Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere

Abstract In this article, the combined magnetohydrodynamic heat, momentum and mass (species) transfer in external boundary layer flow of Casson nanofluid from an isothermal sphere surface with convective condition under an applied magnetic field is studied theoretically. The effects of Brownian motion and thermophoresis are incorporated in the model in the presence of both heat and nanoparticle mass transfer convective conditions. The governing partial differential equations (PDEs) are transformed into highly nonlinear, coupled, multi-degree non-similar partial differential equations consisting of the momentum, energy and concentration equations via appropriate non-similarity transformations. These transformed conservation equations are solved subject to appropriate boundary conditions with a second order accurate finite difference method of the implicit type. The influences of the emerging parameters i.e. magnetic parameter (M), Buoyancy ratio parameter (N), Casson fluid parameter (β), Brownian motion parameter (Nb) and thermophoresis parameter (Nt), Lewis number (Le), Prandtl number (Pr) and thermal slip (ST) on velocity, temperature and nano-particle concentration distributions is illustrated graphically and interpreted at length. Increasing viscoplastic (Casson) parameter decelerates the flow and also decreases thermal and nano-particle concentration. Increasing Brownian motion accelerates the flow and enhances temperatures whereas it reduces nanoparticle concentration boundary layer thickness. Increasing thermophoretic parameter increasing momentum (hydrodynamic) boundary layer thickness and nanoparticle boundary layer thickness whereas it reduces thermal boundary layer thickness. Increasing magnetohydrodynamic body force parameter decelerates the flow whereas it enhances temperature and nano-particle (species) concentrations. The study is relevant to enrobing processes for electric-conductive nano-materials, of potential use in aerospace and other industries.


Introduction
The word "nanotechnology" was probably used for the rst time by the Japanese scientist Norio Taniguchi in 1974. K. Eric Drexler is credited with initial theoretical work in the eld of nanotechnology. The term nanotechnology was used by Drexler in his 1986 book "Engines of creation: The coming era of nanotechnology". Drexler's idea of nanotechnology is referred to as molecular nanotechnology [1]. Earlier the great theoretical physicist Richard Feynman predicted nanotechnology in 1959. In the 1980s and 1990s new nano-materials were discovered and nano uids emerged as a result of the experiments intended to increase the thermal conductivity of liquids. The birth of nano uids is attributed to the revolutionary idea of adding solid particles into uids to increase the thermal conductivity. This innovative idea was put forth by the Scottish physicist J.C. Maxwell as early as 1873.
Nano uids have evolved into a very exciting and rich frontier in modern nano-technology. The excitement can be attributed to the robustness of the concept of nano uid and the plethora of di erent applications of this technology [2]. The properties of nano uid need a lot of ne tuning, many seemingly contradicting studies need clarity and validation. Nano uid have potential applications in micro-electronics, fuel cells, rocket propulsion, and environmental de-toxi cation, spray coating of aircraft wings, pharmaceutical suspensions, medical sprays etc. These applications of nano uid are largely attributable to the enhanced thermal conductivity and Brownian motion dynamics which can be exploited to immense bene t. Nanomaterial work e ciently as new energy materials since they incorporate suspended particles with size as the same as or smaller than the size of de Broglie wave [3].
Recently, nano uid are used in several engineering and industrial applications, including, geothermal reservoir, thermal insulation, oor heating, cancer therapy, nuclear reactor cooling and process industries. The word "nano uid" was rst presented by Choi and Eastman [4], and concluded that heat transfer characteristics in common uids can be increased via suspended nanoparticles. Later on, Buongiorno [5] explored that out of seven slip mechanisms, only Brownian di usion and thermophoresis are important slip mechanism in nano uid. Further, based on his ndings, he proposed a model for mass, momentum and heat transport in nano uid, known as Buongiorno model. Several investigators utilized this model whilst considered di erent types of base uids. Kuznetsov and Nield [6] studied the buoyancy e ects in viscous nano uid induced due to vertical plate. They predicted that heat transfer declined with increase in Brownian motion and thermophoresis parameters. Abdul-Kahar et al. [7] examined the e ects of chemical reaction and thermal radiation on boundary layer ow of Newtonian nano uid over stretching surface.
The study of MHD boundary layer ow of nano uid over wedge has gained considerable attention in the last few years due to its increasing numbers of scienti c, engineering and industrial applications such as MHD pumps, accelerators, generators, high temperature plasmas, cooling of nuclear reactors, biological transportation and drug delivery. The applied magnetic eld is usually used in controlling momentum and heat transfer in the boundary layer ow of di erent uids Turkyilmazoglu [8].
Due to its several practical applications, the boundary layer ow of non-Newtonian uids has gained an immense interest in the recent years. It is well-known that non-Newtonian uids are usually e ective in the manufacturing process of coated sheets, foods, plastic polymers and blood Shit et al. [9]. The complexity of these uids offers a special challenge to engineers, physicist and mathematicians. It is also a common belief that the behavior of theses uids cannot be demonstrated by a single constitutive relation Shehzad et al. [10]. For this purpose, several models have been proposed of which Casson uid model Casson [11] is one of them. Casson uid model is classi ed as non-Newtonian uid based on its solid and liquid interaction behavior. Casson uid possesses shear thinning property, it behaves like solid if the applied shear stress is lesser than yield stress, whereas if applied shear stress exceeds from yield stress, it starts move Abdul Hakeem et al. [12]. Moreover, Casson nano uid has useful applications in polymer processes, manufacturing of plastic or rubber sheets, chemical process equipment, food industries, advanced cooling systems, etc.
The analytical solution of Casson nano uid due to stretching surface in the presence of convective boundary condition was presented by Nadeem et al. [13]. On the other hand, (Imtiaz et al. [14] studied the mixed convection ow of Casson nano uid generated by stretching cylinder under the in uence of convective boundary condition. They noticed that momentum boundary layer thickness dropped with increment in Casson parameters. Iqbal et al. [15] and Mahmood et al. [16] studied theoretically the boundary layer ow of micro polar Casson uid due to stretching surface in the presence and absence of magnetic eld. The numerical solutions were obtained via Kellerbox and Runge-Kutta-Fehelberg schemes, respectively. The above studies considered the nano uid to be electrically non-conducting. However new developments in magnetic nano uid have emerged in recent years which require magnetohydrodynamics to simulate the response of nano uid to applied magnetic elds. Representative systems in which such uids arise include energy devices [17] and medical engineering [18,19]. A number of researchers have simulated various types of multi-physical hydrodynamics problems of magnetic nano uid in di erent congurations, using a diverse range of numerical methods. Qing et al. [20] studied entropy generation in magnetized reactive radiative Casson nano uid over a porous stretching/shrinking surface with successive linearization and Chebyshev spectral collocation methods.The fuselages of aircrafts and hulls of underwater vehicles can also be modeled as cylindrical shells, and the control of vibrations and sound radiation is important. In the low frequency region, active control shows promise for the reduction of vibrations and radiated sound. Very recently Janicki et al. [21] studied that realization of the system for the active control of boundary conditions during the dynamic thermal characterization of electronic components. Gao et al. [22] were the rst authors to introduce passive control of an impinging jet for heat transfer enhancement. The authors found that the addition of triangular tabs at the pipe exit leads to a heat transfer enhancement with an excess of 25 %. Popiel and Boguslawski [23] con rmed that the conguration of the nozzle exit is the most important factor a ecting the heat and mass transfer which occur in the neighborhood of the stagnation point. In spite of these rst very signi cant observations, there are only a few stud-ies dedicated to heat and mass transfer enhancement using jet passive control. Passive control techniques [24,25], which are commonly used to improve jet self-induction and mixing processes. Soomro et al. [26] demonstrate Passive control of nanoparticle due to convective heat transfer over a stretching surface. This recent studies (passive boundary condition) provides one of the motivations for the present research. Nadeem et al. [27,28] presented Optimal Homotopy analysis solutions of micropolar ow in rotating parallel plates. Nadeem et al. [29] explored the convective heating e ects of MHD oblique ow of a Walter's B uid model. Rana et al. studied heat transference in viscoplastic boundary layer ow over a stretching sheet with convective conditions [30,31]. Mehmood et al. [32] investigated the problem of oblique stagnation point ow using Je ery nano uid as a rheological uid model. Mehmood et al. [33] utilized hematite as a heat enhancing agent to investigate the non-aligned stagnation ow and heat transfer of an Ethylene-Glycol and water based nano uid towards an extending surface. They observed that Ethylenebased nano uids have higher local heat ux as compared with water-based nano uids. Mehmood et al. [34] studied mixed convection and thermal radiation on non-aligned Casson uid over a stretching surface. Governing equations of the problem were solved by shooting method coupled with Runge-Kutta-Fehlberg integration technique. Nadeem et al. [35] deliberated steady stagnation point ow of Je ery uid towards a stretching surface. Tabassum et al. [36] explored the in uence of temperature dependent viscosity on Oblique ow of micro polar nano uid. Mahmood et al. [37] analyzed the micro rotation e ects on mixed convection ow induced by a stretching sheet. They carried out numerical investigation by employing Runge-Kutta Fehlberg scheme coupled with shooting technique. M.R. Eid and K.L. Mahny [57], study the heat transfer e ects in Sisko nano uid in the presence of porous medium over a nonlinearly stretching sheet with heat generation/absorption e ects. Eid, M.R. and S.R. Mishra [58], analyzed the non-Newtonian uid ow over a permeable non-linear stretching vertical surface with heat and mass uxes. A.M. Rashad [59] studied the Impact of thermal radiation on MHD slip ow of a Ferro uid over a nonisothermal wedge.
The above studies were generally con ned to internal transport. However external boundary layer convection ows also nd applications in many technological systems including enrobing polymer coating processes, heat exchanger design, solar collector architecture etc. Prasad et al. [38] studied two-dimensional nano uid boundary layer ow from a spherical geometry embedded in porous media with a nite di erence scheme. Subba Rao et al. [39] investigated computationally the heat transfer in magnetized non-Newtonian uid boundary layer ow from an isothermal sphere with Soret and Dufour e ects. Makanda et al. [40] analyzed the radiative heat ux e ect on hydromagnetic dissipative Casson slip uid ow from a horizontal circular cylinder in porous media. Beg et al. [41] derived both homotopy and Adomian decomposition numerical solutions for transient stagnation-point heat and mass transfer from a rotating sphere.
The present work, motivated by applications in enrobing dynamics of magnetic nanomaterial's [42], examines theoretically and computationally the steady-state transport phenomena in magnetohydrodynamic Casson nano uid ow from an isothermal sphere with thermal slip. Mathematical modelling is developed to derive the equations of continuity, momentum, energy and species conservation, based on the Buongiorno nano uid model [6]. The partial di erential boundary layer equations are then transformed into a system of dimensionless non-linear coupled di erential boundary layer equations, which is solved with the robust second order accurate Keller box implicit nite di erence method. The present work extends signi cantly earlier simulations of Hussain et al. [43] (who consider an exponentially stretching surface) to the case of a horizontal circular cylinder with thermal slip condition. An extensive parametric analysis of the in uence of a number of parameters (Brownian motion, thermophoresis, Casson non-Newtonian, thermal slip, stream wise coordinate) on thermo-di usive characteristics is conducted. The simulations are also relevant to calendaring in pseudo-plastic materials fabrication [44].

Magnetohydrodynamic Nanofluid Slip Model
We examine magneto-hydrodynamic (MHD) steady buoyancy-driven convection heat transfer ow of Casson nano uid from an isothermal sphere in the presence of thermal slip. Figure 1 shows the ow model and associated coordinate system. The nano uid uid is taken to be incompressible and a homogenous dilute solution. Magnetic eld of strength, Bo, is applied normally to the ow. The x-coordinate is measured along the circumference of the isothermal sphere from the lowest point and the y-coordinate is measured normal to the surface, and r is the radial distance from symmetric axis to the surface r = a sin x a and 'a' denoting the radius of the sphere. The gravitational acceleration, g acts downwards. Both the isothermal sphere and the uid are maintained initially at the same temperature. Instantaneously they are raised to a temperature Tw > T∞ i.e. the ambient temperature of the uid which remains unchanged. The appropriate constitutive equations for the Casson non-Newtonian model are: in which π = e ij e ij and e ij is the (i,j) th component of deformation rate, π denotes the product of the component of deformation rate with itself, πc shows a critical value of this product based on the non-Newtonian model, µ B the plastic dynamic viscosity of non-Newtonian uid and py the yield stress of uid. The Casson model, although relatively simple, is a robust viscoplastic model and describes accurately the shear stress-strain behavior of certain industrial polymers in which ow is not possible prior to the attainment of a critical shear stress. Unlike the Bingham viscoplastic model which has a linear shear rate, the Casson model has a non-linear shear rate. Casson uid theory was originally propounded to simulate shear thinning (viscosity is reduced with greater shear rates) liquids containing rod-like solids and is equally popular in analysing inks, emulsions, food stu s (chocolate melts), certain gels and paints [45]. More recently it has been embraced in advanced polymeric ow processing [46]. Incorporating the Casson terms and applying the Buongiorno nano uid model, the governing conservation equations, in primitive form, for the regime under investigation i.e. mass continuity, momentum, energy and species, can be written as follows: The boundary conditions imposed at the sphere surface and in the free stream are: At Here u and v are the velocity components in the xand y-directions respectively, ν -the kinematic viscosity of the electrically-conducting nano uid, β -is the non-Newtonian Casson parameter respectively, ρ f is the density of uid, σ is the electrical conductivity of the nano uid, α-the thermal di usivity of the nano uid, Tthe temperature, respectively. Furthermore τ = (ρc) p (ρc) f is the ratio of nanoparticle heat capacity and the base uid heat capacity, D B is the Brownian di usion coe cient; D T is the thermophoretic di usion coe cient; k is the thermal conductivity of nano uid; Tw and C∞ are the ambient uid temperature and concentration, Cp is the speci c heat capacity; Ω and Ω* are the coe cients of thermal expansion and concentration expansion, respectively.
The stream function ψ is de ned by the Cauchy-Riemann equations, u = ∂ψ ∂y and v = − ∂ψ ∂x , and therefore, the continuity equation is automatically satis ed. In order to write the governing equations and the boundary conditions in dimensionless form, the following nondimensional quantities are introduced [39].
The transformed boundary layer equations for momentum, energy and concentration emerge as: The corresponding transformed dimensionless boundary conditions are: At where Pr = γ/α is Prandtl number; M = is the concentration to thermal buoyancy ratio parameter, S T = h k aGr − / is the thermal slip parameter. All other parameters are de ned in the nomenclature. The skin-friction coe cient (sphere surface shear stress function), the local Nusselt number (heat transfer rate) and Sherwood number (mass transfer rate) can be de ned using the transformations described above with the following expressions: Gr Gr Gr

Numerical Solution With Keller Box Implict Method
The strongly coupled, nonlinear conservation equations do not admit analytical (closed-form) solutions. An elegant, implicit di erence nite di erence numerical method developed by Keller [47] is therefore adopted to solve the general ow model de ned by eqns. (8)-(10) with boundary conditions (11). This method is especially appropriate for boundary layer ow equations which are parabolic in nature. It remains one of the most widely applied computational methods in viscous uid dynamics. Recent problems which have used Keller's method include radiative magnetic forced convection ow [48], stretching sheet hydromagnetic ow [49], magnetohydrodynamic Falkner-Skan "wedge" ows [50], magneto-rheological ow from an extending cylinder [51], Hall magneto-gas dynamic generator slip ows [52] and radiative-convective Casson slip boundary layer ows [53]. Keller's method provides unconditional stability and rapid convergence for strongly non-linear ows. It involves four key stages, summarized below.

1)
Reduction of the N th order partial di erential equation system to N rst order equations 2) Finite di erence discretization of reduced equations 3) Quasilinearization of non-linear Keller algebraic equations 4) Block-tridiagonal elimination of linearized Keller algebraic equations Stage 1: Reduction of the N th order partial di erential equation system to N rst order equations Equations (8)- (10) and (11) subject to the boundary conditions are rst written as a system of rst-order equations. For this purpose, we reset eqns. (6)-(7) as a set of simultaneous equations by introducing the new variables.
p Le Where primes denote di erentiation with respect to η.
In terms of the dependent variables, the boundary conditions become At η = :

Fig. 2: Keller Box element and boundary layer mesh
A two-dimensional computational grid (mesh) is imposed on the ξ -η plane as sketched in Fig. 2. The stepping process is de ned by: Where kn and h j denote the step distances in the ξ (stream wise) and η (span wise) directions respectively.

Stage 3: Quasilinearization of non-linear Keller algebraic equations
If we assume f n− j− , u n− j− , v n− j− , p n− j− , s n− j− , t n− j− , to be known for Equations (30) -(36) comprise a system of 6J+6 equations for the solution of 6J+6 unknowns f n j , u n j , v n j , p n j , s n j , t n j ,, j = 0, 1, 2 . . . , J. This non-linear system of algebraic equations is linearized by means of Newton's method as elaborated by Keller [47] and recently this method is used by Subba Rao et al., [54,55].

Stage 4: Block-tridiagonal elimination of linear Keller algebraic equations
The linearized version of eqns. (30) - (36) can now be solved by the block-elimination method, since they possess a block-tridiagonal structure since it consists of block matrices. The complete linearized system is formulated as a block matrix system, where each element in the coe cient matrix is a matrix itself. Then, this system is solved using the e cient Keller-box method. The numerical results are a ected by the number of mesh points in both directions. After some trials in the η-direction (radial coordinate) a larger number of mesh points are selected whereas in the ξ direction (tangential coordinate) signi cantly less mesh points are utilized. ηmax has been set at 12 and this de nes an adequately large value at which the prescribed boundary conditions are satis ed. ξ ma x is set at 1.0 for this ow domain. Mesh independence testing is also performed to ensure that the converged solutions are correct. The computer program of the algorithm is executed in MATLAB running on a PC.

Keller Box Method (KBM) Numerical Results and Discussion
Comprehensive solutions have been obtained with KBM and are presented in Figs. 3 to 11. The numerical problem comprises three dependent thermo-uid dynamic variables (f , θ, ϕ) and seven multi-physical control parameters, Pr, Le, β, M, Nb, Nt, S T ,. The in uence of stream wise space variable ξ is also investigated. In the present computations, the following default parameters are prescribed (unless otherwise stated): Pr = 7.0, Le = 5.0, β = 1.5, M = 0.01, Nb = 0.02 = Nt, S T = 1.0, ξ = 1.0. Figs. 3(a-c) illustrate the e ect of the Casson viscoplastic parameter, β on velocity f , temperature (θ) and concentration (ϕ) pro les. With increasing, β values, initially close to the sphere surface, Fig. 3(a) shows that the ow is strongly decelerated. However, further from the surface, the converse response is induced in the ow. This may be related to the necessity for a yield stress to be attained prior to viscous ow initiation in viscoplastic shearthinning nano uids. Within a short distance of the sphere surface, however a strong acceleration is generated with greater Casson parameter. This serves to decrease momentum boundary layer thickness e ectively. The viscoplastic parameter modi es the shear term f /// in the momentum boundary layer equation (8) with an inverse factor, 1/β, and e ectively assists momentum di usion for β >1. This leads to a thinning in the hydrodynamic boundary layer and associated deceleration. The case β = 0 which corresponds to a Newtonian uid is not considered. An increase in viscoplastic parameter however decreases both temperature and nano-particle concentration magnitudes throughout the boundary layer, although the reduction is relatively weak. Thermal and nanoparticle concentration boundary layer thickness are both suppressed with greater viscoplasticity of the nano uid.
Figs. 4(a-c) depicts the e ect of Prandtl number (Pr) on the velocity f , temperature (θ) and nanoparticle concentration(ϕ)distributions with transverse coordinate (η). Fig. 4(a) shows that with increasing Prandtl number there is a strong deceleration in the ow. The Prandtl number expresses the ratio of momentum di usion rate to thermal di usion rate. When Pr is unity both momentum and heat di use at the same rate and the velocity and thermal boundary layer thicknesses are the same. With Pr > 1 there is a progressive decrease in thermal di usivity relative to momentum di usivity and this serves to retard the boundary layer ow. Momentum boundary layer thickness therefore grows with Prandtl number on the surface of the isothermal sphere. It is also noteworthy that the peak velocity which is achieved close to the sphere surface is systematically displaced closer to the surface with greater Prandtl number. The asymptotically smooth pro les of velocity which decays to zero in the free stream, also conrm the imposition of an adequately large in nity boundary condition. Fig. 4(b) indicates that increasing Prandtl number also suppresses temperatures in the boundary layer and therefore reduces thermal boundary layer thickness. Prandtl number is inversely proportional to thermal With greater Prandtl number, thermal conductivity is reduces and this inhibits thermal conduction heat transfer which cools the boundary layer.
The monotonic decays in Fig. 4(b) are also characteristic of the temperature distribution in curved surface boundary layer ows. Inspection of Fig. 4(c) reveals that increasing Prandtl number strongly elevates the nano-particle concentration magnitudes. Infact a concentration overshoot is induced near the sphere surface. Therefore while thermal transport is reduced with greater Prandtl number, species di usion is encouraged and nano-particle concentration boundary layer thickness grows.
Figs. 5(a-c) illustrate the evolution of velocity, temperature and concentration functions with a variation in the Lewis number, is depicted. Lewis number is the ratio of thermal di usivity to mass (nano-particle) species diffusivity. Le = 1 which physically implies that thermal di usivity of the nano uid and species di usivity of the nanoparticles are the same and both boundary layer thicknesses are equivalent. For Le < 1, mass di usivity exceeds thermal di usivity and vice versa for Le > 1. Both cases are examined in Figs. 5(a-c). In Fig. 5(a), a consistently weak decrease in velocity accompanies an increase in Lewis number. Momentum boundary layer thickness is therefore increased with greater Lewis number. This is sustained throughout the boundary layer. Fig. 5(b) shows that increasing Lewis number also depresses the temperature magnitudes and therefore reduces thermal boundary layer thickness. Therefore judicious selection of nano-particles during doping of polymers has a pronounced in uence on velocity (momentum) and thermal characteristics in enrobing ow, since mass di usivity is dependent on the nature of nano-particle species in the base uid. Fig. 5(c) demonstrates that a more dramatic depression in nanoparticle concentration results from an increase in Lewis number over the same range as Figs. 5(a, b). The concentration pro le evolves from approximately linear decay to strongly parabolic decay with increment in Lewis number.
Figs. 6(a-c) exhibit the pro les for velocity, temperature and concentration, respectively with increasing magnetic body force parameter, M. M = σB a γρGr / de nes the relative in uence of magnetic Lorentzian drag force to viscous hydrodynamic force in the ow. With M < 1 viscous force dominates the magnetic force and the magnetohydrodynamic e ect is weak. However Fig. 6(a) shows that even a weak increase in M induces a marked deceleration in the boundary layer ow and simultaneously thickens the momentum boundary layer. Reverse ow or separation is never induced as testi ed to by the consistently positive values of velocity everywhere transverse to the sphere surface. This concurs with many other studies of magnetized nano uid convection since the radial magnetic eld acts to generate a perpendicular drag force which acts to decelerate the ow. Figs. 6(b) and 6(c) indicate that the dominant e ect of greater magnetic parameter is to elevate both temperatures and nano-particle concentrations. The supplementary work expended in dragging the viscoplastic nano uid against the action of the magnetic eld manifests in kinetic energy dissipation. This energizes the boundary layer since the kinetic energy is dissipated as thermal energy, and this further serves to agitate improved species di usion. As a result both thermal and nano-particle (species) concentration boundary layer thicknesses are increased.
Figs. 7(a-c) depict the response in velocity f , temperature (θ) and concentration (ϕ) functions to a variation in the Brownian motion parameter (Nb). Increasing Brownian motion parameter physically correlates with smaller nanoparticle diameters. Smaller values of Nb corresponding to larger nanoparticles, and imply that surface area is reduced which in turn decreases thermal conduction heat transfer to the sphere surface. This coupled with enhanced macro-convection within the nano uid energizes the boundary layer and accelerates the ow as observed in Fig. 7(a). Similarly the energization of the boundary layer elevates thermal energy which increases temperature in the viscoplastic nano uid. Fig. 7(c) however indicates that the contrary response is computed in the nanoparticle concentration eld. With greater Brownian motion number species di usion is suppressed. E ectively therefore momentum and nanoparticle concentration boundary layer thickness is decreased whereas thermal boundary layer thickness is increased with higher Brownian motion parameter values. Figs. 8(a-c) illustrates the e ect of the thermophoresis parameter (Nt) on the velocity f , temperature (θ) and concentration (ϕ) distributions, respectively. Thermophoretic migration of nano-particles results in exacerbated transfer of heat from the nano uid regime to the sphere surface. This de-energizes the boundary layer and inhibits simultaneously the di usion of momentum, manifesting in a reduction in velocity i.e. retardation in the boundary layer ow and increasing momentum (hydrodynamic) boundary layer thickness, as computed in Fig. 8(a). Temperature is similarly decreased with greater thermophoresis parameter (Fig. 8(b)). Conversely there is a substantial enhancement in nano-particle concentration (and species boundary layer thickness) with greater Nt val- ues. Similar observations have been made by Kunetsov and Nield [14] and Ferdows et al. [19] for respectively, both non-conducting Newtonian and electrically-conducting Newtonian ows.
Figs. 9(a-c) illustrate the variation of velocity, temperature and nano-particle concentration with transverse coordinate (η), for di erent values of thermal slip parameter (S T ). Thermal slip is imposed in the augmented wall boundary condition in eqn. (11). With increasing thermal slip less heat is transmitted to the uid and this deenergizes the boundary layer. This also leads to a general deceleration as observed in Fig. 9(a) and also to a more pronounced depletion in temperatures in Fig. 9(b), in particular near the wall. The e ect of thermal slip is progressively reduced with further distance from the wall (curved surface) into the boundary layer and vanishes some distance before the free stream. It is also apparent from Fig. 9(c) that nanoparticle concentration is reduced with greater thermal slip e ect. Momentum boundary layer thickness is therefore increased whereas thermal and species boundary layer thickness are depressed. Evidently the non-trivial responses computed in gs. 9a-c further emphasize the need to incorporate thermal slip e ects in realistic nano uid enrobing ows.
Figs. 10(a-c) present the e ects of the buoyancy ratio N on the velocity, temperature and nano-particle concentration pro les. In general, increases in the value of N have the prevalent to cause more induced ow along the cone surface. This behavior in the ow velocity increases in the uid temperature and volume fraction species as well as slight decreased in the thermal and species boundary layers thickness as N increases.
Figs. 11(a-c) present the distributions for velocity, temperature and concentration elds with stream wise coordinate ξ , for the viscoplastic nano uid ow. Increasing ξ values correspond to progression around the periphery of the sphere, from the leading edge (ξ = 0). As ξ increases, there is a weak deceleration in the ow (Fig. 11(a)), which is strongest nearer the sphere surface and decays with distance into the free stream. Conversely there is a weak elevation in temperatures ( Fig. 11(b)) and nano-particle concentration magnitudes (Fig. 11(c)) with increasing stream wise coordinate.

Conclusions
A theoretical study has been conducted to simulate the magnetohydrodynamics viscoplastic nano uid boundary layer ow in enrobing processes from an isothermal sphere with thermal slip using the Buongiorno formulation. The transformed momentum, heat and species boundary layer equations have been solved computationally with Keller's nite di erence method. The present study has shown that:

1.
Increasing viscoplastic (Casson) parameter decelerates the ow and also decreases thermal and nanoparticle concentration boundary layer thickness.

2.
Increasing Prandtl number retards the ow and also decreases temperatures and nano-particle concentration values.

3.
Increasing stream wise coordinate decelerates the ow whereas it enhances temperatures and species (nano-particle) concentrations.

4.
Increasing thermal slip strongly reduces velocities, temperatures and nano-particle concentrations.

5.
Increasing Brownian motion accelerates the ow and enhances temperatures whereas it reduces nanoparticle concentration boundary layer thickness. 6.
Increasing thermophoretic parameter increasing momentum (hydrodynamic) boundary layer thickness and nanoparticle boundary layer thickness whereas it reduces thermal boundary layer thickness. 7.
Increasing magnetohydrodynamic body force parameter decelerates the ow whereas it enhances temperature and nano-particle (species) concentrations.
The current study has explored an interesting viscoplastic model for electro-conductive nano materials which are currently of interest in aerospace coating applications. Time-dependent e ects have been neglected. Future studies will therefore address transient enrobing viscoplastic nano uid transport phenomena for alternative geometries (wedges, cones, Cylinder, Plates), also of interest in aerospace materials fabrication and will be communicated imminently.