Correspondence should be addressed to S. M. Abo-Dahab,

Transcription

1 Mathematical Problems in Engineering Volume, Article ID 58, 8 pages doi:.55//58 Research Article Analtical Solution of Thermal Radiation and Chemical Reaction Effects on Unstead MHD Convection through Porous Media with Heat Source/Sink Abdel-Nasser A. Osman, S. M. Abo-Dahab,, and R. A. Mohamed Mathematics Department, Facult of Science, SVU, Qena 8353, Egpt Department of Mathematics, Facult of Science, Taif Universit, Taif 974, Saudi Arabia Correspondence should be addressed to S. M. Abo-Dahab, sdahb@ahoo.com Received 9 March ; Accepted 7 Ma Academic Editor: Moran Wang Copright q Abdel-Nasser A. Osman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. This paper analticall studies the thermal radiation and chemical reaction effect on unstead MHD convection through a porous medium bounded b an infinite vertical plate. The fluid considered here is a gra, absorbing-emitting but nonscattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the energ equation. The dimensionless governing equations are solved using Laplace transform technique. The resulting velocit, temperature and concentration profiles as well as the skin-friction, rate of heat, and mass transfer are shown graphicall for different values of phsical parameters involved.. Introduction Thermal radiation of a gra fluid which is emitting and absorbing radiation in a nonscattering medium has been investigated b Ali et al., Ibrahim and Had, Mansour 3, Hossain et al. 4, 5, Raptis and Perdikis 6, Makinde 7, and Abdus-Sattar and Hamid Kalim 8.All these studies have investigated the unstead flow in a nonporous medium. From the previous literature surve about unstead fluid flow, we observe that few papers were done in a porous medium. The radiative flows of an electricall conducting fluid with high temperature in the presence of a magnetic field are encountered in electrical power generation, astrophsical flows, solar power technolog, space vehicle, nuclear engineering application, and other industrial areas. The analtical solution of unstead MHD laminar convective flow with thermal radiation of a conducting fluid with variable properties through a porous medium in the presence of chemical reaction and heat source or sink has not been investigated.

2 Mathematical Problems in Engineering Combined heat and mass transfer problems with chemical reaction are of importance in man processes and have, therefore, received a considerable amount of attention in the recent ears. In processes such as dring, evaporation at the surface of a water bod, energ transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur simultaneousl. Possible applications of this tpe of flow can be found in man industries, for example, in the electric power industr, among the methods of generating electric power is one in which electrical energ is extracted directl from a moving conducting fluid. Man practical diffusion operations involve the molecular diffusion of a species in the presence of chemical reaction within or at the boundar. There are two tpes of reactions. A homogeneous reaction is one that occurs uniforml throughout a give phase. The species generation in a homogeneous reaction is analogous to internal source of heat generation. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundar of a phase. It can therefore be treated as a boundar condition similar to the constant heat flux condition in heat transfer. Combined heat and mass transfer with chemical reaction in geometric with and without porous media has been studied b others 9 9. This paper deals with the stud of thermal radiation and chemical reaction effects on the unstead MHD convection through a porous medium bounded b an infinite vertical plate with heat source/sink. The governing equations are solved b Laplace transform technique. The results are obtained for velocit, temperature, concentration, skin-friction, rate of heat and mass transfer. The effects of various material parameters are discussed on flow variables and presented b graphs.. Formulation of the Problem Consider the unstead free convection flow of an incompressible viscous fluid due to heat and mass transfer through a porous medium bounded b an infinite vertical plate under the action of an external transfer magnetic field of uniform strength B. The fluid considered is a gra, absorbing-emitting radiation but nonscattering medium. It is assumed that there exists a homogeneous first-order chemical reaction between the fluid and species concentration. Then, b usual Boussinesq s approximation, the unstead flow is governed b the following equations: v, u t v u gβ ( T T gβ ( ( C C u σb ν o ρ ν u, K.. T t v T α T Q( T T q r ρc p,.3 C t For constant and uniform suction,. integrates to v C D C R ( C C..4 v v,.5 where the negative sign indicates that suction is towards the plate.

3 Mathematical Problems in Engineering 3 The initial and boundar conditions are u, T T, C C, for t, u, T T w, C C w, for, t>,.6 u, T T, C C, for,t>. The radiative heat flux under term b using the Rosseland approximation is given b q r 4σ 3k T 4..7 We assume that the temperature differences within the flow are sufficientl small such that T 4 ma be expressed as a linear function of the temperature. This is accomplished b expanding T 4 in a Talor series about T and neglecting higher-order terms, thus T 4 can be expressed as T 4 4T 3 T 3T 4..8 B using.7 and.8,.3 reduces to T t α T Q( T T 6σ T 3 T 3k ρc p..9 To present solutions which are independent of geometr of the flow regime, we introduce the dimensionless variable as follows: u u v o, v o ν, t t v o ν, θ T T T w, φ C C T C w. C. Substituting from. into.,.9, and.4,weobtain u t u ( 4R 3 ( Grθ Gmφ k M u, ( θ θ Pr t θ Prηθ, u ( φ φ Sc t φ Scδφ,.

4 4 Mathematical Problems in Engineering where Gr gβν T w T v 3 o, Gm gβ ν C w C v 3 o, δ R ν, vo k Kν o ν, M σb oν, Sc ν νq, η, ρvo D vo. δ R ν, Pr ρc pν ν vo k α, R 4σ T 3 k. k The initial and boundar conditions in nondimensional form are u, θ, φ,, t, u, θ, φ, at, t>,.3 u, θ, φ as,t>. All the phsical variables are defined in the nomenclature. The solutions are obtained for hdrodnamic flow field in the presence of thermal radiation, chemical reaction and heat source/sink. 3. Analtic Solution In order to obtain the solution of the present problem, we will use the Laplace transform technique. Appling the Laplace transform to the sstem of., and the boundar conditions.3, weget θ F θ F ( s η θ, 3. φ Sc φ Sc s δ φ, 3. u u ( s M u Grθ Gmφ, 3.3 where, M M, k F Pr 4/3 R, 3.4

5 Mathematical Problems in Engineering 5 s is Laplace transformation parameter, u, θ, and φ are Laplace transformation of u, θ, and φ, respectivel, u, θ φ at, t>, s u θ φ, as,t>. 3.5 Solving the sstem of , with the help of the result in 3.5, weget θ (, s s Exp( λ θ, 3.6 φ (, s s Exp( λ φ, u (, s u (, s u (, s u3 (, s u4 (, s, where [ ] ( Gr e λ θ u, s s λ θ λ θ B 4 Gr s ( Gm u, s e λ φ s λ φ λ Gm φ B 4 s [ ] ( Gr e λ u u 3, s s λ θ λ Gr θ B 4 eλ θ e λ φ s eλ u ( αs γ β s B [( ], α s W Q ( α s γ β s B [( ], α s W Q ( αs γ β s B [( ], α s W Q ( ( Gm u 4, s e λ u s λ φ λ Gm φ B 4 s eλ u α s γ β s B [( ], α s W Q 3.9 where B F 4 η, λ θ F F s B, Sc B δ, 4 λ φ Sc Sc s B, B3 4 M, λ u s B 3, α F, β α F, γ F α F η M, W αγ β α,

6 6 Mathematical Problems in Engineering β α Sc, Q γ β B W, Q α γ β B α γ Sc ( Sc δ W, α Sc, M, W α γ β, α B 4 S M. 3. The inverse Laplace transformation of 3.6 is θ (, t e / F e F F /4 η erf c F B t t e F F /4 η erf c F B t t. 3. The analtic solution of 3. can be obtained b taking the inverse transforms of 3.7. So, the solution of the problem for the concentration φ, t for t> φ (, t e / Sc e ScB erf c Sc t B t e ScB erf c Sc t B t. 3. Similarl, the general solution of 3.3 can be obtained b taking the inverse Laplace transform of 3.8. The expressions for velocit, u, velocit gradient, u/, temperature gradient, θ/ and concentration gradient, φ/ are shown in Appendix A. 4. Numerical Results and Discussion In order to get phsical insight into the problem, the numerical calculations are carried out to stud the variations in velocit u, temperature θ and concentration φ. The variation in skin-friction shear stress at the wall, rate of heat and mass transfer are computed. These variations involve the effects of time t, heat generation parameter η, chemical reaction parameter δ, Schmidt number Sc, Prandtl number Pr, thermal radiation parameter R, magnetic field parameter M and permeabilit parameter k. The values of Prandtl number Pr are chosen to be 3, 7, and. The values of Schmidt number Sc are chosen to be.,.6, and.78, which represent hdrogen, water vapour and ammonia, respectivel. Representative velocit, temperature and concentration profiles across the boundar laer for different values of the dimensionless time t are presented in Figure. As the dimensionless time increases, the velocit, temperature and concentration profiles increase.

7 Mathematical Problems in Engineering 7.6 t :=.,.,.3,.4 Velocit u.4. Temperature θ.5 t :=.,.,.3, Concentration.5 t :=.,.,.3, c Figure : Velocit, temperature and concentration profiles against for various values of t with Pr, M., η, k, Sc., Gr Gm 5, δ., and R...6 η := 5, 4,.5, Velocit u.4. Temperature θ.5 η := 5, 4,.5, Figure : Velocit and temperature profiles against for various values of η with t., Pr, M., k, Sc., Gr Gm 5, δ., and R.. Figure describe the behavior of velocit and temperature profiles across the boundar laer for different values of the heat generation parameter η. As the heat source parameter η increases, the velocit and temperature profiles increase. The volumetric heat source term ma exert a strong influence on the heat transfer and as a consequence, also on the fluid flow.

8 8 Mathematical Problems in Engineering.6 δ := 5,, 5 Velocit u.4. Concentration.5 δ := 5,, Figure 3: Velocit and concentration profiles against for various values of δ with t., Pr, M., k, η, Sc., Gr Gm 5, and R...6 Sc :=.,.6,.78 Velocit u.4. Concentration.5 Sc :=.,.6, Figure 4: Velocit and concentration profiles against for various values of Sc with t., Pr, M., k, η, Gr Gm 5, and R.. Figure 3 shows the velocit and concentration profiles for different values of chemical reaction parameter. As the chemical reaction parameter increases, the velocit increases but the concentration profile decreases. Figure 4 displas the effects of Schmidt number on the velocit and concentration profiles, respectivel. As the Schmidt number increases, the velocit and concentration decrease. Reductions in the velocit and concentration profiles are accompanied b simultaneous reductions in the velocit and concentration boundar laers. These behaviors are evident from Figure 4. Figure 5 illustrates the velocit and temperature profiles for different values of Prandtl number. The numerical results show that the effect of increasing values of Prandtl number results in a decreasing velocit. Also, it is shown that an increase in the Prandtl number results tend to a decreasing of the thermal boundar laer and in general it lowers the average temperature through the boundar laer. The reason is that, the smaller values of Pr are equivalent to increase in the thermal conductivit of the fluid and therefore heat is able to diffuse awa from the heated surface more rapidl for higher values of Pr. Hence in the case of smaller Prandtl numbers, the thermal boundar laer is thicker and the rate of heat transfer is reduced.

9 Mathematical Problems in Engineering 9.6 Pr := 3, 7, Pr := 3, 7, Velocit u.4. Temperature θ Figure 5: Velocit and temperature profiles against for various values of Pr with t., Sc., M., k, η, Gr Gm 5, and R...6 Velocit u.4. R :=,.,.,.3 Temperature θ.5 R :=,.,., Figure 6: Velocit and temperature profiles against for various values of R with t., Pr, M., k, η, Gr Gm 5., and Sc.. For different values of the radiation parameter R, the velocit and temperature profiles are shown in Figure 6. It is noticed that an increase in the radiation parameter results an increase in the velocit and temperature within the boundar laer, also it increases the thickness of the velocit and temperature boundar laers. The velocit profiles for different values of the magnetic field parameter and dimensionless permeabilit are shown in Figure 7. It is clear that the velocit decreases with increasing of the magnetic field parameter. It is because that the application of transverse magnetic field will result a restrictive tpe force Lorentz s force, similar to drag force which tends to restrictive the fluid flow and thus reducing its velocit. The presence of porous media increases the resistance flow resulting in a decrease in the flow velocit. This behavior is depicted b the decrease in the velocit as permeabilit decreases and when k i.e., the porous medium effect is vanishes the velocit is greater in the flow field. These behaviors are shown in Figure 7. Figure 8 displas the effect of Sc, t, M, k, δ, and Pr on shear stress u,t with respect to radiation parameter R, it is obvious that there is a slight changes in shear stress, also, it is

10 Mathematical Problems in Engineering.6 Velocit u.4. M :=.,.3,.5 3 Figure 7: Velocit profiles against for various valuesof M and k with t., Pr, M., Sc., R., and η. seen that shear stress increases with an increasing of k and δ but decreases with an increasing values of Sc, t, M,andPr. Figure 9 shows the influence of time, heat source parameter and the radiation parameter on the negative values of gradient temperature i.e., θ,t with respect to the Prandtl number, it is seen that the increasing values of the time, heat source parameter and radiation parameter tend to decreasing in the negative values of the temperature gradient, also, it increases with the increasing of Pr. Figure displas the influence of time and chemical reaction parameter on the negative values of concentration gradient i.e., φ,t respect the Schmidt number, it is concluded that it increases with the increasing of Sc and chemical reaction δ but decreases with an increase of t. 5. Conclusion A mathematical model has been presented for analticall studies the thermal radiation and chemical reaction effect on unstead MHD convection through a porous medium bounded b an infinite vertical plate. The fluid considered here is a gra, absorbing-emitting but nonscattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energ equation. The dimensionless governing equations are solved using Laplace transform technique. The resulting velocit, temperature and concentration profiles as well as the skin-friction, rate of heat and mass transfer are shown graphicall for different values of phsical parameters involved. It has been shown that the fluid is accelerated, that is, velocit u is increased with an increasing values of time t, chemical reaction parameter δ, Prandtl number Pr, radiation parameter R, while the show opposite tends with an increasing values of heat source parameter η, Schmidt number Sc and magnetic field parameter M. Also, it is shown that velocit u does not affected with the various values of the dimensionless permeabilit k. We conclude that, the negative temperature gradient θ,t increases as Pr increases but decreases as t, η, andr increase. Finall, we obvious that, the negative concentration gradient φ,t increases as Sc and δ increases but decreases as t increase. It is hoped that the results obtained here will not onl provide useful information for applications, but also serve as a complement to the previous studies.

11 Mathematical Problems in Engineering.5 Sc :=.,.6,.78 Shear stress 4 Shear stress.5.5 t :=.,.,.3, R R M :=.,.3,.5.5 k :=.3,.5,.7 Shear stress.5 Shear stress R c R d.4 8 δ := 5,, 5.45 Pr := 7, 8, 9 Shear stress 6 4 Shear stress R R e f Figure 8: Variation of shear stress u,t against R for various values of Sc, t, M, k, δ, and Pr with constant Pr, M., k, Sc., Gm Gr 5, and η.

12 Mathematical Problems in Engineering 8 8 t :=.,.,.3,.4 η := 5, 4, 3,.3 -Temperature gradient 6 4 -Temperature gradient Pr Pr 5 R :=,.,.,.3 -Temperature gradient c Figure 9: Variation of θ,t for various values of t, η, andr with Pr, M., k, Sc., Gr Gm 5., and η. Pr 5 t :=.,.,.3,.4 -Concentrationgradient 5 - Concentration gradient 5 5 δ := 5,, Sc Sc Figure : Variation of φ,t for various values of t and δ with Pr, M., k, Sc., Gr Gm 5., and η.

13 Mathematical Problems in Engineering 3 Appendix A. The inverse laplace transformation of 3.8 is u (, t u (, t u (, t u3 (, t u4 (, t, A. where { ( Gr u, t α α t t θ (, u e W t u cos Q t u du γ αw Q θ (, u e W t u sin Q t u du β t θ (, u G t u du β (B t W θ (, u t u G τ e W t u τ cos Q t u τ dτdu β [ W (B Q W (W Q ] [ t θ (, u ]} t u G τ e W t u τ sin Q t u τ dτdu, { ( Gm t u, t α α φ (, u e W t u cos Q t u du γ α W Q t φ (, u t e W t u sin Q t u du β φ (, u G t u du β (B t W φ (, u t u G τ e W t u τ cos Q t u τ dτdu { ( Gr u 3, t α α β [ ( ] W (B t Q W W Q φ (, u t u } G τ e W t u τ sin Q t u τ dτdu, t θ (, u e W t u cos Q t u du γ αw Q t θ (, u e W t u sin Q t u du β t θ (, u G t u du β (B t W ( t u θ, u G τ e W t u τ cos Q t u τ dτdu

14 4 Mathematical Problems in Engineering β [ W (B Q W (W Q ] t ( θ, u t u } G τ e W t u τ sin Q t u τ dτdu, { ( Gm t ( u 4, t α α θ, u e W t u cos Q t u du γ α W Q t ( t θ, u e W t u ( sin Q t u du β θ, u G t u du β (B t W ( t u θ, u G τ e W t u τ cos Q t u τ dτdu β [ ( ] W (B t Q W W ( Q θ, u t u } G τ e W t u τ sin Q t u τ dτdu, A. where G t erf (B t, B G t erf (B t, B { ( ( θ, t e / e B 3 erf c t B ( 3 t e B 3 erf c t B } 3 t. A.3 θ,t F B F F erf(b t πt e B t, A.4 φ,t Sc B Sc Sc erf (B t πt e B t, A.5 u, t u 4 i Gr { α u i, t, i,, 3, 4, α t θ,x e W t x cos Q (t x dx γ αw Q

15 Mathematical Problems in Engineering 5 u u 3 t ( θ,x e W t x sin Q t x dx β β (B t W ( cos Q t x τ β Q Gm { t α α Gr { α α [ W (B W t t x Θ,x G τ e W t x τ dτdx (W Q ] t θ,x t x } ( G τ e W t x τ sin Q t x τ dτdx, t φ,x e W t x cos Q ( t x dx γ α W Q θ,x G (t x dx φ,x e W t x ( sin Q t x t dx β φ,x G (t x dx β (B t t x W φ,x G τ e W t x τ cos Q (t x τ dτdx β [ ( ] W (B t Q W W Q φ,x t x } G τ e W t x τ sin Q (t x τ dτdx, t t θ,x e W t x cos Q (t x dx γ αw Q ( θ,x e W t x sin Q t x dx β β (B t W ( cos Q t x τ β Q [ W (B W t t x θ,x G τ e W t x τ dτdx (W Q ] t θ,x t x } ( G τ e W t x τ sin Q t x τ dτdx, θ,x G (t x dx

16 6 Mathematical Problems in Engineering u 4 Gm { t α θ,x e W t x ( cos Q t x dx γ α W Q α t θ,x xθ,t t x, φ,x xφ,t t x, θ,x e W t x ( sin Q t x t ( dx β θ,x G t x dx β (B t t x W θ,x G τ e W t x τ cos Q (t x τ Θ,t B 3 erf (B 3 t Θ,x xθ,t t x. dτdx β [ ( ] W (B t Q W W Q θ,x t x } G τ e W t x τ sin Q (t x τ dτdx, πt e B 3 t, A.6 Nomenclatures B : Magnetic induction C : Concentration of the fluid for awa from the plate C inthefreestream C : Concentration of the fluid D: Chemical molecular diffusivit g: Gravitational acceleration G m : Solutal Grashof number G r : Grashof number K: Permeabilit of the porous medium k: Dimensionless permeabilit k : Thermal conductivit of the fluid k : Mean absorption coefficient M: Magnetic field parameter Pr: Prandtl number qr: Radiative heat flux Q: Hea tsource/sink coefficient R: Radiation parameter R : First-order chemical reaction rate Sc: Schmidt number t: Dimensionless time

17 Mathematical Problems in Engineering 7 t : Dimensional time T : Temperature of the fluid T w, C w : Surface temperature and concentration T : Temperature of the fluid for awa from the plate inthefreestream u: Dimensionless velocit u, v : Components of dimensional velocities along x and directions x, : Dimensional distances along and perpendicular to the plate : Nondimensional distance α: Thermal diffusivit β: Coefficient of thermal expansion β : Coefficient of expansion with concentration η: Heat source parameter θ: Dimensionless temperature ν: Kinematic coefficient of viscosit δ: Chemical reaction parameter ρ: Fluid densit σ: Electrical conductivit of the fluid σ : Stefan-Boltzmann flux φ: Dimensionless concentration. References M. M. Ali, T. S. Chen, and B. F. Armal, Natural convection-radiation interaction in boundar-laer flow over horizontal surfaces, AIAA Journal, vol., no., pp , 984. F. S. Ibrahim and F. M. Had, Mixed convection-radiation interaction in boundar-laer flow over horizontal surfaces, Astrophsics and Space Science, vol. 68, no., pp , M. A. Mansour, Radiative and free-convection effects on the oscillator flow past a vertical plate, Astrophsics and Space Science, vol. 66, no., pp , M. A. Hossain, M. A. Alim, and D. A. S. Rees, The effect of radiation on free convection from a porous vertical plate, International Journal of Heat and Mass Transfer, vol. 4, no., pp. 8 9, M. A. Hossain, K. Khanafer, and K. Vafai, The effect of radiation on free convection flow of fluid with variable viscosit from a porous vertical plate, International Journal of Thermal Sciences, vol. 4, no., pp. 5 4,. 6 A. Raptis and C. Perdikis, Radiation and free convection flow past a moving plate, International Journal of Applied Mechanics and Engineering, vol. 4, no. 4, pp. 87 8, O. D. Makinde, Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, International Communications in Heat and Mass Transfer, vol. 3, no., pp. 4 49, 5. 8 M. D. Abdus-Sattar and M. D. Hamid Kalim, Unstead free-convection interaction with thermal radiation in a boundar-laer flow past a vertical porous plate, Journal of Mathematical and Phsical Sciences, vol. 3, pp. 5 37, U. N. Das, R. Deka, and V. M. Soundalgekar, Effects of mass transfer on flow past an impulsivel started infinite vertical plate with constant heat flux and chemical reaction, Forschung im Ingenieurwesen, vol. 6, no., pp , 994. A. J. Chamkha, MHD flow of a uniforml streched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction, International Communications in Heat and Mass Transfer, vol. 3, no. 3, pp. 43 4, 3. R. Muthucumaraswam and P. Ganesan, On impulsive motion of a vertical plate with heat flux and diffusion of chemicall reactive species, Forschung im Ingenieurwesen, vol. 66, no., pp. 7 3,. R. Muthucumaraswam and P. Ganesan, First-order chemical reaction on flow past an impulsivel started vertical plate with uniform heat and mass flux, Acta Mechanica, vol. 47, no. -4, pp ,.

18 8 Mathematical Problems in Engineering 3 R. Muthucumaraswam, Effects of suction on heat and mass transfer along a moving vertical surface in the presence of chemical reaction, Forschung im Ingenieurwesen, vol. 67, no. 3, pp. 9 3,. 4 A. Raptis and C. Perdikis, Viscous flow over a non-linearl stretching sheet in the presence of a chemical reaction and magnetic field, International Journal of Non-Linear Mechanics, vol.4,no.4,pp , 6. 5 M. A. Seddeek, A. A. Darwish, and M. S. Abdelmeguid, Effects of chemical reaction and variable viscosit on hdromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation, Communications in Nonlinear Science & Numerical Simulation, vol., no., pp. 95 3, 7. 6 F. S. Ibrahim, A. M. Elaiw, and A. A. Bakr, Effect of the chemical reaction and radiation absorption on the unstead MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction, Communications in Nonlinear Science & Numerical Simulation, vol. 3, no. 6, pp , 8. 7 A. Postelnicu, Influence of chemical reaction on heat and mass transfer b natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Heat and Mass Transfer, vol. 43, no. 6, pp , 7. 8 R. A. Mohamed, I. A. Abbas, and S. M. Abo-Dahab, Finite element analsis of hdromagnetic flow and heat transfer of a heat generation fluid over a surface embedded in a non-darcian porous medium in the presence of chemical reaction, Communications in Nonlinear Science & Numerical Simulation,vol. 4, no. 4, pp , 9. 9 R. A. Mohamed and S. M. Abo-Dahab, Influence of chemical reaction and thermal radiation on the heat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porous medium with heat generation, International Journal of Thermal Sciences, vol. 48, no. 9, pp. 8 83, 9.

19 Advances in Operations Research Volume 4 Advances in Decision Sciences Volume 4 Mathematical Problems in Engineering Volume 4 Journal of Algebra Probabilit and Statistics Volume 4 The Scientific World Journal Volume 4 International Journal of Differential Equations Volume 4 Volume 4 Submit our manuscripts at International Journal of Advances in Combinatorics Mathematical Phsics Volume 4 Journal of Complex Analsis Volume 4 International Journal of Mathematics and Mathematical Sciences Journal of Stochastic Analsis Abstract and Applied Analsis International Journal of Mathematics Volume 4 Volume 4 Discrete Dnamics in Nature and Societ Volume 4 Volume 4 Journal of Journal of Discrete Mathematics Journal of Volume 4 Applied Mathematics Journal of Function Spaces Volume 4 Volume 4 Volume 4 Optimization Volume 4 Volume 4