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Double-Diffusive Natural Convection with Marangoni and Cooling Effects

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Double-Diffusive Natural Convection with Marangoni and Cooling Effects World Academy of Science, Engineering and Technology International Journal of Physical and Mathematical Sciences Vol:7, No:6, 2013 Double-Diffusive Natural Convection with Marangoni and Cooling Effects Norazam Arbin and Ishak Hashim Bourich et al. [6] investigated the effect of buoyancy ratio Abstract—Double-diffusive natural convection in an open top on the dynamic behaviour of the fluid and heat and mass square cavity and heated from the side is studied numerically. transfer for a double diffusive convection in a porous Constant temperatures and concentration are imposed along the right enclosure. Baytas et al. [7] showed that for a non-Darcy flow, and left walls while the heat balance at the surface is assumed to obey where the enclosure filled with a step type porous layer, if the Newton’s law of cooling. The finite difference method is used to fluid/porous interface is not horizontal and contains a step solve the dimensionless governing equations. The numerical results are reported for the effect of Marangoni number, Biot number and change in height, the convection of the heat and mass are Prandtl number on the contours of streamlines, temperature and dramatically changed. concentration. The predicted results for the average Nusselt number Most of the studies were done in closed rectangular or and Sherwood number are presented for various parametric square cavities. Some of the recent researches were conducted conditions. The parameters involved are as follows; the thermal by Li et al. [8], Chen et al. [9] and Teamah et al. [1]. Different Marangoni number, 0 ≤ Ma ≤ 1000 , the solutal Marangoni number, T enclosure geometries were studied by some researchers and 0 ≤ Mac ≤1000 , the Biot number, 0 ≤ Bi ≤ 6 , Grashof number, good reviews were reported by Costa [10] and Papanicolaou & Gr = 105 and aspect ratio 1. The study focused on both flows; thermal Belessiotis [11], where the enclosures are parallelogrammic dominated, N = 0.8 , and compositional dominated, N = 1.3 . and trapezoidal, respectively. The effect of Rayleigh number, the inclination angle and the aspect ratio of the enclosure will Keywords—Double-diffusive, Marangoni effects, heat and mass leave strong changes on the flow structure. transfer. The problems of natural convection in an open cavity are starting to gain interest among researchers due to its wide I. INTRODUCTION application and involve Marangoni/surface tension effect. HE flows generated by buoyancy due to simultaneous Sivasankaran et al. [12] studied numerically the magneto- Ttemperature and concentration gradients, also known as convection of cold water in an open cavity with variable fluid double diffusive convection, arises in a very wide range of properties. They observed that heat transfer rate behaves non- fields such as oceanography, astrophysics, the process of linearly with density inversion parameter and Marangoni chemical vapour transport, drying and crystal growth [1]. This number. Saleh et al. [13] investigated on the effect of cooling type of flow have been studied experimentally and and surface tension on differentially heated open cavity. They numerically, and also received considerable attention by many concluded that global quantity of heat transfer rate decreases researchers. at the cold wall and increases at the hot wall with increasing Nishimura et al. [2] studied numerically double diffusive cooling intensity for any fluid type. convection in a rectangular enclosure subject to opposing Teamah & El-Maghlany [14] found that by increasing horizontal thermal and compositional buoyancies. They found Lewis number, the mass transfer increased but there was no that the oscillatory flow occurs in a limited range of buoyancy significant effect on the heat transfer for a case of double ratio. Costa [3] established the complete mathematical and diffusive convection with insulated moving lid. The results of numerical model for the calculation of double diffusive natural Younis et al. [15] on double diffusive natural convection in convection with heat and mass diffusive walls. The heat and open lid enclosure filled with binary fluids, showed that the masslines analysis were obtained in this study. The effect of rate of heat and mass transfer are high for an enclosure with a magnetic field in the convection was then studied by Chamka free surface. International Science Index, Physical and Mathematical Sciences Vol:7, No:6, 2013 waset.org/Publication/5465 & Al-Naser [4] and Teamah [5]. They obtained a good result It is observed from the most studies available in the and concluded that magnetic field will reduce the heat transfer literature that double diffusive natural convection in open and fluid circulation within the enclosure. cavities with Marangoni effects have not been studied deeply. Therefore the aim of this work is to visualize and analyse Marangoni and convective cooling effects on a double diffusive natural convection in differentially heated open top Norazam Arbin is with the Department of Mathematics, Faculty of square cavity. Computer and Mathematical Sciences, Universiti Teknologi Mara (Perak), 32610 Seri Iskandar, Perak, Malaysia (e-mail: [email protected]). Ishak Hashim is with the School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia (e-mail: [email protected]). International Scholarly and Scientific Research & Innovation 7(6) 2013 1026 scholar.waset.org/1307-6892/5465 World Academy of Science, Engineering and Technology International Journal of Physical and Mathematical Sciences Vol:7, No:6, 2013 II. MATHEMATICAL FORMULATION ∂c ∂ c ∂2 c ∂ 2 c u+ v = D 2 + 2 (7) The schematic diagram of the system under consideration is ∂x ∂ y ∂x ∂ y shown in Fig. 1. Different temperature and concentrations were imposed between the left (Tc, cc) and right walls (Th, ch), The boundary conditions are where Tc < Th and cc < ch. The bottom wall is adiabatic and impermeable. The top of the cavity is open and the u= v =0, T = Tandcc = catx c = 0 (8) temperature gradient is applied along the upper boundary. The u= v =0, T = Tandc = catx = L (9) heat balance at the upper boundary is assumed to obey h h ∂T ∂ c Newton’s law of cooling. Moreover, the top free surface is u= v =0, = = 0 at y = 0 (10) assumed to be insulating with respect to heat, non-diffusive ∂y ∂ y with respect to the solute, flat and non-deformable, which ∂T ∂ u ∂σ ∂ T ∂ σ ∂ c means the surface tension is very high. The surface tension, u= v =0, − k = ha ( T − T c ), µ =− + at y = L(11) ∂y ∂ y ∂ T ∂ x ∂ c ∂ x σ , on the upper boundary is assumed to vary linearly with both T and c, Introducing the following non-dimensional variables for the governing equations, σ= σ0 − ηT()()T − T 0 − η c c − c 0 (1) x y uL vLψPr ω L2 Pr where and are the thermal and solutal coefficients of XYUV=,,,,,, = = = Ψ = Ω = ηT ηC LL α α υ υ surface tension, respectively. The Boussinesq approximation 3 T− Tc c − c c υ βg() Th− T c L equation with opposite and compositional buoyancy force is Θ =,C = , Pr = , Gr = 2 , Th− T c c h − c c α υ used for the body force terms in the momentum equations, (12) which is ∂σ()TTL− ∂ σ ()c − c L Ma = −h c,,Ma = − h c T∂T µα c ∂TD µ ρ= ρ1 − β (T − T ) − β ( c − c ) (2) 0 [ T c c c ] L/ k β c− c () Bi=, N = c h c y ∂T 1/ hβT() T h− T c −k = ha() T − T c ∂y L A set of governing equations in the stream function, ∂u ∂σ ∂ T ∂ σ ∂ c µ = − + vorticity and temperature is obtained after eliminating the ∂y ∂ T ∂ x ∂ c ∂ x pressure between the two momentum equations, T T c c ∂2 Ψ ∂ 2 Ψ + = −Ω (13) ∂XY2 ∂ 2 ∂2 Ω ∂ 2 Ω1 ∂Ψ ∂Ω ∂Ψ ∂Ω ∂Θ ∂C x 2+ 2 = − −GrPr − + N (14) ∂XY ∂ Pr ∂YXXYXX ∂ ∂ ∂ ∂ ∂ L ∂2 Θ ∂ 2 Θ ∂Ψ ∂Θ ∂Ψ ∂Θ 2+ 2 = − (15) Fig. 1 Schematic diagram of the model ∂XY ∂ ∂YXXY ∂ ∂ ∂ ∂2CCCC ∂ 2 ∂Ψ ∂ ∂Ψ ∂ The fluid is assumed to be Newtonian and incompressible. 2+ 2 =Le − (16) The flow is steady and laminar. The direction of the ∂XY ∂ ∂YXXY ∂ ∂ ∂ gravitational force is in the negative y-direction. Following the and the dimensionless boundary conditions are previous assumptions, the equations of mass, momentum, energy and concentration can be written in dimensional form ∂2 Ψ as Ψ =0, Ω = −2 , Θ = 0,C = 0 at X = 0 (17) ∂X 2 International Science Index, Physical and Mathematical Sciences Vol:7, No:6, 2013 waset.org/Publication/5465 ∂u ∂ v ∂ Ψ + = 0 (3) Ψ =0, Ω = − , Θ = 1,C = 1 at X = 1 (18) ∂x ∂ y ∂X 2 ∂u ∂ u1 ∂ p ∂2 u ∂ 2 u ∂2 Ψ ∂Θ ∂C (4) u+ v = − +υ 2 + 2 Ψ =0, Ω = −2 , = =0at Y = 0 (19) ∂x ∂ yρ ∂ x ∂x ∂ y ∂Y ∂YY ∂ 2 2 ∂v ∂ v1 ∂ p ∂ v ∂ v ∂ΘMac ∂C ∂Θ u+ v = − +υ + + g β()() T − T − g β c − c (5) Ω = −MaT + , = Bi Θ at Y = 1 (20) 2 2 TCCC ∂X Le ∂ X ∂ Y ∂x ∂ yρ ∂ y ∂x ∂ y 2 2 ∂TTTT ∂ ∂ ∂ where the stream function, ψ , the vorticity, ω , the u+ v =α 2 + 2 (6) ∂x ∂ y ∂x ∂ y temperature, T, and the concentration, c, as variables are International Scholarly and Scientific Research & Innovation 7(6) 2013 1027 scholar.waset.org/1307-6892/5465 World Academy of Science, Engineering and Technology International Journal of Physical and Mathematical Sciences Vol:7, No:6, 2013 defined as u= ∂ψ ∂ y , v= −∂ψ ∂ x and average Nusselt number and Sherwood number compare well with that obtained by Chen et al. [16] as shown in Table I.
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