Transp Porous Med (2) 87:7 23 DOI 7/s242--966- Unsteady Natural Convection, Heat and Mass Transfer in Inclined Triangular Porous Enclosures in the Presence of Heat Source or Sink: Effect of Sinusoidal Variation of Boundary Conditions M. A. Mansour M. M. Abd-Elaziz R. A. Mohamed Sameh E. Ahmed Received: 2 July 2 / Accepted: 3 September 2 / Published online: 2 February 2 Springer Science+Business Media B.V. 2 Abstract The problem of double-diffusive convection in inclined triangular porous enclosures with sinusoidal variation of boundary conditions in the presence of heat source or sink was discussed numerically. The dimensionless governing equations of the problem were solved numerically by using finite difference method. The effects of governing parameters, namely, the dimensionless time parameter, various values of the inclination angle, Darcy number, the heat generation/absorption parameter, the buoyancy parameter and the amplitude wave length ratio on the streamlines, temperature and concentration contours as well as the velocity component in the x-direction at the triangle mid-section, the average Nusselt and Sherwood numbers at the bottom wall of the triangle for various values of aspect ratio were considered. The present results are validated by favorable comparisons with previously published results. All the results of the problem were presented in graphical and tabular forms and discussed. Keywords Double diffusive Various boundary conditions Heat generation/absorption Triangular enclosures List of Symbols a Dimensional amplitude of sinusoidal change (m) A Amplitude wave length ratio (a/h) Ā Enclosure aspect ratio (L/H) C Dimensional concentration (kg m 3 ) M. A. Mansour Department of Mathematics, Faculty of Sciences, Assuit University, Assuit, Egypt M. M. Abd-Elaziz R. A. Mohamed S. E. Ahmed (B) Department of Mathematics, Faculty of Sciences, South Valley University, Qena, Egypt e-mail: sameh_sci_math@yahoo.com 23
8 M. A. Mansour et al. c p Specific heat of the fluid (J kg K ) D Species diffusivity (m 2 s ) Da Darcy number (k/h 2 ) g Gravitational acceleration (m s 2 ) H Height of the triangle (m) k Permeability of porous medium (m 2 ) L Length of cavity wall (m) Le Lewis number (α e /D) N Buoyancy parameter [β C (C h C c )/β T (T h T c )] Nu Nusselt number, defined in Eq. 9 Pr Prandtl number (v/α e ) Ra Rayleigh number (gβ T TH 3 /υα e ) Ra D Rayleigh Darcy number (Da Ra) Sh Sherwood number, defined in Eq. 9 t Time (s) T Dimensional temperature (K) u,v Dimensional velocity components (m s ) U, V Dimensionless velocity components (uh/α e,vh/α e ) x, y Dimensional coordinates (m) X, Y Dimensionless coordinates (x/h, y/h) Greek Symbols α Inclination angle of the enclosure ( ) α e Effective thermal diffusivity of the porous medium, m 2 s (k/ρc p ) β T Thermal expansion coefficient (K ) β c Compositional expansion coefficient (kg m 3 ) ε Porosity ζ Dimensional vorticity (s ) φ Dimensionless concentration (C C c /C h C c ) ϕ Dimensional heat generation/absorption coefficient δ Dimensionless heat generation/absorption coefficient (ϕ H 2 /α e ρc p ) Ω Dimensionless vorticity (ζ H 2 /α e ) μ Dynamic viscosity (kg m s ) υ Kinamatic viscosity, m 2 s (μ/ρ) ρ Density (kg m 3 ) σ Heat capacity, defined in Eq. 5 τ Dimensionless time (α e t/h 2 ) θ Dimensionless temperature (T T c /T h T c ) Ψ Dimensional stream function (m 2 s ) ψ Dimensionless stream function (Ψ/α e ) Subscripts c Cold h Hot 23
Effect of Sinusoidal Variation of Boundary Conditions 9 Introduction The corresponding problem of convective heat and mass transfer in a saturated porous medium has many important applications in geothermal and geophysical engineering such as the extraction of geothermal energy, the migration of moisture in fibrous insulation, under ground disposal of nuclear waste, and the spreading of chemical pollutants in saturated soil. A literature survey shows that the comprehensive review of these problems was made by Nield and Bejan (26). Goyeau et al. (996) have studied the double diffusive natural convection using Darcy Brinkman formulation in a porous cavity with impermeable boundaries, horizontal temperature and concentration differences. Bourich et al. (24) have studied a double diffusive natural convection in a square porous cavity submitted to cross gradients of heat and solute concentration numerically. Bahloul et al. (24) have investigated the double diffusive convection in a long vertical cavity heated from the below and imposed concentration gradient from the sides both analytically and numerically. Double diffusive steady natural convection in a vertical stack of square enclosures, with heat and mass diffusive walls, was studied numerically by Costa (997). Gobin et al. (25) have focused on the simulation of double diffusive convective flows in a binary fluid, confined in a vertical enclosure, divided into two vertical layers, one porous and the other fluid. The combined heat and mass transfer rates for natural convection driven by the temperature and concentration gradients have been developed in a cavity containing fluid and porous layers by Singh et al. (999) andthey showed that the degree of penetration of the fluid into porous region strongly depended upon the Darcy, thermal, and solutal Rayleigh number. On the other hand, because of its numerous applications in geo-physics and energy-related engineering problems, natural convection heat transfer induced by internal heat generation has recently received considerable attention. Such applications include heat removal from nuclear fuel debris, underground disposal of radioactive waste materials, storage of foodstuff, and exothermic chemical reactions in packed-bed reactor (see, for instance, Kakac et al. (985)). Chamkha and Al-Naser (22) studied hydromagnetic double-diffusive convection in a rectangle enclosure with uniform side heat and mass fluxes and opposing temperature and concentration gradients. They observed that the effect of the magnetic field was found to reduce the overall heat transfer and fluid circulation within the enclosure. The problem of magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium was studied by Grosan et al. (29). It is shown that the intensity of the core convection is considerably affected by the considered parameters. It is also found that the local Nusselt number decreases on the bottom wall as the angle of inclination to the horizontal of applied magnetic field increases. The problem of double- diffusive natural convection in inclined porous cavities with various aspect ratios and temperature- dependent heat source or sink was studied by Chamkha and Al-Mudhaf (28). It was found that the heat and mass transfer and the flow characteristics inside the enclosure depended strongly on the buoyancy ratio, cavity inclination angle and the heat generation or absorption effects. In general, increasing the cavity tilting angle produced reductions in the average Nusselt and Sherwood numbers with the exception of a critical angle for which they become maximum. In all these studies, the authors considered the flows inside rectangular enclosures and this because of the simplicity of solutions for the rectangular enclosures. Also, the natural convection in a non-rectangular enclosures such as triangular enclosures have received a considerable attention, Tzeng et al. (25) studied natural convection in a roof of triangular enclosures. The problem of free convection in both oblique and trapezoidal enclosures filled with a porous medium was discussed by Baytas and Pop (999, 2). Kumar and Kumar (24) analyzed the parallel computation of natural convection in trapezoidal enclosures. 23
M. A. Mansour et al. More applications and good understanding of this subject is given in the recent articles by Basak et al. (28a, b, 2a, b), Oztop et al. (28, 29), Varol et al. (28a, b, c, 29a, b, c), Varol and Oztop (29), and Varol et al. (26, 27a, b). The mean objective of this paper is to study the effect of various boundary conditions on double diffusive convection inside inclined porous triangular cavities in presence of heat source or sink. Numerical solution based on finite difference method was employed to solve the governing equations. Some graphical and tabular results were presented to illustrate the different influences of the problem parameters on fluid motion, heat, and mass characteristics. 2 Mathematical Model Consider unsteady, laminar, double-diffusive natural convection flow inside an inclined porous triangle cavity with the effects of various thermal and concentration boundary conditions in the presence of heat source or sink. In the present problem, the following assumptions have been made I. In the triangle cavity, the bottom wall is non-uniformly heated and concentrated, the inclined wall are cooled and have no mass. In any case, the left wall is assumed to be adiabatic and impermeable to mass transfer. II. Properties of the fluid are isotropic and homogeneous everywhere. III. A uniform source of heat generation/absorption in the flow region with constant volumetric rate is considered. IV. The viscous, radiation, and Joule heating effects are neglected. V. The density is assumed to be a linear function of temperature and concentration (ρ = ρ ( β T (T T c ) β C (C C c ))). The geometric and the Cartesian coordinate system are schematically shown in Fig.. Under the above assumptions, the governing equations for the problem in general form (after eliminating the pressure gradient terms) are 2 ψ = Ω, () Ω ε τ + ( ε 2 U Ω X + V Ω ) = Pr Y ε 2 Ω + Ra ( D θ Pr cos α Da X + N φ ) X Ra ( D θ Pr sin α Da Y + N φ ) Pr Ω, (2) Y Da σ θ τ + U θ X + V θ Y = 2 θ + δθ, (3) ε φ τ + U φ X + V φ Y = Le 2 φ (4) In the above equations, the following dimensionless variables are employed 23 X = x H, Y = y H, τ = α et H 2, = ζ H 2 α e, ψ = α e, u = y, N = β C(C h C c ) β T (T h T c ), θ = T T r T h T r, φ = C C r, C h C r v = x, ζ = ( 2 x 2 + 2 y 2 ), Le = α e D, δ = ϕ H 2 ρc p α e, Ra = gβ T H 3 T υα e (5)
Effect of Sinusoidal Variation of Boundary Conditions Fig. Physical model of the problem Da = k H 2, Ra D = DaRa, σ = ερc p + ( ε)ρ s c s, ρc p where all the parameters appearing in the above equations are given in the list of symbols. The appropriate boundary conditions for the present problem are X = : U = V = ψ =, θ φ =, =, Y, X X Y = : U = V = ψ =,θ = ASin(π X), φ = ASin(π X) X Ā, (6) Y = X, U = V = ψ =,θ = φ =. Ā where Ā = L/H is the enclosure aspect ratio. The boundary condition imposed on the vorticity is written as w = 2 ψ w S 2 (7) where the subscript w refers to the wall condition and S indicates the direction normal to the wall surface. 3 Numerical Method Equations 4 with the boundary conditions (6) were solved numerically by employing the finite difference method (Chamkha and Al-Naser 2; Mansour and Gorla 999; Chamkha et al. 2). Forward difference approach was used to approximate the first derivative for the time and central difference approaches were used to approximate the second derivatives in 23
2 M. A. Mansour et al. Table Comparison of average Nusselt and Sherwood numbers with Chamkha and Al-Mudhaf (28) (rectangular cavity) and Da = 5, Le =, N =,δ = Chamkha and Al-Mudhaf (28) Present results α Nu Sh Nu Sh.496 3.7265.495 3.24658 π/6.5382 3.376.669 3.4357 π/4.55 3.25365.5882 3.29385 π/3.49 2.994.4938 3.2459 Table 2 Accurcy test at A =, Ā =, Da = 4, Le =, N = 5, Ra D =,δ = 5,α = Grid ψ max ψ min Nu Sh 3 3.58338.799334 2.52689 2.6745 4 4.57228.844639 2.24488 2.686397 6 6.558322.89332 2.85948 2.68686 8 8.5546.95265 2.6494 2.679897.54745.928947 2.4467 2.6737 both the X-andY-directions. The details of the present method are presented in the papers (Chamkha and Al-Naser 2; Mansourand Gorla 999; Chamkha et al. 2). The iteration process is terminated if the following condition satisfies χi, n+ j χi, n j 6 (8) i, j where χ is the general dependent variable which can stands for θ, φ, orψ and n denotes the iteration step. The numerical method was implemented in a FORTRAN software. Table 2 shows an accuracy tests using the finite difference method using five sets of grids: 3 3, 4 4, 6 6, 8 8,. There is a good agreement was found between (6 6) and (8 8) grids, so the numerical computations were carried out for (6 6) and (8 8) grid nodal points with a time step of 4. This method was found to be suitable and gave results that are very close to the numerical results obtained by Chamkha and Al-Mudhaf (28). Table shows an excellent agreement between the present results and the results obtained by Chamkha and Al-Mudhaf (28). This favorable comparison lends confidence in the numerical results to be reported subsequently. The heat and mass transfer coefficients at the enclosure bottom wall in terms of the local Nusselt and Sherwood numbers are defined by ( ) ( ) θ φ Nu =, Sh =. (9) Y Y = Y Y = In addition, the average Nusselt number and Sherwood number at the enclosure bottom wall are defined as 23 Nu = Ā Ā NudX, Sh = Ā Ā Sh dx. ()
Effect of Sinusoidal Variation of Boundary Conditions 3 4 Results and Discussion Numerical computations are carried out and a parametric study is performed to illustrate the influence of the physical parameters on the streamlines, temperature, and concentration contours as well as the horizontal velocity component at the enclosure mid-section, the average Nusselt and Sherwood numbers at the bottom wall of the enclosure. The results of this parametric study are shown in Figs. 2, 3, 4, 5, 6, 7, 8, 9,,, 2,and 3 and Tables 2, 3. In all the results to be reported below, the porosity of porous medium was taken to be.6. 4. The Effect of the Inclination Angle of the Enclosure Figure 2 shows the effect of the inclination angle of the enclosure α(,π/4,π/2) on the streamlines, temperature, and concentration contours. In this case, the aspect ratio of the [.2,] [,.25] [,.25].9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9 Fig. 2 Contours of streamlines (top), temperature (middle) and concentration (bottom) contours at α =,π/4,π/2 (increasing from left to right). The referenced case is A =, Ā =, Da = 5, Le =, N =, Ra D =,δ = 5 23
4 M. A. Mansour et al. Y α=, π/6, π/4, π/2 A= Da= -3 Le= N=5 Pr=.7 Ra D = δ=5. -4-2 - -8-6 -4-2 2 4 6 8 2 4 6 8 2 22 Fig. 3 Effect of the inclination angle α on the horizontal velocity component at the enclosure mid-section at Ā = U 6. 5.5 5. 4.5 4. 3.5 3. A= Da= -3 Le= N=5 Pr=.7 Ra D = δ=5 Nu Sh 2.5 2. 2 4 6 8 Fig. 4 Effect of the inclination angle α on the average Nusselt and Sherwood numbers at the enclosure bottom wall at Ā = α triangular cavity was fixed at the value Ā =.. At α =, the fluid moves from the bottom wall towards the top corner and forms one clockwise circular cell with ψ min =.2 inside the triangle. Tilting the triangle by π/4 causes an increasing in the rate of the fluid motion and the fluid flow takes the opposite direction by forming anticlockwise circular cell 23
Effect of Sinusoidal Variation of Boundary Conditions 5 [,.25] [,.25] [,.65].9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9.9.8.7.6.5.5.6.7.8.9 Fig. 5 Contours of streamlines (top), temperature (middle) and concentration (bottom) contours at N =,, 2 (increasing from left to right). The referenced case is A =, Ā =, Le =, N = 5, Ra D =,α = π/4,δ = 5 Table 3 Effect of dimensionless time parameter on the stream function, average Nusselt and Sherwood numbers at A =, Ā =, Da = 4, Le =, N = 5, Ra D =,δ = 5,α = τ ψ max ψ min Nu Sh.5.72842.856734 2.9338 2.86234.8.6245.88644 2.87496 2.7486..58336.8888 2.86643 2.7582 Steady state.558322.89332 2.85948 2.68686 inside the cavity with ψ max =.25. When α = π/2 is considered, the streamlines crowded beside the bottom wall which mean that the fluid is more active in this region. For more understanding for this influence on the fluid flow, the horizontal velocity component at the 23
6 M. A. Mansour et al. Y N=-,,, 2 A= Da= -5 Le= Pr=.7 Ra D = α=π/4 δ=5. -4-2 2 4 6 Fig. 6 Effect of buoyancy parameter N on the horizontal velocity component at the enclosure mid-section Ā = U 3.6 3.4 3.2 3. 2.8 A= Da= -5 Le= N=5 Pr=.7 Ra D = α=π/3 δ=5 Nu Sh 2.6 2.4 2.2 2. - -5 5 Fig. 7 Effect of buoyancy parameter N on the average Nusselt and Sherwood numbers at the enclosure bottom wall at Ā = N enclosure mid-section is plotted in Fig. 3. As expected, increase in the inclination angle α leads to increase in the horizontal velocity component. Regarding the temperature contours, all the temperature lines start at different places from the horizontal wall, and then these lines divided to: the temperature lines θ 5 occur in two regions, one of them at the 23
Effect of Sinusoidal Variation of Boundary Conditions 7 ψ [ 9, 2] ψ [.25,.7] Fig. 8 Contours of streamlines (top), temperature (middle) and concentration (bottom) contours at Da = 4, 5 (decreasing from left to right). The referenced case is A =, Ā = 3, Le =, N = 5, Ra D =,α = π/4,δ = 5 Y A= Le= N=5 Pr=.7 Ra D = α=π/4 δ=5 Da=2x -3, -4, -5. -4-3 -2-2 Fig. 9 Effect of Darcy number Da on the horizontal velocity component at the enclosure mid-section Ā = 3 U bottom left corner in symmetrical form and the other occur at the top half of the triangle. The remaining lines θ.5 start from different places on the horizontal wall and vanish after some distances from this wall. Tilting the triangle by π/4 decreasesnumberof the lines 23
8 M. A. Mansour et al. 3.4 3.2 3. 2.8 A= Le= N=5 Pr=.7 Ra D = α=π/4 δ=5 Nu Sh 2.6 2.4 2.2 2..8.5..5 2. Dax -4 Fig. Effect of Darcy number Da on the average Nusselt and Sherwood numbers at the enclosure bottom wall at Ā = 3 ψ [ 5, 3] ψ [ 9, 2] Fig. Contours of streamlines (top), temperature (middle) and concentration (bottom) contours at γ =, (increasing from left to right). The referenced case is A =, Ā = 3, Da = 3, Le =, N = 5, Ra D =,α = π/4 23
Effect of Sinusoidal Variation of Boundary Conditions 9 3.6 3.4 3.2 3. 2.8 2.6 2.4 2.2 2..8 A= Da= -5 Le= N=5 Pr=.7 Ra D = α=π/4 Nu Sh.6.4 - -5 5 Fig. 2 Effect heat generation/absorption parameter δ on the average Nusselt and Sherwood numbers at the enclosure bottom wall at Ā = δ which arise at the bottom left corner to include all lines θ 5. Increasing the values of the inclination angle leads to decrease the area which covered by the temperature lines. On the other hand, the concentration behaviors are similar to the temperature behaviors. The concentration contours φ.55 for non-inclination cavity (α = ) occur at two places, one of them at the left bottom corner in symmetrical form and the other occur at the top half of the triangle. The remaining lines start from different places on the horizontal wall and vanish after some distances from this wall. Increasing values of the inclination angle of the triangle results in a decreasing in the number of the contour lines which occur in the bottom left corner to become all lines φ 5 when α = π/4, and only contour line φ =.5 when α = π/2. In addition, the effect of the inclination angle α on the average Nusselt and Sherwood numbers at the enclosure bottom wall can be observed from Fig. 4.It is found that, increasing values of the inclination angle α leads to increase both of the average Nusselt and Sherwood numbers, however, further increasing in α leads to opposite behaviors. 4.2 The Effect of the Buoyancy Parameter The effectof buoyancyparameter N on the streamlines, temperature, and concentration contours for fixed value of the inclination angle (α = π/4) is plotted in Fig. 5. In this case, the enclosure aspect ratio was fixed at value Ā =.. It is clear that, the presence of aiding buoyancy force (N = ) leads to decrease the rate of the fluid flow. Also, the rate of the fluid motion increases by increasing the buoyancy parameter, N, and the stream function takes the values ψ(,.25), ψ(,.25), andψ(,.65) which corresponds to the cases of the aiding buoyancy force (N = ), absence of the buoyancy force (N = ), and the opposing buoyancy force (N = 2), respectively. The same behavior is observed from 23
2 M. A. Mansour et al. Fig. 6 for the horizontal velocity component. On the other hand, there is no noticeable effect on the temperature contours during the variations of buoyancy force except that, the region which can be covered by the temperature contour lines in the case N = is bigger than the region when N = 2. Whereas, the slight effect of buoyancy force can be observed on the concentration contours. As the buoyancy parameter increases, the number of the lines which occur at the bottom left corner decreases to become all lines φ.5 and the concentration contours become more asymptotic. On the contrary, as the buoyancy parameter increases, both of the average Nusselt and Sherwood numbers increase. This is clearly shown in Fig. 7 which displays the effect of buoyancy parameter N on the average Nusselt and Sherwood numbers at the enclosure bottom wall at Ā =. 4.3 The Effect of the Darcy Number For fixed value of the inclination angle (α = π/4), the effect of porous medium represented by Darcy number Da on the streamlines, temperature, and concentration contours was investigated. This effect is clearly presented in Fig. 8. In this case, the enclosure aspect ratio was increased to the value Ā = 3. The figure shows that the fluid flow follows the geometry of the enclosure by forming three clockwise and anticlockwise circular cells inside the cavity. When Darcy number takes the value Da = 4, there are strong motions with ψ min = 9 and ψ max = 2. However, when Darcy number decreases to the value Da = 5, the rate of the fluid motion decreases, and the maximum and minimum values of stream function become ψ min =.25 and ψ max =.7, respectively. The same observations can be seen from Fig. 9 for the horizontal velocity component. In addition, the problem boundary conditions show that the temperature and the concentration of the fluid have a slight effect by increasing the aspect ratio of the triangle. The ranges of θ and φ in this case are. θ. and. φ., respectively. The positive temperature and concentration lines occur at two regions. One of them at X. and the other at 2. < X 3., but the negative temperature and concentration lines arise at the region. < X 2.. When Darcy number takes the value Da = 4, the concentration lines become more asymptotic. Moreover, Fig. shows that the corresponding average Nusselt and Sherwood numbers increase with decreasing Darcy number Da. 4.4 The Effect of the Heat Generation/Absorption Parameter Figure illustrates the effect of the heat generation/absorption parameter δ on the streamlines, temperature, and concentration contours for fixed values of the inclination angle (α = π/4) and the enclosure aspect ratio Ā = 3. It is observed that increasing the heat generation/absorption parameter causes an increase in the clockwise contours of streamlines and a decrease in the anticlockwise counters. The clear effect for this parameter appears in the temperature behavior. It increases by increasing the heat generation/absorption parameter δ and the range of θ becomes. θ.2 atδ =. The concentration contours have the same patterns of the temperature contours, but they have no noticeable changes by increasing the heat generation/absorption parameter δ. Figure 2 displays the effect heat generation/absorption parameter δ on the the average Nusselt and Sherwood numbers at the enclosure bottom wall at Ā =. It is found that, the rates of heat and mass transfer decreases by increasing the heat generation/absorption parameter δ. 23
Effect of Sinusoidal Variation of Boundary Conditions 2 3. Da= -5 Le= N=5 2.5 Pr=.7 Ra D = α=π/4 2. δ=5 ψ max ψ min Nu Sh.5..5..6.8. Fig. 3 Effect of amplitude wave length on values of stream function, average Nusselt number and average Sherwood number at Ā = A 4.5 The Effects of the Dimensionless Time Parameter and the Amplitude Wave Length Ratio The effect of the amplitude wave length ratio A on the maximum values of stream function ψ max, the average Nusselt number Nu, and the average Sherwood number Sh is depicted in Fig. 3 at α = π/4 andā =. It is clear that, increasing the amplitude wave length ratio leads to increase in the maximum value of the stream function, the average Nusselt and Sherwood numbers. This is because of stronger buoyant flow for higher heat and mass generation rates. On the other hand, the effect of the dimensionless time parameter τ on the maximum values of the stream function ψ max, the average Nusselt number Nu and the average Sherwood number Sh is presented in Table 3. The table shows that the absolute values of stream function ψ, the average Nusselt number Nu, and the average Sherwood number Sh decreases monotonically as the dimensionless time parameter increases until they take fixed values at the steady state. All these behaviors are seen from Table 3 with the referenced case at A =, Ā =, Da = 4, Le =, N = 5, Ra D =,δ = 5,α =. 5Conclusion In the present paper, the problem of double-diffusive convection in inclined porous triangular enclosures with the effects of sinusoidal variation of boundary conditions in the presence of heat source or sink. The finite difference method was employed for the solution of the present problem. Comparisons with previously published work on special cases of the problem were performed and found to be in excellent agreement. Graphical and tabular results for various parametric conditions were presented and discussed. From this investigation, we can draw the following conclusion: 23
22 M. A. Mansour et al. I. Increasing the inclination angle leads to increase in the horizontal velocity component and the average Nusselt and Sherwood numbers, however, further increasing in the inclination angle leads to opposite behaviors. II. As the buoyancy parameter increases, both the average Nusselt and Sherwood numbers increase, and the horizontal velocity component takes the same behavior. III. A faster motion is considered when Darcy number increases and the average Nusselt and Sherwood numbers take the same behaviors. IV. The average Nusselt and Sherwood numbers must be affected by the presence of the buoyancy force, the presence of buoyancy force leads to increase in both the average Nusselt and Sherwood numbers. V. The presence of heat sink leads to accelerate the fluid motion but the presence of heat source leads to a decay of the fluid motion. VI. The rates of heat and mass transfer decreases by increasing the heat generation/ absorption parameter VII. Both of the maximum value of stream function, the the average Nusselt and Sherwood numbers decrease with increasing either the amplitude wave length ratio or the dimensionless time parameter References Bahloul, A., Kalla, L., Bennacer, R., Beji, H., Vasseur, P.: Natural convection in a vertical porous slot heated from below and with horizontal concentration gradients. Int. J. Therm. Sci. 43, 653 663 (24) Basak, T., Roy, S., Babu, S.K.: Natural convection and flow simulation in differentially heated isosceles triangle enclosures filled with porous medium. Chem. Eng. Sci. 63, 3328 334 (28a) Basak, T., Roy, S., Babu, S.K., Pop, I.: Finite element simulations of natural convection flow in an isosceles triangular enclosure filled with a porous medium: effects of various boundary conditions. Int. J. Heat Mass Transf. 5, 2733 274 (28b) Basak, T., Roy, S., Ramakrishna, D., Pop, I.: Visualization of heat transport due to natural convection for hot materials confined within two entrapped porous triangular cavities via heatline concept. Int. J. Heat Mass Transf. 53, 2 22 (2a) Basak, T., Roy, S., Ramakrishna, D., Pop, I.: Visualization of heat transport during natural convection within porous triangular cavities via heatline approach. Numer. Heat Transf. A 57, 43 452 (2b) Baytas, A.C., Pop, I.: Free convection in oblique enclosures filled with a porous medium. Int. J. Heat Mass Transf. 42, 47 57 (999) Baytas, A.C., Pop, I.: Natural convection in a trapezoidal enclosure filled with a porous medium. Int. J. Eng. Sci. 39, 25 34 (2) Bourich, M., Amahmid, A., Hasnaoui, M.: Double diffusive convection in a porous enclosure submitted to cross gradients of temperature and concentration. Energy Convers. Manag. 45, 655 67 (24) Chamkha, A.J., Al-Mudhaf, A.: Double-diffusive natural convection in inclined porous cavities with various aspect ratios and temperature-dependent heat source or sink. Heat Mass Transf. 44, 679 693 (28) Chamkha, A.J., Al-Naser, H.: Double-diffusive convection in an inclined porous enclosure with opposing temperature and concentration gradients. Int. J. Therm. Sci. 4, 227 244 (2) Chamkha, A.J., Al-Naser, H.: Hydromagnetic double-diffusive convection in a rectangular enclosure with uniform side heat and mass fluxes and opposing temperature and concentration gradients. Int. J. Therm. Sci. 4, 936 948 (22) Chamkha, A.J., Mohamed, R.A., Ahmed, S.E.: Unsteady MHD natural convection from a heated vertical porous plate in a micropolar fluid with Joule heating, chemical reaction and radiation effects. mec (2). doi:7/s2--932- Costa, V.A.F.: Double diffusive natural convection in a square enclosure with heat and mass diffusive walls. Int. J. Heat Mass Transf. 4, 46 47 (997) Gobin, D., Goyeau, B., Neculae, A.A.: Convective heat and solute transfer in partially porous cavities. Int. J. Heat Mass Transf. 48, 898 98 (25) Goyeau, B., Songbe, D., Gobin, J.-P.: Numerical study of double-diffusive natural convection in a porous cavity using the Darcy Brinkman formulation. Int. J. Heat Mass Transf. 39, 363 378 (996) 23
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