Effect of Viscous dissipation on Heat Transfer of Magneto- Williamson Nanofluid

IOSR Journal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 11, Issue 4 Ver. V (Jul - Aug. 015), PP 5-37.iosrjournals.org Effect of Viscous dissipation on Heat Transfer of Magneto- Williamson Nanofluid K. Lakshmi Narayana 1, K. Gangadhar and M J. Subhakar 3 1, Department of Mathematics, Acharya Nagarjuna University, Ongole, Andhra Pradesh -53001, India 3 Department of Mathematics, Noble College, Machilipatnam, A.P. 51001.India. 1 Corresponding Author Email ID: lnarayanak1@gmail.com, kgangadharmaths@gmail.com Abstract: This study investigates the effect of the nano particle effect on magnetohydrodynamic boundary layer flo over a stretching surface ith the effect of viscous dissipation. The governing partial differential equations are transformed to a system of ordinary differential equations and solved numerically using fifth order Runge- Kutta method integration scheme and Matlab bvp4c solver. The effects of the Non-Netonian Williamson parameter, Prandtl number, Leis number, the diffusivity ratio parameter, heat capacities ratio parameter, Eckert number, Schmidt number on the fluid properties as ell as on the skin friction and Nusselt number coefficients are determined and shon graphically. Keyords: MHD, nanoparticle, viscous dissipation, Williamson fluid model, heat transfer. I. Introduction Nanoparticles have one dimension that measures 100 nanometers or less. The properties of a lot of conventional materials change hen formed from nanoparticles. This is typically because nanoparticles contain a greater surface area per eight than larger particles hich causes them to be more reactive to some other molecules. Choi [1] investigated the theoretical learn of the thermal conductivity of nanofluids ith Copper nanophase meterials and he estimated the potential profit of the fluids and also he shon that one of the benefits of nanofluids ill be dramatic reductions in heat exchanger pumping poer. The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. []. This phenomenon suggests the opportunity of using nanofluids in advanced nuclear systems [3]. A comprehensive survey of convective transport in nanofluids as made by Buongiorno [4], ho says that a satisfactory explanation for the abnormal increase of the thermal conductivity and viscosity is yet to be found. He focused on added heat transfer enhancement observed in convective situations. Kuznetsov and Nield [5] have examined the influence of nanoparticles on natural convection boundary-layer flo past a vertical plate using a model in hich Bronian motion and thermophoresis are accounted for. The authors have assumed the simplest possible boundary conditions, namely those in hich both the temperature and the nanoparticle fraction are constant along the all. Furthermore, Nield and Kuznetsov [6, 7] have studied the Cheng and Minkoycz [8] problem of natural convection past a vertical plate in a porous medium saturated by a nanofluid and used for the nanofluid incorporates the effects of Bronian motion and thermophoresis for the porous medium. The problem of viscous flo and heat transfer over a stretching sheet has important industrial applications, for example, in metallurgical processes, such as draing of continuous filaments through quiescent fluids, annealing and tinning of copper ires, glass bloing, manufacturing of plastic and rubber sheets, crystal groing, and continuous cooling and fiber spinning, in addition to ide-ranging applications in many engineering processes, such as polymer extrusion, ire draing, continuous casting, manufacturing of foods and paper, glass fiber production, stretching of plastic films, and many others. During the manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. The final product ith the desired characteristics strictly depends upon the stretching rate, the rate of cooling in the process, and the process of stretching. In vie of these applications, Sakiadis [9, 10] investigated the boundary layer on a moving continuous flat surface and a moving continuous cylindrical surface, for both laminar and turbulent flo in the boundary layer. Syahira Mansur and Anuar Ishak [11] found that the local Nusselt number and the local Sherood number as ell as the temperature and concentration profiles for some values of the convective parameter, stretching/shrinking parameter, Bronian motion parameter, and thermophoresis parameter. Nadeem and Hussain [1] analyze the nano particle effect on boundary layer flo of Williamson fluid over a stretching surface. Dissipation is the process of converting mechanical energy of donard-floing ater into thermal and acoustical energy. Viscous dissipation is of interest for many applications. Significant temperature rises are observed in polymer processing flos such as injection modelling or extrusion at high rates. Aerodynamic heating in the thin boundary layer around high speed aircraft raises the temperature of the skin. In a completely DOI: 10.9790/578-1145537.iosrjournals.org 5 Page

different application, the dissipation function is used to characterize the viscosity of dilute suspensions Einstein [13]. Viscous dissipation for a fluid ith suspended particles is equated to the viscous dissipation in a pure Netonian fluid, both being in the same flo (same macroscopic velocity gradient). Vajravelu and Hadjinicolaou [14] studied the heat transfer characteristics over a stretching surface ith viscous dissipation in the presence of internal heat generation or absorption. Recently, Yohannes et al [15] analyzes the thermal boundary layer thickness increases ith increasing the values of Eckert number. More recently, Zaimi et al. [16] presents a similarity solution of the boundary layer flo and heat transfer over a nonlinearly stretching/shrinking sheet immersed in a nanofluid ith suction effect. Hoever, the interactions of magnetohydrodynamic boundary layer flo on heat transfer of Williamson nano fluid flo ith viscous dissipation. 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 the fourth order Runge-Kutta method along ith shooting technique. The effects of various governing parameters on the fluid velocity, temperature, nanoparticle volume friction, reduced Nusselt number and nanoparticle volume friction gradient are shon in figures and analyzed in detail. II. Mathematical Formulation Let us consider the to-dimensional steady flo of an incompressible, viscous dissipative nano Williamson fluid over a stretching surface. The plate is stretched along x-axis ith a velocity Bx, here B>0 is stretching parameter. The fluid velocity, temperature and nanoparticle concentration near surface are assumed to be U, T and C, respectively. For Williamson fluid model is defined in (Dapra [17]) as 0 0 1 (.1) 1 here is extra stress tensor, 0 is limiting viscosity at zero shear rate and is limiting viscosity at infinite shear rate, 0is a time constant, 1 is the first Rivlin Erickson tensor and 1 (.) 1 trace (.3) Here e considered the case for hich 0 0 1 or by using binomial expansion e get 1 1 0 1 and 1 The above model reduces to Netonian for 0 The equations governing the flo are Continuity equation. Thus Eq. (.1) can be ritten as is defined as follos: DOI: 10.9790/578-1145537.iosrjournals.org 6 Page (.4) (.5) u v 0 (.6) x y Momentum equation u u v u u u u (.7) x y y y y Energy equation u v D T T T pcp C T DT T u B x y y c y y T y c y (.8)

Volumetric species equation C C C DT T u v DB x y y T y The boundary conditions are u U, v v, T T, C C at y 0 (.9) u 0, T T, C C as y (.10) Since the surface is stretched ith velocity Bx, thus U = Bx and u and v are horizontal and vertical components of velocity, is kinematic viscosity, v is the suction or injuction velocity ith v 0 for suction and v 0 for injection. is the nanofluid thermal diffusivity. is nanofluid density, c and pcp are heat capacities of nanofluid and nanoparticles, respectively, T is temperature, k is nanofluid thermal conductivity, D B is Bronian diffusion coefficient, C is nanoparticle volumetric fraction, D T is thermophoretic diffusion coefficient and T is the ambient fluid temperature. In order to transform the equations (.6) to (.10) into a set of ordinary differential equations, the folloing similarity transformations and dimensionless variables are introduced. B u Bxf ', v B f, y T T C C ( ), ( ),Pr T T C C 3 B U Le, x, Sc, Ec D D c T T B B pcp ( C C ) DBT( C C) Nc, Nt c DT( T T ) here f ( ) is the dimensionless stream function, θ - the dimensionless temperature, - the dimensionless (.11) nanoparticle volume fraction, η - the similarity variable, - the Non-Netonian illiamson parameter, Le - the Leis number, Nc - the heat capacities ratio, Nt - the diffusivity ratio, Ec-the Eckert number, Pr - the Prandtl number, Sc - the Schmidt number. In vie of the equation (.11), the equations (.7) to (.10) transform into f ''' ff '' f ' f '' f ''' 0 (.1) Nc Nc " Pr f ' ' ' ' Ecf '' 0 Le LeNt (.13) 1 " Scf ' '' 0 (.15) Nt The transformed boundary conditions can be ritten as f S, f ' 1, 1, 1 at 0 f ' 0, 0, 0 as (.16) Where the constant parameter / S v B corresponds to the suction (S>0) and injection (S<0) or the ithdraal of the fluid, respectively. If e put 0, our problem reduces to the one for Netonian nano and for D B = D T = 0 in Eq. (.8) our heat equation reduce to the classical boundary layer heat equation in the absence of viscous dissipation. Physical quantities of interest are Local skin friction coefficient C, Local Nusselt number Nu and Local Sherood number Sh. f DOI: 10.9790/578-1145537.iosrjournals.org 7 Page

x T x C C, Nu, Sh f U T T y y 0 C C y y0 or by introducing the transformations (15), e have Nu Sh Re C f f '' f '', ' 0, ' 0 Re Re 0 Bx Where Re is the local Reynolds number. (.17) (.18) III. Solution Of The Problem The set of non-linear coupled differential Eqs. (.1)-(.15) subject to the boundary conditions Eq. (.16) constitute a to-point boundary value problem. In order to solve these equations numerically e follo most efficient numerical shooting technique ith fifth-order Runge-Kutta-integration scheme. In this method it is most important to choose the appropriate finite values of. To select e begin ith some initial guess value and solve the problem ith some particular set of parameters to obtain f '', ' and '. The solution process is repeated ith another large value of until to successive values of f '', ' and ' differ only after desired digit signifying the limit of the boundary along. The last value of is chosen as appropriate value of the limit for that particular set of parameters. The four ordinary differential Eqs. (.1)-(.15) ere first formulated as a set of seven first-order simultaneous equations of seven unknons folloing the method of superposition [18]. Thus, e set y f, y f ', y f '', y, y ', y, y ' 1 3 4 5 6 7 y ' y, y ' y y 1 3 1 0 0 0 0 S, y 0 1 1 y ' y y y y 3 1 3 1 y3 3 1 4 5 4 y ' y, y 0 1 Nc Nc y ' Pr y y Ecy y y y Le LeNt y 5 1 5 3 5 7 5 5 6 7 6 y ' y, y 0 1 1 y ' Scy y y Nt y 7 1 6 5 7 3 ' Eqs. (.1)-(.15) then reduced into a system of ordinary differential equations, i.e., here 1 3 are determined such that it satisfies y 0, y4 0 and 6 0, until the boundary conditions y 0, y 0 and and 3 DOI: 10.9790/578-1145537.iosrjournals.org 8 Page, and y. The shooting method is y are used to guess 1 4 satisfied. Then the resulting differential equations can be integrated by fourth-order Runge-Kutta scheme. The above procedure is repeated until e get the results up to the desired degree of accuracy, 6 10 6 0 IV. Results And Discussion In order to get a clear insight of the physical problem, the velocity, temperature and nanoparticle volume friction have been discussed by assigning numerical values to the governing parameters encountered in the problem. Numerical computations are shon from figs.1-13.

Figs. 1(a)-(c) shos the effect of the Non-Netonian Williamson parameter on the velocity, temperature and mass volume fraction profiles. It is observed that the velocity of the fluid decreases ith an increase the Non-Netonian Williamson parameter and temperature of the fluid as ell as mass volume friction of the fluid increases the Non-Netonian Williamson parameter. Figs. (a)-(c) shos the effect of the suction parameter on the velocity, temperature and mass volume fraction profiles. It is observed that the velocity of the fluid decreases ith an increase the suction parameter and temperature of the fluid as ell as mass volume friction of the fluid increases the suction parameter. The effect of viscous dissipation on temperature and mass volume friction is shon in figs. 3(a)&(b). Increases in Ec the temperature of the fluid is decreases as ell as opposite results ere found in mass volume friction. The effect of Nc on temperature and mass volume friction is shon in figs. 4(a)&(b). An increase in Nc the temperature of the fluid is increases as ell as opposite results ere found in mass volume friction. From fig. 5(a) & 5(b) sho that the effect of Leis number (Le) on temperature and mass volume friction. Since Leis number is the ratio of nanoparticle thermal diffusivity to Bronian diffusivity. It is observe that the temperature of the fluid decreases here as mass volume friction increases ith an increase in the Leis number. The effect of the Prandtl number (Pr) on temperature is shon in fig.6. Since the Prandtl number is the ratio of momentum diffusivity to the nanofluid thermal diffusivity. It is noticed that temperature of the fluid increases ith an increases the Prandtl number. The variation of mass volume friction verses diffusivity ratio parameter (Nt) is plotted in fig. 7. Since the diffusivity ratio is the ratio of Bronian diffusivity to the thermophoretic diffusivity. It is seen that as Nt increases the mass volume friction of the fluid decreases. From fig.8 sho that the variation of the Schmidt number (Sc) to the nanoparticle volume friction. Since Schmidt number is the ratio of the momentum diffusivity to Bronian diffusivity. It is seen that nanoparticle volume friction decreases ith increases the Schmidt number. Fig.9 shos the effects of S and λ on skin friction. From fig.9 it is seen that the skin friction increases ith an increase S or λ. The variations of Ec and λ on reduced Nusselt number is shon in fig.10. It is observed that the reduced Nusselt number increases ith an increase the parameter Ec and decrease ith an increasing the parameter λ. The effect of Ec and λ on Sherood number is shon in fig.11. it is found that the Sherood number reduces ith an increase in the parameters Ec or λ. Fig.1 shos the effects of Nt and Nc on local Nusselt number. From fig.1 it is seen that the local Nusselt number increases ith an increase Nt hereas decreases ith the influence of Nc. The variations of Nt and Nc on reduced Sherood number is shon in fig.13. It is observed that the reduced Sherood number increases ith an increase the parameter Nt or Nc. Table 1 is shos to compare our results for the viscous case in the absence of nanoparticles and viscous dissipation. These results are found to be in good agreement. V. Conclusions In this paper numerically investigated the nanoparticle effect on magnetohydrodynamic boundary layer flo of Williamson fluid over a stretching surface in the presence of viscous dissipation. The important findings of the paper are: The velocity of the fluid decreases ith an increase of the Non-Netonian Williamson parameter. The fluid temperature and mass volume friction increases ith the influence of Non-Netonian Williamson parameter. The nanoparticle volume friction enhances the viscous dissipation. The influence of heat capacities ratio parameter or viscous dissipation reduces the heat transfer coefficient. nanoparticle volume friction gradient enhances the diffusivity ratio parameter and heat capacities ratio parameter. DOI: 10.9790/578-1145537.iosrjournals.org 9 Page

Fig.1(a) Velocity for different values of λ Fig.1(b) Temperature for different values of λ Fig.1(c) Nanoparticle volume fraction for different values of λ DOI: 10.9790/578-1145537.iosrjournals.org 30 Page

Fig. (a) Velocity for different values of S Fig.(b) Temperature for different values of S Fig.(c) Nanoparticle volume fraction for different values of S DOI: 10.9790/578-1145537.iosrjournals.org 31 Page

Fig.3(a) Temperature for different values of Ec Fig.3(b) Nanoparticle volume fraction for different values of Ec Fig.4(a) Temperature for different values of Nc DOI: 10.9790/578-1145537.iosrjournals.org 3 Page

Fig.4(b) Nanoparticle volume fraction for different values of Nc Fig.5(a) Temperature for different values of Le Fig.5(b) Nanoparticle volume fraction for different values of Le DOI: 10.9790/578-1145537.iosrjournals.org 33 Page

Fig.6 Temperature for different values of Pr Fig.7 Nanoparticle volume fraction for different values of Nt Fig.8 Nanoparticle volume fraction for different values of Sc DOI: 10.9790/578-1145537.iosrjournals.org 34 Page

Fig.9 Effect of λ and S on the reduced skin friction Fig.10 Effect of λ and Ec on the reduced Nusselt number Fig.11 Effect of λ and Ec on the local Sherood number DOI: 10.9790/578-1145537.iosrjournals.org 35 Page

Fig.1 Effect of Nt and Nc on the reduced Nusselt number Pr 0.07 0. 0.7.0 Fig.13 Effect of Nt and Nc on the local Sherood number Table 1 Comparison for viscous case Present results RKF5 bvp4c 0.066 0.073 0.1695 0.1695 0.453916 0.4539 0.911358 0.9114 Nadeem and Hussain [1] 0.066 0.169 0.454 0.911 '(0) '(0) ith Pr for λ =Nc=Nt=Le=Ec=Sc=0 Khan and Pop [19] 0.066 0.169 0.454 0.911 Golra and Sidai [0] 0.066 0.169 0.454 0.911 Wang [1] References [1]. Choi, S.U.S., (1995), Enhancing thermal conductivity of fluid ith nanoparticles, developments and applications of non-netonian flo. ASME FED, Vol.31, pp.99-105. []. Masuda, H., Ebata, A., Teramae, K., and Hishinuma, N., (1993), Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles, Netsu Bussei, Vol.7, pp.7-33. [3]. Buongiorno, J., Hu, W., (005), Nanofluid coolants for advanced nuclear poer plants, Proceedings of ICAPP 05: May 005 Seoul. Sydney: Curran Associates, Inc; pp.15-19. [4]. Buongiorno, J., (006), Convective transport in nanofluids, ASME J Heat Transf, Vol.18, pp.40-50. 0.066 0.169 0.454 0.911 DOI: 10.9790/578-1145537.iosrjournals.org 36 Page

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