Transient Flow and Mass Transfer of a Third Grade Fluid Past a Vertical Porous Plate in the Presence of Chemical Reaction

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1 Nigerian Journal of Science Vol 48 (04): Transient Flow and Mass Transfer of a Third Grade Fluid Past a Vertical Porous Plate in the Presence of Chemical Reaction Abstract *I.G. BAOKU AND B.I. OLAJUWON Department of Phsical Sciences, Bells Universit of Technolog, P.M.B. 05, Ota, Ogun State, Nigeria. Department of Mathematics, Federal Universit of Agriculture, P.M.B. 40, Abeokuta, Ogun State, Nigeria. *Corresponding Author baokuismail@gmail.com, olauwonbi@funaab.edu.ng The effects of chemical reaction and suction, on the transient flow of an incompressible third grade fluid past a vertical porous plate, are studied. The governing time-based coupled partial differential equations are discretized using an efficient and unconditionall stable Crank-Nicolson finite difference scheme and the resulting algebraic nonlinear equations are solved b the modified Newton's method. The velocit and species concentration profiles are computed, graphicall presented and discussed in detail for various values of embedded pertinent flow parameters. It is concluded that the fluid velocit decreases with increasing suction and viscoelastic second grade parameters while it increases with an increase in the Schmidt number, solutal Grashof number and third grade parameter. However, as the Schmidt number, suction and rate of chemical reaction parameters increase, the decrease the species concentration of the flow field. Introduction Concentration differences, applied external forces, chemical reaction and the medium, in which man transport processes exist in natural and technological applications, affect the mass and momentum transport. The stud of mass transfer, with some phsical variables, is useful for improving a number of chemical technologies such as polmer production, aerodnamic extrusion of plastic sheets, manufacturing of ceramics and cooling of nuclear reactors, e.t.c. The effect of homogeneous first-order chemical reaction in a laminar boundar laer flow in the neighbourhood of a horizontal flat plate for destructive and generative chemical reaction was investigated b Chambre and Young (958). Apelblat (980) Kewords: Mass Transfer, Third Grade Fluid, Porous Plate, Chemical Reaction studied the problem of mass transfer with chemical reaction of first-order. He proposed analtical solutions and asmptotic expressions for homogeneous and heterogeneous chemical reactions and gave some extensions for mass transfer in non-newtonian liquids or in the case of permeable surfaces. Das et al. (994) examined the influence of firstorder chemical reaction on the flow past an impulsivel started infinite vertical plate with uniform heat flux and mass transfer. The emploed an exact solution b using Laplace transform technique to obtain solutions to velocit and concentration profiles with a concluding remark that the velocit decreases due to the presence of first order chemical reaction. Chen and Arce (997) presented the convective-diffusive mass transfer problem with 47

2 I.G. Baoku & B.I. Olauwon: Transient Flow & Mass Transfer of a Third Grade Fluid Past a Vertical Porous Plate in the Presence of CR chemical reaction in a Couette planar flow in terms of the integral-spectral methods. The discussed the effects of relevant parameters for the sstem on the computational algorithm with respect to convergence and numerical stabilit characteristics. Also, the chemical reaction effect on heat and mass transfer of an MHD fluid flow past a semi-infinite plate was analzed b Anali Devi and Kandasam (00). The solved the non-linear governing equation using Runge- Kutta Gill's method and concluded that in the presence of chemical reaction, the velocit and concentration decrease with increase of the chemical reaction parameter. The velocit also decreases and the concentration is uniform with increase of the magnetic parameter. Muthucumaraswam and Ganesan (00a) investigated the effect of chemical reaction and inection as well as flow characteristics in an unstead upward motion of an isothermal plate with mass flux through the plates. The showed that the fluid velocit decreases with increasing chemical reaction parameter. The (00b) further examined numericall the transient natural convection flow of an incompressible viscous fluid past an impulsive started semi-infinite isothermal vertical plate with mass diffusion and homogeneous chemical reaction of first-order. The observed that owing to the presence of the first-order chemical reaction, the velocit increases during generative chemical reaction and decreases in the case of destructive chemical reaction. Muthucumaraswam and Meenakshisundaram (006) studied the influence of the homogeneous chemical reaction of first order on a viscous incompressible unstead flow past an infinite vertical oscillating plate with variable temperature and mass diffusion. Appling Laplace transform method for solving dimensionless governing equations, the observed that the velocit increases with decreasing phase angle or chemical reaction parameter. The effect of third-order fluid on the peristaltic transport in a circular clindrical tube was established b Haat et al. (00). The compared solutions of both analtic (perturbation) and 48 numerical solution that could have applications to a range of peristaltic flows for variet of non- Newtonian fluids. The influence of magnetic field on transient free convection flow of an electricall conducting fluid over an impulsivel started isothermal vertical plate was numericall considered b Al-Odat and Al-Azab (007). Using an implicit finite-difference scheme of Crank-Nicolson tpe, the found that the velocit as well as concentration decreases with increasing chemical reaction parameter. Said and Haat (007) presented the non-similar solution to the boundar laer flow problem of a third-order fluid over a stretching surface. The obtained the analtical solution of the problem using homotop analsis method and compared their solutions with the existing results in the literature. Stepanak and Achwal (009) analzed the problem of absorption with first-order chemical reaction into a liquid film in laminar flow. The results obtained in terms of sum of a hperbolic function and infinite series of hpergeometric function showed that the length of the entrance region could var substantiall depending upon the reaction rate constant. El-Arabaw (009) examined and made an analsis to the effects of suction/inection and chemical reaction on mass transfer characteristics over a stretching surface. He obtained analtical solutions and determined variations of embedded parameters on the problem. Also, a numerical solution attempt for unstead convective flow of a third grade fluid past an infinite vertical porous plate was made b Naak et al. (0). The studied the effects of phsical parameters on the flow and concluded that the velocit increases with an increase of the second grade elastic parameter whereas the velocit decreases with an increase of third grade elastic parameter. Haat et al. (0) investigated the effect on mass transfer with higher order chemical reaction at a time-dependent three-dimensional boundar laer flow of an elasticoviscous fluid over a stretching surface. The computed the results b using the homotop analsis method (HAM) and

3 Nigerian Journal of Science Vol 48 (04): variations of embedding parameters on the velocit and concentration were graphicall discussed. Other works of Haat et al. (00a, 00b) on flow and mass transfer focused respectivel on the homotop solution for unstead three-dimensional MHD flow over a porous space and similar solution of stretching flow with mass transfer. The boundar flow and mass transfer of a viscoelastic fluid immersed in a porous medium over a stretching surface in the presence of surface slip, chemical reaction and variable viscosit had been reported b Mahmoud (00). Solving the problem numericall b means of the fourth order Runge-Kutta integration scheme coupled with the shooting technique, he examined the effects of various involved parameters on the velocit and concentration fields as well as the local skin-friction coefficient and local Sherwood number. Also, Raptis et al. (0) studied the stead two-dimensional freeconvection and mass transfer flow of an incompressible viscous fluid through a porous medium bounded b an infinite vertical limiting surface. The discussed the rate of heat transfer when the surface is subected to a constant suction velocit. Elbashbesh et al. (0) examined the effects of suction/inection and variable chemical reaction on mass transfer characteristics over unstead stretching surface embedded in a porous medium. The established that the velocit as well as concentration decreases with increasing suction/inection and unsteadiness parameters while the concentration decreases with increasing chemical reaction parameter. Okoa (008) investigated the thermal transition of a reactive flow of a third-grade fluid with viscous heating and chemical reaction between two horizontal flat plates in the presence of imposed pressure gradient. He focused on the disappearance of criticalit of the solution set for various parameters. Okoa (0) also considered how a model for the motion with exponential viscosit of a third grade fluid flowing between parallel plates affects the full developed and laminar reactive flow. He further discussed the criticalit and disappearance of criticalit of phsical parameter with other competing parameters. Das et al. (0) studied the effects of mass transfer on an unstead free convective flow past a vertical porous plate in a porous medium with constant suction. In their work, the also considered the interaction of an electricall conducting fluid flow with magnetic field and heat source. The established that a growing Hartmann number decreases the transient temperature of the flow fields whereas a growing permeabilit or heat source parameter reverses the effect. Das et al. (0) further provided solutions to the problem of an unstead hdromagnetic convective flow past an infinite vertical porous flat plate in a porous medium. Emploing multi-parameter perturbation technique to obtain solutions for velocit and temperature profiles, the concluded that the permeabilit enhances the skin-friction as well as the rate of heat transfer at the wall. Toki and Tokis (007) gave exact solutions for the unstead free convection flows on a porous plate with time-dependent heating. Keimanesh et al. (0) studied the flow of a third grade non- Newtonian fluid between two parallel plates. The used the multi-step differential transform method to obtain the solution to the problem. Khan et al. (04) investigated the effects of an arbitrar wall shear stress on unstead magnetohdrodnamic conugate flow in a porous medium with ramped wall temperature. The concluded that velocit solutions are more general and could produce a huge number of exact solutions correlative to various fluid motions. Ahmed et al. (04) examined the transport modelling for magnetohdrodnamic unstead flow over a vertical plate in Darcian porous media. The emploed the finite difference approach for the solutions of the unstead flow. The aim of the present paper is to investigate the effect of mass transfer on transient free convective flow of an incompressible third grade fluid past a vertical porous plate in the presence of a first-order chemical reaction and suction. The work exhibits velocit distributions when a verticall upward plate suddenl sets in motion in its own plane and species concentration distributions of the fluid. The investigation is based on time-based boundar laer dimensionless variables. The numerical 49

4 I.G. Baoku & B.I. Olauwon: Transient Flow & Mass Transfer of a Third Grade Fluid Past a Vertical Porous Plate in the Presence of CR solutions are then obtained using Crank-Nicolson finite difference scheme with modified Newton's iteration technique. Results are presented graphicall and discussed quantitativel for parameter of practical interest from phsical point of view. Mathematical Formulation Consider a transient, laminar, incompressible and viscoelastic third grade fluid flow with mass transfer past an infinite vertical porous plate in the presence of a first-order chemical reaction. The fluid properties are assumed to be constant in the limited temperature range. The x - axis is taken along the plate verticall upwards and - axis is perpendicular to it. The plate is set in motion in its own plane with a velocit U(t). The concentration of diffusing species is ver small in comparison to the other chemical species. The species concentration far awa from the plate C is infinitesimall small. The chemical reactions are taking place in the flow regime, and all other phsical properties are assumed to be constant. Since the plate is infinitel long, the phsical variables are functions of and t onl. Hence, from the continuit equation, the velocit field is obtained as: u u, t, v V0 () where u and v are the velocities of the fluid along x and axes respectivel, and V 0 >0 indicates suction velocit. Flow Analsis for Mass Transfer The constitutive equation for Cauch's stress tensor of an incompressible homogeneous third grade fluid, according to Rivlin and Ericksen (955), Coleman and Noll (965) and Truesdell and Noll (004), is: p () I A A A A A A A A tra A where pi is the spherical part of the stress due to the constraint of incompressibilit; p is the pressure; I is an identit tensor, and are normal stresses, and, and are material constants. The Rivlin-Ericksen tensors A, A gradv gradv T A A n T V A gradv A A gradv n n n, n, () t where is the gradient operator, is the material time derivative and V is the velocit. If all the motions of the fluid are to be compatible with thermodnamics in the sense that these motions meet the Clausius-Duhem inequalit and if it is assumed that the specific Helmholtz free energ is a minimum when the fluid is locall at rest, then 0, 0, 4, 0 d dt, 0. (4) For a detailed thermodnamic analsis on the model represented b equation (), one can refer to the work of Fosdick and Raagopal (980). Therefore, the constitutive relation for a thermodnamicall compatible third grade fluid becomes: pi (5) A A A tra A The stress components of () b virtue of equation () are: x x u u u p t, (6) u u u p 6, (7) t 50

5 Nigerian Journal of Science Vol 48 (04): z z p, (8) u u u t u u t x V 0 (9) 0 (0) x z z Appling the similarit transformation for scaling species concentration equations in mass transfer analsis where the species concentration, C C C C to obtain: w where x x, x z z x, z z u u u u u Gc t t (9) Inserting the stress components (6-0) and velocit field given b () and using (5) in the equation of motion while neglecting bod force, we have: u u u u V t t u u u 0 V 0 6 g C C () C Mass Transfer Analsis For the thermodnamicall compatible third grade fluid using (), the species concentration equation with first-order homogeneous chemical reactions for the flow field is: C C C V D k C C 0 () t where D and k are mass diffusivit and rate of chemical reaction respectivel. The initial and boundar conditions are: t 0: u 0, C 0, 0 () n U t n t> 0: u U t e a t, when 0 n (4) v C C w (5) u 0 (6) C C, when (7) The following dimensionless variables are introduced: u u, U U a, a, U tu t (8) 5 t Sc where (0) V 0, k, U 4 6, U Cg Cw C, Gc D, Sc U U U are suction, rate of chemical reaction, elastic second and third grade parameters, solutal Grashof and Schmidt numbers respectivel. The boundar conditions () - (7) are also nondimensionalized using (8) with the dimensionless variables to obtain the following initial and U boundar conditions: t 0 : 0, u 0, 0; () a t n t > 0: u e t, when 0 ; (), when 0 ; () u u 0, when, 0, when ; (4) 0, when ; (5) Computational Procedure C C, when 0 The governing nonlinear coupled partial differential equations (9) and (0) with the initial u 0, when,, when and boundar conditions (-5) are solved using Crank-Nicolson finite difference scheme, which has been discussed b Conte and De Boor (980), Jain et al. (999), Ganesan and Palani (00) and Baoku et al. (0). Therefore, the governing equations based on the transient state conditions are discretized. The numerical method of Crank- Nicolson tpe does not restrict the value of r to be chosen. The finite difference equations corresponding to the governing equations are given as:

6 I.G. Baoku & B.I. Olauwon: Transient Flow & Mass Transfer of a Third Grade Fluid Past a Vertical Porous Plate in the Presence of CR u rh r r r rh r u u,, u i i i, 4 h h 4 h h i, rh r r rh r r u i, u i, u i, u i, 4 4 h h h u h r u u i, i, i, i, i, i, i, h (7) r i, h i, i i, i, u u u u rh Gc i, i,, i, i, i, i, i, i, i, i, i, i, i, i, i, i, i, i, i, code as described b Heck (00). The convergence of the process was quite satisfactor and the numerical stabilit of the method was guaranteed b the implicit nature of the scheme. Hence, the scheme is consistent; stabilit and consistenc ensure convergence. Results and Discussion Sch r r r Sch r i, i, Sc Sc 4r Sch r h r i, i, Sc i, (8) where i designates the grid point along - direction, along t -direction and r t h. Hence, the equations of motion and species concentration are reduced to a sstem of algebraic nonlinear coupled-equations. The mesh size h is 0.05 with time step t =0.. The values of u (,t) and (,t) are known for all grid points when t = 0 from the initial conditions. Modified Newton s iterative technique is emploed to solve the sstem of coupled nonlinear algebraic equations. Computations are carried out b moving along - direction. After computing values corresponding to each i at a time level, the values at the next time level are determined in the similar manner. The effect of transient free convection flow on mass transfer of an incompressible third grade fluid past a vertical porous plate in the presence of a first-order chemical reaction was investigated. The research exhibits velocit distributions when a verticall upward plate suddenl sets in motion at a velocit U in its own plane and concentration distributions. The governing equations of the flow field are solved b emploing an efficient Crank-Nicolson finite difference technique with modified Newton s iterative technique and approximate solutions are obtained for velocit and species concentration distributions. The influences of some controlling parameters on the flow field are analzed and discussed with the aid of velocit profiles (Figures -) and species concentration profiles (Figures 4 and 5). The implicit nature of Crank-Nicolson method is unconditionall stable and has local truncation error O t,h which tends to zero as t and h tend to zero. There is no drawback of conditional stabilit from one level to the next. The implicit method gives stable solutions and requires the iterative procedures which were done at each step forward in time because this problem is an initialboundar value problem with a finite number of spatial grid points. Though, the corresponding difference equations do not automaticall guarantee the convergence of the mesh h. To achieve maximum numerical efficienc, the tridiagonal and pentadiagonal procedures were used to solve the two-point conditions for (0) and four-point conditions for (9) respectivel. The above procedure was transformed into Maple 0 Figure : Velocit distribution when n=0.8 and n=.0 for values of at =5.0, =.0, Gc=0. 5

7 Nigerian Journal of Science Vol 48 (04): Figure : Velocit distribution when n=0.8 and n =.0 for values of at =.0, =5.0, Gc=0. Figure 4: Concentration distribution for values of Sc at =.0 and =. Figure : Velocit distribution when n=0.8 and n =.0 for values of at =.0, =.0, Gc=0. During numerical calculations, the values of the some parameters have been realisticall chosen while the values of other parameters are chosen arbitraril. The effect of the suction parameter on the transient velocit distributions are shown in Figure. It is observed from Figure that the velocit fluid decreases with the increase in suction parameter w for both cases n= and n = 0.8, that is Figure 5: Concentration distribution for values of at Sc =7.0 and =.0. constant and variable acceleration respectivel, at an point within the fluid. Figure displas the influence of the elastic parameter on the velocit distribution. It is clear from Figure that the fluid velocit increases for both constant and variable accelerations when the third grade parameter increases. However, the reverse is the case for the second grade parameters. This 5

8 I.G. Baoku & B.I. Olauwon: Transient Flow & Mass Transfer of a Third Grade Fluid Past a Vertical Porous Plate in the Presence of CR observation is similar to that made b Naak et al. (0). Figure analzed the effect of mass diffusivit shown b Schmidt number Sc on the velocit distribution for both constant and variable accelerations along the vertical plate. It is obvious from Figure that an increase in Sc leads to a rise to the velocit of an point within the fluid. The species concentration of the flow field suffers a substantial change with the variation of the flow parameters such as the Schmidt number Sc and the rate of chemical reaction parameter. It is observed in Figure 4 that when there is an increase in Schmidt number, there is a reduction in the concentration field. Finall, it is evident from Figure 5 where destructive chemical reactions are considered, that the consequence of increasing the rate of chemical reaction has the effect of decreasing the species concentration distribution. Conclusion The transient flow and mass transfer of a third grade fluid past a vertical porous plate in the presence of chemical reaction is studied. The governing time-based partial differential equations are solved using an efficient numerical scheme of implicit Crank-Nicolson finite difference technique with the modified Newton's iterative method. Simulation of the scheme is done with the aid of Maple codes to obtain approximate solutions of the problem. The solution procedure is valid for all values of elastic parameters unlike perturbation and power series methods that are valid for small values of the elastic parameters. The conclusions of the stud are:. The velocit increases significantl with increasing solutal Grashof number and Schmidt number.. The fluid velocit also increases as the value of second grade parameter increases. However, the third grade parameter has opposite effect on the velocit.. An increase in the suction parameter has the influence of decreasing the velocit distribution as well as the species concentration distribution As the rate of chemical reaction and Schmidt number increase, it results into a decrease in the species concentration in all part of the fluid. References Ahmed, S., Batin, A. and Chamkha, A.J. (04). Finite Difference Approach in Porous Media Transport Modeling for Magnetohdrodnamic Unstead Flow over a Vertical Plate: Darcian Model. International Journal for Numerical Methods in Heat and Fluid Flow 4: 04-. Al-Odat, M. Q. and Al-Azab, T. A. (007). Influence of chemical reaction on transient MHD free convection over a moving vertical plate. Emirates J. for Engineering Research : 5-. Anali Devi, S. P. and Kandasam, R. (00). Effects of chemical reaction, heat and mass transfer on MHD flow past a semi-infinite plate. Z. Angew. Math. Mech. 80: Apelblat, A. (980). Mass transfer with a chemical reaction of first order: Analtical Solution. The Chemical Engineering Journal 9, 9-7. Baoku, I. G., Israel-Cooke, C. and Olauwon, B.I. (0). Influence of thermal radiation on a transient MHD Couette flow through a porous medium. Journal of Applied Fluid Mechanics 5: Chambre, P. L. and Young, J. D. (958). On the diffusion of chemicall reactive species in a laminar boundar laer flow. Phsics of Fluids : Chen, Z. and Arce, P. (997). An integral-spectral approach for convective-diffusive mass transfer with chemical reaction in Couette flow-mathematical formulation and numerical illustrations. Chemical Engineering Journal 68: -7. Conte, S. D. and De Boor, C. (980). Elementar Numerical Analsis: An Algorithmic Approach, McGraw-Hill Inc., New York. Coleman, B.D. and Noll, W. (965). An Approximation Theorem for Functionals, with Applications in Continuum Mechanics. Arch. Rat.

9 Nigerian Journal of Science Vol 48 (04): Mech. Anal. 56, Das, S.S., Biswal, S.R., Tripath, U.K. and Das, P. (0). Mass transfer effects on unstead hdromagnetic convective flow past a vertical porous plate in a porous medium with heat source. J. Appl Fluid Mech 4: Das, U. N., Deka, R. K. and Soundalgekar, V. M. (994). Effects of mass transfer on flow past infinite vertical plate with constant heat flux and chemical reaction. Forsch. Ingenieurwes 60: Das, S.S., Mait, M. and Das, J.K. (0). Unstead hdromagnetic convective flow past an infinite vertical porous flat plate in a porous medium. Int'l J. Energ Environ : El-Arabaw, H. A. M. (009). Exact solution of mass transfer over stretching surface with chemical reaction and suction/inection. Journal of Mathematics and Statistics 5: Elbashbesh, E. M. A., Emam, T. G. and Abdel- Wahed, M. S. (0). Mass transfer over unstead stretching surface embedded in a porous medium in the presence of variable chemical reaction and suction/inection. Applied Mathematical Sciences 5: Fosdick, R. L. and Raagopal, K. R. (980), Thermodnamics and stabilit of fluids of third grade, Proc. R. Soc., London. Ganesan, P. and Palani, G. (00). Natural convection effects on an impulsivel started isothermal inclined plate. Acta Mechanica 5: 7. Haat, T., Awais, M. and Said, M. (00). Similar solutions of stretching flow with mass transfer. Int'l J. Num. Meth. Fluids 64: Haat, T., Mustapha, M. and Hendi, A. A. (0). Time-dependent three-dimensional flow and mass transfer of elasticoviscous fluid over unstead stretching sheet. Applied Mathematics and Mechanics : Haat, T., Qasim, M. and Abbas, Z. (00). Homotop solution for unstead three-dimensional MHD flow and mass transfer in a porous 55 space. Comm. Nonlinear Sci. Num. Simul. 5: Haat, T., Wang, Y., Siddiqui, A. M., Hutter, K. and Asghar, S. (00). Peristaltic transport of a third-order fluid in a circular clindrical tube. J. Math. Models Methods Appl. Sci., rd Heck, A. (00), Introduction to Maple, Edn., Springer-Verlag, German. Jain, M. K., Iengar, S.R.K. and Saldanha, J.S.V. (977). Numerical Solution of a Fourthorder Ordinar Differential equations. Journal of Engineering Math. : Khan, A., Khan, I., Ali, F., Ulhaq, S. and Shafie, S. (04). Effects of wall shear stress on unstead MHD conugate flow in a porous medium with ramped wall temperature. Plos One 9 (), e9080. Keimanesh, M., Rashidi, M., Chamkha, A.J. and Jafari, R. (0). Stud of a Third-Grade Non-Newtonian Fluid Flow between Two Parallel Plates Using the Multi-Step Differential Transform Method. Computers and Mathematics with Applications 6: Mahmoud, M. A. A. (00). Chemical reaction and variable viscosit effects on flow and mass transfer of a non-newtonian viscoelastic fluid past a stretching surface embedded in a porous medium. Meccanica 45: Muthucumarswam, R. and Ganesan, P. (00). Effect of the chemical reaction and inection on flow characteristics in an unstead upward motion of an isothermal plate. J. of Applied Mechanics and Technical Phsics 4: Muthucumarswam, R. and Ganesan, P. (00). Natural convection on a moving isothermal vertical plate with reaction. Journal of Engineering Phsics and Thermophsics 7: -9. Muthucumaraswam, R. and Meenakshisundaram, S. (006). Theoretical stud of chemical reaction effects on vertical oscillating plate with variable temperature. Theoret. Appl. Mech. : Naak, I., Naak, A.K. and Pard, S. (0). Numerical solution for the flow and heat

10 I.G. Baoku & B.I. Olauwon: Transient Flow & Mass Transfer of a Third Grade Fluid Past a Vertical Porous Plate in the Presence of CR transfer of a third grade fluid past a porous vertical plate. Advanced Studies in Theoretical Phsics 6: Okoa, S.S. (0). Disappearance of criticalit in thermal explosion for reactive third-grade fluid with Renold's model viscosit in a flat channel. International Journal of Non- Linear Mechanics 46: 0 5. Okoa, S.S. (008). On the transition of a generalized Couette flow of a reactive third-grade fluid with viscous dissipation. International Communication in Heat and Mass Transfer 5: Raptis, A., Tzivanidis, G. and Kafousias, N. (98). Free convection and mass transfer flow through a porous medium bounded b an infinite vertical limiting surface with constant suction. Letters in Heat and Mass Transfer 8: Rivlin, R. S. and Ericksen, J. L. (955). Stress Deformation Relation for Isotropic Materials. J. Rat. Mech. Anas 4: -45. Said, M. and Haat, T. (007). Non-similar solution for boundar laer flow of a third-order fluid over a stretching sheet. Appl. Math. Comput. 89: Stepanek, J. B. and Achwal, S. K. (009). Absorption with first-order chemical reaction into a liquid film in laminar flow. The Canadian J. of Chemical Engineering 54: Toki, C.J. and Tokis, J.N. (007). Exact solutions for the unstead free convection flows on a porous plate with time-dependent heating. Z. Angew. Math. Mech. 87: 4. Truesdell, C. and Noll, W. (004), The Nonlinear rd Field Theories of Mechanics. Edn. Springer, Berlin. I.G. Baoku and B.I. Olauwon Nigerian Journal of Science Vol. 48 (04): ISSN