Effects of Chemical Reaction and Joule Heating on MHD Generalized Couette Flow between Two Parallel Vertical Porous Plates with Induced Magnetic Field and Newtonian Heating/Cooling

In this article, the efects of chemical reaction and Joule heating on MHD generalized Couette fow between two vertical porous plates with induced magnetic feld and Newtonian heating/cooling have been investigated. Te mathematical model used for the MHD generalized Couette fow takes into account the efect of viscous dissipation. Te system of nonlinear partial diferential equations governing the fow was solved numerically using the fnite diference method. Te resulting numerical schemes are simulated in MATLAB to obtain the profles of the fow variables such as velocity, induced magnetic feld, temperature, and concentration profles graphically. Also, the efects of the fow parameters on the skin-friction coefcient, Nusselt number, and Sherwood number are obtained and discussed numerically through tabular forms. Te fndings show that an increase in the chemical reaction parameter leads to a decrease in the concentration profles. Also, increase in the Joule heating parameter and heat generation parameter leads to an increase in the temperature profles. Induced magnetic feld profles increase with an increase in Reynold’s number. Te fndings of this study are important due to its application in developing a variety of chemical technologies, including polymer manufacturing, MHD pumps, food processing, chemical catalytic reactors, astronomy, MHD fow meters, and lubrication.


Introduction
Magnetohydrodynamic (MHD) generalized Couette fow continues to be of interest to researchers due to its various range of practical applications in lubrication, manufacturing processes, MHD power generator, polymer processing, astrophysical fuid dynamics, plasma aerodynamics, and geophysical fuid dynamics.Te study of magnetohydrodynamic generalized Couette fows of diferent non-Newtonian fuids has been attracted by various researchers in fuid dynamics due to their applications in engineering and industry.Te laminar fow of a viscous fuid between two parallel surfaces, one of which travels tangentially with respect to the other, is known as the Couette fow.Tis fow can be motivated by a pressure gradient that is present in the fow direction.Several scholars have investigated the magnetohydrodynamic generalized Couette fow between two parallel vertical plates.Khan et al. [1] analyzed the second law to determine how Newtonian heating afects the Couette fow of a viscoelastic dusty fuid and the transfer of heat in a rotating frame.Umavathi et al. [2] studied the Poiseuille-Couette fow with heat transfer in the inclined channel.Tey concluded that increases in the Grashof number, angle of inclination, and height ratio increase velocity, while increases in the Hartmann number, viscosity, and conductivity ratios decrease the velocity profles.Bég et al. [3] analyzed the oblique magnetic feld in a rotating highly permeable medium.Barikbin et al. [4] analyzed non-Newtonian fuid for MHD Couette fow using the Ritz-Galerkin method.Sreekala and Reddy [5] studied the efect of inclined magnetic feld with the steady MHD Couette fow.Joseph et al. [6] investigated heat transfer in inclined magnetic feld with unsteady MHD Couette fow between two infnitely parallel porous plates.Te temperature-dependent transient MHD Couette fow and heat transfer of dusty fuid were studied by Mosayebidorcheh et al. [7].Jha et al. [8] investigated thermal radiation with unsteady MHD free convective Couette fow.Ngiangia and Okechukwu [9] discussed the impact of variable electroconductivity and radiation.Tey indicated that increases in electroconductivity, Prandtl number, Reynolds number, and Grashof number result in an increase in velocity distribution, while increases in the magnetic feld result in a drop in velocity.Raju et al. [10] discussed the infuence of difusion thermo and thermal difusion on a natural convection Couette fow using the fnite element method.Ali et al. [11] investigated convective cooling of the nanofuids in a rotating system.Job and Gunakala [12] studied the thermal radiation for vertical permeable plates using Galerkin's fnite element method.Also, Ali et al. [13] analyzed the Couette fow of a maxwell fuid for three dimensional with periodic injection/suction.Hussain et al. [14] analyzed instability of the MHD Couette fow of an electrically conducting fuid.Anyanwu et al. [15] discussed the infuence of radioactive and a constant pressure gradient on an unsteady MHD Couette fow.
Chemical reaction can be categorized as homogeneous or heterogeneous reaction which can happen in a single phase or at the surface.Te efects of chemical reaction depend on the order of the reaction rate for both heterogeneous and homogeneous reaction.Te study of chemical reaction processes is benefcial in developing a variety of chemical technologies, including polymer manufacturing, MHD pumps, MHD fow meter, chemical catalytic reactors, and food processing.Te chemical reactions have numerous applications in many sectors of science, engineering, and technology.Te two-step exothermic chemical reaction of generalized Couette hydromagnetic fow with thermal criticality and entropy generation was presented by Kareem and Gbadeyan [16].Jha and Sarki [17] analyzed the efects of Dufour and chemical reaction on a moving vertical porous plate.Soumya et al. [18] studied the efects of chemical reaction, variable thermal conductivity, and nanoparticles injection on the MHD three-dimensional free convective Couette fow.Reddy et al. [19] discussed the efects of chemical reaction and thermal radiation on an unsteady MHD Couette fow through a porous medium with periodic wall temperature.Te study reveals that increasing efect of chemical reaction parameter leads to decrease in the concentration profles.Increasing value of magnetic parameter leads to decrease in the velocity profles.Ajibade and Umar [20] analyzed the efect of chemical reaction and radiation absorption on the unsteady MHD free convective Couette fow in a vertical channel.Apelblat [21] discussed analytically the efects of frst order chemical reaction with mass transfer.Prakash et al. [22] analyzed the efects of chemical reaction and thermo-difusion on the MHD free convective Couette fow.Abbaszadeh et al. [23] investigated the efects of two step exothermic reaction on generalized Couette hydromagnetic fow using the direct meshless local Petrov-Galerkin method.Chen and Arce [24] discussed the mathematical and numerical approach for convective-diffusive mass transfer with chemical reaction in the Couette fow.Kareem and Gbadeyan [16] studied entropy generation and thermal criticality of generalized Couette hydromagnetic fow of two-step exothermic chemical reaction in a channel.Te study concluded that low-dissipation rates can promote irreversibility processes and low-heat source terms can reduce thermal explosion.
An electric current fows through a resistance, it can be said to "Joule heating" the object being passed through.Joule heating is important due to its various applications including portable fan heater, bulb that emits light, clothes iron, and the hot fow produced by a hairdryer.Sarkar [25] investigated the MHD Couette-Poiseuille fow with laminar forced convection and viscous and Joule dissipations.Zueco et al. [26] studied the efects of Joule heating, Hall and ion slip in a non-Darcian porous media channel with a nonlinear transient hydromagnetic partially ionized the dissipative Couette fow.Ramesh [27] analyzed the efects of Joule heating and viscous dissipation on a Jefrey fuid's Couette and Poiseuille fows by taking into account the slip boundary conditions.Bég et al. [28] analyzed numerically the efects of Hall current, Joule heating, viscous fow, and Ion slip on magnetohydrodynamic Hartmann-Couette fow and heat transfer in a Darcian channel.Saleel et al. [29] discussed the microfuidics fow control using the electro-osmotic Couette fow and Joule heating.Shamshuddin et al. [30] investigated the efects of Joule heating and viscous dissipation in non-Fourier MHD squeezing fow, as well as heat and mass transfer between Riga plates with thermal radiation.Shamshuddin and Satya Narayana [31] studied the efect of viscous dissipation and Joule heating combined with a Cattaneo-Christov heat fux on MHD fow past a Riga plate.
Induced magnetic on the hydromagnetic generalized Couette fow is benefcial in MHD pumps and MHD fow meters.Te efects of induced magnetic feld have numerous applications in many sectors of science, engineering, and technology.Te magnetic feld is modifed by the induced magnetic feld, which also creates its own magnetic feld in the fuid.Te fow of the fuid through the magnetic feld also generates mechanical forces that change the fuid's motion.Consequently, it is necessary to include the infuence of the induced magnetic feld in the hydromagnetic equations in a number of physical conditions.In the metallurgical sector, magnetic felds are employed for stirring, heating liquid metals, and levitating.Sarveshanand and Singh [32] discussed the efects of induced magnetic feld on magnetohydrodynamic free convection between vertical parallel porous plates.Te study concluded that increasing suction parameter leads to decrease the induced current density and velocity feld and while it has increasing efect on the induced magnetic feld.Mng'ang'a et al. [33] investigated the efects of induced magnetic feld in the direction of the fuid fow on hydromagnetic surface driven fow.Kwanza and Balakiyema [34] analyzed the magnetic induction on MHD free convective fow past an infnite vertical porous plate.Te study reveals that the increase of the Prandtl number and the 2 International Journal of Mathematics and Mathematical Sciences suction velocity lead to a decrease in the temperature, induced magnetic feld and velocity profles in the boundary layer region whereas increase in the magnetic feld parameter and Grashof number leads to increased induced magnetic feld profles.Singh et al. [35] discuss the efects of induced magnetic feld on hydromagnetic free convection.Akbar et al. [36] investigated the efects of induced magnetic feld on interaction of nanoparticles for the peristaltic fow in an asymmetric channel.Sher Akbar et al. [37] discuss the efects of induced magnetic feld and heat fux in a permeable channel.Te novelty of this work is to investigate the combined efects of chemical reaction and Joule heating on the MHD generalized Couette fow between two parallel vertical porous plates with induced magnetic feld and Newtonian heating/cooling by taking into the account the efects of viscous dissipation.Te system of nonlinear partial diferential equation of the governing equations was obtained using the fnite diference method and implemented in MATLAB.Te efects of various dimensionless parameters on velocity profles, temperature profles, concentration profles, and induced magnetic feld profles are discussed in detail.Te results of this problem can be helpful in various science and engineering.Te results of this problem can be helpful in various devices subject to signifcant variations in the gravitational force, its application on heat exchanger designs, wire and glass fber drawings, and its application in nuclear engineering in connection with reactor cooling.In line with the aforementioned objectives and aim, this study proposes answers to the following research questions. (

Mathematical Formulation
Let us consider the bidimensional unsteady laminar fow of incompressible, electrically conducting, and viscous between two infnite parallel vertical porous plates with magnetic feld strength vector ( B → � (B x , B y )), constant suction velocity u � u o , and a constant magnetic feld B 0 applied perpendicular to the plates as depicted in Figure 1.Fluid fow between two parallel vertical porous plates with infnite lengths at x � 0 and x � h.
Te fuid is fowing through a porous medium that is assumed to obey Darcy's law.Electrically conducting fuid produce induced currents then produce an induced magnetic feld (B x �→ and B y �→ ) in x and y direction, respectively, that opposite the applied feld, resisting the conductor's motion.
At initial concentration C � C sp and temperature T � T sp at time t ≤ 0, the fuid is initially stationary, as are the plates at x � 0 and x � h.When t > 0, the porous plate at x � 0 begins to move impulsively in its own plane at a constant velocity of U, and its concentration and temperature rise to T � T mp and C � C mp , respectively.In contrast, the other porous plate, which is located at a distance of h from it, remains fxed and maintains its concentration and temperature at C � C sp and T � T sp .Te concentration, induced magnetic profle, temperature, and velocity are functions of x and t since the porous plates have an unlimited length.In the x− direction, the fuid is suctioned with constant velocity u 0 .
Te unsteady MHD Couette fow between two vertical porous plates is governed by the following nonlinear partial diferential equations.Using Boussinesq's approximation, the governing equations for the current physical system in dimensional form are as follow [8]:-Continuity equation Momentum equation ( Energy Equation International Journal of Mathematics and Mathematical Sciences Concentration equation Equation of magnetic induction Te corresponding initial and boundary conditions are as follows [8]:- To nondimensionalize the governing equations, we use the following dimensionless quantities for the present hydromagnetic problem Using the dimensionless variables defned in equations ( 2)-( 6) and ( 8) in nondimensional forms are as follows:- where J � B 2 0 σh 2 u 0 /]ρc p represents Joule heating parameter, M � B 2 0 σh 2 /]ρ represents magnetic parameter, S � u 0 /U represents injection/suction parameter, X � h 2 /k p represents the permeability parameter, Re � Uh/] represents the Reynolds number, Te corresponding initial and boundary conditions are as follows:

Numerical Solution
Te governing ( 9)-( 13) are nonlinear partial diferential equations, therefore cannot be solved analytically.Te numerical method for nonlinear partial diferential equations of momentum, concentration, energy, and induced given in ( 9)-( 13) are solved using the fnite diference method subject to the initial and boundary condition (14).Te condition for time stability is the Courant-Friedrichs-Lewy or CFL condition which depends on space and time discretization.A mesh is fxed at Δx � 0.3 and Δt � 0.0001 to ensure the stability and convergence.Te transport equations ( 9)-( 13) at the grid point (i, j) are expressed in diference form using Taylor's series expansion.Tus, the values of V, C, Hx, Hy, and T at grid point t � 0 are known; hence the velocity, induced magnetic feld, concentration, and temperature felds have been solved at time t i+1 � t i + Δt using the known values of the previous time t � t i for all i � 1, 2, . . .M − 1.

(17)
For practical engineering application and design, the quantities of interest are Sherwood number (Sh), skin friction coefcient (C f ), and Nusselt number (Nu) which are defned as where q m � − (zC/zx) x�0,1 , q h � − (zT/zx) x�0,1 , and Substituting equation ( 8) into equation ( 18), we obtain 6 International Journal of Mathematics and Mathematical Sciences Both temperature, velocity, concentration, and induced magnetic feld profles were obtained and utilized to compute numerical values of the Sherwood number, skin friction coefcient, and Nusselt number in equation (19).

Results and Discussion
To investigate the physical signifcance of the modeled MHD generalized Couette fow, the infuence of various physical parameters involved in the problem such as Joule heating parameter, heat generation parameter, Reynold's number, Prandtl number, Schmidt number, Grashof numbers for heat and mass transfer, Eckert number, magnetic Prandtl number, suction/injection parameter, magnetic parameter, chemical reaction parameter, and permeability parameter on the nondimensional temperature, velocity, concentration, and induced magnetic feld profles are evaluated numerically and executed graphically and discussed to get the physical insight into the problem, whilst the numerical values for skin friction coefcient, Nusselt, and Sherwood numbers are represented in the tabular form.Te numerical solutions have been conducted by adopting the default values of various involved physical parameters in the problem such as Re � 0.2, Pr � 0.71, and X � 2 until otherwise specifed particularly.
Figure 2 depicts the efects of Grashof number for mass transfer (Gc) on velocity profles.Graphically, it is observed that the velocity of the fuid increases with an increase in the Grashof number for mass transfer (Gc), since Grashof number for mass transfer represents the ratio of the species buoyancy force to the viscous force.Physically, increasing Grashof number leads to a decrease in the viscosity of the fuid which results to a decrease in the viscous force which leads to an increase in the species Bouyant force and thus increase the velocity of the fuid.
Figure 3 depicts the efects of Grashof number for heat transfer (Gr) on velocity profles.Graphically, it is observed that increasing the values of Grashof number for heat transfer (Gr) leads to an increase in the velocity profles.By defnition, Grashof number for heat transfer represents the ratio of the thermal buoyancy force to the viscous force.Physically, increasing the Grashof number reduces the viscous force and increase thermal buoyancy force thus leads to an increase in the velocity profles.
Te efect of magnetic parameter (M) on velocity profles is shown in Figure 4. Graphically, it is observed that increasing the values of the magnetic parameter leads to a decrease in the velocity profles, since magnetic parameter represents the ratio of electromagnetic force to an inertia force.Physically, increasing magnetic parameter leads to an increase in the electromagnetic force which caused by the magnetic feld that control the fuid fow characteristics.Te presence of a magnetic feld in an electrically conducting fuid enhance the Lorentz force which acts against to the fuid fow when the magnetic feld is applied normal to the direction of the fuid fow which leads to a slowly the motion fuid and thus decrease the velocity of the fuid.
Figure 5 illustrates the efects of permeability parameter (X) on velocity profles.Graphically, it is observed that increasing the values of the permeability parameter (X) leads to a decrease in the velocity profles.Physically, increasing the permeability parameter which opposes the fow and leads to enhance deceleration of the fow.Terefore,  International Journal of Mathematics and Mathematical Sciences with an increase in the permeability parameter causes the resistance to the fuid motion which resulted to decrease the velocity boundary layer and thus the velocity profles decrease.
Te efects of suction/injection parameter (S) on velocity profles are shown in Figure 6.Graphically, it is observed that increasing the values of the suction parameter (S < 0) leads to a decrease in the velocity of the fuid.Te injection parameter (S > 0) increases with a decrease in the velocity of the fuid.Physically, increasing suction parameter (S) implies the removal of the fuid from the fow system which causes destabilization the velocity boundary layer leads to a decrease in the motion of the fuid and thus decrease the velocity profles.
Figure 7 exhibits the efects of the Reynolds number (Re) on velocity profles.Graphically, it is observed that increasing the values of the Reynolds number (Re) leads to an increase in the velocity profles, since the Reynolds number represents the ratio of inertia force to the viscous force.Physically, increasing the Reynolds number reduces the viscous forces in the fuid and thus result to an increase in the velocity profles.
Figure 8 presents the efects of chemical reaction parameter (K) on concentration profles.Graphically, it is observed that increasing the values of the chemical reaction parameter (K) leads to a decrease in the fuid concentration.Te positive values of K indicate the destructive type of the chemical reaction.Te infuence of a destructive chemical reaction parameter (K > 0) caused a decrease in the concentration difusion species.Physically, increasing chemical reaction for destructive results to molecular motion become higher, which enhances the transport phenomenon, thus decreases the concentration profles.
Figure 9 illustrates the efects of suction/injection (S) on concentration profles.Graphically, it is observed that increasing the values of the suction (S) leads to a decrease in the concentration profles.Physically, increasing suction/ injection parameter (S) results in the thinning of the concentration boundary layer of the fuid fow which leads to a decrease in the concentration profles.
Te efects of Schmidt number (Sc) on concentration Profles are shown in Figure 10.Graphically, it is observed that the concentration profles decrease as the Schmidt number (Sc) increases.By the defnition, the Schmidt number represents the ratio of kinematic air viscosity to the mass difusivity.Physically, increasing Schmidt number   reduces the mass difusivity, which results to a decrease in the concentration profles.Te efects of magnetic Prandtl number (Prm) on induced magnetic feld profles are shown in Figures 11 and 12. Graphically, it is observed that increasing the values of the magnetic Prandtl number (Prm) leads to a decrease in the induced magnetic feld profles.Physically, increasing the magnetic Prandtl number reduces the magnetic difusivity, which leads to a decrease in the induced magnetic feld by the motion of a conducting medium and thus the induced magnetic feld profles decrease.
Te efects of suction/injection parameter (S) on induced magnetic profles are shown in Figures 13 and 14.Graphically, it is observed that increasing the values of the suction parameter (S) leads to a decrease in the induced magnetic feld profles.Increasing suction/injection parameter (S) leads to a decrease in the fuid the fow system and thus decreases the velocity of the fuid which reduces the interaction between the fuid and magnetic feld and thus decreases the induced magnetic feld profles.
Te efects of the Reynolds number (Re) on an induced magnetic feld profles along x− direction are shown in Figure 15.Graphically, it is observed that increasing the values of the Reynolds number (Re) leads to an increase in the induced magnetic feld.By the defnition, the Reynolds number represents the ratio of inertia force to the viscous International Journal of Mathematics and Mathematical Sciences forces.Physically, increasing the Reynolds number reduces the viscous force which leads to an increase of interaction between the fuid and magnetic feld and thus increases induced magnetic feld profles.
Te efects of the Reynolds number (Re) on induced magnetic feld profles along y− direction are shown in Figure 16.Graphically, it is observed that increase in the Reynolds number (Re) leads to a decrease in the induced magnetic feld.Te Reynolds number represents the ratio of inertia force to the viscous forces.Physically, increasing the Reynolds number leads to a decrease of interaction between the fuid and magnetic feld and thus decreases induced magnetic feld profles.
Figure 17 demonstrates the efects of suction/injection parameter (S) on temperature profles.Graphically, it is observed that increase in the suction/injection parameter (S) leads to an decrease in the fuid temperature.Physically, increasing suction/injection parameter (S) results in the thinning of the thermal boundary layer of the fuid fow which leads to decrease the temperature profles.
Te efects of the Eckert number (Ec) on temperature profles are shown in Figure 18.Graphically, it is observed increasing the values of the Eckert number (Ec) leads to an increase in the temperature profle.By the defnition, the Eckert number represents the ratio of the fuid fow kinetic energy to the enthalpy of the fuid.Physically, increasing the Eckert number reduces the enthalpy of the fuid and thus increases the temperature of the fuid.Te motion of the fuid increases as an increase of the Eckert number which transforms into kinetic energy which lead to an increase in the temperature profles.Te efects of the Prandtl number (Pr) on temperature profles are shown in Figure 19.Graphically, it is observed that the temperature of the fuid decreases as the values of the Prandtl number (Pr) increases.By the defnition, the Prandtl number represents the ratio of viscous force/momentum difusivity to the thermal difusivity.Physically, increasing the Prandtl number reduces the thermal difusivity and increase viscosity of the fuid which leads to an increase of temperature of the fuid.
Figure 20 illustrates the efects of varying heat generation parameter Q on temperature profles.Graphically, it is observed that temperature profle is greatly infuenced by heat generation parameter Q. Physically, increasing heat generation parameter Q leads to an increase in the temperature of the fuid.By increasing heat generation parameter Q leads to an increase in kinetic energy which leads to an increase in the thermal boundary layer and thus results to an increase in the temperature profles.
Te impacts of varying Joule heating parameter (J) on temperature profles are shown Figure 21.Graphically, it is noticed that the temperature profles increase with an increase in the Joule heating parameter (J).Physically, increasing the Joule heating parameter leads to an increase in the friction, and thus the mechanical energy converted into thermal energy which results to an increase in the temperature of the fuid and thus temperature profles increase.
Table 1 shows the efects of the Prandtl number, Eckert number, heat generation parameter, and Joule heating parameter on the rate of heat transfer (Nusselt number) at the stationary and moving plates.It is observed from the table that the Nusselt number θ′(1) at the stationary plate decreases with an increase in the Prandtl number whereas increase with an increase in Joule heating parameter, heat generation parameter, and Eckert number.Te Nusselt number θ′(0) at the moving plate increases with an increase in Joule heating parameter, Eckert number, and heat generation parameter whereas decrease with an increase in the Prandtl number.
Table 2 shows the efects of magnetic parameter, Prandtl number, Reynolds number, heat generation parameter, and Joule heating parameter on the wall shear stress (skin friction coefcient) at the moving and stationary plates.It is observed from the table that the wall shear stress V * ′ (1) at the stationary plate increases with an increase in the Prandtl number, magnetic parameter, and Reynolds number whereas decrease with an increase in heat generation parameter and Joule heating parameter.Te skin friction coefcient V * ′ (0) at the moving plate increases with an increase in the Prandtl number, magnetic parameter, and Reynolds number whereas decrease with an increase in heat generation parameter and Joule heating parameter.International Journal of Mathematics and Mathematical Sciences moving and stationary plates decrease with an increase in the chemical reaction parameter and Schmidt number.

Concluding Remarks
We examine the efects of chemical reaction and Joule heating on the MHD generalized Couette fow between two vertical porous plates with an induced magnetic feld and Newtonian heating/cooling.Te system of the governing equations of partial diferential equations of the model is solved numerically using the fnite diference method.Te numerical results for the temperature, velocity, concentration, and induced magnetic feld profles are presented graphically for the pertinent parameters.Te numerical values for skin friction coefcient, Nusselt, and Sherwood numbers are represented in the tabular form.Te major fndings of the present analysis can be summarized in the following points: (i) Te velocity profles increase with rising the values of Gr, Re, and Gc while increasing M and S have an opposite efect on the velocity profles (ii) Te temperature of the fuid increase with an increasing the values of Q, J, and Ec while increasing Pr and S leads to a decrease in the temperature fuid (iii) Te rates of heat transfer at the moving and stationary plates increases with an increase in the J, Q, and Ec while decrease with an increase in Pr (iv) Te skin friction coefcient (wall shear stress) at the stationary and moving plates increases with an increase in M, Pr, and Re while decrease with an increase in the J and Q (v) Te rate of mass transfer at the moving plate ϕ′(0) decreases with an increase in the K and Sc.However, at the stationary plate ϕ′(1) decreases with an increase in Sc and K (vi) An increasing K, S, and Sc decrease the concentration profles (vii) An induced magnetic feld profles decrease with rising the values of Prm, Re, and S Future research can be conducted on the MHD generalized Couette fow between two parallel vertical porous plates in presence of an inclined variable magnetic feld.Nondimensional temperature ϕ: Nondimensional concentration ]: Kinematic viscosity (m 2 s − 1 ) μ e : Magnetic permeability (Hm − 1 ) Q: Internal heating generation parameter/Newtonian heating/cooling parameter K: Chemical reaction parameter k r : Reaction coefcient J: Joule heating parameter.

Figure 1 :
Figure 1: Schematic diagram of the physical system.

Figure 4 :
Figure 4: Velocity profles for diferent values of M.

Figure 8 :
Figure 8: Concentration profles for diferent values of K.

Table 3
shows the efects of Schmidt number and chemical reaction parameter on the Sherwood number at the moving and stationary plates.It is observed from the table that the rates of mass transfer (Sherwood number) at the

Table 1 :
Results of the Nusselt number for various values of physical parameters.

Table 2 :
Results of skin-friction coefcient for various values of physical parameters.
Bold values represent the variation of physical parameters.

Table 3 :
Results of the Sherwood number for various values of physical parameters.Bold values represent the variation of physical parameters.12 International Journal of Mathematics and Mathematical Sciences T mp : Temperature of the fuid in the moving plate (k) T sp : Temperature of the fuid in the stationary plate (k) C mp : Concentration of the fuid in the moving plate (mole/kg) C sp : Concentration of the fuid in the stationary plate