Alexandria Engineering Journal (2015) xxx, xxx–xxx HOSTED BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com REVIEW Numerical study of flow and heat transfer of non-Newtonian Tangent Hyperbolic fluid from a sphere with Biot number effects S. Abdul Gaffar a, V. Ramachandra Prasad b,*, O. Anwar Be´g c a Department of Mathematics, Salalah College of Technology, Salalah, Oman b Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle 517325, India c Gort Engovation Research (Aerospace), 15 Southmere Avenue, Great Horton, Bradford BD7 3NU, West Yorkshire, UK Received 1 September 2014; revised 23 March 2015; accepted 4 July 2015 KEYWORDS Abstract In this article, we investigate the nonlinear steady boundary layer flow and heat transfer Non-Newtonian Tangent of an incompressible Tangent Hyperbolic fluid from a sphere. The transformed conservation equa- Hyperbolic fluid; tions are solved numerically subject to physically appropriate boundary conditions using implicit Skin friction; finite-difference Keller Box technique. The numerical code is validated with previous studies. The Weissenberg number; influence of a number of emerging non-dimensional parameters, namely Weissenberg number Power law index; (We), power law index (n), Prandtl number (Pr), Biot number (cÞ and dimensionless tangential Boundary layer flow; coordinate (nÞ on velocity and temperature evolution in the boundary layer regime is examined Biot number in detail. Furthermore, the effects of these parameters on heat transfer rate and skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation is achieved. It is found that the velocity, Skin friction and the Nusselt number (heat transfer rate) are decreased with increasing Weissenberg number (We), whereas the temperature is increased. Increasing power law index (n) increases the velocity and the Nusselt number (heat transfer rate) but decreases the temperature and the Skin friction. An increase in the Biot number (cÞ is observed to increase velocity, temperature, local skin friction and Nusselt number. The study is relevant to chemical materials processing applications. ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/). * Corresponding author. E-mail address: [email protected] (V.R. Prasad). Peer review under responsibility of Faculty of Engineering, Alexandria University. http://dx.doi.org/10.1016/j.aej.2015.07.001 1110-0168 ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: S.A. Gaffar et al., Numerical study of flow and heat transfer of non-Newtonian Tangent Hyperbolic fluid from a sphere with Biot number effects, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.07.001 2 S.A. Gaffar et al. Nomenclature a radius of the sphere Greek Cf skin friction coefficient a thermal diffusivity f non-dimensional steam function g the dimensionless radial coordinate Gr Grashof number l dynamic viscosity g acceleration due to gravity m kinematic viscosity k thermal conductivity of fluid h non-dimensional temperature n power law index q density of non-Newtonian fluid Nu local Nusselt number n the dimensionless tangential coordinate Pr Prandtl number w dimensionless stream function rðxÞ radial distance from symmetrical axis to surface of c Biot number the sphere C time dependent material constant T temperature of the fluid P second invariant strain tensor u; v non-dimensional velocity components along the x- and y-directions, respectively Subscripts V velocity vector w conditions at the wall (sphere surface) We Weissenberg number 1 free stream conditions x stream wise coordinate y transverse coordinate Contents 1. Introduction . 00 2. Non-Newtonian constitutive Tangent Hyperbolic fluid model . 00 3. Mathematical flow model. 00 4. Numerical solution with Keller Box implicit method . 00 5. Numerical results and interpretation . 00 6. Conclusions . 00 References . 00 1. Introduction environmental technologies including energy storage, gas turbines, nuclear plants, rocket propulsion, geothermal reser- The dynamics of non-Newtonian fluids has been a popular voirs, and photovoltaic panels. The convective boundary area of research owing to ever-increasing applications in chem- condition has also attracted some interest and this usually ical and process engineering. Examples of such fluids include is simulated via a Biot number in the wall thermal boundary coal-oil slurries, shampoo, paints, clay coating and suspen- condition. Recently, Ishak [6] discussed the similarity solu- sions, grease, cosmetic products, custard, and physiological tions for flow and heat transfer over a permeable surface liquids (blood, bile, synovial fluid). The classical equations with convective boundary condition. Aziz [7] provided a employed in simulating Newtonian viscous flows i.e. the similarity solution for laminar thermal boundary layer over Navier–Stokes equations fail to simulate a number of critical a flat surface with a convective surface boundary condition. characteristics of non-Newtonian fluids. Hence several consti- Aziz [8] further studied hydrodynamic and thermal slip flow tutive equations of non-Newtonian fluids have been presented boundary layers with an iso-flux thermal boundary condi- over the past decades. The relationship between the shear tion. The buoyancy effects on thermal boundary layer over stress and rate of strain in such fluids is very complicated in a vertical plate subject with a convective surface boundary comparison with viscous fluids. The viscoelastic features in condition were studied by Makinde and Olanrewaju [9]. non-Newtonian fluids add more complexities in the resulting Further recent analyses include Makinde and Aziz [10]. equations when compared with Navier–Stokes equations. Gupta et al. [11] used a variational finite element to simulate Significant attention has been directed at mathematical and mixed convective–radiative micropolar shrinking sheet flow numerical simulation of non-Newtonian fluids. Recent investi- with a convective boundary condition. Makinde et al. [12] gations have implemented, respectively the Casson model [1], studied cross diffusion effects and Biot number influence second-order Reiner–Rivlin differential fluid models [2], on hydromagnetic Newtonian boundary layer flow with power-law nanoscale models [3], Eringen micro-morphic mod- homogenous chemical reactions and MAPLE quadrature els [4] and Jeffreys viscoelastic model [5]. routines. Be´g et al. [13] analyzed Biot number and buoyancy Convective heat transfer has also mobilized substantial effects on magnetohydrodynamic thermal slip flows. interest owing to its importance in industrial and Subhashini et al. [14] studied wall transpiration and cross Please cite this article in press as: S.A. Gaffar et al., Numerical study of flow and heat transfer of non-Newtonian Tangent Hyperbolic fluid from a sphere with Biot number effects, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.07.001 Numerical study of flow and heat transfer 3 diffusion effects on free convection boundary layers with a a mushroom-shaped plume which was observed to detract in convective boundary condition. length and thickness with increasing Grashof number. He fur- An interesting non-Newtonian model developed for chemi- ther computed flow separation at high Grashof number and an cal engineering systems is the Tangent Hyperbolic fluid model. associated recirculation vortex arising in the wake of the This rheological model has certain advantages over the other sphere. Furthermore this study showed that local Nusselt num- non-Newtonian formulations, including simplicity, ease of ber along the sphere surface initially falls, attaining a mini- computation and physical robustness. Furthermore it is mum, and thereafter rises markedly in the vicinity of sphere deduced from kinetic theory of liquids rather than the empiri- rear. Sharma and Bhatnagar [30] used the Van Dyke method cal relation. Several communications utilizing the Tangent of matched asymptotic expansions to obtain solutions for Hyperbolic fluid model have been presented in the scientific lit- creeping heat transfer (viscous-dominated flow) from a spher- erature. There is no single non-Newtonian model that exhibits ical body to power-law fluids. Be´g et al. [31] examined the free all the properties of non-Newtonian fluids. Among several convection magnetohydrodynamic flow from a sphere in por- non-Newtonian fluids, hyperbolic tangent model is one of ous media using network simulation, showing that tempera- the non-Newtonian models presented by Pop and Ingham tures are boosted with magnetic field and heat transfer is [15]. Nadeem and Akram [16] made a detailed study on the enhanced from the lower stagnation point towards the upper peristaltic transport of a hyperbolic tangent fluid in an asym- stagnation point. Potter and Riley [32] used a perturbation metric channel. Nadeem and Akram [17] investigated the peri- expansion approach to evaluate analytically the eruption of staltic flow of a MHD hyperbolic tangent fluid in a vertical boundary layer into plume arising from free convection asymmetric channel with heat transfer. Akram and Nadeem