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Natural Convection of Fluid with Variable Viscosity from a Heated

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Natural Convection of Fluid with Variable Viscosity from a Heated Z. angew. Math. Phys. 53 (2002) 48{52 0044-2275/02/010048-10 $ 1.50+0.20/0 Zeitschrift fÄur angewandte c 2002 BirkhÄauser Verlag, Basel Mathematik und Physik ZAMP ° Natural convection of fluid with variable viscosity from a heated vertical wavy surface M. A. Hossain, S. Kabir and D. A. S. Rees Abstract. In the present paper, the e®ect of a temperature dependent viscosity on natural convection flow of viscous incompressible fluid from a vertical wavy surface has been investigated using an implicit ¯nite di®erence method. Here we have focused our attention on the evaluation of the local skin-friction and the local Nusselt number. The governing parameters are the Prandtl number, Pr, ranging from 1 to 100, the amplitude of the waviness of the surface, ®, ranging from 0.0 to 0.4 and the viscosity variation parameter, ", ranging from 0.0 to 6. Keywords. Natural-convection, variable-viscosity, wavy-surface. 1. Introduction The objective of the present paper is to study the natural convection flow from a vertical wavy surface for a fluid with temperature dependent viscosity. Rough- ened surfaces are encountered in several heat transfer devices such as flat plate solar collectors and flat plate condensers in refrigerators. Larger scale surface nonuniformities are encountered in cavity wall insulating systems and grain stor- age containers, for example. The only papers to date which study the e®ects of such nonuniformities on the thermal boundary layer flow of Newtonian fluid are those of Yao (1983) and Moulic and Yao (1989). Hossain and Pop (1996) inves- tigated the magnetohydrodynamic boundary layer flow and heat transfer from a continuous moving wavy surface and the problem of free-convection flow from a wavy vertical surface in presence of a transverse magnetic ¯eld was studied by Hossain et al. (1997). On the other hand, Rees and Pop (1995) investigated the free convection induced by a horizontal surface exhibiting small-amplitude waves and which is embedded in a porous medium. Recently, Hossain and Rees (1999) have investigated the combined e®ect of thermal and mass di®usion on the natu- ral convection flow of a viscous incompressible fluid from a vertical wavy surface. The e®ect of waviness of the surface on the heat-flux and mass flux had been in- vestigated in combination with the species concentration for a fluid with Prandtl number equal to 0.7. Vol. 53 (2002) Natural convection of fluid 49 In all the above studies the viscosity of the fluid had been assumed to be constant. However, it is known that this physical property may change signi¯cantly with temperature. For instance, the viscosity of water decreases by almost 60 percent when the temperature increases from 1000C(¹=0:0131g=cms) to 5000C (¹ =0:00548g=cms). It is necessary to take into account this variation of viscosity in order to predict accurately the flow behaviour. Recently Gary et al. (1982), and Mehta and Sood (1992) have shown that when this e®ect is included, the flow characteristics may be changed substantially compared with the constant viscosity case. Recently, Kafoussias and Williams (1997) and Kafoussias and Rees (1998) investigated the e®ect of temperature dependent viscosity on the mixed convection flow from a vertical flat plate in the region near the leading edge using the local nonsimilarity method. Very recently, Hossain and Munir (2000) have investigated the same problem in the regions both near and far from the leading edge using perturbation methods and in the intermediate region by means of a ¯nite di®erence method. From the above investigations it is found that variation of viscosity with temperature is an interesting macroscopic physical phenomenon in fluid mechanics. In this investigation the focus is on the boundary-layer regime caused by a uni- form temperature wavy surface immersed in a fluid with a temperature dependent viscosity. The transformed boundary-layer equations are solved numerically using the Keller-box technique (Keller 1973), an implicit ¯nite di®erence method. Con- sideration is given to the situation where the buoyancy forces assist the natural convection flow for various values of the viscosity-variation parameter,", with the Prandtl number Pr ranging between 1 to 100. The results allow us to predict the di®erent behavior that can be observed when the relevant parameters are varied. 2. Formulation of the problem Consider the steady laminar free convective boundary layer flow of a viscous in- compressible fluid along a semi-in¯nite wavy vertical plate where the viscosity of the fluid depends on its temperature. The surface of the plate may be described by y^ =^σ(^x) and the temperature of the surface is held uniform at Tw which is higher than the ambient temperature T . The boundary layer analysis outlined below allows σ to be arbitrary, but our detailed1 numerical work will assume that the surface exhibits sinusoidal deformations. The dimensionless governing equations are the Navier-Stokes equation, the en- ergy equation and the continuity equation in two-dimensional Cartesian coordi- nates (see Fig. 1). Under the usual Boussinesq approximation, the flow is governed by the follow- 50 M. A. Hossain, S. Kabir and D. A. S. Rees ZAMP x$ T¥ u$ L g v$ T a $ $ $ $ yxw==saafsin a kx f Tw y$ Figure 1. Physical model and coordinate system. ing boundary layer equations: @u^ @v^ + = 0 (1) @x^ @y^ @u^ @u^ 1@p^ 1 u^ +^v = + :(¹ u^)+g¯(T T ) (2) @x^ @y^ ¡½@x^ ½r r ¡ 1 @v^ @v^ 1@p^ 1 u^ +^v = + :(¹ v^) (3) @x^ @y^ ¡½@y^ ½r r @T @T k u^ +^v = 2T (4) @x^ @y^ ½Cp r whereu ^ andv ^ are the velocity components parallel tox ^ andy ^, 2 (= @2=@x^2 + @2=@y^2) is Laplacian, g is the acceleration due to gravity, ½ is ther density, · is the thermal conductivity, Cp is the speci¯c heat at constant pressure, and ¹(T ) is the viscosity of the fluid depending on the temperature T of the fluid in the boundary layer region. Out of the many forms of viscosity variation which are available in the literature we have considered only the form introduced by Ling and Dybbs (1987) and which was also used by Kafoussias and Rees (1998): ¹ = ¹0 [1 + γ(T T )] (5) ¡ 1 Here γ is a constant and ¹0 is the viscosity of the ambient fluid. The boundary conditions for the present problem are u^ =0; v^=0;T=T aty ^ =^y =σ(^x) w w (6) u^ 0;T T;p p asy ^ ! !1 !1 !1 Vol. 53 (2002) Natural convection of fluid 51 where p is constant pressure of the ambient fluid. Now1 we introduce the following nondimensional dependent and independent variables: 2 x^ y^σ^ 1=4 L 3=4 x = ;y=¡Gr ;p= 2 Gr¡ (^p p ) L L ½º0 ¡ 1 ½L 1=2 ½L 1=4 u = Gr¡ u;^ v = Gr¡ (^v σxu^) ¹0 ¹0 ¡ T T dσ^ dσ g¯(Tw T ) 3 θ = ¡ 1 ,σx= = ;Gr= 2¡ 1 L (7) Tw T dx^ dx º ¡ 1 where L is the characteristic length associated with the wavy surface, and º0 =(¹0=½) is the reference kinematic viscosity. Introducing the above transforma- tions into the Eqs. (1)-(6) and after ignoring terms of small orders in the Grashof number Gr, we obtain the following equations: @u @v + = 0 (8) @x @y @u @u 1 @p @p 1+σ2 @2u " @u ∂θ u + v = + Gr1=4σ + x + θ (9) @x @y ¡½ @x x @y 1+εθ @y2 ¡ 1+εθ @y @y @u @u @p σ (1 + σ2) @2u σ u + v + σ u2 = Gr1=4 + x x (10) x µ @x @y¶ xx ¡ @y 1+εθ @y2 ∂θ ∂θ 1 @2θ u + v = (1 + σ2) (11) @x @y Pr x @y2 The boundary conditions for the present problem are u =0;v=0,θ=1 aty=0 (12) u 0,θ 0;p 0asy ! ! ! !1 In Eq. (7) " = γ(Tw T ) is the viscosity-variation parameter. From (12) it ¡ 1 can be seen clearly that the dimensionless viscosity ¹=¹0 lies in the range 1=(1+") and 1; its value decreasing with increasing temperature when ">0. The convection induced by the wavy surface is described in equations (8)- (11). Equation (10) indicates that the pressure gradient along the y direction is 1=4 O(Gr¡ ), which implies that lowest order pressure gradient along x direction can be determined from the inviscid-flow solution. For the present problem this pressure gradient is zero since there is no externally induced free stream. Equation (9) also shows that Gr1=4@p=@y is O(1) and is determined by the left hand side of this equation. Thus, the elimination of @p=@y between Eqs. (9) and (10) leads to 2 2 @u @v 1+σx @ u σxσxx 2 " @u ∂θ 1 u + v = 2 + 2u 2 + 2 θ (13) @x @y 1+εθ @y 1+σx ¡(1 + εθ) @y @y µ1+σx¶ 52 M. A. Hossain, S. Kabir and D. A. S. Rees ZAMP Now we introduce the further transformations into the Eqs. (13), (8)-(9) as described below: 3=4 1=4 à = x f(x; y);´=yx¡ ,θ=θ(x; y) where à is the stream function which satis¯es the equation of continuity (6), and which is de¯ned according to u = @Ã=@y and v = @Ã=@x. The boundary layer equations now become ¡ 2 1+σx 3 1 xσxσxx 2 " 1 f 000 + ff00 + 2 f0 2 θ0f 00 + 2 θ 1+εθ 4 ¡ µ2 1+σx ¶ ¡(1 + εθ) µ1+σx¶ @f0 @f =x f0 f00 (14) µ @x ¡ @x¶ 1 2 3 ∂θ @f 1+σx θ00 + fθ0 = x f0 θ0 (15) Pr ³ ´ 4 µ @x ¡ @x¶ The boundary conditions to be satis¯ed are that f = f 0,θ=1 at ´=0 (16) f0 =0,θ=0 as ´ !1 The quantities of physical interest are the surface shear-stress and rate of heat transfer which may be described in terms of the skin-friction, cf and the local Nusselt number, Nu, respectively, from the relations given below: 2 3=4 1+σx C (Gr=4) = f00(x; 0) (17) f µ 1+" ¶ 1=3 1=4 2 Nu(Gr=4)¡ = 1+σx θ0(x; 0) (18) ¡ ³ ´ Here we integrate the locally nonsimilar partial di®erential equations (14)-(15) subject to the boundary conditions (16) by means of the Keller box method.
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