ + sθ′ + N b f ′ θ′ + N t θ′2 = ξ(s′ 2 2 ∂ξ ∂ξ
(22)
1 N t ′′ 1 ∂f ∂s f ′′ + Le sf ′ + θ = Le ξ(s′ − f ′ ), 2 Nb 2 ∂ξ ∂ξ
(23)
where N r, N b, N t, Le and ξ denote a buoyancy ratio, a Brownian motion parameter, a thermophoresis parameter, a Lewis number and suction/injection parameter, respectively are defined by (ρp − ρf ∞ )(ϕw − ϕ∞ ) , ρf ∞ β(Tw − T∞ )(1 − ϕ∞ )
(24)
Nb =
ε(ρc)p DB (ϕw − ϕ∞ ) , (ρc)f αm
(25)
Nt =
ε(ρc)p DT (Tw − T∞ ) , (ρc)f αm T∞
(26)
Nr =
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Le =
αm , εDB
(27)
the boundary conditions (5) and (6) becomes at η = 0 : s = ξ, θ = 1, f = 1,
(28)
as η → ∞ : s′ = 0, θ = 0, f = 0.
(29)
Integrating Eq. (21) once and using boundary conditions (29) we have s′ − θ + N rf = 0
(30)
Equations (30), (22), and (23) are solved subject to boundary conditions given by Eq. (28) and the following boundary conditions as η → ∞ : θ = 0, f = 0
(31)
When N r, N b, ξ and N t are all zero, the boundary-value problem, Eqs. (21) and (22), reduces to the classical problem solved by Cheng and Minkowycz [7]. When υ0 is zero, Eqs. (21) and (22) reduce to the problem solved by Nield and Kuznetsov [13]. The Nusselt number N u is defined by Nu =
q ′′ x , km (Tw − T∞ )
(32)
where q ′′ is the wall heat flux defined by ∂T q = −km |y=0 ∂y
1/2
Rax = −km (Tw − T∞ )θ (0, ξ) , x
′′
′
(33)
substituting from Eq.(33) in Eq.(32) we get 1
N u/Rax2 = −θ′ (0, ξ)
3.
(34)
Numerical Method
To solve the problem, the first step is to write the ODEs (22), (23) and (30) under conditions (28) and (31) as a system of first order ODEs. The basic idea is to introduce new variables, one for each variable in the original problem plus one for each of its derivatives up to one less than the highest derivative appearing. We used the Finite-difference method (we used NAG library for this purpose) for the solution of boundary value problem of the form y ′ = f (x, y),
a ≤ x ≤ b,
(35)
subject to general nonlinear, two-point boundary conditions g(y(a), y(b)) = 0.
(36)
Effect of Suction/Injection on Natural Convective Boundary-Layer Flow of . . .
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Finite-difference equations are set up on a mesh of points and estimated values for the solution at the grid points are chosen. Using these estimated values as starting values a Newton iteration is used to solve the Finite-difference equations. The accuracy of the solution is then improved by deferred corrections or the addition of points to the mesh or a combination of both. The technique used is described fully in [17]. The absolute error tolerance for this method is 10−4 .
4.
Results and Discussion
We have solved Eqs. (22), (23) and (30) under conditions (28) and (31) for a typical case, chosen as that for Le = 10, N r = 0.5, N b = 0.5, N t = 0.5, ξ = 0, are shown in Fig. 2. In order to test the accuracy of the present results, we have compared these results with those of Nield and Kuznetsov [13] when we neglect the effects of the suction/injection parameter ξ. We notice that the comparison shows an excellent agreement, as presented in Table 1. In Table 2, we calculated for 125 sets of values of Nr, Nb, Nt in the range [0.1, 0.2, 0.3, 0.4, 0.5] and a linear regression was performed on the results for various values of ξ and Le. So, the reduced Nusselt number N ur can be obtained from the correlation N ur = 0.444 + Cr N r + Cb N b + Ct N t.
(37)
where the constants Cr , Cb and Ct are the coefficients in the linear regression. For example, when Le = 10 the reduced Nusselt number N ur can be obtained from the correlation N urestξ=−0.3 = 0.444 − 0.246N r − 0.283N b − 0.228N t,
(38)
N urestξ=0 = 0.444 − 0.111N r − 0.2446N b − 0.1498N t,
(39)
N urestξ=0.3 = 0.444 + 0.0712N r − 0.2008N b − 0.0457N t,
(40)
valid for N r, N b, N t each taking values in the range [0, 0.5]. The Eq. (39) is identical to the equation which obtained by Nield and Kuznetsov [13]. Clearly an increase in any of the buoyancy-ratio number N r, the Brownian motion parameter N b, or the thermophoresis parameter N t at ξ = −0.3, 0.0 leads to a decrease in the value of the reduced Nusselt number (corresponding to an increase in the thermal boundary-layer thickness) and this effect increases in the case of injection. At ξ = 0.3 the reduced Nusselt number decreases with the increase of N b and N t, but it increases with the increase of N r. So, in the case of suction the thermal boundary-layer thickness decreases with the increase of the buoyancy-ratio number N r. The correlation formula overestimates the reduction from the standard fluid value 0.444. These results show that the coefficient of N t varies little as Le varies. On the other hand, the coefficients of N r and N b increase markedly as Le increases. Finally from Table 2 we observe that the accuracy of the linear regression estimate increases as Le increases. We believe that for most practical purposes the simple linear regression formula in Eq. (37) should be adequate. If one wants a more accurate formula then one can perform a quadratic regression. The reduced Nusselt number N ur can be obtained from the correlation N ur = 0.444 + Cr1 N r + Cb1 N b + Ct1 N t + Cr2 N t + Cb2 N t +Ct2 N t + Cbr N t + Ctr N t + Crb N t,
(41)
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where Cr1 , Cr2 , Cb1 , Cb2 , Ct1 , Ct2 , Cbt , Ctr and Crb are the coefficients in the quadratic regression estimate. For the case Le = 10, for example, we obtained instead of Eqs. (38) and (40) the formulas N urestξ=−0.3 = 0.444 − 0.409N r − 0.4765N b − 0.4104N t + 0.2043N r2 +0.2214N b2 + 0.2586N t2 + 0.2952N bN t +0.2269N tN r + 0.3403N rN b,
(42)
N urestξ=0.3 = 0.444 + 0.2784N r − 0.0902N b + 0.1055N t − 0.2975N r2 −0.2434N b2 − 0.2454N t2 − 0.076N bN t −0.2898N tN r − 0.1584N rN b,
(43)
The relatively large interactions between N r and N b (displayed by the coefficient of the last term in Eq. (41)), and between N r and N t (displayed by the coefficient of the third to last term in Eq. (41)) are of interest. In Tables 3 and 4 we presented values of the coefficients in the quadratic regression estimate for various values of Le and ξ. In the case of injection, where ξ = −0.3 we found numerical difficulty with very large values of Le so the variation with Le is circumscribed and the results are presented in Table 3. On the other hand, for the case of suction the results are presented in Table 4. The variations of the field variables with suction and injection parameter ξ = −0.3, −0.2, −0.1, 0.0, 0.1, 0.2, 0.3 in the case Le = 10 are shown in Figs. 3-6. Figs. 3 and 4 show the effect of suction and injection parameter ξ on velocity distributions. It is clear from Fig. 3 the nanofluid velocity in the y direction at the wall has the value of suction and injection parameter ξ and it begins to increase through the momentum boundary layer until it reaches to a constant value which depends on the value of ξ. On the other hand, the profile of nanofluid velocity in the x direction with variation of ξ is shown in Fig. 4. We noticed that the x-velocity with various values of ξ starts from a constant value (0.5) and begins to increase until it reaches to the peak value and then it decreases and tends to zero at the edge of the momentum boundary layer. It is clear from Fig. 4 that the thickness of momentum boundary layer increases in the case of injection while it decreases in the case of suction. Figs. 5 and 6 are presented to show the effect of stretching plate parameter ξ on temperature and nanoparticles distributions through the boundary layer. It is obvious that the thermal boundary layer thickness decreases in the suction case while it increases in the injection case, and that what we expected before from regression correlations. Further, we noticed from Fig. 6 that the volume fraction of nanoparticles just varies in a layer near the wall and the thickness of this layer increases in suction while it decreases in injection. This interpret that heat transfer coefficient in the case of suction is bigger than it in the case in injection because when the nanoparticles are closer together they tend to agglomerate due to molecular forces such as Van Der Waals force, thus that causes the effective surface area to volume ratio of the nanoparticles to decrease, which reduces the thermal conductivity of the fluid.
5.
Conclusions
We studied the problem of natural convection boundary-layer flow in a porous medium past a vertical porous plate saturated by a nanofluid using a model incorporates the effects of Brownian motion and thermophoresis. In this study we have employed the Darcy model for the momentum equation. A nonsimilar solution is presented to solve the problem which depends on four dimensionless parameters, namely a Lewis number Le, a buoyancy-ratio parameter N r, a Brownian motion parameter N b, and a thermophoresis parameter N t in addition to the suction and injection parameter ξ. We used a linear and a quadratic regression for various values of ξ to study the way in which the heat transfer coefficient,
REFERENCES
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represented by a Nusselt number N u, depends on these four parameters. We have discussed graphicly the way in which the velocity, temperature and nanoparticles profiles, as well as the local Nusselt number (heat transfer coefficient) depend on these parameters. We found that the nanoparticles in the fluid is more stable in the case of suction and that enhance the heat transfer property of the fluid.
References [1] J. A. Eastman, S. U. S. Choi, S. Li, W.Yu, and L. J. Thompson. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett., 6:718:721, 2001. [2] S. K. Das, N. Putra, P. Thiesen, and W. Roetzel. Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J. Heat Transfer, 125:567:575, 2003. [3] S. Choi. Enhancing thermal conductivity of fluids with nanoparticle. in: D. A. Siginer, H. P. Wang, (Eds.), Developments and Applications of Non-Newtonian Flows, ASME FED, 66:99:105, 1995. [4] H. Masuda, A. Ebata, K. Teramae, and N. Hishinuma. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei, 7:227:233, 1993. [5] M. Shahi, A. H Mahmoudi, and F. Talebi. Numerical study of mixed convective cooling in a square cavity ventilated and partially heated from the below utilizing nanofluid. Int. comm. Heat Mass Tran., 37:201:213, 2010. [6] J. Buongiorno. Convective transport in nanofluids. ASME J. Heat Tran., 128:240:250, 2006. [7] P. Cheng and W. J. Minkowycz. Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophys. Res., 82:2040:2044, 1977. [8] A. Bejan. Convection Heat Transfer. Wiley, Hoboken, third edtition edition, 2004. [9] A. Bejan and K. R. Khair. Heat and mass transfer by natural convection in a porous medium. Int. J. Heat. Mass. Tran., 28:909:919, 1985. [10] D. A. Nield and A. Bejan. Convection in Porous Media. Springer, New York, 2006. [11] A. Mahdy and F.M. Hady. Effect of thermophoretic particle deposition in non-newtonian free convection flow over a vertical plate with magnetic field effect. J. of Non-Newton. Fluid, 161:37:41, 2009. [12] X. Q. Wang and A. S. Mujumdar. Heat transfer characteristics of nanofluids: a review. Int. J. Thermal Sci., 46:1:19, 2007. [13] D. A. Nield and A. V. Kuznetsov. The chengminkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int. J. Heat Mass Tran., 52:5792:5795, 2009. [14] D. Y. Tzou. Instability of nanofluids in natural convection. ASME J. Heat Tran., 130:372:401, 2008. [15] D. Y. Tzou. Thermal instability of nanofluids in natural convection. Int. J. Heat Mass Tran., 51:2967:2979, 2008. [16] D. A. Nield and A. V. Kuznetsov. Thermal instability in a porous medium layer saturated by a nanofluid. Int. J. Heat Mass Tran., 52:5796:5801, 2009. [17] V. Pereyra. An adaptive finite-difference Fortran program for first order nonlinear ordinary boundary problems Codes for Boundary Value Problems in Ordinary Differential Equations, volume 76 of Lecture Notes in Computer Science. 1979.
Table 1.: Comparison of results for the linear regression coefficients for the reduced Nusselt number 1/2 at ξ=0. Here Cr , Cb , Ct , are the coefficients in the linear regression estimate N uest /Rax = 0.444 + Cr N r + Cb N b + Ct N t and ε is the maximum relative error defined by ε =| (N uest − N u)/N u |, applicable for N r, N b, N t each in [0,0.5]. Le
1 2 5 10 20 50 100 200 500 1000
Cr ——————— Present Ref[13] -0.310 -0.309 -0.230 -0.230 -0.148 -0.148 -0.111 -0.111 -0.086 -0.086 -0.064 -0.064 -0.053 -0.053 -0.046 -0.045 -0.039 -0.039 -0.036 -0.036
Cb ——————— Present Ref[13] -0.060 -0.060 -0.129 -0.129 -0.210 -0.209 -0.245 -0.245 -0.268 -0.268 -0.288 -0.288 -0.298 -0.298 -0.305 -0.304 -0.310 -0.310 -0.313 -0.313
Ct ——————— Present Ref[13] -0.166 -0.166 -0.162 -0.162 -0.152 -0.152 -0.150 -0.150 -0.150 -0.149 -0.149 -0.149 -0.149 -0.148 -0.149 -0.148 -0.149 -0.148 -0.149 -0.148
ε ——————— Present Ref[13] 0.214 0.154 0.279 0.147 0.126 0.126 0.119 0.119 0.114 0.114 0.110 0.110 0.108 0.108 0.107 0.107 0.106 0.106 0.105 0.107
Table 2.: Linear regression coefficients for the reduced Nusselt number at ξ=-0.3,0.3. Here Cr , Cb , 1/2 Ct , are the coefficients in the linear regression estimate N uest /Rax = 0.444 + Cr N r + Cb N b + Ct N t and ε is the maximum relative error defined by ε =| (N uest − N u)/N u |, applicable for N r, N b, N t each in [0,0.5]. Le
1 2 5 10 20 50 100 200 500 1000
Cr ——————— -0.3 0.3 -0.180 -0.332 -0.061 -0.275 0.032 -0.246 0.071 -0.218 0.095 0.095 0.117 0.120 0.121 0.121
Cb ——————— -0.3 0.3 0.091 -0.209 -0.04 -0.259 -0.152 -0.283 -0.201 -0.302 -0.232 -0.254 -0.262 -0.266 -0.269 -0.270
62
Ct ——————— -0.3 0.3 -0.081 -0.236 -0.059 -0.230 -0.047 -0.228 -0.046 -0.230 -0.046 -0.046 -0.046 -0.046 -0.046 -0.046
ε ——————— -0.3 0.3 0.410 0.618 0.398 0.583 0.185 0.566 0.183 0.548 0.182 0.183 0.184 0.184 0.184 0.184
Table 3.: Quadratic regression coefficients and error bound for the reduced Nusselt number at ξ=-0.3. Here Cr1 , Cr2 , Cb1 , Cb2 , Ct1 , Ct2 , Cbt , Ctr and Crb are the coefficients in the linear regression estimate 1/2 N uest /Rax = 0.444+Cr1 N r +Cb1 N b+Ct1 N t+Cr2 N r2 +Cb2 N b2 +Ct2 N t2 +Cbt N bN t+Ctr N tN r + Crb N rN b and ε is the maximum relative error defined by ε =| (N uest − N u)/N u |, applicable for N r, N b, N t each in [0,0.5]. Le 2 5 10 20
Cr1 -0.5075 -0.4396 -0.4090 -0.3977
Cb1 -0.3628 -0.4419 -0.4765 -0.4939
Ct1 -0.4252 -0.4142 -0.4104 -0.4081
Cr2 0.1783 0.1874 0.2043 0.2693
Cb2 0.0420 0.1692 0.2214 0.2509
Ct2 0.2616 0.2595 0.2586 0.2591
Cbt 0.3745 0.3135 0.2952 0.2935
Ctr 0.1705 0.2169 0.2269 0.2154
Crb 0.4946 0.3918 0.3403 0.2769
ε 0.2233 0.2172 0.2139 0.2110
Table 4.: Quadratic regression coefficients and error bound for the reduced Nusselt number at ξ=0.3. Here Cr1 , Cr2 , Cb1 , Cb2 , Ct1 , Ct2 , Cbt , Ctr and Crb are the coefficients in the linear regression estimate 1/2 N uest /Rax = 0.444+Cr1 N r +Cb1 N b+Ct1 N t+Cr2 N r2 +Cb2 N b2 +Ct2 N t2 +Cbt N bN t+Ctr N tN r + Crb N rN b and ε is the maximum relative error defined by ε =| (N uest − N u)/N u |, applicable for N r, N b, N t each in [0,0.5]. Le 1 2 5 10 20 50 100 200 500 1000
Cr1 -0.0192 0.1312 0.2273 0.2784 0.3119 0.3357 0.3444 0.3484 0.3509 0.3515
Cb1 0.3665 0.1435 -0.0126 -0.0902 -0.1411 -0.1788 -0.1931 -0.2003 -0.2046 -0.206
Ct1 0.0496 0.1085 0.1047 0.1055 0.1054 0.1051 0.1049 0.1049 0.1047 0.1047
Cr2 -0.3758 -0.3655 -0.3067 -0.2975 -0.2948 -0.2938 -0.2936 -0.2935 -0.2936 -0.2935
Cb2 -0.9928 -0.644 -0.35 -0.2434 -0.1786 -0.1326 -0.1153 -0.1066 -0.1013 -0.0996
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Ct2 -0.2243 -0.2862 -0.2468 -0.2454 -0.2446 -0.2442 -0.2438 -0.2439 -0.2437 -0.2437
Cbt 0.2603 0.1213 -0.047 -0.076 -0.0876 -0.0935 -0.0952 -0.0964 -0.0968 -0.0969
Ctr -0.5889 -0.4669 -0.3186 -0.2898 -0.2774 -0.2696 -0.2674 -0.2658 -0.2647 -0.2645
Crb 0.4624 0.1973 -0.0723 -0.1584 -0.2076 -0.2408 -0.2527 -0.2585 -0.2621 -0.2631
ε 0.0662 0.1006 0.0345 0.0142 0.0069 0.0067 0.0066 0.0062 0.0061 0.0063
2
1.5
1 s
ξ=−0.3,−0.2,−0.1,0,0.1,0.2,0.3 0.5
0
−0.5
0
2
4
η
6
8
10
Figure 3.: Plots of dimensionless similarity function s(η, ξ) for the case Le = 10 and ξ = −0.3, −0.2, −0.1, 0.0, 0.1, 0.2, 0.3.
0.8 0.7 0.6 0.5 s’
ξ=0.3,0.2,0.1,0,−0.1,−0.2,−0.3 0.4 0.3 0.2 0.1 0
0
2
4
η
6
8
10
Figure 4.: Plots of dimensionless similarity function s′ (η, ξ) for the case Le = 10 and ξ = −0.3, −0.2, −0.1, 0.0, 0.1, 0.2, 0.3.
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1 0.9 0.8 0.7
θ
0.6
ξ=−0.3,−0.2,−0.1,0,0.1,0.2,0.3
0.5 0.4 0.3 0.2 0.1 0
0
2
4
η
6
8
10
Figure 5.: Plots of dimensionless similarity function θ(η, ξ) for the case Le = 10 and ξ = −0.3, −0.2, −0.1, 0.0, 0.1, 0.2, 0.3.
1 0.9 0.8 0.7
f
0.6 ξ=−0.3,−0.2,−0.1,0,0.1,0.2,0.3
0.5 0.4 0.3 0.2 0.1 0
0
2
4
η
6
8
10
Figure 6.: Plots of dimensionless similarity function f (η, ξ) for the case Le = 10 and ξ = −0.3, −0.2, −0.1, 0.0, 0.1, 0.2, 0.3.
65
