Impact of Arrhenius activation energy in viscoelastic nanomaterial flow subject to binary chemical reaction and nonlinear mixed convection

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1 Impact of Arrhenius activation energy in viscoelastic nanomaterial flo subject to binary chemical reaction and nonlinear mied convection Salman Ahmad 1,*, Muhammad Ijaz Khan 1, M. Waleed Ahmed Khan 1, Tufail A. Khan, Tasaar Hayat 1, and Ahmed Alsaedi 1 Department of Mathematics, Quaid-I-Azam University 450, Islamabad 44000, Pakistan Department of Basic Sciences, University of Engineering & Technology, Peshaar, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University P. O. Bo 8007, Jeddah 1589, Saudi Arabia * Corresponding author; salmanuom06@gmail.com Abstract The computational investigations on mied convection stagnation point flo of Jeffrey nanofluid over a stretched surface is presented herein. The sheet is placed vertical over hich nanomaterials floing upard direction. Arrhenius activation energy and binary chemical reaction are accounted. Nonlinear radiative heat flu, MHD, viscous dissipation, heat source/sink and Joule heating are considered. Initially the nonlinear flo epressions are converted to ordinary one and then tackled for series solutions by HAM. Consider flo problem are discussed for velocity, temperature and concentration through various flo variables. Furthermore, coefficient of skin friction, Sherood number and heat transfer rate are computed graphically. Keyords: Activation energy; Non-linear radiative heat flu; Jeffrey nanofluid; Viscous dissipation and Joule heating. 1. Introduction Studies of non-netonian fluids have great interest due to their ide range applications in various field like physiology, pharmaceutical, fiber technology, coating of ires, food products, crystal groth and so forth. Properties of non-netonian fluid cannot characterize by single constitutive relation. Therefore, various non-netonian fluid models have been proposed (see [1-14]). Generally, these models are divided into rate, differential and integral types. Here e focused on Jeffrey fluid model. Which is a rate type fluid model and it is describe both relaation and retardation times behavior. Studies associated to Jeffrey fluid can be seen in Refs. [15-]. Suspension of nano-scale sized particles in base fluid is knon as nanofluid. Nanoparticles in base fluid used to improve thermal conductivity of base fluid. There are ample demands of nanofluids in various fields like medical, industries and engineering etc. For heating and cooling purpose nanofluids used in industries, modern drug delivery system, electronic devices batteries and hyperthermia etc. Heat transfer enhancement by nanoparticles as first addressed Choi [] Buongiorno [4] developed a model to describe thermal conductivity enhancement in nanomaterials. He addressed seven slip phenomena i.e. 1

2 Bronian diffusion, thermophoresis, Magnus, gravity, fluid drainage and inertia. He concluded that thermophoresis and Bronian diffusion are major ruling slip phenomena in the nanomaterials. Shekholeslami et al. [5] studies the impact of MHD on flo of CuO H O nanomaterials ith mied convection. Farooq et al. [6] disclosed influences of nonlinear thermal radiation and MHD on stagnation point flo of viscoelastic nanofluid. Abbasi et al. [7] eplored flo of nanomaterial over a moving surface. The effect of magnetic dipole on flo of Maell nanofluids is investigated by Hayat et al. [8]. Lin et al. [9] eplored impact of MHD on flo of pseudo-plastic nanomaterial. Studies associated to nanofluid can be seen in Refs. [0-5]. The objective of this investigation is to eamine the influences of activation energy, heat source/sink, viscous dissipation, Joule heating, magnetic field on nonlinear mied convective stagnation point flo of nonlinear radiative Jeffrey nanofluids over a stretching sheet. Transformations procedure is implemented to transform the governing partial differential equations into ordinary ones. Series solution is pointed out by homotopy algorithm [6-8]. The outcome of flo variable on concentration, velocity, temperature, Sherood number, Nusselt number and skin friction is analyzed and discussed through graphs.. Modeling Mied convection stagnation point flo of Jeffrey nanofluid over a stretchable surface is investigated. Arrhenius activation energy and binary chemical reaction are considered. Electrically conducting fluid is considered. Energy equation is discussed in the presence of nonlinear radiative heat flu and heat source/sink. Furthermore, dissipation and Joule heating are taken. Flo diagram is presented in Fig. 1. The flo epressions are [9]: u v 0, y (1) e 1 y 1 y y y u u ue Bo u u u u u u u u v y ue u u u v y y g 1 T T T T C C 4 C C, () Implementing [40]: DT Q0 B T T k T 16 T C T T u v y c p y kcp y T y D y y T y c T T p B o u u u u u c u 1 1, p cp y u y y v y y n D T r ep, C C C T T Ea u v DB k C C y y T y T T ith ( ), 0, T, C D T u u T 0 at 0, a v k y hf T T DB y T y y u ue( ) b, T T, C C as y. (4) (5) ()

3 TT v a f, u af,, T T (6) a CC y,. C C The flo epressions take the folloing form ( iv) f 1 ff f Ha A f A f ff 1 Gr 1 1 0, Re t Gr c (7) 1 Pr 1 f Nb Nt EcHaf Tr 1 Tc Tc Pr Ec f f f f ff, (8) 1 1 Nt E f 1 ep 0, Sc Sc Nb Tc Tc (10) f (0) 0, f (0) 1, Nb (0) Nt (0) 0, (0) i 1 (0), (11) f( ) A, ( ) 0, ( ) 0. In the above epressions Ha B o represents Hartmann number, a a1 material variable, o b 16 A a ratio parameter, T T Tr o kk radiation variable, T T t 1 convection variable due a to temperature, Re 4C local Reynolds number, c convection variable due to gc concentration, Gr g1t T Grashof number due to concentration, Gr Grashof c number due to temperature, Pr p DTT T k Prandtl number, N the thermophoresis t T a B parameter, Ec c the Eckert number, Nb p TT DC Bronian motion parameter, T Tc T dimensionless temperature, T k r reaction rate, Sc a D B Schmidt number, Q0 Ea ac heat source/sink variable, E p T denotes the dimensionless activation energy and T h f Biot number. i k a. Physical quantities.1. Coefficient of skin friction (surface drag force) Mathematically, it is defined as C f, (1) u here denotes the all shear stress and defined as 1 u u u 1 u v y y y y 0 Putting Eq. (1) in Eq. (1), one has 0.5 C f Re f (0) f (0) f (0) f (0) f (0). 1. (1) (14)

4 .. Heat transfer rate (Nusselt number) We have Nu q k T, T here q represents the all heat flu and mathematically epressed as o 16 T T q k1. o kk y Invoking Eq. (16) in Eq. (15), e have 0.5 Re 1 r (0) c (0). Nu T T (17) (15) (16).. Sherood number (mass transfer rate) It is defined as J Sh, (18) DC B here J indicates the all mass flu and mathematically J C DB y y0. (19) From Eq. (19) and Eq. (18), e get the folloing form 0.5 Sh Re (0), (0) a here Re signifies the local Reynolds number, C f the skin friction, Nu the Nusselt number and Sh the Sherood number. 4. HAM solution In order to obtained the series solutions of nonlinear ordinary differential equations by homotopy analysis method it is compulsory to define the linear operator and initial guesses. The linear operator and initial guesses for momentum, temperature and nanoparticles concentration are defined as f0( ) 1 A(1 ) (1 A)ep( ), i 0( ) 1 ep( ), (1) i i Nt ( ) ep( ), 0 1i Nb ith L f L L d d d d, d 1, d d 1, d () 4

5 f in hich 1 7 i L c1 c ep( ) c ep( ) 0, 4ep( ) 5ep( ) L c c 0, L c6ep( ) c6ep( ) 0, c signifies the arbitrary constant. () 5. Convergence analysis The auiliary variables f, and plays a noteorthy role in convergence series solutions. These variables control and adjust the convergent portion of series solutions. The curves for momentum, temperature and nanoparticles volume concentration are plotted in Fig.. It has been eamined that the suitable estimations for f ), ( ) and ( ) are 1.8 f 0.1, and Table 1 is constructed for the convergence of series solutions hen Nb 0.7, T c 0.5, n E A 0.1, Nt 0., Gr G r 0.4, Ha t i 0., E 0., Pr Sc 1.0 and T 0.4. c r Table 1. Numerical results for momentum, temperature and nanoparticles volume concentration. Order of approimation f Discussion In this section e eamined the effects of flo variables on velocity, concentration, temperature, skin friction, Sherood and Nusselts numbers. Figs. (-7) eamined the behavior of A,, Gr, Gr and Ha on velocity ( f ( )). Impact of A on ( f ( )) is presented in Fig.. It is noted that ( f ( )) enhances for larger A. Fig. 4 depict the influence of on ( f ( )). For larger estimation of velocity sho decreasing behavior. Physically is the relation of relaation to observation times. By increase in relaation time is higher and generates more resistance to flo due to hich f reduces. Fig. 5 captured the effect of Gr on ( f ( )). Clearly ( f ( )) is increasing function of. The outcome of ( f ( )) ith variation of Gr is described in Fig. 6. ( f ( )) decreased through Gr. Fig. 7 is sketched for ( f ( )) ith variation in Ha. This figure sho that ( f ( )) is decays for higher estimation of Ha. Physically Ha is an increasing function of resistive force (Lorentz force) therefore ( f ( )) diminished. Figs. (8-1) described the influences of Ha,, Tr,Pr, Ec and i on ( ). Impact of Ha on ( ) is plotted in Fig. 8. It is noticed that ( ) boosts via Ha Fig. 9 demonstrated the behavior of on ( ). ( ) enhances ith larger variation in. Fig. 10 sho the effect of Tr on ( ). For higher values of Tr temperature is enhanced. Variation of ( ) through Pr is portrayed in Fig. 11. ( ) is decreasing function of Pr. Fig. 1 is focused to describe the impact of Ec on ( ). Clearly ( ) boosts for larger estimation of Ec. Fig. 1 captured the influence of i on ( ). 5

6 This Fig. sho that ( ) enhanced for larger values of i. Figs. (14-18) disclosed the characteristics of, Sc, Nt, Nb and E on ( ). Impact of on ( ) is portrayed in Fig. 14. It is noticed that is dominant for higher values of Fig. 15 sketched for ( ) through variation in Sc. ( ) boosts ith Sc Fig. 16 captured the impact of Nt on ( ). Clearly ( ) is a decreasing function of Nt Variation of ( ) through Nb is presented in Fig. 17. ( ) is dominant for larger values of Nb. Fig. 18 depict the characteristic of E on ( ). ( ) boosts via E. The effects of, Ha, Gr and Gr on Cf are presented in Figs. (19-0). In these Figs. e noted that Cf is boosts via, Ha and Gr hile reduces through Gr. Influences of Ha,,Pr and Ec on Nu are reported in Figs. (1-). Clearly Nu is a decreasing function of Ha and hoever enhanced ith Pr and Ec. Characteristics of Nt, Nb, Ec and Tc on Sh are disclosed in Figs. (-4). It is noticed that Sh is boosts via Nb and Tc hile decays for larger values of Nt and Sc. fig.. -curves for, and. fig. 1: Systematic diagram fig.. via. fig.. via. 6

7 fig. via. fig. via. fig. via. fig. via. fig.. via. fig.. via. 7

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10 fig.. via and. fig.. via and. 7. Concluding remarks Here e investigated the effects of activation energy, Joule heating, viscous dissipation, and magnetic field on mied convective radiative flo of Jeffrey nanofluid over a sheet. Main outcomes are listed belo: ( f ( )) is enhanced through A and Gr hile decays ith, Ha, Gr. ( ) boosts via, Ha, Tr, Ec and i hile reduces ith Pr. ( ) is dominant for larger, Sc, Nb and E hoever decreased through Nt. Cf and Nu sho opposite behavior against Ha and. Sh increased for larger Nb and Tc hile decreased ith Nt and Sc. Nomenclature u and v velocity components and y space coordinates σ electric conductivity u stretching velocity B 0 strength of magnetic field u e free stream velocity μ dynamic viscosity ν kinematic viscosity χ 1 coefficient of linear thermal epansion χ coefficient of nonlinear thermal epansion χ coefficient of linear χ 4 coefficient of nonlinear concentration epansion concentration epansion D B Bronian diffusion T Temperature coefficient k mean absorption coefficient k thermal conductivity σ Stefan-Boltzmann coefficient ρ Density c p specific heat D T thermophoresis diffusion coefficient C concentration Q 0 coefficient of heat source/sink T ambient temperature n fitted rate constant C ambient concentration k r reaction rate h f coefficient of heat transport q r heat flu κ Boltzmann constant E a activation energy 10

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