IOSR Journal o Mathematics (IOSR-JM) e-issn:78-578, p-issn: 39-765X.Volume,Issue Ver. III (Jan.-Feb.6)PP 88- www.iosrjournals.org Eect o Chemical Reaction on MHD Boundary Layer Flow o Williamson Nanoluid Embedded In a Porous Medium in the Consideration o Viscous Dissipation and Newtonian Heating M.Lavanya, Dr.M.Sreedhar Babu, G.Venkata Ramanaiah 3,3 Research scholar, Dept. o Applied Mathematics, Y.V.University, Kadapa Andhra Pradesh, India. Dept. o Applied Mathematics, Y.V.University, Kadapa Andhra Pradesh, India. Abstract: The consideration o nanoluids has been paid a good attention on the ree convection; the analysis ocusing nanoluids in porous media are limited in literature. Thus, the use o nanoluids in porous media would be very much helpul in heat and mass transer enhancement. In this paper, the inluence o magnetohydrodynamic, thermal radiation, Newtonian heating and chemical reaction on heat and mass transer over a horizontal linearly stretching sheet embedded in a porous medium illed with a nanoluid is discussed in detail. The solutions o the nonlinear equations governing the velocɨty, temperature and concentration proiles are solved numerically using Runge-Kutta Gill procedure together with shooting method and graphical results or the resulting parameters are displayed and discussed. The inluence o the physical parameters on skinriction coeicient, local Nusselt number and local Sherwood number are shown in a tabulated orm. Keywords: MHD, nanoparticle, Newtonian heating, porous medium, viscous dissipation, Williamson luid model. I. Introduction Nanoluids are the luid suspensions o nanoparticles showing many interesting properties and distinctive eatures. Al, Cu, Fe and Titanium or their oxides are the most requently used nanoparticles. The materials with the sizes o nanometers possess unique physical and chemical properties. This led to the study o nanoluids in a variety o processes which resulted in notable applications in engineering and sciences. On account o its practical applications in industries, a great deal o attention is being paid to the research works on convective heat transer in nanoluids in porous medium. [3] introduced the term naanoluids to demote to the luid with incomplete nanoparticles. [] conirmed that the adding o a small quantity o nanoparticles to conventional heat transer liquids increases the thermal conductivity o the luid up to in the region o two times.[] discussed convective transport in nanoluids.he reported that only Brownin diusion and thermophoresis are essential slip Mechanisms in nanoluids. In many realistic cases, the heat transer rom the surace is relative to the local surace temperature. Such an eect is known as Newtonian heating eect. [9] investigated conjugate convective low with Newtonian heating. Due to various applications researchers are attracted to consider the Newtonian heating condition within their problems. [6] presented an exact examination o mass and heat transer past a vertical plate with Newtonian heating. Recently, the unsteady boundary layer low and heat transer o a Casson luid under the Newtonian heating boundary condition or non-newtonian luid was studied by[7]. [] concluded that the thermal radiation and Newtonian heating, both have raising eects on the luid temperature and concentration decreases with increasing chemical reaction and Schmidt number. However, consideration o nanoluids has been paid a good attention on the ree convection; the analysis ocusing nanoluids in porous media are limited in literature. Thus, the use o nanoluids in porous media would be very much helpul in heat and mass transer enhancement. In this paper, the inluence o magnetohydrodynamic, thermal radiation, Newtonian heating and chemical reaction on heat and mass transer over a horizontal linearly stretching sheet embedded in a porous medium illed with a nanoluid is discussed in detail. The solutions o the nonlinear equations governing the velocɨty, temperature and concentration proiles are solved numerically using Runge-Kutta Gill procedure together with shooting method and graphical results or the resulting parameters are displayed and discussed. The inluence o the physical parameters on skinriction coeicient, local Nusselt number and local Sherwood number are shown in a tabulated orm. II. Mathematical Formulation Consider a two-dimensional steady viscous low o an electrically conducting viscous incompressible Williamson nanoluid through a stretching surace embedded with porous medium. The plate is stretched along x-axis with a velocity ax, where a> is stretching coeicient. The luid velocity, temperature and nanoparticle concentration near surace are taken to be U w, T w and C w, respectively, which are as illustrated in igure A. DOI:.979/578-388 www.iosrjournals.org 88 Page
Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded Fig. A: Schematic diagram o the boundary layer low over Stretching sheet With the usual Boussinesq and the boundary layer approximations, the governing equations o continuity, momentum, energy and volumetric species are written as ollows u + v = (.) x y u u u + v = υ u u x y y + υ y u T T + v = α T x y + τ D y B u σb y C y T + D T y T T T u υ u (.) ρ k T q y r + υ ( u ρc y ρc y ) (.3) u C C + v = D C x y B + D T k y y C (.) The appropriate boundary conditions or the low model are written as ollows u = U w, v =, T y = h st, C y = h cc at y = u, T T, C C as y (.5) Since the surace be stretched with velocity Bx, thus U w = Bx and u and v are horizontal and vertical components o velocity, be kinematic viscosity, q, q are the heat and mass luxes unit area at the surace, respectively. w m B be magnetic ield, be nanoluid thermal diusivity, h s be heat transer coeicient, h c be concentration coeicient, be nanoluid density, q be heat source or sink constant, c be heat capacities o the luid, T be temperature, k be nanoluid thermal conductivity, D B be Brownian diusion coeicient, C be nanoparticle volumetric raction, D T be thermophoretic diusion coeicient and T be the ambient luid temperature. The radiative heat lux term q r is simpliied by the Rosseland approximation and is expressed as q r = σ T (.6) 3k y Since the Resseland approximation is valid only, optically thick luids can be considered or the present investigation. It is assumed that the temperature dierences within the low are signiicantly small, then (.6) can be linearized by expanding T into the Taylor series about T and neglecting higher order terms as T T 3 T 3T (.7) Then, the radiation term in Equation (.3) becomes as below q r y 3 T = 6σ T 3k y (.8) Substituting o this equation (.8) into the Equation (.3), we have u T T + v T x y C T 6σ T 3 y B y y + T ρc 3k y υ ( u ρc y ) (.9) T y + D T T The equation o continuity (.) is satisied or the choice o a stream unction ψ (x, y) such that u = ψ y, v = ψ x (.) In order to transorm the equations (.), (.) and (.9) into a set o ordinary dierential equations, the ollowing similarity transormations and dimensionless variables are introduced. ψ = aυ x η, η = y a υ, θ η = T T T w T, β η = C C C w C DOI:.979/578-388 www.iosrjournals.org 89 Page
Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded Pr = υ, M = σb, K = υ, Le = υ, Nb = τd B C w C α ρa k a D B Nt = τd T T w T, γ = h υ T υ s, β a = h υ c, kr = υk a a Ec = U w c T w T, Rd = σ T k c 3 υ (.) where ( ) be dimensionless stream unction, θ be dimensionless temperature, be dimensionless nanoparticle volume raction, η be similarity variable, M be magnetic parameter, be non-newtonian williamson parameter, Le be Lewis number, Nb be Brownian motion parameter, Nt be thermophoresis parameter, Ec be Eckert number, Pr be Prandtl number, be conjugate parameter or Newtonian heating, be conjugate parameter or concentration. Ater the substitution o these transormations (.) (.) into the equations (.), (.9) and (.), the resulting non-linear ordinary dierential equations are written as η + η η η + λ η η M + K η = (.) Pr + 3 Rd θ η + η θ η + Nbθ η β η + Ntθ η + Ec η = (.3) β η + Le η β η + Nt Nb θ η krβ η = (.) together with the boundary conditions η =, η =, θ η = γ + θ η, β η = β + β η at η = η, θ η, β η as η (.5) Where the prime denote dierentiation with respect to η. The Physical quantities o interest like skin riction coeicientc, local Nusselt number Nu x and local Sherwood number Sh x are deined as C = τ w ρu, Nu x = xq w w k T T x = x q m w D B C C w (.6) Where the shear stress τ w, surace heat lux q w and surace mass lux q m are given by τ w = μ u y + Γ u, q y w = k T, q y m = D C B (.7) y= y= y y= Using the non-dimensional variables, we obtain C Re x / = + λ, Nu x Re x / = θ, Sh x Re x / = β () (.8) Where Re x = xu w (x) υ is the local Reynolds number. III. Solution O The Problem For solving equations (.) (.), a step by step integration method i.e. Runge Kutta method has been applied. For carrying in the numerical integration, the equations are reduced to a set o irst order dierential equation. For perorming this we make the ollowing substitutions: y =, y =, y 3 =, y = θ, y 5 = θ, y 6 = β, y 7 = β y 3 = y y y 3 + M + K y y 5 = +λy 3 Pr + 3 Rd y y 5 + Nby 5 y 7 + Nty 5 + Ecy 3 (3.) y 7 = Ley y 7 Nt Nb Pr + 3 Rd y y 5 + Nby 5 y 7 + Nty 5 + Ecy 3 + kry 6 together with the boundary conditions are given by (.5) taking the orm y =, y =, y 5 = γ + y, y 7 = β + y 6 y, y, y 6 (3.) In order to carry out the step by step integration o equations (.9) (.), Gills procedures as given in Ralston and Wil (96) have been used. To start the integration it is necessary to provide all the values o y, y, y, y, y, y at rom which point, the orward integration has been carried out but rom the 3 5 6 boundary conditions it is seen that the values o y, y, y are not known. So we are to provide such values o 3 7 y, y, y along with the known values o the other unction at as would satisy the boundary conditions 3 7 as to a prescribed accuracy ater step by step integrations are perormed. Since the values o 3 7,, y y y which are supplied are merely rough values, some corrections have to be made in these values in order that the DOI:.979/578-388 www.iosrjournals.org 9 Page
Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded boundary conditions to are satisied. These corrections in the values o y, y, y are taken care o by 3 7 a sel-iterative procedure which can or convenience be called Corrective procedure. IV. Results And Discussion In order to acquire physical understanding, the velocɨty, temperature & concentration distributions have been illustrated by varying the numerical values o the various parameters demonstrated in the present problem. The numerical results are tabulated and exhibited with the graphical illustration. Figures (a), (b) & (c) respectively, demonstrate the dierent values o magnetic parameter (M) on dimensionless velocɨty, temperature and concentration proiles. The values o M is taken to be M =,, 3, 6 and the other parameters are kept constant at K =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. It is noticed that, with the hype in the values o M rom to 6 then the velocity signiicantly decreases consequently decreases the thickness o momentum boundary layer but the temperature and concentration o the luid increases. This is due to the act that, the introduction o transverse magnetic ield has a tendency to create a drag, known as the Lorentz orce which tends to resist the low. Figures (a), (b) & (c) establish the dierent values o permeability o the porous medium parameter (K) on velocɨty, temperature and concentration proiles, respectively. The values o K is taken to be K =,,, and the other parameters are kept constant at M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. It is noticed that, with the hype in the values o K rom to then the velocity decreases consequently decreases the thickness o momentum boundary layer but the temperature and concentration o the luid increases. The reason or this, the porous medium obstructs the luid to move reely through the boundary layer. This leads to the raise in the thickness o thermal and concentration boundary layer. Figures 3(a), 3(b) & 3(c) illustrate the eect o non-newtonian williamson parameter (λ) on velocɨty, temperature and concentration proiles, respectively. The values o λ is taken to be λ =,,,.3 and by setting the values o other parameters to be constant at or K =, M =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. From these igures it is observed that as λ raises rom to, the velocity remarkably decreases. While the non-newtonian Williamson parameter increases, the temperature proile is increased. This act is illustrated in igure 3(b). In the concentration proile, increase the value o λ result in a signiicant increase in the boundary layer thickness, which is shown in igure 3(c). The phenomenon in which the particles can diuse under the eect o a temperature gradient is called thermophoresis. The inluence o thermophoresis parameter (Nt) on the temperature and concentration ields is shown in igures (a) and (b) respectively. The values o Nt is taken to be Nt =,,.3,. and the other parameters are ixed as K =, M =, λ =, Pr =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. It is conormed that enhances the values o Nt rom to., the temperature & concentration o the luid increases. This leads to enhance the thickness o thermal and concèntration boundary layer. The random motion o nanoparticles within the base luid is called Brownian motion which occurs due to the continuous collisions between the nanoparticles and the molecules o the base luid. The eect o Brownian motion parameter (Nb) on the temperature and concentration ields is illustrated in igures 5(a) & 5(b) respectively. The values o Nb is taken to be Nb =,,.3,. and the other parameters are ixed as K =, M =, λ =, Pr =, Nt =, Le =, Ec =, kr =, Rd =, γ = & β =. It is conormed that enhances the values o Nb rom to., the temperature proile signiicantly increases but concentration decreases. In the system nanoluid, the Brownian motion takes a place in the presence o nànoparticles. When hype the values o Nb, the Brownian motion is aected and the concèntration boundary layer thickness reduces and accordingly the heat transer characteristics o the luid changes. The eect o Eckert number (Ec) on temperature and concentration ields is depicted in igures 6(a) and 6(b), respectively. The values o Ec is taken to be Ec =,,,.5 and the other parameters are ixed as K =, M =, λ =, Pr =, Nt =, Nb =, Le =, kr =, Rd =, γ = & β =. It is observed that the thickness o thermal and concentration boundary layer signiicantly increases with raising the values o Ec rom to.5. For the constant parameters K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, γ = & β =, Figures 7(a) & 7(b) illustrate the eect o radiation parameter (Rd) in the presence o viscous dissipation on the dimensionless temperature and nanoparticle volume raction proiles. From igures 7(a) & 7(b), it can be understood that thermal and concentration boundary layer thickness increases with an increase in thermal radiation. Thereore higher values o radiation parameter imply higher surace heat lux which will increase the temperature within the boundary layer region.figures 8(a) and 8(b) depicts the various values o conjugate parameter or Newtonian heating (γ) on the dimensionless temperature and concentration proiles, respectively. The values o γ is taken to be γ =,,,.3 and the other parameters are ixed as K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd = & β =. It can be understood that raising the values o γ rom to. the resultant temperature and concentration increases consequently thickness o thermal and concèntration boundary layer enhances. DOI:.979/578-388 www.iosrjournals.org 9 Page
Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded Figures 9(a) and 9(b) depicts the various values o conjugate parameter or concentration (β ) on the temperature and concentration ields, respectively. The values o β is taken to be β =,.3,,.8 and the other parameters are ixed as K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd = & γ =. It is obtained that raising the values o β rom to., the resulting dimensionless temperature and concentration o the luid increases. This leads to enhance the thermal and concentration boundary layer thickness. The eect o Prandtl number Pr on dimensionless temperature ields is shown in igures respectively. The values o Pr is taken to be Pr =, 6, 8, and the other parameters are ixed as K =, M =, λ =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. Physically, increasing Prandtl number becomes a key actor to reduce the thickness o thermal boundary layer. The eect o Lewis number Le verses concentration proile is shown in igure respectively. The values o Le is taken to be Le =, 6, 8, and the other parameters are ixed as K =, M =, λ =, Pr =, Nt =, Nb =, Ec =, kr =, Rd =, γ = & β =. It can be obtained that the concentration o the luid decreases with raising the values o Le rom to. The eect o chemical reaction parameter (kr) verses concentration proile is shown in igure respectively. The values o kr is taken to be kr =,,, and the other parameters are ixed as K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, Rd =, γ = & β =. It can be obtained that the concentration o the luid decreases with raising the values o kr rom to. In order to standardize the method used in the present study and to decide the accuracy o the present analysis and to compare with the results available (Nadeem and Hussain (3), Khan and Pop (), Gorla and Sidawi (99) and Wang (989)) relating to the local Nusselt number, and ound in an agreement (Table.). Table Shows that magnitude o skin-riction coeicient ''( ) ''( ), local Nusselt number '( ) and local Sherwood number '( ) on M, K, λ, Pr, Nt, Nb, K, Ec, kr, Rd, γ and β, respectively. As M increases rom to then then ''( ) ''( ), '( ) and '( ) increases. As K increases rom to 3 ''( ) ''( ), '( ) and '( ) increases. As λ increases rom to. then ''( ) ''( ), '( ) and '( ) increases. As Pr increases rom to 5 then '( ) and '( ) decreases. As Nt increases rom to then increases. As Nb increases rom to then ''( ) ''( ) increases '( ) decreases and '( ) ''( ) ''( ), '( ) and '( ) increases. As Le increases rom to then '( ) and '( ) decreases. As Nt increases rom to then ''( ) ''( ) increases '( ) decreases and '( ) increases. As Ec increases rom to.5 then '( ) increases and '( ) decreases. As kr increases rom to then '( ) and '( ) Rd increases rom to then then decreases. As ''( ) ''( ), '( ) and '( ) increases. As γ increases rom to ''( ) ''( ), '( ) and '( ) increases. As β increases rom to.8 then '( ) and '( ) increases. V. Conclusions In this paper numerically investigated radiation and chemical reaction eects on the steady boundary layer lowo MHD Williamson luid through porous medium toward a horizontal linearly stretching sheet in the presence o nanoparticles in the presence o viscous dissipation and Newtonian heating. The important indings o the paper are: DOI:.979/578-388 www.iosrjournals.org 9 Page
() ' () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded The velocity o the luid decreases with an increase o the Non-Newtonian Williamson parameter or magnetic parameter or permeability parameter. The luid temperature and concentration increases with the inluence o non-newtonian Williamson parameter or magnetic parameter or permeability parameter. The temperature and concentration enhance with the raising the values o Eckert number or radiation parameter or conjugate parameter or Newtonian heating or conjugate parameter or concentration. Both the local Nusselt number and local Sherwood number decreases with the raising the values o non- Newtonian Williamson parameter or magnetic parameter..8.6. M =,,, 6 3 5 6 Fig.(a) Velocity or a range o values o M with K =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =..5.5 M = 6,,, 6 8 Fig.(b) Temperature or a range o values o M with K =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. DOI:.979/578-388 www.iosrjournals.org 93 Page
() ' () () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded.3 5 5 M = 6,,,.5 6 8 Fig.(c) Nanoparticle volume raction or a range o values o M with K =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =..8.6. K =,,, 3 5 6 Fig. (a) Velocity or a range o values o K with M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =..8.6. K =,,, 6 8 Fig.(b) Temperature or a range o values o K with M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. DOI:.979/578-388 www.iosrjournals.org 9 Page
() ' () () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded 5 5 K =,,,.5 6 8 Fig.(c) Nanoparticle volume raction or a range o values o K with M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =..8.6. =,,,.3 3 5 6 Fig.3(a) Velocity or a range o values o λ with K =, M =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =..6..3 =,,,.3 3 5 6 Fig.3(b) Temperature or a range o values o λ with K =, M =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. DOI:.979/578-388 www.iosrjournals.org 95 Page
() () () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded 5 5.5 =,,,.3 3 5 6 7 8 Fig.3(c) Nanoparticle volume raction or a range o values o λ with K =, M =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =..8.6. Nt =.,.3,, 3 5 6 7 8 Fig.(a) Temperature or a range o values o Nt with K =, M =, λ =, Pr =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =...3 Nt =.,.3,, 3 5 6 7 8 Fig.(b) Nanoparticle volume raction or a range o values o Nt with K =, M =, λ =, Pr =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. DOI:.979/578-388 www.iosrjournals.org 96 Page
() () () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded.6..3 Nb =.,.3,, 3 5 6 Fig.5(a) Temperature or a range o values o Nb with K =, M =, λ =, Pr =, Nt =, Le =, Ec =, kr =, Rd =, γ = & β =. 5 5 Nb =,,.3,..5 3 5 6 Fig.5(b) Nanoparticle volume raction or a range o values o Nb with K =, M =, λ =, Pr =, Nt =, Le =, Ec =, kr =, Rd =, γ = & β =. 5 3 Ec =.5,,, 3 5 6 Fig.6(a) Temperature or a range o values o Ec with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, kr =, Rd =, γ = & β =. DOI:.979/578-388 www.iosrjournals.org 97 Page
() () () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded..3 Ec =.5,,, 3 5 6 7 8 Fig.6(b) Nanoparticle volume raction or a range o values o Ec with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, kr =, Rd =, γ = & β =..8.6. Rd =,,, 6 8 Fig.7(a) Temperature or a range o values o Rd with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, γ = & β =. 5 5.5 Rd =,,, 6 8 Fig.7(b) Nanoparticle volume raction or a range o values o Rd with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, γ = & β =. DOI:.979/578-388 www.iosrjournals.org 98 Page
() () () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded.8.6. =.3,,, 3 5 6 Fig.8(a) Temperature or a range o values o γ with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd = & β =..3 5 5 =.3,,,.5 3 5 6 7 Fig.8(b) Nanàparticle volume raction or a range o values o γ with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd = & β =..8.7.6..3 =.8,,.3, 3 5 6 Fig.9(a) Temperature or a range o values o β with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd = & γ =. DOI:.979/578-388 www.iosrjournals.org 99 Page
() () () Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded.5 =.8,,.3, 3 5 6 Fig.9(b) Nànoparticle volume raction or a range o values o β with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, kr =, Rd = & γ =..6..3 Pr =, 6, 8, 3 5 6 Fig. Temperature or a range o values o Pr with K =, M =, λ =, Nt =, Nb =, Le =, Ec =, kr =, Rd =, γ = & β =. 5 5 Le =, 6, 8,.5 3 5 6 7 8 Fig. Nanoparticle volume raction or a range o values o Le with K =, M =, λ =, Pr =, Nt =, Nb =, Ec =, kr =, Rd =, γ = & β =. DOI:.979/578-388 www.iosrjournals.org Page
() Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded.3 5 5 Kr =,,, 3 5 6 7 8 Fig. Nanoparticle volume raction or a range o values o Kr with K =, M =, λ =, Pr =, Nt =, Nb =, Le =, Ec =, Rd =, γ = & β =. Table Comparison or viscous case '( ) with Pr or β = Nt = Nb = Le = Ec = γ = Pr.5 '( ) Present results Nadeem and Hussain (3) Khan and Pop () Golra and Sidawi (99) Wang (989).7.7..66 695.5396.9358.66 69.5.9 Table Computations or viscous case.66 69.5.9.66 69.5.9.66 69.5.9 ''( ) ''( ), '( ), '( ). M K λ Pr Nt Nb Le Ec kr Rd γ β ''( ) ''( ).5 3.3.35. 8 5 6 8.5.5 3.3..59756.873.86.3595.99975.869.8885.799737.9639.9.59756.59756.59756.59756.59758.59758.59756.59758.59758.59756.59756.59756.59756.59756.59756.59756.59756.59756.59756.59756.59756.59756.59756 '( ).3.33976.3633.3868.357983.39696.538.35.3866.3357 87677 79773 778.3895.3866 7.3895.3866 7.39766.3936.3995.995.7676.3397.3967.3935.395.375.5 77.653.9756 '( ) 6938 8637 535 575 967 563 556 73 739 755 86 7 76.393.3536.685.393.3536.685 3635 339 593 3389 65 9933 378 3857 7687 993 586 55556 6698.3369 DOI:.979/578-388 www.iosrjournals.org Page
Eect O Chemical Reaction On MHD Boundary Layer Flow O Williamson Nanoluid Embedded.59758 8.778.69335..6.8.59756.59756.59756.379.339.3573 8577.7683.857 Reerences [] Buongiorno, J., (6), Convective transport in nanoluids, Journal o heat transer, Vol. 8, No.3, pp.-5. [] Choi SUS, Zhang, ZG, Yu, W., Lockwood, F.E & Gruike, EA, (), Anomalously thermal conductivity enhancement in nano tube suspensions, Applied Physics Letters, Vol.79, No., pp.5-5. [3] Choi, S. U. S., (995) Enhancing thermal conductivity o luids with nanoparticles in developments and applications o non- Newtonian lows, ASME, FED-vol. 3/MD- vol. 66, pp. 99-5. [] Das M., Mahato, R., Nandkeolyar, R., (5), Newtonian heating eect on unsteady hydromagnetic Casson luid low past a lat plate with heat and mass transer, Alexandria Engineering Journal, Vol.5, pp.87 879. [5] Gorla, R.S.R., and Sidawi, I., (99), Free convection on a vertical stretching surace with suction and blowing, Appl Sci Res, Vol.5, pp.7-57. [6] Hussanan, A. Khan, I. Shaie, S., (3), An exact analysis o heat and mass transer past a vertical plate with Newtonian heating, J. Appl. Math., Article ID: 357. [7] Hussanan, A., Salleh, M.Z., Tahar, R.M. Khan, I., (), Unsteady boundary layer low and heat transer o a Casson luid past an oscillating vertical plate with Newtonian heating, PloS One., Vol. 9(), e8763. [8] Khan, W.A., and Pop, I., (), Boundary-layer low o a nanoluid past a stretching sheet, Int J Heat Mass Trans, Vol. 53, pp.77-83. [9] Merkin, J.H., (99), Natural-convection boundary-layer low on a vertical surace with Newtonian heating, Int. J. Heat Fluid Flow, Vol.5 (5), pp.39 398. [] Nadeem, S., and Hussain, S. T., (3), Flow and heat transer analysis o Williamson nanoluid, Appl Nanosci., DOI 7/s3-3-8-. [] Wang, Y., (989), Free convection on a vertical stretching surace, J Appl Math Mech., Vol.69, pp.8-. DOI:.979/578-388 www.iosrjournals.org Page