Mixed convection in a vertical channel subject to Robin boundary condition

Flavius Patrulescu
(original), ISI/JCR, Numerical Analysis, paper

Abstract

The steady mixed convection flow in a vertical channel is investigated for laminar and fully developed flow regime. In the modelling of the heat transfer the viscous dissipation term was also considered. Temperature on the right wall is assumed constant while a mixed boundary condition (Robin boundary condition) is considered on the left wall. The governing equations are expressed in non-dimensional form and then solved both analytically and numerically. It was found that there is a decrease in reversal flow with an increase in the mixed convection parameter.

Authors

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Teodor Groşan
(Babeş-Bolyai University Faculty of Mathematics and Computer Sciences)

Adrian Vasile Lar
(Școala Cojocna, Romania)

Keywords

viscous fluid; forced convection; heat transfer; fully developed flow

Cite this paper as:

F. Pătrulescu, T. Groşan, A.V. Lar, Mixed convection in a vertical channel subject to Robin boundary condition, Studia Universitatis Babeş-Bolyai, seria Mathematica, vol. 55, no. 2 (2010), pp. 167-176

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https://ictp.acad.ro/patrulescu/papers/2010_Patrulescu_STUDIA_Mixed_convection_Robin_boundary.pdf

About this paper

Journal

Studia Universitatis Babeş-Bolyai. Mathematica

Publisher Name

Universitatea Babeş-Bolyai, Cluj-Napoca; Cluj University Press, Cluj-Napoca

Paper on the journal website

http://www.cs.ubbcluj.ro/~studia-m/2010-2/patrulescu-final.pdf

Print ISSN

0252-1938

Online ISSN

2065-961X

MR

2661883

ZBL

1224.76037

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https://scholar.google.ro/scholar?oi=bibs&hl=ro&cites=13494721152361498203

References

[1] Aung, Fully developed laminar free convection between vertical plates heated asymmetrically, Int. J. Heat Mass Transfer 15 (1972), 1577-1580.
[2] Aung, L.S. Fletcher, V. Sernas, Developing laminar free convection between vertical flat plates with asymmetric heating, Int. J. Heat Mass Transfer, 15 (1972), 2293-2308.
[3] Aung, G. Worku, Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures, J. Heat Transfer, 108 (1986), 299-304.
[4] Aung, G. Worku, Theory of fully developed, combined convection including flow reversal, J. Heat Transfer, 108 (1986), 485-488.
[5] Barletta, Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal-isoflux boundary conditions, J. Heat Transfer,  121 (1999), 349-356.
[6] Barletta, Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux, Int. J. Heat Mass Transfer,  45 (2002),  641-654.
[7] Boulama, N. Galanis, Analytical solution for fully developed mixed convection between parallel vertical plates with heat and mass transfer, J. Heat Transfer,  126 (2004), 381-388.
[8] Bejan, Convection Heat Transfer (2nd edition), Wiley, New York (1995).
[9] Barletta, Laminar mixed convection with viscous dissipation in a vertical channel, Int. J. Heat Mass Transfer, 41 (1988),  3501-3513.
[10] Pop, D. B. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford (2001).
[11] Kohr, I. Pop, Viscous Incompressible Flow for Low Reynolds Numbers, WIT Press, Southampton (2004).

Paper (preprint) in HTML form

2010_Patrulescu_STUDIA_Mixed_convection_Robin_boundary

Mixed convection in a vertical channel subject to Robin boundary condition

F. Pătrulescul, T. Groşan 2 2 ^(2){ }^{2}2, A.V. Lar 3 3 ^(3){ }^{3}3 1 Tiberiu Popoviciu Institute of Numerical Analysis 1 Tiberiu Popoviciu Institute of Numerical Analysis  1^("Tiberiu Popoviciu Institute of Numerical Analysis ")1^{\text {Tiberiu Popoviciu Institute of Numerical Analysis }}1Tiberiu Popoviciu Institute of Numerical Analysis  P.O. Box 68-1, 400110 Cluj-Napoca, Romania, fpatrulescu@ictp.acad.ro 2 2 ^(2){ }^{2}2 Faculty of Mathematics and Computer Science Babes-Bolyai University, Cluj-Napoca, Romania 3 3 ^(3){ }^{3}3 Scoala Cojocna, Cluj, Romania

Abstract

The steady mixed convection flow in a vertical channel is investigated for laminar and fully developed flow regime. In the modelling of the heat transfer the viscous dissipation term was also considered. Temperature on the right wall is assumed constant while a mixed boundary condition (Robin boundary condition) is considered on the left wall. The governing equations are expressed in non-dimensional form and then solved both analytically and numerically. It was found that there is a decrease in reversal flow with an increase in the mixed convection parameter.

2010 Mathematics Subject Classification : 76D05,80A20
Keywords: viscous fluid, forced convection, heat transfer, fully developed flow

1 Introduction

Heat transfer in channels occurs in many industrial processes and natural phenomena. It has been, therefore, the subject of many detailed, mostly numerical studies for different flow configurations. Most of the interest in this subject is due to its practical applications, for example, in the design of cooling systems for electronic devices and in the field of solar energy collection. Some of the published papers, such as by Aung [1], Aung et al. [2], Aung and Worku [3, 4], Barletta [5, 6], and Boulama and Galanis [7], are concerned with the evaluation of the temperature and velocity profiles for the vertical parallel-flow fully developed regime. As is well known, heat exchangers technology involves convective flows in vertical channels. In most cases, these flows imply conditions of uniform heating of a channel, which can be modelled either by uniform wall temperature (UWT) or uniform wall heat flux (UHF) thermal boundary conditions. In the present paper, new types of boundary conditions are considered. The right wall is kept at constant temperature while a convective heat flux is considered on the left wall (see, Bejan[8]):
(1) ( k ∂ T ∂ y ) y = 0 + h a ( T a − T ) y = 0 = 0 (1) k ∂ T ∂ y y = 0 + h a T a − T y = 0 = 0 {:(1)(k(del T)/(del y))_(y=0)+h_(a)(T_(a)-T)_(y=0)=0:}\begin{equation*} \left(k \frac{\partial T}{\partial y}\right)_{y=0}+h_{a}\left(T_{a}-T\right)_{y=0}=0 \tag{1} \end{equation*}(1)(k∂T∂y)y=0+ha(Ta−T)y=0=0
where k k kkk is the thermal conductivity, h a h a h_(a)h_{a}ha is the external heat transfer coefficient and T a T a T_(a)T_{a}Ta is the external temperature (see Figure 1). This kind of boundary condition is appropriate to express matematically heat loosing in insullation problems.In addition we have taken in account in this paper the effect of viscous dissipation, see Barletta[9].
Figure 1: Geometry of the problem and the co-ordinate system

2 Basic Equations

Consider a viscous and incompressible fluid, which steadily flows between two infinite vertical and parallel plane walls. At the entrance of the channel the fluid has an entrance velocity U 0 U 0 U_(0)U_{0}U0 parallel to the vertical axis of the channel. The geometry of the problem, the boundary conditions, and the coordinate system are shown in Fig. 1. The variation of density with temperature is given by the Boussinesq approximation and the fluid rises in the duct driven by buoyancy forces and initial velocity. Hence, the flow is due to difference in temperature and in the pressure gradient. The flow being fully developed the following relations apply here v = 0 , ∂ v / ∂ y = 0 , ∂ p / ∂ y = 0 v = 0 , ∂ v / ∂ y = 0 , ∂ p / ∂ y = 0 v=0,del v//del y=0,del p//del y=0v=0, \partial v / \partial y=0, \partial p / \partial y=0v=0,∂v/∂y=0,∂p/∂y=0, where v v vvv is the velocity in the transversal direction and p p ppp is the pressure. Thus, from the continuity equation, we get ∂ u / ∂ x = 0 ∂ u / ∂ x = 0 del u//del x=0\partial u / \partial x=0∂u/∂x=0 so that the velocity component along x x xxx-axis depends only by y , u = u ( y ) y , u = u ( y ) y,u=u(y)y, u=u(y)y,u=u(y). Based on the fact that the flow is fully developed we can assume that the temperature T = T ( y ) T = T ( y ) T=T(y)T=T(y)T=T(y). Under these assumptions the momentum and energy equations for the flow and heat transfer have the following form:
(2) ν d 2 u d y 2 − 1 ρ d p d x + g β ( T − T 0 ) = 0 (3) α d 2 T d y 2 + ν c p ( d u d y ) 2 = 0 (2) ν d 2 u d y 2 − 1 ρ d p d x + g β T − T 0 = 0 (3) α d 2 T d y 2 + ν c p d u d y 2 = 0 {:[(2)nu(d^(2)u)/(dy^(2))-(1)/(rho)(dp)/(dx)+g beta(T-T_(0))=0],[(3)alpha(d^(2)T)/(dy^(2))+(nu)/(c_(p))((du)/(dy))^(2)=0]:}\begin{gather*} \nu \frac{d^{2} u}{d y^{2}}-\frac{1}{\rho} \frac{d p}{d x}+g \beta\left(T-T_{0}\right)=0 \tag{2}\\ \alpha \frac{d^{2} T}{d y^{2}}+\frac{\nu}{c_{p}}\left(\frac{d u}{d y}\right)^{2}=0 \tag{3} \end{gather*}(2)νd2udy2−1ρdpdx+gβ(T−T0)=0(3)αd2Tdy2+νcp(dudy)2=0
subject to the boundary condition given by Eq. (1), noslip condition for for velocity at the walls and constant temperature at the left wall:
(4) u ( 0 ) = 0 , u ( L ) = 0 , T ( L ) = T w (4) u ( 0 ) = 0 , u ( L ) = 0 , T ( L ) = T w {:(4)u(0)=0","u(L)=0","T(L)=T_(w):}\begin{equation*} u(0)=0, u(L)=0, T(L)=T_{w} \tag{4} \end{equation*}(4)u(0)=0,u(L)=0,T(L)=Tw
where α α alpha\alphaα is the thermal diffusivity of the viscous fluid, ρ ρ rho\rhoρ is the fluid density and c p c p c_(p)c_{p}cp is the specific heat at constant pressure. In the sistem (2) and (3) there is an additional unknown, the gradient of pressure, d p / d x d p / d x dp//dxd p / d xdp/dx. In order to close the above system subject to the boundary conditions (1) and (4) it is necessary to consider the equation of the mass flux conservation:
(5) U 0 = 1 L ∫ 0 L u ( y ) d y (5) U 0 = 1 L ∫ 0 L   u ( y ) d y {:(5)U_(0)=(1)/(L)int_(0)^(L)u(y)dy:}\begin{equation*} U_{0}=\frac{1}{L} \int_{0}^{L} u(y) d y \tag{5} \end{equation*}(5)U0=1L∫0Lu(y)dy
where L L LLL is the channel width. Further, we introduce the following dimensionless variables (see Pop and Ingham[10] or Kohr and Pop[11]):
(6) U = u U 0 , X = x R e L , Y = y L , θ = T − T 0 T w − T 0 , P = L 2 ρ ν 2 p (6) U = u U 0 , X = x R e L , Y = y L , θ = T − T 0 T w − T 0 , P = L 2 ρ ν 2 p {:(6)U=(u)/(U_(0))","X=(xRe)/(L)","Y=(y)/(L)","theta=(T-T_(0))/(T_(w)-T_(0))","P=(L^(2))/(rhonu^(2))p:}\begin{equation*} U=\frac{u}{U_{0}}, X=\frac{x R e}{L}, Y=\frac{y}{L}, \theta=\frac{T-T_{0}}{T_{w}-T_{0}}, P=\frac{L^{2}}{\rho \nu^{2}} p \tag{6} \end{equation*}(6)U=uU0,X=xReL,Y=yL,θ=T−T0Tw−T0,P=L2ρν2p
where Re = U 0 L / ν Re = U 0 L / ν Re=U_(0)L//nu\operatorname{Re}=U_{0} L / \nuRe=U0L/ν is the Reynolds number and T 0 = ( T a + T w ) / 2 T 0 = T a + T w / 2 T_(0)=(T_(a)+T_(w))//2T_{0}=\left(T_{a}+T_{w}\right) / 2T0=(Ta+Tw)/2 is a characteristic temperature. Using (6) in the equations (2)-(3), in the boundary conditions (1) and (4) and in the mass flux conservation (5) we obtain:
(7) d 2 U d Y 2 + λ θ − γ = 0 (8) d 2 θ d Y 2 + B r ( d U d Y ) 2 = 0 (9) U ( 0 ) = 0 , U ( 1 ) = 0 , ( d θ d Y ) Y = 0 = κ ( 1 + θ ) Y = 0 , θ ( 1 ) = 1 (10) ∫ 0 1 U ( Y ) d Y = 1 (7) d 2 U d Y 2 + λ θ − γ = 0 (8) d 2 θ d Y 2 + B r d U d Y 2 = 0 (9) U ( 0 ) = 0 , U ( 1 ) = 0 , d θ d Y Y = 0 = κ ( 1 + θ ) Y = 0 , θ ( 1 ) = 1 (10) ∫ 0 1   U ( Y ) d Y = 1 {:[(7)(d^(2)U)/(dY^(2))+lambda theta-gamma=0],[(8)(d^(2)theta)/(dY^(2))+Br((dU)/(dY))^(2)=0],[(9)U(0)=0","U(1)=0","((d theta)/(dY))_(Y=0)=kappa(1+theta)_(Y=0)","theta(1)=1],[(10)int_(0)^(1)U(Y)dY=1]:}\begin{gather*} \frac{d^{2} U}{d Y^{2}}+\lambda \theta-\gamma=0 \tag{7}\\ \frac{d^{2} \theta}{d Y^{2}}+B r\left(\frac{d U}{d Y}\right)^{2}=0 \tag{8}\\ U(0)=0, U(1)=0,\left(\frac{d \theta}{d Y}\right)_{Y=0}=\kappa(1+\theta)_{Y=0}, \theta(1)=1 \tag{9}\\ \int_{0}^{1} U(Y) d Y=1 \tag{10} \end{gather*}(7)d2UdY2+λθ−γ=0(8)d2θdY2+Br(dUdY)2=0(9)U(0)=0,U(1)=0,(dθdY)Y=0=κ(1+θ)Y=0,θ(1)=1(10)∫01U(Y)dY=1
In Eqs ( 7 ) − ( 10 ) γ Eqs ( 7 ) − ( 10 ) γ Eqs(7)-(10)gamma\operatorname{Eqs}(7)-(10) \gammaEqs(7)−(10)γ is the pressure gradient in X X XXX direction, B r B r BrB rBr is the Brinkman number, λ λ lambda\lambdaλ is the mixed convection parameter and κ κ kappa\kappaκ is the convection heat transfer parameter given by
(11) γ = d P d X , B r = Pr E c = μ U 0 2 k ( T w − T 0 ) , λ = G r R e = g β ( T w − T 0 ) L 2 U 0 ν , κ = h a L k (11) γ = d P d X , B r = Pr E c = μ U 0 2 k T w − T 0 , λ = G r R e = g β T w − T 0 L 2 U 0 ν , κ = h a L k {:(11)gamma=(dP)/(dX)","Br=Pr Ec=(muU_(0)^(2))/(k(T_(w)-T_(0)))","lambda=(Gr)/(Re)=(g beta(T_(w)-T_(0))L^(2))/(U_(0)nu)","kappa=(h_(a)L)/(k):}\begin{equation*} \gamma=\frac{d P}{d X}, B r=\operatorname{Pr} E c=\frac{\mu U_{0}^{2}}{k\left(T_{w}-T_{0}\right)}, \lambda=\frac{G r}{R e}=\frac{g \beta\left(T_{w}-T_{0}\right) L^{2}}{U_{0} \nu}, \kappa=\frac{h_{a} L}{k} \tag{11} \end{equation*}(11)γ=dPdX,Br=PrEc=μU02k(Tw−T0),λ=GrRe=gβ(Tw−T0)L2U0ν,κ=haLk
and Pr , E c , G r Pr , E c , G r Pr,Ec,Gr\operatorname{Pr}, E c, G rPr,Ec,Gr and Re are the Prandtl number, Eckert number, Grashoff number and Reynolds number,respectively, defined as:
(12) Pr = ν α , E c = U 0 2 c p ( T w − T 0 ) , G r = g β ( T w − T 0 ) L 3 ν 2 , Re = U 0 L ν (12) Pr = ν α , E c = U 0 2 c p T w − T 0 , G r = g β T w − T 0 L 3 ν 2 , Re = U 0 L ν {:(12)Pr=(nu )/(alpha)","Ec=(U_(0)^(2))/(c_(p)(T_(w)-T_(0)))","Gr=(g beta(T_(w)-T_(0))L^(3))/(nu^(2))","Re=(U_(0)L)/(nu):}\begin{equation*} \operatorname{Pr}=\frac{\nu}{\alpha}, E c=\frac{U_{0}^{2}}{c_{p}\left(T_{w}-T_{0}\right)}, G r=\frac{g \beta\left(T_{w}-T_{0}\right) L^{3}}{\nu^{2}}, \operatorname{Re}=\frac{U_{0} L}{\nu} \tag{12} \end{equation*}(12)Pr=να,Ec=U02cp(Tw−T0),Gr=gβ(Tw−T0)L3ν2,Re=U0Lν
The physical quantity of interest in this problem are the skin friction coefficient C f C f C_(f)C_{f}Cf and the Nusselt number N u N u NuN uNu, which are defined as:
(13) C f = μ ρ U 0 2 ( d u d y ) y = 0 , L , N u = ( h f L k ) y = 0 , L (13) C f = μ ρ U 0 2 d u d y y = 0 , L , N u = h f L k y = 0 , L {:(13)C_(f)=(mu)/(rhoU_(0)^(2))((du)/(dy))_(y=0,L)","Nu=((h_(f)L)/(k))_(y=0,L):}\begin{equation*} C_{f}=\frac{\mu}{\rho U_{0}^{2}}\left(\frac{d u}{d y}\right)_{y=0, L}, N u=\left(\frac{h_{f} L}{k}\right)_{y=0, L} \tag{13} \end{equation*}(13)Cf=μρU02(dudy)y=0,L,Nu=(hfLk)y=0,L
In Eq. (13) h f h f h_(f)h_{f}hf is the internal heat transfer coefficient which can be calculated from the heat transfer balance at the wall:
( k ∂ T ∂ n ) wall = h f ( T wall − T fluid ) ) k ∂ T ∂ n wall  = h f T wall  − T fluid  {:(k(del T)/(deln))_("wall ")=h_(f)(T_("wall ")-T_("fluid ")))\left.\left(k \frac{\partial T}{\partial \mathbf{n}}\right)_{\text {wall }}=h_{f}\left(T_{\text {wall }}-T_{\text {fluid }}\right)\right)(k∂T∂n)wall =hf(Twall −Tfluid ))
where n n n\mathbf{n}n is the normal to the wall. Using dimensionless varibles (6) we obtain:
(14) C f R e = ( d U d Y ) Y = 0 , 1 , N u | Y = 0 = κ ( θ ( 0 ) + 1 θ ( 0 ) − 1 ) , N u | Y = 1 = − ( d θ d Y ) Y = 1 θ ( 0 ) − 1 (14) C f R e = d U d Y Y = 0 , 1 , N u Y = 0 = κ θ ( 0 ) + 1 θ ( 0 ) − 1 , N u Y = 1 = − d θ d Y Y = 1 θ ( 0 ) − 1 {:(14)C_(f)Re=((dU)/(dY))_(Y=0,1)","Nu|_(Y=0)=kappa((theta(0)+1)/(theta(0)-1))","Nu|_(Y=1)=-(((d theta)/(dY))_(Y=1))/(theta(0)-1):}\begin{equation*} C_{f} R e=\left(\frac{d U}{d Y}\right)_{Y=0,1},\left.N u\right|_{Y=0}=\kappa\left(\frac{\theta(0)+1}{\theta(0)-1}\right),\left.N u\right|_{Y=1}=-\frac{\left(\frac{d \theta}{d Y}\right)_{Y=1}}{\theta(0)-1} \tag{14} \end{equation*}(14)CfRe=(dUdY)Y=0,1,Nu|Y=0=κ(θ(0)+1θ(0)−1),Nu|Y=1=−(dθdY)Y=1θ(0)−1

3 Results and Discussions

Equations (7) to (10) admit an analytical solution in two particular cases:
i) Case Br = 0 = 0 =0=0=0
In this case the system (7) and (8) becomes:
(15) d 2 U d Y 2 − d P d X + λ θ = 0 (15) d 2 U d Y 2 − d P d X + λ θ = 0 {:(15)(d^(2)U)/(dY^(2))-(dP)/(dX)+lambda theta=0:}\begin{equation*} \frac{d^{2} U}{d Y^{2}}-\frac{d P}{d X}+\lambda \theta=0 \tag{15} \end{equation*}(15)d2UdY2−dPdX+λθ=0
(16) d 2 θ d Y 2 = 0 (16) d 2 θ d Y 2 = 0 {:(16)(d^(2)theta)/(dY^(2))=0:}\begin{equation*} \frac{d^{2} \theta}{d Y^{2}}=0 \tag{16} \end{equation*}(16)d2θdY2=0
subject to the boundary conditions (9). Further, from Eq.(15), (16) and condition (10) we obtain
θ ( Y ) = 2 κ 1 + κ Y + 1 − κ 1 + κ (17) U ( Y ) = − κ λ 1 + κ Y 3 3 + ( γ + 1 − κ 1 + κ λ ) Y 2 2 + ( κ λ 3 ( 1 + κ ) − 1 2 ( γ + 1 − κ 1 + κ ) ) Y γ = − 12 + λ 1 + κ θ ( Y ) = 2 κ 1 + κ Y + 1 − κ 1 + κ (17) U ( Y ) = − κ λ 1 + κ Y 3 3 + γ + 1 − κ 1 + κ λ Y 2 2 + κ λ 3 ( 1 + κ ) − 1 2 γ + 1 − κ 1 + κ Y γ = − 12 + λ 1 + κ {:[theta(Y)=(2kappa)/(1+kappa)Y+(1-kappa)/(1+kappa)],[(17)U(Y)=-(kappa lambda)/(1+kappa)(Y^(3))/(3)+(gamma+(1-kappa)/(1+kappa)lambda)(Y^(2))/(2)+((kappa lambda)/(3(1+kappa))-(1)/(2)(gamma+(1-kappa)/(1+kappa)))Y],[gamma=-12+(lambda)/(1+kappa)]:}\begin{gather*} \theta(Y)=\frac{2 \kappa}{1+\kappa} Y+\frac{1-\kappa}{1+\kappa} \\ U(Y)=-\frac{\kappa \lambda}{1+\kappa} \frac{Y^{3}}{3}+\left(\gamma+\frac{1-\kappa}{1+\kappa} \lambda\right) \frac{Y^{2}}{2}+\left(\frac{\kappa \lambda}{3(1+\kappa)}-\frac{1}{2}\left(\gamma+\frac{1-\kappa}{1+\kappa}\right)\right) Y \tag{17}\\ \gamma=-12+\frac{\lambda}{1+\kappa} \end{gather*}θ(Y)=2κ1+κY+1−κ1+κ(17)U(Y)=−κλ1+κY33+(γ+1−κ1+κλ)Y22+(κλ3(1+κ)−12(γ+1−κ1+κ))Yγ=−12+λ1+κ
ii) Case λ = 0 λ = 0 lambda=0\lambda=0λ=0
For λ = 0 λ = 0 lambda=0\lambda=0λ=0 the forced convection only is considered. The system (7) and (8) takes the following form:
(18) d 2 U d Y 2 − γ = 0 (19) d 2 θ d Y 2 + B r ( d U d Y ) 2 = 0 (18) d 2 U d Y 2 − γ = 0 (19) d 2 θ d Y 2 + B r d U d Y 2 = 0 {:[(18)(d^(2)U)/(dY^(2))-gamma=0],[(19)(d^(2)theta)/(dY^(2))+Br((dU)/(dY))^(2)=0]:}\begin{gather*} \frac{d^{2} U}{d Y^{2}}-\gamma=0 \tag{18}\\ \frac{d^{2} \theta}{d Y^{2}}+B r\left(\frac{d U}{d Y}\right)^{2}=0 \tag{19} \end{gather*}(18)d2UdY2−γ=0(19)d2θdY2+Br(dUdY)2=0
Taking in account that γ γ gamma\gammaγ is constant, using the boundary conditions (9) and mass flux conservation (10) we have:
U ( Y ) = − 6 Y 2 + 6 Y (20) θ ( Y ) = − 12 B r Y 4 + 24 B r Y 3 − 18 B r Y 2 + 2 κ 1 + κ ( 1 + 3 B r ) Y + 1 + 6 B r − κ 1 + κ γ = − 12 U ( Y ) = − 6 Y 2 + 6 Y (20) θ ( Y ) = − 12 B r Y 4 + 24 B r Y 3 − 18 B r Y 2 + 2 κ 1 + κ ( 1 + 3 B r ) Y + 1 + 6 B r − κ 1 + κ γ = − 12 {:[U(Y)=-6Y^(2)+6Y],[(20)theta(Y)=-12 BrY^(4)+24 BrY^(3)-18 BrY^(2)+(2kappa)/(1+kappa)(1+3Br)Y+(1+6Br-kappa)/(1+kappa)],[gamma=-12]:}\begin{gather*} U(Y)=-6 Y^{2}+6 Y \\ \theta(Y)=-12 B r Y^{4}+24 B r Y^{3}-18 B r Y^{2}+\frac{2 \kappa}{1+\kappa}(1+3 B r) Y+\frac{1+6 B r-\kappa}{1+\kappa} \tag{20}\\ \gamma=-12 \end{gather*}U(Y)=−6Y2+6Y(20)θ(Y)=−12BrY4+24BrY3−18BrY2+2κ1+κ(1+3Br)Y+1+6Br−κ1+κγ=−12
Equations (7) and (8) subject to (9) and (10) were solved numerically for different values of the parameters, λ , κ λ , κ lambda,kappa\lambda, \kappaλ,κ and Br ( λ = 0 , 100 , 250 , 500 ; κ = 0.01 , 0.1 , 1 , 10 ; Br = 0 , 0.001 , 0.01 , 0.025 ) Br ( λ = 0 , 100 , 250 , 500 ; κ = 0.01 , 0.1 , 1 , 10 ; Br = 0 , 0.001 , 0.01 , 0.025 ) Br(lambda=0,100,250,500;kappa=0.01,0.1,1,10;Br=0,0.001,0.01,0.025)\operatorname{Br}(\lambda=0,100,250,500 ; \kappa=0.01,0.1,1,10 ; \mathrm{Br}=0,0.001,0.01,0.025)Br(λ=0,100,250,500;κ=0.01,0.1,1,10;Br=0,0.001,0.01,0.025) using an implicit finite-difference method for velocity and a Gauss-Seidel iteration for temperature. Dimensionless velocity profiles, U ( Y ) U ( Y ) U(Y)\mathrm{U}(\mathrm{Y})U(Y), and temperature profiles, θ ( Y ) θ ( Y ) theta(Y)\theta(Y)θ(Y), are presented in Figs. 2 to 7 for different values of the above parameters. Analytical solutions ( λ = 0 , Br = 0 λ = 0 , Br = 0 lambda=0,Br=0\lambda=0, \mathrm{Br}=0λ=0,Br=0 ) are also presented on figures with a circle marker.
The variation of the velocity U ( Y ) U ( Y ) U(Y)U(Y)U(Y) and temperature θ ( Y ) θ ( Y ) theta(Y)\theta(Y)θ(Y) with the mixed convection parameter λ λ lambda\lambdaλ is presented in Figs. 2 and 5. We notice
Acknowledgements. The work of the second author was supported from UEFISCU Grant PN-II-ID-PCE-2007-1/525 (Romanian Ministry of Education and Research).
λ λ lambda\lambdaλ κ κ kappa\kappaκ B r = 0 B r = 0 Br=0B r=0Br=0 B r = 0.001 B r = 0.001 Br=0.001B r=0.001Br=0.001 B r = 0.01 B r = 0.01 Br=0.01B r=0.01Br=0.01
0 0.1 5.940594 5.940594 5.940594
1 5.940594 5.940594 5.940594
10 5.940594 5.940594 5.940594
100 0.1 4.470594 4.501963 4.799567
1 -2.144405 -2.134825 -2.047795
10 -8.759405 -8.784130 -9.001301
500 0.1 -1.409405 -1.27279 -0.125966
1 -34.484405 -32.905781 -25.265381
10 -67.559405 -68.912398 -68.791864
lambda kappa Br=0 Br=0.001 Br=0.01 0 0.1 5.940594 5.940594 5.940594 1 5.940594 5.940594 5.940594 10 5.940594 5.940594 5.940594 100 0.1 4.470594 4.501963 4.799567 1 -2.144405 -2.134825 -2.047795 10 -8.759405 -8.784130 -9.001301 500 0.1 -1.409405 -1.27279 -0.125966 1 -34.484405 -32.905781 -25.265381 10 -67.559405 -68.912398 -68.791864| $\lambda$ | $\kappa$ | $B r=0$ | $B r=0.001$ | $B r=0.01$ | | :---: | :---: | ---: | ---: | ---: | | 0 | 0.1 | 5.940594 | 5.940594 | 5.940594 | | | 1 | 5.940594 | 5.940594 | 5.940594 | | | 10 | 5.940594 | 5.940594 | 5.940594 | | 100 | 0.1 | 4.470594 | 4.501963 | 4.799567 | | | 1 | -2.144405 | -2.134825 | -2.047795 | | | 10 | -8.759405 | -8.784130 | -9.001301 | | 500 | 0.1 | -1.409405 | -1.27279 | -0.125966 | | | 1 | -34.484405 | -32.905781 | -25.265381 | | | 10 | -67.559405 | -68.912398 | -68.791864 |
Table 1: Friction coefficient C f R e | Y = 0 C f R e Y = 0 C_(f)Re|_(Y=0)\left.C_{f} R e\right|_{Y=0}CfRe|Y=0
λ λ lambda\lambdaλ κ κ kappa\kappaκ B r = 0 B r = 0 Br=0B r=0Br=0 B r = 0.001 B r = 0.001 Br=0.001B r=0.001Br=0.001 B r = 0.01 B r = 0.01 Br=0.01B r=0.01Br=0.01
0 0.1 -5.940594 -5.940594 -5.940594
1 -5.940594 -5.940594 -5.940594
10 -5.940594 -5.940594 -5.940594
100 0.1 -7.410594 -7.370828 -6.999151
1 -14.025594 -13.992628 -13.700667
10 -20.640594 -20.606004 -20.288425
500 0.1 -13.290594 -13.053063 -11.199059
1 -46.365594 -43.025955 -27.986534
10 -79.440594 -74.606586 -42.694129
lambda kappa Br=0 Br=0.001 Br=0.01 0 0.1 -5.940594 -5.940594 -5.940594 1 -5.940594 -5.940594 -5.940594 10 -5.940594 -5.940594 -5.940594 100 0.1 -7.410594 -7.370828 -6.999151 1 -14.025594 -13.992628 -13.700667 10 -20.640594 -20.606004 -20.288425 500 0.1 -13.290594 -13.053063 -11.199059 1 -46.365594 -43.025955 -27.986534 10 -79.440594 -74.606586 -42.694129| $\lambda$ | $\kappa$ | $B r=0$ | $B r=0.001$ | $B r=0.01$ | | :---: | :---: | ---: | ---: | ---: | | 0 | 0.1 | -5.940594 | -5.940594 | -5.940594 | | | 1 | -5.940594 | -5.940594 | -5.940594 | | | 10 | -5.940594 | -5.940594 | -5.940594 | | 100 | 0.1 | -7.410594 | -7.370828 | -6.999151 | | | 1 | -14.025594 | -13.992628 | -13.700667 | | | 10 | -20.640594 | -20.606004 | -20.288425 | | 500 | 0.1 | -13.290594 | -13.053063 | -11.199059 | | | 1 | -46.365594 | -43.025955 | -27.986534 | | | 10 | -79.440594 | -74.606586 | -42.694129 |
Table 2: Friction coefficient C f R e | Y = 1 C f R e Y = 1 C_(f)Re|_(Y=1)\left.C_{f} R e\right|_{Y=1}CfRe|Y=1
λ λ lambda\lambdaλ κ κ kappa\kappaκ B r = 0 B r = 0 Br=0B r=0Br=0 B r = 0.001 B r = 0.001 Br=0.001B r=0.001Br=0.001 B r = 0.01 B r = 0.01 Br=0.01B r=0.01Br=0.01
0 0.1 -0.999999 -1.032983 -1.451739
1 -1.000000 -1.005839 -1.059969
10 -1.000000 -1.003203 -1.032116
100 0.1 -0.999999 -1.027827 -1.377468
1 -1.000000 -1.006256 -1.063425
10 -1.000000 -1.009053 -1.092552
500 0.1 -0.999999 -1.031473 -1.360956
1 -1.000000 -1.133834 -2.040945
10 -1.000000 -1.276971 -3.552673
lambda kappa Br=0 Br=0.001 Br=0.01 0 0.1 -0.999999 -1.032983 -1.451739 1 -1.000000 -1.005839 -1.059969 10 -1.000000 -1.003203 -1.032116 100 0.1 -0.999999 -1.027827 -1.377468 1 -1.000000 -1.006256 -1.063425 10 -1.000000 -1.009053 -1.092552 500 0.1 -0.999999 -1.031473 -1.360956 1 -1.000000 -1.133834 -2.040945 10 -1.000000 -1.276971 -3.552673| $\lambda$ | $\kappa$ | $B r=0$ | $B r=0.001$ | $B r=0.01$ | | :---: | :---: | ---: | ---: | ---: | | 0 | 0.1 | -0.999999 | -1.032983 | -1.451739 | | | 1 | -1.000000 | -1.005839 | -1.059969 | | | 10 | -1.000000 | -1.003203 | -1.032116 | | 100 | 0.1 | -0.999999 | -1.027827 | -1.377468 | | | 1 | -1.000000 | -1.006256 | -1.063425 | | | 10 | -1.000000 | -1.009053 | -1.092552 | | 500 | 0.1 | -0.999999 | -1.031473 | -1.360956 | | | 1 | -1.000000 | -1.133834 | -2.040945 | | | 10 | -1.000000 | -1.276971 | -3.552673 |
Table 3: Nusselt number on the left wall N u | Y = 0 N u Y = 0 Nu|_(Y=0)\left.N u\right|_{Y=0}Nu|Y=0
λ λ lambda\lambdaλ κ κ kappa\kappaκ B r = 0 B r = 0 Br=0B r=0Br=0 B r = 0.001 B r = 0.001 Br=0.001B r=0.001Br=0.001 B r = 0.01 B r = 0.01 Br=0.01B r=0.01Br=0.01
0 0.1 1.000000 0.967016 0.548260
1 1.000000 0.994160 0.940030
10 1.000000 0.996796 0.967883
100 0.1 1.000000 0.959813 0.498805
1 1.000000 0.981394 0.813917
10 0.999999 0.978624 0.787469
500 0.1 1.000000 0.906917 0.033431
1 1.000000 0.810584 -0.328300
10 0.999999 0.683490 -0.854004
lambda kappa Br=0 Br=0.001 Br=0.01 0 0.1 1.000000 0.967016 0.548260 1 1.000000 0.994160 0.940030 10 1.000000 0.996796 0.967883 100 0.1 1.000000 0.959813 0.498805 1 1.000000 0.981394 0.813917 10 0.999999 0.978624 0.787469 500 0.1 1.000000 0.906917 0.033431 1 1.000000 0.810584 -0.328300 10 0.999999 0.683490 -0.854004| $\lambda$ | $\kappa$ | $B r=0$ | $B r=0.001$ | $B r=0.01$ | | :---: | :---: | ---: | ---: | ---: | | 0 | 0.1 | 1.000000 | 0.967016 | 0.548260 | | | 1 | 1.000000 | 0.994160 | 0.940030 | | | 10 | 1.000000 | 0.996796 | 0.967883 | | 100 | 0.1 | 1.000000 | 0.959813 | 0.498805 | | | 1 | 1.000000 | 0.981394 | 0.813917 | | | 10 | 0.999999 | 0.978624 | 0.787469 | | 500 | 0.1 | 1.000000 | 0.906917 | 0.033431 | | | 1 | 1.000000 | 0.810584 | -0.328300 | | | 10 | 0.999999 | 0.683490 | -0.854004 |
Table 4: Nusselt number on the right wall N u | Y = 1 N u Y = 1 Nu|_(Y=1)\left.N u\right|_{Y=1}Nu|Y=1
Figure 2: Velocity profiles for different values of λ λ lambda\lambdaλ
Figure 3: Velocity profiles for different values of B r B r BrB rBr
Figure 4: Velocity profiles for different values of κ κ kappa\kappaκ
Figure 5: Temperature profiles for different values of λ λ lambda\lambdaλ
Figure 6: Temperature profiles for different values of B r B r BrB rBr
Figure 7: Temperature profiles for different values of κ κ kappa\kappaκ

References

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  2. Aung, W., Fletcher, L.S., Sernas, V.: Developing laminar free convection between vertical flat plates with asymmetric heating. In: Int. J. Heat Mass Transfer, vol.15, 1972), p. 2293-2308.
  3. Aung, W., Worku, G.: Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures. In: J. Heat Transfer, vol. 108, 1986, p. 299-304.
  4. Aung, W., Worku, G.: Theory of fully developed, combined convection including flow reversal. In: J. Heat Transfer, vol. 108, 1986, p. 485-488.
  5. Barletta, A.: Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal-isoflux boundary conditions. In: J. Heat Transfer, vol. 121, 1999, p. 349-356.
  6. Barletta, A.: Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux. In: Int. J. Heat Mass Transfer, vol. 45, 2002, p. 641-654.
  7. Boulama, K., Galanis, N.: Analytical solution for fully developed mixed convection between parallel vertical plates with heat and mass transfer. In: J. Heat Transfer, vol. 126, 2004, p. 381-388.
  8. Bejan, A., 1995. Convection Heat Transfer (2nd edition), Wiley, New York.
  9. Barletta, A.: Laminar mixed convection with viscous dissipation in a vertical channel. In: Int. J. Heat Mass Transfer, vol. 41, 1988, p. 3501-3513.
  10. Pop, I., Ingham, D.B., 2001. Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford.
  11. Kohr, M., Pop, I., 2004.Viscous Incompressible Flow for Low Reynolds Numbers, WIT Press, Southamton.
2010

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