Effect of chemical reaction on slip flow of MHD Casson fluid over a stretching sheet with heat and mass transfer

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1 Available online at.pelagiaresearchlibrary.com Advances in Applied Science Research, 5, 6(8):5-3 ISSN: CODEN (USA): AASRFC Effect of chemical reaction on slip flo of MHD Casson fluid over a stretching sheet ith heat and mass transfer P. Sathies Kumar and K. Gangadhar Acharya Nagarjuna University Ongole Campus, Ongole, Andhra Pradesh, India ABSTRACT The to-dimensional magnetohydrodynamic (MHD) stagnation-point flo of electrically conducting non-netonian Casson fluid and heat transfer toards a stretching sheet in the presence of mass transfer and chemical reaction is considered. Implementing similarity transformations, the governing momentum, and energy equations are transformed to self-similar nonlinear ODEs and numerical computations are performed to solve those. The investigation reveals many important aspects of flo and heat and mass transfer. If velocity ratio parameter (B) and momentum slip parameter (S), magnetic parameter (M) increase, then the velocity boundary layer thickness becomes thinner. On the other hand, for Casson fluid it is found that the velocity boundary layer thickness is larger compared to that of Netonian fluid. The magnitude of all skin-friction coefficient reduces ith Casson parameter () and also all skin-friction coefficient increases ith S for B< hereas decreases ith S for B>. The heat and mass transfer rate is enhanced ith increasing values of velocity ratio parameter. The rate of heat transfer is reduces ith increasing thermal slip parameter b. Moreover, the presence of chemical reaction reduces concentration. Keyords: MHD, Casson Fluid, Heat and mass Transfer, Stretching Sheet, Chemical Reaction, Momentum Slip, Thermal Slip. INTRODUCTION Due to the increasing importance of non-netonian fluids in industry, the stretching sheet concept has more recently been extended to fluids obeying non-netonian constitutive equations (Prasad et al. []). Khan [] and Sanjayanand and Khan [3] studied the viscous-elastic boundary layer flo and heat transfer due to an exponentially stretching sheet. Nadeem et al. [4] analysed the flo of Jeffrey fluid and heat transfer past an exponentially stretching sheet. Sahoo and Poncet [5] analysed the effects of slip on third grade fluid past an exponentially stretching sheet. Hayat et al. [6] investigated the Soret and Dufour effects on the magnetohydrodynamic (MHD) flo of the Casson fluid over a stretched surface using homotopy procedure. Bhattacharyya [7] studied the boundary layer stagnation-point flo of Casson fluid and heat transfer toards a shrinking/stretching sheet and concluded that the velocity and thermal boundary layer thicknesses are larger for Casson fluid than that of Netonian fluid. Ogunsola and Peter [8] studied the effect of thermal radiation on unsteady gravity flo of a poer-la fluid ith viscous dissipation through a porous medium. Hayat et al. [9] investigated the mixed convection stagnation-point flo of an incompressible non-netonian fluid over a stretching sheet under convective boundary conditions. Kirubhashankar et al. [] investigated the unsteady to-dimensional flo of a non-netonian fluid over a stretching porous surface having a prescribed surface temperature. Prasanna Kumar and Gangadhar [] investigated the momentum and thermal slip flo of MHD Casson fluid over a stretching sheet ith viscous dissipation. Processes involving the mass transfer effect have been recognized as important principally in chemical processing equipment (Cortell []). The driving force for mass transfer is the difference in concentration (Hayat et al. [3]). There are some fluids hich react chemically ith some other ingredients present in them (Raptis and Perdikis 5

2 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 [4]). The transport of mass and momentum ith chemical reactive species in the flo caused by a linear stretching sheet is discussed by Andersson et al. [5]. Akyildiz et al. [6] analyzed the transport and diffusion of chemically reactive species over a stretching sheet in a non-netonian fluid (fluid of differential type, second grade). Haritha and Sarojamma [7] investigated the effect of variable thermal conductivity and thermal radiation on heat and mass transfer in the MHD flo of a Casson fluid over a porous stretching sheet and concluded that the presence of Casson parameter decreases the all shear stress due to the fact that the presence of Casson parameter and increase in the value of β reduce velocities and hence the skin friction coefficient reduces and similarly the Nusselt number and Sherood number reduce. Sarojamma et al.[8] investigated the heat and mass transfer characteristics of a magneto hydrodynamic Casson fluid in a parallel plate channel ith stretching alls subject to a uniform transverse magnetic field and this study revealed that ith increase in the strength of the magnetic field, the fluid velocity decrease hoever an enhancement in temperature is noticed and also increase in the Casson parameter the idth of the central core region is observed to reduce. Chemical reaction can be codified as either heterogeneous or homogeneous processes. This depends on hether they occur at an interface or as a single phase volume reaction. The effect of the presence of foreign mass on the free convection flo past a semi-infinite vertical plate as studied by Gebhart et al. [9]. The presence of a foreign mass in air or ater causes some kind of chemical reaction. During a chemical reaction beteen to species, heat is also generated (Byron Bird R.et al. []). In most of cases of chemical reaction, the reaction rate depends on the concentration of the species itself. A reaction is said to be first order if the rate of reaction is directly proportional to concentration itself (Cussler []). Mukhopadhyay and Gorla [] obtained the numerical solutions for a class of nonlinear differential equations arising in diffusion of chemically reactive species of a non-netonian fluid toards an exponentially stretching surface and considered the first order constructive/destructive chemical reaction and concluded that the increasing values of the Casson parameter is to suppress the velocity field and the concentration is enhanced ith increasing Casson parameter. Hunegna and Kishan [3] investigated the unsteady magnetohydrodynamic heat and mass transfer flo over stretching sheet embedded in porous medium ith variable viscosity and thermal conductivity in presence of viscous dissipation and chemical reaction and concluded that the concentration decreases ith the increase of chemical reaction and Schmidt parameter but increases ith the increase of unsteadiness parameter. Prasanna kumar and Gangadhar [4] studied the magneto-convective non- Netonian nanofluid ith momentum and temperature dependent slip flo from a permeable stretching sheet ith porous medium and chemical reaction. Hoever, the interactions of magnetohydrodynamic (MHD) stagnation-point flo of electrically conducting non- Netonian Casson fluid and heat and mass transfer toards a stretching sheet in the presence of chemical reaction, momentum and thermal slip flo is considered. The governing boundary layer equations have been transformed to a to-point boundary value problem in similarity variables and the resultant problem is solved numerically using bvp4c MATLAB solver. The effects of various governing parameters on the fluid velocity, temperature, Concentration, Skin-friction, local Nusselt number and local Sherood number are shon in figures and analyzed in detail.. MATHEMATICAL FORMULATION Consider the steady to-dimensional incompressible flo of electrically conducting and chemically reactive Casson fluid bounded by a stretching sheet at =, ith the flo being confined in >. It is also assumed that the rheological equation of state for an isotropic and incompressible flo of a Casson fluid can be ritten as [5, 6] τ ij = p y µ B + eij, π > π c π p y µ B + eij, π < π c π c (.) here is plastic dynamic viscosity of the non-netonian fluid, is the yield stress of fluid, is the product of the component of deformation rate ith itself, namely, =, is the (, ) th component of the deformation rate, and is critical value of based on non-netonian model. The governing equations of motion, energy equation and spices equation may be ritten in usual notation as Continuity equation u v + = x y (.) 6

3 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 Momentum equation u u du s u σ H u + v = Us + υ + u U x y dx β y ρ Energy equation ρcp u v k x y y y Spices equation T T T qr + = u v D K C C x y y C C C + = ( ) ( ) s (.3) (.4) (.5) The boundary conditions are u T u = U + Nυ, v =, T = T + D, C = C at y = y y u Us, T T, C C as y (.6) here and v are the velocity components in and directions, respectively, = is the straining velocity of the stagnation-point flo ith (>) being the straining constant, = c is the stretching velocity of the sheet ith c (>) being the stretching constant, is the kinematic fluid viscosity, is the fluid density, = / p is β µ π B c y the non-netonian or Casson parameter, N is the velocity slip factor, D is the thermal slip factor, is the electrical conductivity of the fluid, T is the temperature, C is the concentration, k is the thermal conductivity, c is the specific heat, K is the chemical rate constant, qr is the radiative heat flux, T is the constant temperature at the sheet, is the constant concentration at the sheet, T is the free stream temperature assumed to be constant, C is the free stream concentration assumed to be constant, and is the strength of magnetic field applied in the direction, ith the induced magnetic field being neglected. Using the Rosseland approximation for radiation [7], 4 4 σ * T qr = is obtained, here σ * the Stefan-Boltzmann constant is and k is the absorption coefficient. 3k y We presume that the temperature variation ithin the flo is such that Expanding 4 T about T and neglecting higher-order terms e get p C 4 T may be expanded in a Taylor s series = 4 3 T T T T No (.4) reduces to σ ρcp u v k 3 T T T 6 * T T + = + x y y 3kρc p y We introduce also a stream functionsψ is defined by ψ u = y and v ψ = x Equations (.3), (.5) and (.7) becomes 3 ψ ψ ψ ψ du s ψ σ H ψ = U s + υ + U 3 s y x y x y dx β y ρ y ψ T ψ T T qr ρc p = k y x x y y y (.7) (.8) (.9) (.) 7

4 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 ψ y x x y y C ψ C C = D K ( C C ) (.) The boundary conditions are ψ ψ, ψ, T = U, + Nυ = T = T + D C = C at y = y y x y ψ U,, s T T C C as y (.) y No, the dimensionless variable for the stream function is implemented as T T C C ψ = cvxf ( ), θ =, φ = T T C C (.3) here the similarity variable is given by = y c / υ. Using relation (.3) and similarity variable, (.9) to (.) finally takes the folloing self-similar form: + f ''' + ff '' f ' M ( f ' B) + B = β (.4) 3R + 4 θ " + Pr f θ ' = 3R (.5) " f ' kr Sc φ φ φ (.6) The transformed boundary conditions can be ritten as f =, f ' = + Sf '', θ = + bθ '', φ = at = f ' B, θ, φ = as (.7) here primes denote differentiation ith respect to, M = σ H ρc is the magnetic parameter, b = D c / υ / p is the thermal slip parameter, B = a / c is the velocity ratio parameter, S = N c / is the momentum slip 3 parameter, Pr = µc / k is the Prandtl number R = k * k / 4 T is the thermal radiation parameter and p kr = K / c is the chemical reaction parameter,. The physical quantities of interest are the all skin friction coefficient x, the local Nusselt number Nu, and the local Sherood number Sh, hich are defined as τ xq xqm C fx =, Nu, x = Shx = ρu ( x) α T T D C C ( ) ( ) σ here τ is the shear stress or skin friction along the stretching sheet, the mass flux from the sheet and those are defined as υ υ q is the heat flux from the sheet and (.8) q m is 8

5 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 py u τ = µ B + π y q T = α y C = qm D y y= y= y= (.6) Thus, e get the all skin friction coefficient x, the local Nusselt number Nu and the local Sherood number Sh as follos: C fx Rex = + f ''( ) β Nux = θ '( ) (.7) Re Sh x Re x x ( ) = φ ' here Re x U x υ = is the local Reynolds number. 3 SOLUTION OF THE PROBLEM The set of equations (.4) to (.7) ere reduced to a system of first-order differential equations and solved using a MATLAB boundary value problem solver called bvp4c. This program solves boundary value problems for ordinary differential equations of the form y' f ( x, y, p), a x b general nonlinear, to-point boundary conditions ( ( ), ( ), ) =, by implementing a collocation method subject to g y a y b p. Here p is a vector of unknon parameters. Boundary value problems (BVPs) arise in most diverse forms. Just about any BVP can be formulated for solution ith bvp4c. The first step is to rite the ODEs as a system of first order ordinary differential equations. The details of the solution method are presented in Shampine and Kierzenka[8]. RESULTS AND DISCUSSION The abovementioned numerical scheme is carried out for various values of physical parameters, namely, the velocity ratio parameter (B), the magnetic parameter (), the Casson parameter (), momentum slip parameter (S), thermal slip parameter (b), the Prandtl number (Pr), Schmidt number (Sc), thermal radiation parameter (), and chemical reaction parameter (kr) to obtain the effects of those parameters on dimensionless velocity and temperature distributions. The obtained computational results are presented graphically in Figures -5 and the variations in velocity, temperature and concentration are discussed. Table Comparison for f ''() for M=S= B f ''() Present Study Prasanna Kumar and Gangadhar [4] Bhattacharyya [3] Nazar et al. [3] Mahapatra and Gupta [9] Firstly, a comparison of the obtained results ith previously published data is performed. The values of all skin friction coefficient f ''( ) for Netonian fluid case ( ) β = in the absence of momentum slip parameter and external magnetic field for different values of velocity ratio parameter (B) are compared ith those obtained by 9

6 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 Prasanna Kumar and Gangadhar [4], Mahapatra and Gupta [9], Nazar et al. [3] and Bhattacharya [3] in Table in order to verify the validity of the numerical scheme used and those are found in excellent agreement. M =.5, β =, S=R=b=kr=, Pr =.7, Sc =.4 B =.5 B =.5 f ' B =.5 B =.5 B = Fig. Velocity for different B.9.8 M =.5, β =, S=R=b=kr=, Pr =.7, Sc =.4 θ B =.,.5,,.5, 5 5 Fig. Temperature for different B The velocity, temperature and concentration profiles for various values of velocity ratio parameter B are plotted in Figs., and 3, respectively. Depending on the velocity ratio parameter, to different kinds of boundary layers are obtained as described by Mahapatra and Gupta [9] for Netonian fluid. In the first kind, the velocity of fluid inside the boundary layer decreases from the surface toards the edge of the layer (for B < ) and in the second kind the

7 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 fluid velocity increases from the surface toards the edge (for B > ). Those characters can be seen from velocity profiles in Fig.. Also, it is important to note that if B = ( = ), that is, the stretching velocity and the straining velocity are equal, then there is no boundary layer of Casson fluid flo near the sheet, hich is similar to that of Chiam s [3] observation for Netonian fluid and also similar to that of Bhattacharya [3]. From Figure, it is seen that in all cases thermal boundary layer is formed and the temperature at a point decreases ith B, hich is similar to that of Bhattacharya [3] observation for Casson fluid. From figure 3, concentration of the fluid decreases ith an increasing the velocity ratio parameter. M =.5, β =, S=R=b=kr=, Pr =.7, Sc = φ.4 B =.,.5,,.5, Fig. 3 Concentration for different B β =.5,,, 5 B = M =.5, S=R=b=kr= Pr =.7, Sc =.4.5 f '.5 β =.5,,, 5 B = Fig.4 Velocity for different B and β

8 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8): M=S=R=b=kr=, Pr =.7, Sc = θ.4.3. B =.3 B =.. β =.5,,, Fig.5 Temperature for different β and B S=M=R=b=, kr =., Pr =.7, Sc = B =. B =.3 φ.4 β = 5,,, Fig.6 Concentration for different β and B The effects of Casson parameter on the velocity, temperature and concentration fields are depicted in Figs. 4, 5 and 6. It is orthhile to note that the velocity increases ith the increase in values of for B = and it decreases ith for B =.. Consequently, the velocity boundary layer thickness reduces for both values of B. Due to the increase of Casson parameter, the yield stress falls and consequently velocity boundary layer thickness decreases, these results are similar to that of Bhattacharya [3]. The influences of Casson parameter on the

9 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 temperature profiles are different in to cases, B =.3 and B =.. Temperature and concentration at a point increases ith increasing for both B =. and B =.3. M =,, 4, 6 B = β =, S=R=b=kr= Pr =.7, Sc =.4.5 f '.5 M =,, 4, 6 B = Fig.7 Velocity for different B and M θ β =, M=S=b=kr=R=, Pr =.7, Sc =.4 M = 6, 4,, B =. B = Fig.8 Temperature for different B and M In Figs. 7, 8 and 9, the velocity, temperature and concentration profiles are presented for several values of magnetic parameter. Similar to that of Casson parameter, due to the increase of magnetic parameter the dimensionless velocity at fixed increases for B = and for B =. the velocity decreases. Consequently, for both types of boundary layers, the thickness decreases. The Lorentz force induced by the dual actions of electric and magnetic 3

10 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 fields reduces the velocity boundary layer thickness by opposing the transport phenomenon. Also, the temperature and concentration increases ith for both B =. and B =.3. β =, S=R=b=, kr =., Pr =.7, Sc = B =. B =.3 φ.4 M = 6, 4,,. 5 5 Fig.9 Concentration for different B and M S =,.5,, B = β =, M=R=b=kr= Pr =.7, Sc =.4.5 f '.5 S =,.5,, B = Fig. Velocity for different S and B The effects of momentum slip parameter S on the velocity, temperature and concentration is depicted in Figs., and, respectively. It is orthhile to note that the velocity increases ith the increase in values of S for B = and it decreases ith S for B =.. Also, the temperature and concentration decreases ith S for both B =. and B =.3. 4

11 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8): β =, M=S=R=kr=, Pr =.7, Sc = θ.5.4 S = 3,,,.3. B =.. B = Fig. Temperature for different S and B β =, M=R=b=, kr =., Pr =.7, Sc = B =. B =.3 φ.4 S = 3,,,. 5 5 Fig. Concentration for different S and B The dimensionless temperature profiles for several values of Prandtl number Pr, thermal radiation parameter and thermal slip parameter b are exhibited in Figs. 3, 4 and 5, respectively, for to values of B. In both cases (B =. and.3), the temperature decreases ith increasing values of Prandtl number, radiation parameter and thermal slip parameter and the thermal boundary layer thickness becomes smaller in all cases. The dimensionless concentration profiles for several values of Schmidt number Sc, chemical reaction parameter kr are exhibited in 5

12 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8): β =, M=S=b=kr=R=, Sc =.4 B =. B =.3 θ.5.4 Pr =.7, 5., Fig.3 Temperature for different B and Pr.8.7 β =, M=S=b=kr=, Pr =.7, Sc = θ.4 R = 3, 5, B =.3 B =. 5 5 Fig.4 Temperature for different B and R Figs. 6 and 7, respectively, for to values of C. In both cases (B =. and.3), the concentration decreases ith increasing values of Schmidt number and chemical reaction parameter and the solutal boundary layer thickness becomes smaller in all cases. 6

13 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3.9 β =, M=S=R=kr=, Pr =.7, Sc = θ.5.4 b =,,, 3.3. B =.. B = Fig.5 Temperature for different B and b.8.6 β =, M=S=kr=R=b=, Pr =.7 B =. B =.3 φ.4 Sc =.4,.6,.78,.6. Fig.6 Concentration for different B and Sc The physical quantities, the all skin friction coefficient C fx, local Nusselt number Nu x, and local Sherood number Sh hich have immense engineering applications, are proportional to the values of + f ''( ) x β, 7

14 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 θ '( ),and φ '() β respectively. The values of + f ''( ) and φ '() against the momentum, θ '( ) slip parameter S are plotted in Figs.8, 9 and, respectively for different values of B. From the figures, it is observed that the magnitude of all skin friction coefficient, local Nusselt number and local Sherood number decreases ith increasing values of velocity ratio parameter B and the magnitude of all skin friction coefficient, local Nusselt number and local Sherood number increases ith increasing values of momentum slip parameter S hen B <, opposite results ere found hen B >. β =, M=S=R=b=, Pr =.7, Sc =.4 φ kr =,.5,,.5 B =. B = Fig.7 Concentration for different B and kr β=, M=R=b=, Pr=.7, Sc=.4, kr=. (+/β)f ''() - B =,.5,.5, S Fig.8 Effect of S and B on ( + / β ) f ''( ) 8

15 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8): β=, M=R=b=, Pr=.7, Sc=.4, kr=. B =,.5,.5,.35 -θ'() S Fig.9 Effect of S and B on θ '().65.6 β=, M=R=b=, Pr=.7, Sc=.4, kr=. B =,.5,.5,.55 -φ'() S Fig. Effect of S and B on φ '() The effects of magnetic parameter and Casson parameter on local skin-friction, local Nusselt number and local Sherood number is shon in figs, and 3 respectively. It is noticed that the magnitude of the skin-friction coefficient decrease ith the influence of M and increase ith raising β. It is also observed that the local Nusselt 9

16 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8):5-3 number and local Sherood number reduces ith the influence of M or β. The local Nusselt number (Figure 4) increases ith R, hereas reduced ith the influence of thermal slip parameter. The effects of Schmidt number and chemical reaction parameter on local Sherood number is plotted in Fig. 5. It is noticed that the local Sherood number decreases ith in increases the Schmidt number or chemical reaction parameter (+/β)f ''() M =,, R=b=S=, B=., Pr=.7, Sc=.4, kr= β Fig. Effect of β and M on ( + / β ) f ''( )... M =,, 4 -θ'() R=b=S=, B=., Pr=.7, Sc=.4, kr= β Fig. Effect of M and β on ( + / β ) f ''( )

17 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8): R=b=S=, B=., Pr=.7, Sc=.4, kr=..39 M =,, 4 -φ'() β Fig.3 Effect of M and β on φ '( ) -θ'() R = 3, 5, 7 β=, M=S=, B=kr=., Pr= b Fig.4 Effect of b and R on θ '( )

18 P. Sathies Kumar and K. Gangadhar Adv. Appl. Sci. Res., 5, 6(8): β=, M=R=b=, Pr=.7, B=. -φ'() Sc =.4,.6,.78, kr Fig.5 Effect of Sc and kr on φ '( ) CONCLUSION The MHD stagnation-point flo of Casson fluid and heat and mass transfer over a stretching sheet ith radiation are investigated taking into consideration the chemical reaction, momentum and thermal slip effects. Using similarity transformations, the governing equations are transformed to self-similar ordinary differential equations hich are then solved using Bvp4c MATLAB solver. From the study, the folloing remarks can be summarized. (a) The velocity boundary layer thickness reduces ith velocity ratio parameter, magnetic parameter and momentum slip parameter. (b) The velocity boundary layer thickness for Casson fluid is larger than that of Netonian fluid. (c) Temperature and concentration of the fluid increases ith an increase Casson parameter. (d) The magnitude of all skin-friction coefficient increases ith Casson parameter, but local Nusselt number and local Sherood number decreases ith Casson parameter β. REFERENCES [] Prasad K. V., Abel S., and Datti P. S., (3), Int. J. of Non-Linear Mech., Vol.38, pp [] Khan S. K., (6), Int. J. of Appl. Mech. and Engng., Vol., Vol [3] Sanjayanand E., and Khan S. K., (6), Int. J. of Ther. Sci., Vol.45, pp [4] Nadeem S., Zaheer S., and Fang T., (), Numer. Algor., DOI.7/s [5] Sahoo B., and Poncet S., (), Int. J. of Heat and Mass Transfer, Vol54, pp [6] Hayat T., Shehzad S. A., and Alsaedi A., (), Appl. Math. Mech. -Engl. Ed., Vol.33(), pp.3 3. [7] Krishnendu Bhattacharyya, (3), Boundary layer stagnation-point flo of casson fluid and heat transfer toards a shrinking/stretching sheet, Frontiers in Heat and Mass Transfer (FHMT), Vol.4, 33. [8] Ogunsola A.W., Peter B.A., (4), IOSR Journal of Computer Engineering (IOSR-JCE) e-issn: 78-66, p- ISSN: , Vol. 6, Issue 4, pp [9] Hayat T., Shehzad S. A., Alsaedi A., Alhothuali M. S., (), Chin. Phys. Lett. Vol. 9, No., 474. [] Kirubhashankar C. K., Ganesh S., and Mohamed Ismail A., (5), Applied Mathematical Sciences, Vol. 9, No. 7, pp [] Prasanna Kumar, T., and Gangadhar, K., (5), International Journal Of Modern Engineering Research (IJMER), Vol. 5, Iss. 5, pp.4-37.

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