Effect of chemical reaction and viscous dissipation on MHD nanofluid flow over a horizontal cylinder : analytical solution

Transcription

1 Effect of cheical reaction and viscous dissipation on MHD nanofluid flo over a horizontal cylinder : analytical solution hukla, N, Rana, P, Beg, OA and ingha, B Title Authors Type URL Published Date 17 Effect of cheical reaction and viscous dissipation on MHD nanofluid flo over a horizontal cylinder : analytical solution hukla, N, Rana, P, Beg, OA and ingha, B Conference or Workshop Ite This version is available at: UIR is a digital collection of the research output of the University of alford. Where copyright perits, full text aterial held in the repository is ade freely available online and can be read, donloaded and copied for non coercial private study or research purposes. Please check the anuscript for any further copyright restrictions. For ore inforation, including our policy and subission procedure, please contact the Repository Tea at: usir@salford.ac.uk.

2 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 1 EFFECT OF CHEMICAL REACTION AND VICOU DIIPATION ON MHD NANOFLUID FLOW OVER A HORIZONTAL CYLINDER: ANALYTICAL OLUTION Nisha hukla 1, Puneet Rana 1 *, O. Anar Bég and Bani ingh 1 1 Departent of Matheatics, JIIT Noida, Uttar Pradesh, India. Fluid Mechanics and Nanosystes, Aeronautical/Mechanical Engineering, alford University, England, UK *Corresponding and presenting author : Eail: puneetranaiitr@gail.co, puneet.rana@jiit.ac.in Abstract: An analytical study of the MHD boundary layer flo of electrically conducting nanofluid over a horizontal cylinder ith the effects of cheical reaction and viscous dissipation is presented. iilarity transforations have been applied to transfor the cylindrical for of the governing equations into the syste of coupled ordinary differential equations and then hootopy analysis ethod has been ipleented to solve the syste. HAM does not contain any sall or large paraeter like perturbation technique and also provides an easiest approach to ensure the convergence of the series of solution. The effects of cheical reaction paraeter, agnetic paraeter and other iportant governing paraeters ith no flux nanoparticles concentration is carried out to describe iportant physical quantities. Keyords: Hootopy analysis ethod; Nanofluid; MHD; Cheical reaction; tretching cylinder; Noenclature q Wall heat flux (W) v, u Velocity coponents along x- axis and radial direction (/s) K Cheical reaction T Abient teperature (K) coefficient (s -1 ) T Nanofluid teperature T Nanofluid teperature (K) at surface (K) Bronian otion DT Therophoresis paraeter Nb DBC Nt T T paraeter T V Ec c f T T Eckert nuber Diensionless strea function Pr Vx Re Prandtl nuber Reynolds nuber

3 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) M B V K VD B rv Diensionless Magnetic field Diensionless teperature Cheical paraeter reaction v c D Curvature paraeter c c B p f chidt nuber trea function iilarity variable Ratio of heat capacities Diensionless nanopa Theral diffusivity( /s) rticle concentration ubscript Abient condition f Fluid Condition on surface p Nanoparticle 1. Introduction The study of heat and ass transfer of nanofluid flo ith effects of cheical reactions over the stretching surface has a ide range of applications in cheical industries like that production of polyers and food processing, evaporation, cooling and drying processes, nuclear reactors cooling and also in petroleu industries. The ter nanofluid [1-3] represents a base fluid ith suspension of etallic nanosized particles. The cheical reactions can change the property and quality of any product. Hence, any researchers are considering the effects of cheical reactions in different types of probles. Krishnaurthy et al.[4] have presented a nuerical study of MHD boundary layer flo of Williason nanofluid over a horizontally linear stretching sheet ith cheical reaction and radiation effects. They observed that the concentration of nanoparticles in boundary layer is decreased ith increase in the value of cheical reaction paraeter. Mohaed [5] has carried out the effects of cheical reactions and absorption on a ixed convective boundary layer flo past an exponentially stretching sheet and obtained that herood nuber is an increasing function of cheical reaction paraeter. Venkadtesarlu and Narayana [6] have provided the sae study for the rotating syste. Reddy et al.[7] have investigated the effects of cheical reaction and theral radiation on an unsteady flo of nanofluid bounded by a oving vertical flat plate ith convective and diffusive boundary conditions.

4 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 3 The present article deals ith the analytical study of cheical reactive nanoparticles on flo and heat transfer over a horizontal stretching cylinder, hich is not entioned in any above literatures. We have applied hootopy analysis ethod [8-11] to solve the syste of ordinary nonlinear differential equations hich are the transfored for of governing equations.. Nanofluid Dynaic Model In present proble, e analyze the effects of cheical reactions on a steady to diensional boundary layer flo of an incopressible electrically conducting nanofluid past a horizontal stretching cylinder. The cylindrical coordinates (r, x) are taken along the radial and axial directions of cylinder, respectively and the stretching velocity of cylinder is V V x. A constant agnetic field B is assued to be applied in radial direction of cylinder [Fig.1]. The boundary layer equations, hich represent such type of flo, can be ritten as [6, 1]: u u v, r r x u v v 1 v u v B v, () r x r r r T T k T T v T C T u v D, (3) r x c r r r c r r r T r f 1 DT B f C C C 1 C D T T 1 T u v DB K( C C ). (4) r x r r r T r r r The associated boundary conditions are: C DT T v( x, r) V, u, T T, DB, at r r, r T r vx, r, T T, C C, as r. (5) We introduce folloing siilarity transforations to reduce the eqs. ()-(5) into diensionless for [1]: (1) r r V r, V xr, v V x, u V r T T, r T T, C C, C (6)

5 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 4 The non diensional syste of ordinary differential equations is 1 M, (7) 1 Nb Nt Pr 1 Ec, (8) 1 Nt Nt c Nb Nb, (9) and transfored boundary conditions are, 1, ( ) 1, Nb Nt, at,,. as (1).1. The physical quantities of interest In this study, the iportant physical quantities are the easureent of coefficient of skin friction local Nusselt nuber and herood nuber, hich are defined as: C v Nu x xq k T T xq, hx, (11) DC B here v r rr, q T k r rr and q C DB r rr. (1) Applying siilarity transforations (6) on the eqs. (11)-(14), e obtain 1 Re C 1 1, Re Nu, Re h x x. (13) 3. HAM eries olution To solve eqs. (7)-(1), e have selected folloing initial guesses, linear operators and auxiliary functions : ( ) 1 exp( ), ( ) exp( ), ( ) Nt exp( ), (14) Nb 3 ( ) L, 3 L ( ), L ( ), H 1 H H The th order deforation equations L ( ( ) 1 1( )) h HR ( ), L ( ( ) 1 1( )) h H R ( ), 1 1 L ( ( ) ( )) h H R ( )

6 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 5 ith boundary conditions: (), (), (), Nb () + Nt (), here as ( ), ( ), ( ), (15) 1 ( ) (1 ) 1 1 ( i 1 i i 1 i ) 1 i R M, R Nb Nt Ec ( ) (1 )( 1 ( i1 i i1 i )) Pr (1 ) i 1 i i1 i i i i 1 +, Nt 1 R ( ) (1 ) c i 1 i 1. Nb i The th ters ( ), ( ) and ( ) are obtained by the folloing equations: ( ) ( ) C C C e, (16) * 1 3 * ( ) ( ) C4 C5e, (17) ( ) ( ) C C e, (18) * 6 7 here Ci( i 1..7) are the constants, hich can be obtained by the boundary conditions (15) and * ( ), * ( ) and * ( ) are represents the particular integrals. Hence the HAM series solutions are obtained in folloing for of equations: (, q) ( ) ( ), q Convergence of HAM (, ) ( ) ( ), (, q) ( ) ( ). (19) The convergence of the eqs. (16)-(18) are dependent upon the auxiliary paraeters To find out the adequate values of these paraeters, e have sketched the h-curves ith and 1 1 h, h and h [9]., for different values of agnetic paraeter M at 15-th order of approxiations hich are displayed in Fig.. These figures indicate that the acceptable ranges of h, h and h are h [.7.1], h [.5.5], h [.5.5] and Table-1 ensure that the series solutions (19) are convergent upto four places of decial for h.35, h.15 and h.15.

7 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 6 4. Discussion of Results In this section, e exaine the effects of governing paraeters such as agnetic paraeter M, curvature paraeter, cheical reaction paraeter and Eckert nuber Ec on velocity ( ), skin friction coefficient, teperature, rate of heat transfer, concentration { } and herood nuber { }. The default values of paraeters are taken as Nt.3, Nb.1, Pr 7, M.5, Ec.5, c 1, =.1, =.5. We have copared the HAM results ith shooting results hich are presented in Table- & 3. These Tables represent a good agreeent aong the both results. The influence of leading paraeters on physical quantities is presented graphically. Fig. 3 is sketched to exaine the effect of M on ( ), shos that ( ) decreases ith an increase the value of M but and and, hich are increasing function of M. It is due to the physical fact that as agnetic paraeter increases, the effect of Lorentz forces also increases hich can reduce the otion, but teperature of syste and nanoparticles species are increased due to these forces. Fig. 4(a,b) illustrates the effects of and on { } and { } hich indicates that reduces for the increasing values of and but { } { } increases ith these paraeters. The cobined effects of M and on is presented in Fig. 4c hich represents that decreases as both these paraeters increases. It eans that cheical reactions can reduces the transfer of heat in the syste but increases the ass transfer. Fig. 5 is plotted to inspect the effects of Eckert nuber and agnetic paraeter M on and { }. { } { } decreases ith an increase the value of Ec and M but { } increases ith these paraeters. 5. Conclusions This article presents an analytical study of the effects of cheical reaction and viscous dissipation on a MHD nanofluid flo over a horizontal stretching cylinder. It is conclude that ( ) and decreasing functions M hereas increases ith it. are reduces ith an increase the value of M and. decreases for the increasing value of cheical reaction paraeter but { } { }

8 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 7 shos an opposite behavior ith this paraeter. { } decreases ith these paraeters. { } reduces as Ec and M increase hereas References 1..U.. Choi, J.A. Eastan, Enhancing theral conductivity of fluids ith nanoparticles, Materials cience 31 (1995) R. aidur, K.Y. Leong, H.A. Mohaad, Revie on applications and challenges of nanofluids, Reneable and ustainable Energy Revies, 15 (11) J. Buongiorno, Convective transport in nanofluids, AME Journal of Heat Transfer, 18 (6) M. R. Krishnaurthy, B. C. Prasannakuara, B. J. ireesha, R.. R. orla, Effect of cheical reaction on MHD boundary layer flo and elting heat transfer of Williason nanofluid in porous ediu, Engineering cience and Technology an International Journal, R. E. Mohaed, Cheical reaction effect on MHD boundary-layer flo of to-phase nanofluid odel over an exponentially stretching sheet ith a heat generation, Journal of Molecular Liquids, (16) B. Venkatesarlu, P. V.. Narayana, Cheical reaction and radiation absorption effects on the flo and heat transfer of a nanofluid in a rotating syste, Applied Nanoscience, 5 (15) J.V. R. Reddy, V. ugunaa, N. andeep, C. ulochana, Influence of cheical reaction, radiation and rotation on MHD nanofluid flo past a pereable flat plate in porous ediu, Journal of the Nigerian Matheatical ociety, 35 (16) J. Liao, Beyond Perturbation: Introduction to Hootopy Analysis Method, Chapan & Hall/CRC Press, London/Boca Ratton, (3). 9. T. Hayat, M. Itiaz, A. Alsaedi, Unsteady flo of nanofluid ith double stratification and agnetohydrodynaics, International Journal of Heat and Mass Transfer, 9 (16) T. Zubair, M. Usan, U. Ali,. T. Mohyud-Din, Hootopy analysis ethod for syste of partial differential equations, International Journal of Modern Engineering ciences, (1)

9 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) B. Raftari, F. Parvaneh, K. Vajravelu, Hootopy analysis of the agnetohydrodynaic flo and heat transfer of a second grade fluid in porous channel, Energy, 59( 14) R. Dhanai, P. Rana, L. Kuar, MHD ixed convection nanofluid flo and heat transfer over an inclined cylinder due to velocity and theral slip effects: Buongiornos odel, Poder Technology, 88 (16) Table-1 Order of convergence of HAM for the values of (), () and () for the fixed values of paraeters Nt = Nb =.1, c = 1, Ec =.1, M =.1, Pr = 1,.1, =.5, h =.35, h =.15, h =.15. Order () () () Table- Nuerical Coparison of HAM and hooting results of..1. () for the different values of M and M HAM hooting Table-3 Nuerical Coparison of HAM and hooting results of { } and { } for the different values of Ec and and fixed values of paraeters Nt = Nb =.1, c = 1, Pr = 1, M =.1,

10 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 9 =.5. Ec { } { } HAM hooting HAM hooting Figure 1. Physical Model ith co-ordinate axes Figure. h-curves for diensionless variables

11 International Conference on Recent Advances in Matheatical ciences and its Applications RAMA, Noida, India (8-1 DEC 16) 1 Figure 3. Effect of Magnetic field paraeter on velocity, teperature and concentration profiles. Figure 4. Effect of cheical reaction, curvature and agnetic field on heat and ass transfer. Figure 5. Cobined effect of viscous dissipation and agnetic paraeter on heat and ass transfer.