Slip Effects on Electrical Unsteady MHD Natural Convection Flow of Nanofluid over a Permeable Shrinking Sheet with Thermal Radiation

Transcription

1 Engineering Letters, 6:, EL_6 3 Slip Eects on Electrical Unsteady MHD Natural Convection Flo o Nanoluid over a Permeable Shrinking Sheet ith hermal Radiation Yahaya Shagaiya Daniel, Zainal Abdul Aziz, Zuhaila Ismail, and Faisal Salah Abstract he unsteady magnetohydrodynamic (MHD) natural convection lo and heat transer o an electrical conducting incompressible viscous nanoluid over a linear permeable shrinking sheet in the presence o electric ield, thermal radiation, viscous dissipation, chemical reaction, slip and passively controlled conditions at the all is studied. he boundary layer governing equations hich are partial dierential equations are converted into a system o nonlinear ordinary dierential equations by applying a suitable similarity transormation. Implicit inite dierence scheme is applied to investigate the numerical results ho the various physical embedded parameters aect the nanoluid lo and heat transer ith the aid o dierent graphical presentations and tabular orms. he nanoluid lo is due to a decelerating shrinking sheet as the electric ield reduced the nanoluid velocity, and the irst solution is stable compared to the second solution. hermal radiation and viscous dissipation boost the nanoluid temperature hereas thermal slip reduces. hermal convective parameter and mass convective parameter demonstrated opposite behavior. he magnetic ield, unsteadiness parameter, and the suction parameter iden the range or the solution existence. Comparisons ith previously published orks seen in literature ere perormed and ound to be in excellent agreement. Index erms Magnetic nanoluid, unsteady lo, shrinking sheet, suction, dual solutions, thermal radiation I. INRODUCION HE conventional heat transer luids due to poor nature o thermal conductivity cannot meet up ith the Manuscript received Feb 0, 07; revised Aug 09, 07. (Submitted Oct. 09, 07.) his ork as supported by Ministry o Higher Education and Research Management Centre. Y.S. Daniel is a Postgraduate student ith the Department o Mathematical Sciences, Faculty o Science, Universiti eknologi Malaysia, 830 UM Johor Bahru, Johor, Malaysia ( shagaiya@gmail.com). Z. A. Aziz is a Proessor o Applied Mathematics and Director o the UM Centre or Industrial and Applied Mathematics, Institute Ibnu Sina or Scientiic and Industrial Research, 830 UM Johor Bahru, Johor, Malaysia ( zainalaz@utm.my). Z. Ismail is a Senior Lecturer and sta member in the UM Centre or Industrial and Applied Mathematics, Institute Ibnu Sina or Scientiic and Industrial Research, 830 UM Johor Bahru, Johor, Malaysia (corresponding author phone number: ; zuhaila@utm.my). F. Salah is Assoc. Proessor at the Department o Mathematics, Faculty o Science, University o Kordoan Elobied, 5, Sudan ( aisal9999@yahoo.com). expectations and requirement or cooling rate. As such leads to a ne class o luid knon as nanoluid. It is colloidal suspensions o ultraine nanoparticles into a base luid. he application o additives enhance the convective heat transer perormance o the conventional luids and also increase the thermal conductivity. Magnetic nanoluid contains the luid and magnetic nanoparticle, eatures as result o its uniqueness. It plays a vital role as results o broad applications such as the processing o using metals due to the electric urnace through a magnetic ield, and cooling o the initial plate enclosed nuclear reactor involving vessel here the hot plasma is separated rom the plate by means o the magnetic ield [-]. he nanoluids can be enriched through engineering system by dierent techniques involving plasma, synthesis and sheet processing. Some o these applications via aerodynamic extrusion o plastic sheets, the boundary layer against luid ilm enclosed condensation mechanism and heat treated material that los beteen ind-up rolls and eed, cooling o metallic all seen in cooling bath, the boundary layer against material handling conveyors. he innovative ork o Choi and Eastman [], the enhancement o convective heat transer according to Buongiorno [3] assumed that the volume raction o nanoparticles in the nanoluid may not be uniorm. Buongiorno [3] developed a to-phase model or convective transport in nanoluids incorporated the eects o Bronian motion and thermophoresis. iari and Das [4] considered that thermophysical properties ere vieed as unctions o nanoparticle volume. Ater these mentioned orks, dierent researchers have explored on nanoluid [5-5] due to its substantially signiicant and applications. Boundary layer lo due to shrinking sheet in connection ith nanoluid is an important type o los as results o its crucial role. Conversely to stretching sheet, the shrinking case, the lo on the boundary is toards a ixed point, or instance, transorms the loose sleeves are rapping o plastic that irmly it into the shape o the sealed o contains in shrinking sheet, and rising and shrinking balloon. It is designed or various kinds o materials involving shrinking transparency, dierent strength, and luster. here are to conditions that the lo due to shrinking sheet can be noticed such as adequate suction [6] on the boundary or stagnation lo [7] so that the lo o the shrinking sheet is conined in the boundary layer. here is an extensive (Advance online publication: 0 February 08)

2 Engineering Letters, 6:, EL_6 3 literature material on the boundary layer lo against shrinking sheet as result o its modern industrial applications [8-34]. It has been extensively used in dierent engineering ields and industries or expanding and contracting o suraces via shrinking rapping, hot rolling, bundle rapping, ire rolling, extrusion o sheet material and glass iber. he aim o the present investigation is to study the lo and heat transer problem o unsteady MHD natural convection lo and heat transer, a to-dimensional laminar lo o a viscous nanoluid due to a linear permeable shrinking sheet, ith slip eects in the presence o electric ield, thermal radiation, viscous dissipation, and chemical reaction. he momentum and energy ields at the all are the slip conditions. he no-slip assumption is inconsistent ith physical behavior that is more practically lo situations. It is o paramount importance to replace the no-slip boundary condition ith partial slip conditions. he nanoparticle volume raction on the boundary is passively controlled rather than actively [35]. he combination o nanoparticles and conventional luids depends on the intention and purpose, our base luid is ater [7]. In the analysis, the constitute boundary layer governing equations have been converted to a to-point boundary layer value problem ith the aids o deined similarity variables. he resultant nonlinear ordinary dierential equations are solved using implicit inite dierence scheme knon as Keller box method [36]. he impacts o the entrenched physical parameters on the nanoluid velocity, temperature and concentration have been discussed and displayed in graphs and tables. Unlike stretching sheet, the solutions or shrinking sheet are not unique. Furthermore, the combined eects o the embedded parameters on the boundary layer lo and heat transer due to nanoluid have been examined. Based on the author s knoledge, the present investigation is o essential values in the modern industries and not earlier reported in the literature. II. MAHEMAICAL FORMULAION Consider the unsteady electrical magnetohydrodynamic (MHD) natural convective, to-dimensional incompressible electrically conducting viscous and laminar lo o a aterbased nanoluid over a permeable shrinking sheet in the presence o thermal radiation, viscous dissipation, and chemical reaction. he lo is subjected to applied electric and magnetic ields E and B hich are assumed to be applied in the direction y 0, normal to the surace see Fig.. he electric and magnetic ields conirm the Ohms la J E V B, here J stand or the Joule current, V is the luid velocity and denote the electrical conductivity. he induced magnetic ield and Hall current impacts are overlooked due to small magnetic Reynolds number. he velocity o the linearly shrinking sheet is u x, t, here 0 or a shrinking assumed in the orm sheet and the velocity o the mass transer denoted v t, here x and y axes are measured along the shrinking sheet and t are the time. It is also assumed that the temperature at the surace o the sheet denoted by and the ambient temperature and concentration denoted by and respectively. Using the Buongiorno model ith aorementioned assumptions, the boundary layer governing equations are expressed as ollos: Slit Boundary Layer permeable Shrinking Sheet Nanoluid Magnetic ield Electric ield Magnetic ield Fig. Physical coniguration o the geometry Force Continuity equation v u 0 () x y x Momentum equation u u u p u u u v EB B u t x y x x y p () g y Momentum equation v v v p v v u v EB B v t x y y x y p (3) g Energy equation k u qr u v t x y c x y c y c y D (4) DB x x y y x y Concentration equation D (5) u v DB k t x y x y x y he boundary conditions at the sheet or the physical model are presented by u y 0 : u u x, t l, v v t, x, t l, y y D DB 0 y y y : u 0,, (6) Here u x, t bx at denotes the velocity o the shrinking sheet, v t v0 at is the all mass transer, hen v 0 represent the injection v 0 indicates the suction. Where u and v stands or the velocity components along the x and y axes respectively. For g, p, k c,,,, and p is the gravitational acceleration, the pressure, the thermal diusivity, the (Advance online publication: 0 February 08)

3 Engineering Letters, 6:, EL_6 3 kinematic viscosity, the density, the luid density and particles density respectively. We also have l l at,, B,, p l l at D D c c hich represents the velocity slip actor, temperature slips actor, Bronian diusion coeicient, the thermophoresis diusion coeicient, the ratio beteen the eective heat transer capacity o the ultraine nanoparticle material and the heat k k at is the rate o capacity o the luid and 0 chemical reaction respectively. he radiative heat lux q via Rosseland approximation [7] r is applied to equation (4), such that, * 4 4 qr (7) * 3k y * Where represent the Stean-Boltzmann constant and k * 4 denote the mean absorption coeicient. Expanding by using aylor s series about and neglecting higher order terms, e have, (8) Using equation (8) into (7), e get, qr 6 (9) * y 3k y Use equation (9) in equation (4), e have, * 6 * 3 k u u v t x y c x y c y c k y D (0) DB x x y y x y Using the order o magnitude analysis or the y direction momentum equation hich is normal to the shrinking sheet and boundary layer approximation [37], such as u v u u v v v,,, () y x t x y p 0 y Ater the analysis, the boundary layer equations ()-(5) are reduced to the olloing as: v u 0 () x y u u u p u u v EB B u t x y x y p g * 6 * 3 (3) k u u v t x y c y c y c k y D (4) DB y y y D (5) u v DB k t x y y y he resulted equations are reduced into the dimensionless orm by introducing the olloing dimensionless quantities [38-4]. bv x b, y at,, v at W bx (6), W ( x, t), 0 v at he stream unction can be deined as: u, v y (7) x he equations o momentum, energy and nanoparticle concentration in dimensionless orm become: M E M 0 (8) 4 Rd Nb Nt Ec 0 (9) Pr 3 Nt Sc Sc Sc 0 (0) Nb he boundary conditions are given by s, L, L, Nb Nt 0, at 0 0, 0, 0, as () Here,, and is the dimensionless velocity, temperature, and concentration respectively. We have the olloing parameters: abdenote the unsteadiness parameter, L l b v is the velocity slip parameter, L l b v depicts the thermal slip parameter, Gr Re thermal convective parameter Gr g l 3 is the Grasho number, X x l is the dimensionless constant, M Gm Re mass convective parameter Gm g l stand or the mass Grasho, 3 p Pr represent the Prandtl number, is the Bronian motion, Nb c D c p B Sc D B denote the Schmidt number, Nt c D c is the thermophoresis, 0 p W M B b represent the magnetic ield, 0 W 0 E E u B denote the electric ield, Ec u c indicate the Eckert number, W P W s v0 vb is the suctions 0 /injections 0 parameter, 3 Rd 4 * k * k denote the radiation parameter, k0 b is the chemical reaction, or 0 associates to destructive chemical reaction hile 0 corresponds to generative chemical reaction respectively. Where prime represents dierentiation ith respect to. he skin riction coeicient and the local Nusselt number are W, xqw c Nu u x, t k, () W W Where 6 * u qw k, W, (3) 3 k* y y y0 y0 Here is the shear stress or the shrinking surace, q is the surace heat lux, hile Re ulis the Reynolds number and k the thermal conductivity o the nanoluid. For the local skin-riction coeicient and local Nusselt are presented in non-dimensional orm as (Advance online publication: 0 February 08)

4 Engineering Letters, 6:, EL_6 3 Re c 0, Nu 4 3 Re Rd 0, III. RESULS AND DISCUSSION (4) Folloing Cebeci and Bradsha [36] the system o ordinary dierential equations (8)-(0) subject to the boundary conditions () are solved numerically using implicit inite dierence scheme knon as Keller box method or dierent values o the parameters. It is orth mentioning that the step size along ith the boundary layer thickness is selected rendering to the values o parameters. hese calculations are repeated until 5 convergence criteria are satisied at 0 is used. he numerical values o the skin riction ith the available published data by Yasin et al. [3], Bhattacharyya [4] and Dhanai et al. [43] are displayed in ables and. In able is the comparison o the numerical values or the skin riction 0, hereas able supporting the existence o multiple solutions o the present study. he present computation scheme in some limiting sense to that o the previous investigation noticed a perect agreement. he eects o the sundry parameters due to the decelerating sheet 0 on the velocity, temperature, and nanoparticle concentration are given in Figs. -7. Fig. reveal the variation o magnetic ield M on the velocity proile. he ater-based nanoluid velocity along the shrinking sheet increases ith M in the irst solution, hereas in the second solution its decreases. he nanoluid due to the resistive orce associated ith Lorentz orce has the tendency to retard the lo o the nanoluid. Intense magnetic ield contributes resistance to lo. Physically, the eect o magnetic ield is such that it gives rise to Lorentz orce in a direction hich opposes the lo in either direction as such leads to a reduction in the nanoluid velocity. In Figs. 3 and 4 demonstrate the eects o electric ield E and unsteadiness parameter on the velocity proiles. In the irst solution, the ater-based nanoluid velocity shrinkages ith an increase in electric ield and unsteadiness parameters toards the sheet surace. he second solutions, close to the all the nanoluid velocity intensiied ith the electric ield and along the surace, ater some distance, it s eakening. he Lorentz orce acts as accelerating body orce hich accelerates the lo behavior due to interaction ith electrically conducting nanoluid. he unsteadiness parameter designates a reduction in both solutions near the sheet due to decelerating lo. his depicts that irst solution is stable as related to the second solution. he eect o a suction parameter s on the nanoluid velocity proile is portrayed in Fig. 5. In the irst solution, the velocity is a decreasing unction hereas in the second solution illustrates an increasing unction. It means that augmentation o suction leads to more separation due to the dual solution on the ater base nanoluid velocity ith decelerating shrinking. In Fig. 6, the impact o velocity slip parameter L on the nanoluid velocity proile is publicized. he proile reveals that the velocity produces resistance to lo o nanoluid by virtue o higher values o L or both solutions hoever ater some distance along the shrinking sheet second solution supplements or smaller values o L. hese thicknesses are higher or the second solution than the associated thicknesses o the irst solutions. he eects o thermal radiation Rd, Eckert number Ec, thermal convective parameter, unsteadiness parameter and thermal slip L parameter on the temperature proiles are displayed in Figs. 7-. Figs. 7 and 8 established ho the thermal radiation Rd and Eckert number Ec aects the nanoluid temperature proile. It is observed that the eect o thermal radiation enhances the ater base nanoluid temperature or an increase in the values o thermal radiation Rd and Eckert number Ec. he reason is a result o emission due to heat transer in the boundary layer region ith a magnitude o decelerating shrinking. Consequently, the thermal boundary layer becomes thicker in the second solution compared to the irst solution. he variation o thermal convective parameter on the nanoluid temperature proile is revealed in Fig. 9. emperature is an increasing unction ith thermal convective parameter or the case o the second solution hereas or the irst solution there is an insigniicant behavior on the ater-based nanoluid. It s represents the relative strength o thermal buoyancy orce to viscous orce. Increases hen thermal buoyancy orce upsurge. his implies that thermal buoyancy orce tends to accelerate the nanoluid temperature in the thermal layer region as result o high density. he thermal boundary layer thickness becomes larger or greater values o thermal convective parameter due to a decelerating shrinking sheet. Fig. 0 represent the result o unsteadiness parameter on the temperature proile. It orth noticed that the unsteadiness parameter gain the ater base nanoluid temperature. he second solution ampliied and ater some distance aay rom the all shrinkage, that is crossing over. he irst solution is more stable to the second solution. In Fig. exhibit the outcome o thermal slip parameter L on the nanoluid temperature proile. Both solutions diminish or higher values o thermal slip parameter. Higher values o thermal slip parameters resulted to thinner thermal boundary layer thickness due to aterbased nanoluid temperature. he eects on thermophoresis parameter Nt, Bronian motion Nb, Leis number Le, chemical reaction, mass convective parameter, and unsteadiness parameter on M the nanoparticles concentration proiles are presented in Figs. -7. In Fig. unveil the variation o thermophoresis parameter Nt on the concentration proile. It is noticed that along the surace o the shrinking sheet, distance aay rom the all both solutions increases or higher values. Due to a decelerating shrinking sheet o the ater base nanoluid, or larger values o thermophoresis parameter creates a greater mass lux that enhances the nanoparticle volume raction proile. Bronian motion (Advance online publication: 0 February 08)

5 Engineering Letters, 6:, EL_6 3 parameter Nb on the concentration proile has a decreasing upshot see Fig. 3. In both solutions, it reduced along the surace or higher values o Bronian motion. he increment in Bronian motion parameter enhances the temperature hich leads to the decrement in nanoparticle concentration proile o the ater-based nanoluid. hus, it tends to reduce the separating, due to the nanoparticle concentration at the all is passively controlled by mass transer parameter and decelerating shrinking sheet. In Fig. 4 illustrates the inluence o Leis number Le on the concentration proile. It orth noticed that thermal boundary layer thickness is thicker to the solutal boundary layer thickness, hich resulted in a reduction in the nanoparticle concentration o the ater-based nanoluid. Leis number denotes the ratio o the viscous diusion rate to the molecular diusion rate. Physically, deals ith the virtual thickness o the momentum and concentration boundary layers. So, intense Leis number substantially decreases solutal boundary layer thickness. his implies that there is a much aster viscous diusion rate compared ith nanoparticle mass diusion rate. In the decelerating shrinking sheet due to increase in Leis number, the concentration distribution decreases in both solutions (that is the irst and second solutions). Figs. 5 and 6 are the impacts o chemical reaction and mass convective parameter on the concentration proiles M. he nanoparticle concentration condenses ith chemical reaction and mass convective parameter or higher values. It s represent the virtue strength o solutal buoyancy orce to viscous orce. hus, higher values decrease solutal buoyancy orce. his implies that viscous orce tends to shrink the ater base nanoluid lo in the solutal layer region. his is more pronounced in the second solution ith an insigniicant eect in the irst solution. In Fig. 7 demonstrated the poer o unsteadiness parameter on the nanoparticle concentration. In the second solution near to the all it groths and ater some distance along the sheet, surace reduces substantially or higher values o unsteadiness parameter ithin the ater-based nanoluid and the shrinking sheet. While in the case o the irst solution insigniicant drops consequence due to a decelerating shrinking sheet and passively controlled behavior. able : Numerical values o shear stress at the all 0 or dierent values o a suction parameter hen M, M E L 0 and. s Re [4] Re [3] Present results able : Numerical values o shear stress 0 or dierent values o s and M hen M E L 0 and. s M First Second Present Present solution Re[43] solution Re[43] irst solution second solution Fig. Inluence o M on the velocity proile Fig.3 Inluence o E on the velocity proile (Advance online publication: 0 February 08)

6 Engineering Letters, 6:, EL_6 3 Fig.4 Inluence o on the velocity proile Fig.7 Inluence o Rd on the temperature proile Fig.5 Inluence o s on the velocity proile Fig.8 Inluence o Ec on the temperature proile Fig.6 Inluence o L on the velocity proile Fig.9 Inluence o on the temperature proile (Advance online publication: 0 February 08)

7 Engineering Letters, 6:, EL_6 3 Fig.0 Inluence o on the temperature proile Fig.3 Inluence o Nb on the concentration proile Fig. Inluence o L on the temperature proile Fig.4 Inluence o Sc on the concentration proile Fig. Inluence o Nt on the concentration proile Fig.5 Inluence o on the concentration proile (Advance online publication: 0 February 08)

8 Engineering Letters, 6:, EL_6 3 Fig.6 Inluence o M on the concentration proile mass suction parameter iden the range o the solution existence. 3. hermal radiation and Eckert number heighten the temperature and thermal boundary layer thickness. 4. he skin riction is sensitive to an increase in magnetic ield and suction parameters. 5. Electric ield and unsteadiness parameters decrease the velocity near the surace o the sheet, the irst solution (upper solution branch) are stable to compare ith the second solution (loer solution branch). 6. he thermal slip parameter reduces the nanoluid temperature and thermal boundary layer thickness in both solutions hile opposite trend occurred ith unsteadiness parameter. 7. Opposite behavior o the nanoparticle concentration is noticed ith Bronian motion and thermophoresis parameters. 8. he thermal convective parameter and mass convective parameter exhibit opposite behavior hereas chemical reaction reduced the nanoparticle concentration ith insigniicant eect or the irst solutions. ACKNOWLEDGMEN he authors ould like to express their sincere thanks to Ministry o Higher Education and Research Management Centre, UM through GUP ith vote number H90, Flagship vote number 03G50, 3H8 and 03G53 or this research. Fig.7 Inluence o on the concentration proile IV. CONCLUSION he unsteady magnetohydrodynamic (MHD) natural convective lo o electrical conducting nanoluid over a permeable shrinking sheet in the presence o electric ield, thermal radiation, viscous dissipation and chemical reaction ith slips and passively controlled conditions at the all are investigated. he decelerating shrinking sheet, slips and passively controlled conditions have been employed. he boundary layer governing the lo are partial dierential equations are transormed into nonlinear ordinary dierential equations and then solved numerically using implicit inite dierence scheme. he eects o various physical parameters involved in the system o the equations namely: electric ield, magnetic ield, unsteadiness parameter, suction, velocity slip, thermal radiation, Eckert number, thermal convective parameter, thermal slip, thermophoresis, Bronian motion, Leis number, mass convective parameter, and chemical reaction are obtained. he olloing conclusions are dran in this investigation.. he eects o the all suction and decelerating shrinking sheet revealed dual solutions.. he magnetic ield, unsteadiness parameter, and the REFERENCES [] Z. Hedayatnasab, F. Abnisa, and W. M. A. W. Daud, "Revie on magnetic nanoparticles or magnetic nanoluid hyperthermia application," Materials & Design, vol. 3, pp , 07. [] L. Mohammed, H. G. Gomaa, D. Ragab, and J. Zhu, "Magnetic nanoparticles or environmental and biomedical applications: A revie," Particuology, vol. 30, pp. -4, 07. [3] Y. S. Daniel, Z. A. Aziz, Z. Ismail, and F. Salah, "Eects o slip and convective conditions on MHD lo o nanoluid over a porous nonlinear stretching/shrinking sheet," Australian Journal o Mechanical Engineering, pp. -7, 07. [4] S. Shabestari Khiabani, M. Farshba, A. Akbarzadeh, and S. Davaran, "Magnetic nanoparticles: preparation methods, applications in cancer diagnosis and cancer therapy," Artiicial cells, nanomedicine, and biotechnology, vol. 45, no., pp. 6-7, 07. [5] Y. S. Daniel, Z. A. Aziz, Z. Ismail, and F. Salah, "Entropy analysis in electrical magnetohydrodynamic (MHD) lo o nanoluid ith eects o thermal radiation, viscous dissipation, and Chemical reaction," heoretical and Applied Mechanics Letters, [6] Y. S. Daniel, Z. A. Aziz, Z. Ismail, and F. Salah, "Eects o thermal radiation, viscous and Joule heating on electrical MHD nanoluid ith double stratiication," Chinese Journal o Physics, vol. 55, no. 3, pp , 07. [7] Y. S. Daniel and S. K. Daniel, "Eects o buoyancy and thermal radiation on MHD lo over a stretching porous sheet using homotopy analysis method," Alexandria Engineering Journal, vol. 54, no. 3, pp , 05. [8] F. Ismagilov, I. Khayrullin, V. Vavilov, and A. Yakupov, "Generalized Mathematic Model o Electromechanical Energy ransducers ith Non-contact Bearings," Engineering Letters, vol. 5, no., pp , 07. (Advance online publication: 0 February 08)

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Ismail, and F. Salah, "Numerical study o Entropy analysis or electrical unsteady natural magnetohydrodynamic lo o nanoluid and heat transer," Chinese Journal o Physics, vol. 55, no. 5, pp , 07. [4] K. Bhattacharyya, "Eects o heat source/sink on MHD lo and heat transer over a shrinking sheet ith mass suction," Chemical Engineering Research Bulletin, vol. 5, no., pp. - 7, 0. [43] R. Dhanai, P. Rana, and L. Kumar, "Multiple solutions o MHD boundary layer lo and heat transer behavior o nanoluids induced by a poer-la stretching/shrinking permeable sheet ith viscous dissipation," Poder echnology, vol. 73, pp. 6-70, 05. Yahaya Shagaiya Daniel is currently Ph.D student in Applied Mathematics, Department o Mathematical Sciences, Universiti eknologi Malaysia (UM), Johor, Malaysia. Zainal Abdul Aziz graduated Ph.D in Applied Mathematics, 997, Universiti Kebangsaan Malaysia (UKM), Bangi, Malaysia and is currently Proessor at the Department o Mathematical Sciences, UM and Director o UM Centre or Industrial and Applied Mathematics (UM-CIAM), (Advance online publication: 0 February 08)

10 Engineering Letters, 6:, EL_6 3 Institute Ibnu Sina or Scientiic and Industrial Research, UM, Johor, Malaysia. Zuhaila Ismail graduated Ph.D in Applied Mathematics, 03, University o Southampton, Southampton, United Kingdom and is currently Senior Lecturer at the Department o Mathematical Sciences and a ello o UM-CIAM, Universiti eknologi Malaysia (UM), Johor, Malaysia. Faisal Salah graduated Ph.D in Applied Mathematics, 0, Universiti eknologi Malaysia (UM), Johor, Malaysia and is currently Assoc. Pro. at the Department o Mathematics, University o Kordoan, Elobied, Sudan. (Advance online publication: 0 February 08)