Thermophoresis and Brownian motion effects on

Journal of Molecular Liquids 221 (2016) 733–743

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution O.D. Makinde a,⁎, I.L. Animasaun b a b

Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa Department of Mathematical Sciences, Federal University of Technology, Akure, Ondo State, Nigeria

a r t i c l e

i n f o

Article history: Received 3 May 2016 Received in revised form 22 May 2016 Accepted 13 June 2016 Available online 16 June 2016 Keywords: Bioconvection MHD flow Nanofluid Thermophoresis Brownian motion Paraboliod of revolution Quartic chemical reaction

a b s t r a c t In this paper, the combined effects of buoyancy force, Brownian motion, thermophoresis and quartic autocatalytic kind of chemical reaction on bioconvection of nanofluid containing gyrotactic microorganism over an upper horizontal surface of a paraboloid of revolution are analyzed. The case of unequal diffusion coefficients of reactant A (bulk-fluid) and reactant B (catalyst at the surface) in the presence of nonlinear thermal radiation is presented. The fluid under consideration contains random movement of microscopic nanoparticles (i.e. Brownian motion). Moreover, it is assumed that the thermo-physical of the nanofluid does not vary with volume fraction. The governing nonlinear partial differential equations are obtained and transformed into a system of coupled nonlinear ordinary differential equations using similarity transformations and then tackled numerically using the Runge–Kutta fourth order method with shooting technique. Good agreement is obtained between the solutions of RK4SM and MATLAB bvp5c for a limiting case. The influence of pertinent parameters are illustrated graphically and discussed. The presence of Brownian motion and thermophoresis decreases and increases concentration of reactant B (heterogeneous) respectively. Brownian motion boosts concentration of bulk fluid while thermophoresis reduces it. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Thermophoresis is of practical importance in many industrial applications, such as in aerosol collection (thermal precipitator), micro contamination control, removing small particles from gas streams, nuclear reactor safety. Tsai et al. [1] pinpointed the application of the knowledge of thermophoresis in studying the particulate material deposition on turbine blades and determination of exhaust gas particle trajectories from combustion devices. Kowalkowski et al. [2] stated that the socalled thermal field flow fractioning (TFFF) in industry for the separation of (large) molecules or small particles from their solvent is built on the practical knowledge of thermophoresis. Aerosol can be described as a suspension of fine liquid droplets or solid particles in air as either a fog or a smoke. Practically speaking, small particle(s) can be driven from a hot surface towards a cold surface. For instance, small particles (e.g. dry dust) when suspended in a gas (e.g. air) with a temperature gradient, experience a force in the direction of the temperature gradient. The velocity of such particle(s) which drives it from the region of hot surface to the region of cold surface and the force experienced by the ⁎ Corresponding author. E-mail addresses: [email protected] (O.D. Makinde), [email protected] (I.L. Animasaun).

http://dx.doi.org/10.1016/j.molliq.2016.06.047 0167-7322/© 2016 Elsevier B.V. All rights reserved.

suspended particle(s) due to the temperature gradient are called thermophoretic velocity and thermophoretic force respectively. This concept was first observed and reported by John Tyndall in the year 1870. The report was further buttressed by John William Strutt (3rd Baron Rayleigh); see Davis [3]. A common example of thermophoresis is the blackening of glass globe of a kerosene lantern; the temperature gradient established between the flame and the globe drives the carbon particles produced in the combustion process towards the globe where they deposit; Talbot et al. [4]. Animasaun [5] reported the effects of thermophoresis within boundary layer formed in Casson fluid flow along a vertical porous plate. In the report of Sandeep and Sulochana [6] on heat and mass transfer in thermophoretic radiative hydromagnetic nanofluid flow, it was concluded that thermophoretic parameter may not show significant influence on velocity fields but it proves mass transfer rate. In another related research, Tsai [7] stated explicitly that the deposition mechanisms for particles include Brownian diffusion, convection, thermophoresis and other mechanisms, i.e. electrophoresis. Random motion of particle(s) that is suspended in a fluid (either gas or liquid) is known as Brownian motion. In 1785, Jan. Ingenhousz noticed the irregular movement of coal dust on the surface of alcohol. Historically, his description and explanation can be referred to as the first report on the concept of Brownian motion. Norbert Wiener explained

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O.D. Makinde, I.L. Animasaun / Journal of Molecular Liquids 221 (2016) 733–743

what is called Wiener process which has applications in mathematical sciences. In Physics, Wiener process is used to study the diffusion of minute particles suspended in fluid and other types of diffusion. In the year 1872, literature reveals that Robert Brown (Scottish Botanist from Mantrose, Scotland) reported what he discovered while looking through a microscope at particles trapped in cavities inside pollen grains in water. Thereafter, Albert Einstein presented equations describing Brownian motion. Thereafter, Einstein and Cowper [8] reported his investigation on the theory of the Brownian movement. Jang and Choi [9] embarked on a research with the major aim to know the fundamental role of the movement of nanoparticles in a nanofluids. It was found that the Brownian motion of nanoparticles at the molecular and nanoscale level is a major key mechanism governing the thermal behavior of nanoparticle in nanofluids. Buongiorno [10] considered inertia, Brownian diffusion, thermophoresis, diffusiophoresis, magnus effect, fluid drainage and gravity as seven slip mechanisms that could produce a relative velocity between the nanoparticles and the base fluid. In the article, it is concluded that out of all these seven slip mechanisms, only Brownian diffusion and thermophoresis are important slip mechanisms in nanofluids. Hence, explained the concepts of Brownian diffusion as the random motion of nanoparticles within the base fluid and thermophoresis as diffusion of nanoparticles under the effect of a temperature gradient. In view of this, the effects of Brownian motion of nanoparticles and thermophoresis are considered in this research. Zaimi et al. [11] investigated steady boundary layer flow of a nanofluid past a nonlinearly permeable stretching/shrinking, explained the effects of thermophoresis and Brownian motion parameters then concluded that a rising value in Nb and the decreasing in Nt produce a decrease in the nanoparticle concentration. Chemical reaction can be explained as an interaction between two or more chemicals which produces either one or more new chemical compounds. Many chemical reactions actually require heat and accelerator (i.e. catalyst). A chemical reaction in which catalyst is in the same phase (i.e. in the same state of matter) as the reactant(s) is/are known as homogeneous catalytic reaction. Reactions between two gases, between two liquids and mixture of household cooking gas with oxygen gas leading to flame are typical examples of homogeneous catalytic reactions. In heterogeneous catalytic reaction, catalyst and reactants are in different phases (i.e. different states of matter). Examples of heterogeneous catalytic reactions are chemical reactions between a gas and a liquid, a gas and a solid, and liquid and a solid. In addition to this example, a chemical reaction in which one or more reactants undergo chemical change at an interface (i.e. on the surface of a solid catalyst) is called heterogeneous catalytic reaction. When fluid moves along a surface, a thin layer is formed in the vicinity of a surface bounding the fluid; Ludwig Prandtl called it “boundary layer”. Chaudhary and Merkin [12] introduced a simple model for homogeneous-heterogeneous reactions in stagnation-point boundary-layer flow in which the homogeneous (bulk) reaction is assumed to be given by isothermal cubic autocatalator kinetics and the heterogeneous (surface) reaction by first order kinetics. Scott [13] considered quadratic and cubic autocatalytic together with rates of their chemical reaction; then coupled the equation with the diffusion of the reactants through a permeable boundary from an external reservoir where the concentrations are held constant. Motsa et al. [14] investigated the case of equal diffusion coefficients of chemical species A and B in the stagnation point nanofluid flow in the presence of homogenous-heterogeneous reactions using successive linearization method (SLM). Sandeep et al. [15] explained the effects of induced magnetic field on cubic autocatalytic reaction which often occurs in viscoelastic fluid flows towards a stagnation point. In most cases when the concentration of homogeneous species B is highly substantial, the cubic autocatalator kinetics may not be sufficient; hence isothermal quartic autocatalytic reaction is needed in order to unravel the motion and behavior of the flow within boundary layer. The term “nanofluid” was proposed by Choi [16], referring to dispersions of nanoparticles in the base fluid such as water. Nguyen et al. [17]

reported that sedimentation, shear stress and agglomeration are few problems that can be referred to as an agent which may limit the mixtures of nanoparticles in a nanofluid. In nanofluid flow, the existence of free-swimming organisms. In a research carried out by James Henry Platt, the physics of streaming patterns observed in dense cultures of free-swimming organisms was termed bioconvection. According to Platt [18], the moving polygonal patterns in dense cultures of Tetrahymena and other ciliates and flagellates look like “Benard cells” but are not due to thermal convection. There exist many bacteria (organisms) and it is a well-known fact that many bacteria can be damaged and sometimes killed if exposed to high temperature. In contrary, thermophile is an organism commonly found in various heated regions of the earth. Ghorai and Hill [19] further explained that bioconvection is the term used to describe the phenomenon of spontaneous pattern formation in suspensions of micro-organisms such as bacteria and algae. Bioconvection can now be described as the macroscopic convective motion of fluid caused by the density gradient and is created by collective swimming of motile microorganisms. Like natural convection, bioconvection is caused by unstable density stratification. Kuznetsov and Avramenko [20] explained that if bioconvection develops, it enhances mixing and slows down the settling of the particles which may be important in pharmaceutical applications. Considering the contribution of French physicist Jean Claude Eugène Péclet to the body of knowledge on advective and diffusive transport rates, Khan and Makinde [21] investigated nanofluid bioconvection due to gyrotactic microorganisms and remarked that these self-propelled motile microorganisms increase the density of the base fluid by swimming in a particular direction. Related literatures on nanofluid in the presence or absence of Lorentz force can be found in [22–28]. Sandeep et al. [29] reported stagnation point flow, heat and mass transfer behavior of MHD Jeffrey nanofluid in the absence of bioconvection. Recently, Raees et al. [30] reported that bioconvection in nanofluids has great potential in Colibri microvolumes spectrometer and also to improve the stability of nanofluids. In all the literatures mentioned above, it is worth noticing that various researchers has investigated the flow of nanofluid along horizontal, vertical and inclined surfaces with uniform thickness but there exist no report on the flow of nanofluid containing both spherical nanoparticles and gyrotactic microorganism over a horizontal surface of paraboloid of revolution in the presence of quartic autocatalytic kind of homogeneous-heterogeneous chemical reaction. Moreover, in the industry, this kind of chemical reaction often occurs within the boundary layer formed on the surface in the flow of nanofluid. In addition, various researchers has investigated nanofluid flow along horizontal surface with no consideration of thermophoresis effect and Brownian motion of nanofluid as it flows past horizontal surface of paraboloid of revolution. It is paramount to remark that this contribution to the body of knowledge will enhance efficiency and productivity in the industry. 2. Description of the boundary layer flow and formulation of governing equation The mathematical formulation which models the boundary layer flow of nanofluid over an upper surface of an object that can be described as paraboloid of revolution [31,32,34] in the presence of space dependent internal heat source and nonlinear thermal radiation is considered. It is assumed that quartic autocatalytic chemical reaction with catalyst decay between reactants A and B occurs as the steady two-dimensional nanofluid flows. Since chemical reactant B is of higher concentration than that of cubic schemes described in [35] and [12], hence the suitable schemes can be described as isothermal quartic autocatalytic reaction and first order reaction. The concentration of chemical reactant A is “a”. The concentration of chemical reactant b is ℓ. The nanofluid flow under consideration is assumed to occupy the domain A ðx þ bÞ

1−m 2

≤yb∞ as shown in Fig. 1. The immediate fluid layers on the


735

Fig. 1. The coordinate system of nanofluid (containing spherical nanoparticles and gyrotactic-microorganisms) flows past horizontal surface of paraboloid of revolution.

upper surface of horizontal paraboloid of revolution are stretched with velocity Uw = Uo(x + b)m. Moreover, the horizontal paraboloid of revolution is assumed to be non-porous and non-melting. In this case, xaxis is taken along the direction of the horizontal surface and y-axis is normal to it. The origin of x-axis and y-axis are not the starting point of the flow. Hence, the starting point of the flow at the slot is a function 1−m

y ¼ Aðx þ bÞ 2 . Consequently, the velocity along x-direction, velocity along y-direction, temperature, concentration of reactant A, concentration of reactant B and density of motile microorganisms at the horizontal surface are u(x, y), v(x, y), T(x, y), a(x, y), ℓ(x, y) and N(x, y) respectively. As shown in Fig. 1, the temperature of the horizontal sur1−m

face with variable thickness is of the form T w ¼ Aðx þ bÞ 2 ; here parameter “m” is known as velocity power index and “b” is known as parameter related to stretching sheet. Meanwhile, the temperature at free stream is a constant function of temperature. It is assumed that the nanofluid is diluted so that the bioconvection instability can be avoided. It is also assumed that the spherical nanoparticles are suspended in the nanofluid using surfactant and hence, prevents nanoparticles from agglomeration. The microorganisms are assumed to have constant distributions on the horizontal surface of paraboloid of revolution. It is worth mentioning that the base fluid is water so that the gyrotactic microorganisms can survive. In this study, it is also assumed that the existence of spherical nanoparticles has few effects on the motion of the microorganisms. Following the idea and mathematical formulation stated in Kuznetsov and Nield [36], Kuznetsov and Nield [37] and Raees et al. [30] the flux of microorganisms through the boundaries is equal to zero. Mathematically, this can be expressed as ∇ j ¼ 0:

ð1Þ

Adopting the concept of homogeneous-heterogeneous reaction model proposed by Chaudhary and Merkin [12] and Lynch [38], isothermal quartic autocatalytic reaction within the boundary layer when chemical reactant B is of high concentration at the surface is proposed as A þ 3B→4B;

rate of chemical reaction ¼ k1 aℓ3 :

ð2Þ

and on the horizontal surface of paraboloid of revolution in the presence of catalyst, there exist single isothermal first order reaction of the

form A→B;

rate of chemical reaction ¼ ks a;

ð3Þ

where “a” and “ℓ” are the concentrations of chemical reactants A and B. Here, k1 and ks are known as the reaction rate coefficient which may not be actually referred to as a constant because it includes all the likely parameters that may affect reaction rate except concentration which we have explicitly accounted for in Eqs. (2) and (3). Following Pedle et al. [39] and Kuznetsov [40], it is assumed that the random component of microorganisms motion in the nanofluid can be approximated by a diffusion process as ~ −Dm ∇N; j ¼ Nυ þ N υ

~¼ υ

ch W c ∇a: Δa

ð4Þ

where a is the concentration of the homogeneous chemical reactant A, υ is the velocity vector of the flow with u and v being the velocity ~ is the average components in the x− and y− directions respectively, υ swimming velocity vector of oxytactic microorganisms, ch is the chemotaxis constant, Wc is the maximum cell swimming speed and Dm is the ! diffusivity of microorganisms. Considering a magnetic field ( B ) is applied perpendicular to the flow of an electrically conducting fluid with ! velocity vector ( V ) in the x direction. Following the illustrations of ! Devi and Prakash [33] the interaction of magnetic field ( B ) with the in! ! duced current ( j ), the Lorentz force ( F ) is of the form ! ! ! F ¼ j B ¼ σ nf ½BðxÞ2 u

ð5Þ

2.1. Buoyancy-induced model for fluid flow past a surface that can be described as upper paraboloid of revolution (variable thickness when (mb 1)) According to the report of Joseph Valentine Boussinesq in 1987 and 1903, if ρ∞ denotes the density of the fluid at free stream where free stream temperature is T∞. For temperature difference between the wall (Tw) and free stream layer (T∞), the density model is presented as

ρ ¼ ρ∞ ½1−βðT−T ∞ Þ

where

T w NT ∞ :

ð6Þ

736


Here β is known as the coefficient of volume expansion. Considering moderate difference in temperature, this often leads to negligible change in density δρ ¼ jρ−ρ∞ j Substituting ρ from Eq. (6) δρ ¼ jρ∞ ½1−βðT−T ∞ Þ−ρ∞ j δρ ¼ j−ρ∞ βðT−T ∞ Þj δρ ¼ ρ∞ βðT−T ∞ Þ

ð7Þ

For more details on this, see Boussinesq [41,42,44]. The buoyancy term g ðρ−ρ∞ Þ≈gδρ or g ðρ−ρ∞ ÞCosðα Þ≈gδρ

ð8Þ

is of the same order of magnitude as the inertia term or the viscous term; so it is not negligible when formulating the buoyancy model suitable to describe fluid flow over a vertical surface with uniform thickness or inclined surface with uniform thickness. Where α is known as the angle of inclination. Upon equating the approximation in Eq. (8) with Eq. (7), the buoyancy term can be simplified as g ðρ−ρ∞ Þ ¼ gβρ∞ ðT−T ∞ Þ; or g ðρ−ρ∞ ÞCosðα Þ ¼ gβρ∞ ðT−T ∞ ÞCosðα Þ:

ð9Þ

In free convection over a vertical surface with uniform thickness or inclined surface with uniform thickness, the body force term (buoyancy force) and pressure gradient term are algebraically incorporated into the momentum equation as − −

∂p þ ρg x ¼ 0 þ gβρ∞ ðT−T ∞ Þ; ∂x

or

∂p þ ρg x ¼ 0 þ gβρ∞ ðT−T ∞ ÞCosðα Þ: ∂x

ð10Þ

¼ 0). It is known that Here, the pressure gradient term is zero (i.e. ∂p ∂x the forced convection requires excessive stretching of some layers of the fluid; this might either be layers of fluid situated at the free stream or fluid layers adjacent to the wall. For forced convection along a vertical surface with uniform thickness or inclined surface with uniform thickness, the body force term (buoyancy force) and pressure gradient term are algebraically incorporated into the momentum equation as −

∂p du∞ þ 0: þ ρg x ¼ u∞ dx ∂x

ð11Þ

Here, ρgx = 0. Also, for mixed convection along a vertical surface with uniform thickness or inclined surface with uniform thickness, the body force term (Buoyancy force) together with pressure gradient term are algebraically incorporated into the momentum equation as ∂p du∞ þ gβρ∞ ðT−T ∞ Þ or þ ρg x ¼ u∞ dx ∂x ∂p du∞ þ gβρ∞ ðT−T ∞ ÞCosðα Þ: − þ ρg x ¼ u∞ dx ∂x −

ð12Þ

Oberbeck [43,44] presented a model which governs the flow of a classical linearly viscous fluid when pressure changes but the volume is constant (i.e. a kind of isochoric motion), but its volume could change due to changes in temperature. Thereafter, Boussinesq [42] delineated on the theory known as Boussinesq approximation. It states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g (the acceleration due to gravity). The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight

appreciably different between any two layers of fluids. Within past few years, modification of theory or model has been something researchers in the field of boundary layer analysis together with heat and mass transfer embrace and investigate in order to test the validity as presented in academic discourse. In order to demonstrate the effects of temperature dependent physical properties for natural convection in a concentric annulus, a modified Boussinesq approximation was developed and used successfully by Shahraki [45]. Diaz et al. [46] reported that in the Boussinesq approximation the main coupling term is the generation of momentum due to temperature gradients together with viscous heating (heat production due to internal friction). For proper and correct analysis of fluid flow along vertical surface with a temperature lesser than that of the free stream (i.e. Tw b T∞), Boussinesq approximation model was modified and discussed explicitly by Animasaun [47]. It is important to recall that Tw N T∞ in this case of fluid flow over an upper horizontal surface of paraboloid of revolution as shown in Fig. 1. Moreover, mb 1 in the study of boundary layer flow over a surface with variable thickness corresponds with the flow along upper surface of horizontal paraboloid of revolution. This calls for modification in order to avoid misleading effect of Grashof number Grm and bioconvection Rayleigh number Rb. It is worth mentioning that the upper horizontal surface of paraboloid of revolution resembles the pointed upper surface of an aircraft and is neither vertical nor inclined surface. In this study, the modified approximation for free convective and heat transfer for flow past horizontal surface of paraboloid of revolution, accurately of the same order with inertial and viscous terms is proposed as −

h i m þ 1 ∂p ∂ mþ1 ∂ ðT−T ∞ Þ þ þ ρg x ¼ gβx gω ρm −ρ f x 2 2 ∂x ∂x ∂x ð13Þ ðN−N∞ Þ þ 0:

ω is known as average volume of a microorganism, ρm is known as density of the motile microorganism and ρf is known as density of the nanofluid. Meanwhile the concentration of reactant A (homogeneous bulk fluid) and concentration of reactant B (catalyst at the surface) does not satisfies aw N a∞, aw b a∞, ℓw b ℓ∞ and ℓw N ℓ∞; where a and ℓ are dimensional concentrations of homogeneous bulk fluid and catalyst at the surface respectively. In view of this, the buoyancy-induced model is not dependent on chemical reaction between reactant species A and B. Moreover, for free convective heat and mass (concentration of single specie where Cw N C∞ or Cw b C∞) transfer within boundary layer formed in a flow over horizontal surface of paraboloid of revolution, the buoyancy induced model can be considered as −

∂p ∂ mþ1 ∂ mþ1 ðT−T ∞ Þ þ ðC−C ∞ Þ þ 0: þ ρg x ¼ gβx g βx 2 2 ∂x ∂x ∂x ð14Þ

2.2. Governing equation Following the formulation of Chaudhary and Merkin [12], the governing boundary-layer equations are of the form: ∂u ∂v þ ¼ 0; ∂x ∂y u

2 ∂u ∂u μ bf ∂ u ∂ mþ1 þ ðT−T ∞ Þ þv ¼ gβx 2 ∂x ∂y ρbf ∂y2 ∂x h i m þ 1 ∂ σ ½BðxÞ2 þ ðN−N∞ Þ− u; gω ρm −ρ f x ρbf 2 ∂x

ð15Þ

ð16Þ m−1

The special form of magnetic field is defined as BðxÞ ¼ Bo ðx þ bÞ 2 . Using the idea of Rosseland [48], the energy equation in which


nonlinear thermal radiation, thermophoresis and space dependent internal heat source are accounted for is of the form " 2 # 2 κ bf ∂ T ∂T ∂T 1 ∂qr ∂a ∂T DT ∂T u − þv ¼ þ þ τ DA ∂x ∂y ðρCpÞbf ∂y2 ðρCpÞbf ∂y ∂y ∂y T ∞ ∂y " # rffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffi m−1 Q o ½T w ðxÞ−T ∞ m þ 1 Uo 2 þ : ð17Þ Exp −ny ðx þ bÞ ϑ bf 2 ρC p bf The Rosseland approximation which is a simplification of the radiative transport equation (RTE) for the case of optically thick media is adopted to account for the radiative heat flux in the nanofluid as qr ¼ −

4σ ∂T 4 ; 3k ∂y

ð18Þ

where σ⁎ and k⁎ are the Stefan-Boltzman constant and the mean absorption coefficient respectively. In this case of nanofluid, herein (optically thick) the thermal radiation travels only a short distance before being scattered or absorbed. Eq. (18) introduces a new diffusion term into the energy transport equation which was developed based on the theory of conservation of energy. In this study, it is assumed that the temperature differences within the flow are not sufficiently small. In view of this, it may not be realistic to simplify the radiative heat flux by expanding T4 in a Taylor series expansion about T∞ then neglecting higher order terms. Implicit differentiation may be adopted to simplify Eq. (18) and then substitute into energy Eq. (17). The modified energy equation is now u

2 κ bf ∂ T ∂T ∂T 1 ∂ 4σ 3 ∂T þ 4T þv ¼ ∂x ∂y ðρCpÞbf ∂y2 ðρCpÞbf ∂y 3k ∂y " 2 # ∂a ∂T DT ∂T þ τ DA þ ∂y ∂y T ∞ ∂y " # rffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffi m−1 Q ½T w ðxÞ−T ∞ m þ 1 Uo þ o Exp −ny ðx þ bÞ 2 : ð19Þ ϑ bf 2 ρC p bf

Considering the influence of thermophoresis, homogeneous-heterogeneous reaction model for the concentrations of reactants A and B, as stated in the reaction scheme Eqs. (2) and (3) are formulated as

737

for homogeneous (reactant A which is known as bulk fluid) Eq. (20), concentration equation for heterogeneous (reactant B which is known as catalyst at the surface) Eq. (21), and density of motile microorganisms Eq. (22) subject to boundary Conditions (23) and (24) we consider the following variables m

u ¼ U o ðx þ bÞ ; ¼ −K s a;

v ¼ 0;

N ¼ Nw

at

∂a ¼ K s a; ∂y 1−m y ¼ Aðx þ bÞ 2 :

T ¼ T w;

DA

DB

∂l ∂y ð25Þ

The continuity equation is satisfied automatically. The dimensionless governing equation (locally transformed coupled O.D.E. based on local similarity transformation) is 3 2 d f 2m df df d f 2 df − −ζ þ f þ Grm θ þ Rb mm ¼ 0; m þ 1 dη dη3 m þ 1 dη dη dη2 ð1 þ θθw −θÞ3 R þ Pr f

!

ð26Þ

2

d θ dg dθ dθ dθ 1−m df þ Nb þ Nt −P r θ dη dη dη dη m þ 1 dη dη2

dθ ð1 þ θθw −θÞ2 dθ dθ ðθw −1Þ þ3 þ R dη dη dη

2

ð27Þ

2

d g dg 2 Nt d θ 3 þ Sc f ¼ 0; −Sc K gh − N b dη2 dη mþ1 dη2 2

δ

ð28Þ

2

d h dh 2 Nt d θ 3 þ Sc f ¼ 0; þ Sc K gh þ Nb dη2 dη mþ1 dη2 2

ð29Þ 2

d mm 1−m df dmm d g dg dmm −mm P e 2 −P e −Smm mm þ Smm f dη m þ 1 dη dη dη dη2 dη ¼ 0:

ð30Þ

In order to non-dimensionalize the boundary conditions, it is pertinent to note that the minimum value of y is not the starting point of the slot. This implies that the conditions in Eq. (23) are not imposed at y = 0. As shown in Fig. 1, it is obvious that it may not be realistic to say that y = 0 at all point on the horizontal surface with variable thickness. Hence, not valid to set y= 0 in variable η. Upon substituting y ¼ A 1−m

ð20Þ

ðx þ bÞ 2 which is the starting point of the flow at the slot the minimum value of y which accurately corresponds to minimum value of similarity variable η is denoted as.

∂ℓ ∂ℓ ∂ ℓ DT ∂ T ; þv ¼ DB 2 þ K 1 aℓ3 þ T ∞ ∂y2 ∂x ∂y ∂y

ð21Þ

m þ 1 U o 1=2: χ¼A 2 ϑ

2 ∂N ∂N ∂ c W c ∂a ∂ N ¼ Dn 2 : þv þ N h Δa ∂y ∂x ∂y ∂y ∂y

ð22Þ

This implies that at the wall, the boundary condition suitable to scale the boundary layer flow is η =χ. The boundary condition becomes

2

2

u

∂a ∂a ∂ a DT ∂ T ; þv ¼ DA 2 −K 1 aℓ3 − T ∞ ∂y2 ∂x ∂y ∂y

u

u

2

2

Based on the fact that reactant A and reactant B undergo chemical changes at an interface, the heterogeneous catalytic reaction is properly accounted for. Eqs. (15), (16), (19), (20), (21) and (22) are subject to boundary conditions ∂a u ¼ U o ðx þ bÞ ; v ¼ 0; T ¼ T w ; DA ¼ K s a; ∂y 1−m ¼ −K s a; N ¼ Nw at y ¼ Aðx þ bÞ 2 : m

u→0;

T→T ∞ ;

a→ao ;

ℓ→0;

N→N∞

as

∂ℓ DB ∂y

df →0; dχ

θðχ Þ→0;

g ðχ Þ→1;

hðχ Þ→0;

dg ¼ Λgðχ Þ; dχ

mm →0;

δ

dh dχ

ð31Þ

as χ→∞: ð32Þ

ð23Þ y→∞:

df 1−m ¼ 1; f ðχ Þ ¼ χ ; θðχ Þ ¼ 1; dχ mþ1 ¼ −Λgðχ Þ; mm ¼ 1; at χ ¼ 0:

ð24Þ

2.3. Non-dimenzionalization and parametrization In order to non-dimenzionalization and parametrization continuity Eq. (15), momentum Eq. (16), energy Eq. (19), concentration equation

In the dimensionless equation m velocity power index (herein b1), Grm modified local thermal Grashof parameter, κbf is the thermal conductivity of the basefluid, (ρCp)sp is known as the heat capacity of the nanoparticles, (ρCp)bf is known as the heat capacity of the basefluid, Nt is known as thermophoresis parameter, Nb is known as Brownian motion parameter, δ ratio of diffusion coefficients chemical specie A to specie B, θw temperature parameter, Rb is known as the bioconvection Rayleigh number, R is the radiation parameter, Pr is known as Prandtl

738


number, γ is known as space dependent internal heat source parameter, n is the intensity of internal heat generation parameter, Sca is the ratio of viscous diffusion rate to molecular (mass) diffusion rate of specie A, Scb is the ratio of viscous diffusion rate to molecular (mass) diffusion rate of specie B, K is known as strength of the homogeneous reaction, Λ is known as strength of the heterogeneous reaction, Smm is the Schmidt number for diffusing motile microorganism, ζ is known as the modified magnetic field parameter and Pe is known as the bioconvection Peclet number. Where κ bf c W Ks ; P e ¼ ao h c ; Λ ¼ Grm ¼ 2 ; α bf ¼ 1=2 2m−1 m−1 ΔaDn ρC p bf Uo U o ðx þ bÞ DA mþ1 ðx þ bÞ 2 2 ϑbf gω ρm −ρ f ½Nw −N∞ Qo m U w ¼ U o ðx þ bÞ ; Rb ¼ ; γ¼ ; 2m−1 m−1 ρC p bf U o ðx þ bÞ U 2o ðx þ bÞ gβ ðT w −T ∞ Þ

Pr ¼ Sca ¼

ρC p

bf κ

ϑbf ; DA

ϑbf

¼

ϑbf ; α bf

Nb ¼

Scb ¼

ϑbf ; DB

K¼

τDA ao ; α bf

Nt ¼

K 1 ao a2o U o ðx þ bÞ

m−1

;

τðT w −T ∞ Þ DT ; T∞ α bf

θw ¼

DB ; DA

ϑbf ; Dn

δ¼

Smm ¼

Tw ; T∞ ζ¼

R¼

3k κ bf 16σ T 3∞

;

σ B2o U o ρbf

ð33Þ The dimensionless governing Eqs. (26), (27), (28), (29) and (30) are depending on η while the boundary conditions Eqs. (31) and (32) are functions and/or derivatives depending on χ. In order to transform the domain from [χ, ∞) to [0, ∞) it is valid to adopt F(ς) = F(η − χ) = f(η), Θ(ς) = Θ(η − χ) = θ(η), G(ς) = G(η − χ) = g(η), H(ς) = H(η − χ) = h(η) and Mm(ς) = Mm(η − χ) = mm(η). The final dimensionless governing equations (coupled system of nonlinear ordinary differential equation) for spherical 36 nm Al2O3-water nanofluid is 3

2

d F 2m dF dF d F 2 dF − þ F 2 −ζ þ Grm Θ þ Rb M m ¼ 0; m þ 1 dς dς 3 m þ 1 dς dς dς ð1 þ θw Θ−ΘÞ3 R

!

2

d Θ dG dΘ dΘ dΘ 1−m dF þ Nb þ Nt −P r Θ ð35Þ dς dς dς dς m þ 1 dς dς2 dΘ ð1 þ θw Θ−ΘÞ2 dΘ dΘ ðθw −1Þ þP r F þ3 R dς dς dς 2 þP r γ Exp½−nς ¼ 0; mþ1

2

2

d G dG 2 Nt d Θ þ Sca F ¼ 0; −Sca K GH3 − Nb dς2 dς mþ1 dς2 2

δ

ð34Þ

ð36Þ

2

d H dH 2 Nt d Θ þ Scb F ¼ 0; þ Scb K GH3 þ N b dς 2 dς mþ1 dς 2 2

ð37Þ 2

d Mm 1−m dF dMm d G dG dMm −Mm P e 2 −P e −Smm Mm þ Smm F dς m þ 1 dς dς dς dς 2 dς ¼ 0:

ð38Þ

Subject to boundary conditions dF 1−m ¼ 1; F¼χ ; θ ¼ 1; dς mþ1 ¼ −ΛG; Mm ¼ 1; at ς ¼ 0:

dG ¼ ΛG; dς

δ

dH dς

ð39Þ

dF →0; dς

Θ→0;

G→1;

Mm →0;

H→0;

as ς→∞:

ð40Þ

The quantities of interest are the skin friction coefficient Cf, Nusselt number Nux and local density number of the motile microorganism Jnx which are defined as τw ðx þ bÞqw rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi ; ; Nux ¼ mþ1 mþ1 2 ρbf ðU w Þ κ bf ½T w ðxÞ−T ∞ 2 2 ðx þ bÞLw rffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ mþ1 Dn ½Nw ðxÞ−N ∞ 2

Cf ¼

Jnx

where τw is the shear stress (skin friction) on the stretching sheet along the horizontal surface of paraboloid of revolution, qw is the heat flux from the sheet and Lw denotes the wall motile microorganisms flux ∂u τw ¼ ðμ bf qw 1−m ; ∂y y¼AðxþbÞ 2 ! 3 16σ T ∂T ¼ − κ bf þ 1−m ; ∂y y¼AðxþbÞ 2 3k

∂N Lw ¼ −ðDn 1−m ∂y y¼AðxþbÞ 2

Upon using the similarity variables Eq. (25) we obtain Rex C f ¼ F ″ ð0Þ; 1=2

¼ −Mm0 ð0Þ:

−1=2

Nux Rex

R

ð1 þ Θθw −ΘÞ3

¼ −Θ0 ð0Þ;

−1=2

Jnx Rex

ð41Þ

. Where local Reynold number is Rex ¼ U wϑðxþbÞ bf 3. Numerical solution The governing equations with the associated boundary conditions Eqs. (34)–(40) are numerically solved using classical Runge-Kutta method with shooting techniques. The boundary value problem cannot be solved on an infinite interval and it would be impractical to solve it for even a very large finite interval. The infinite boundary condition at a finite point ς at infinity is 6. The set of coupled nonlinear ordinary differential equations along with boundary conditions have been reduced to a system of eleven simultaneous equations of first order for eleven unknowns following the method of superposition (see Na [49]). It is worth mentioning that there exist no related published articles that can be used to validate the accuracy of the numerical results. Meanwhile, the same dimensionless governing equations can easily be solved using ODE solvers such as MATLABs bvp5c. 3.1. Classical Runge-Kutta method along with shooting techniques In order to integrate the corresponding I.V.P., the values of F″(ς = 0), Θ′(ς = 0), G(ς =0), H(ς = 0) and Mm′(ς = 0) are required, such values does not exist after the non-dimenzionalization of the boundary Conditions (23)–(24). Although, at known values of δ, Λ and G(ς = 0) the corresponding values of G′(ς = 0) and H′(ς = 0) can be easily

Table 1 : Comparison between the solutions of classical Runge-Kutta together with shooting (RK4SM) and MATLAB solver bvp5c for the limiting case. χ

F″(ς = 0) (RK4SM)

Mm′(ς = 0) RK4SM

F″(ς = 0) (bvp5c)

Mm′(ς = 0) (bvp5c)

0.1 0.2 0.3 0.4

−0.144534003430829 −0.179449218399145 −0.215196480474511 −0.251745046213699

1.262014732569137 1.313719517482142 1.366449769693500 1.420119747195705

−0.144534003431141 −0.179449218399781 −0.215196480474507 −0.251745046213696

1.262014732561248 1.313719517482416 1.366449769693513 1.420119747195705


estimated. The suitable guess values for F″(ς = 0), Θ′(ς = 0), G(ς = 0), H(ς = 0) and Mm′(ς = 0) are chosen and then integration is carried out. The calculated values of F ' (ς), Θ(ς), G(ς), H(ς) and Mm(ς) at infinity (ς =6) are compared with the given boundary conditions in Eq. (40) and the estimated values F″(ς = 0), Θ′(ς = 0), G(ς = 0), H(ς = 0) and Mm′(ς = 0) are adjusted to give a better approximation for the solution. Series of values for F″(ς = 0), Θ′(ς = 0), G(ς = 0), H(ς = 0) and Mm′(ς = 0) are considered and applied with fourth-order classical Runge-Kutta method using step size Δς = 0.01. The above procedure is repeated until asymptotically converged results is obtained within a tolerance level of 10− 4. It is very important to remark that setting ς∞ = 6, all profiles are compatible with the boundary layer theory and asymptotically satisfies the conditions at free stream as suggested by Pantokratoras [50]. 3.2. MATLAB package (bvp5c) The Matlab package (bvp5c) integrates a system of ordinary differential equations of the form y′ = f(x, y) on the interval ς ∈ [a, b] subject to two-point boundary value conditions bc(y(a),y(b)) = 0; see Gökhan [51]. The bvp5c is a finite difference code that implements the fourstage Lobatto IIIa formula. This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fifth-order accurate uniformly in ς ∈ [a,b]. The formula is implemented as an implicit Runge-Kutta formula. The bvp5c solves the algebraic equations directly while bvp4c uses analytical condensation. The bvp4c handles unknown parameters directly; while bvp5c augments the system with trivial

739

differential equations for unknown parameters; for details, see Shampine et al. [52]. Following Gökhan [51], the Matlab package (bvp5c) is adopted because in terms of scalar evaluation, the performance of bvp5c solver is better than bvp4c. 3.3. Verification of the results In order to verify the accuracy of the present numerical analysis, the results of classical Runge-Kutta together with shooting have been compared with that of bvp5c solution for the limiting case when Pr = 6.8, Smm = 1, Sca = 0.62, Scb = 1.3, K = 0.4, γ = 0.07, n = 0.3, m = 0.25, Nb = 0.1, Nt = 0.3, ζ = 0.2, Rb = 1, Λ = 0.4, Grm = 1, δ = 1.2, R = 0.7, θw = 1.2 and Pe = 0.3 at various values of temperature parameter χ. As shown in Table 1, the comparison in the above case is found to be in good agreement. This good agreement is an encouragement for further study of the effects of other pertinent parameters. 4. Results and discussion The numerical computations have been carried out to investigate effects of increasing Brownian motion parameter (Nb), thermophoresis parameter (Nt) and some pertinent parameters on the motion of nanofluid past upper surface of horizontal paraboloid revolution. The effects of pertinent parameters on the velocity profiles F′(ς), temperature profiles Θ(ς), temperature gradient profiles Θ′(ς), concentration of reactant A (i.e. homogeneous bulk fluid) G(ς), concentration gradient of reactant A G′(ς), concentration of reactant B (i.e. heterogeneous catalyst

Fig. 2. Effect of Nb on G(ς). b: Effect of Nb on G'(ς). c: Effect of Nb on H(ς). d: Effect of Nb on H'(ς).

740


at the surface) H(ς), concentration gradient of reactant B H′(ς) and diffusion of motile microorganism Mm(ς) are considered and illustrated. 4.1. Effects of Brownian motion parameter When Pr= 6.8, Smm = 1, Sca = 0.62, Scb = 1.3, K = 0.4, γ = 0.07, n = 0.3, m= 0.25, Nt = 0.4, ζ = 0.2, Rb = 1, χ = 0.3, Λ =0.4, Grm = 1, δ = 1.2, R = 0.7, θw = 1.2 and Pe = 0.3, effects of Brownian motion parameter (Nb) on concentration of reactant A (homogeneous) is illustrated in Fig. 2a. It is observed that G(ς) increases near the wall. Physically, increase in the magnitude of Brownian motion parameter corresponds to increase in the rate at which nanoparticles move with different velocities in different random direction as the nanofluid climbs horizontal surface of paraboloid of revolution. In the research conducted by Zaimi et al. [11], it was confirmed that this movement enhances transfer of heat and consequently increases the temperature profiles. The same result is observed in this study and not presented for brevity and replication. As temperature profile increases with Nb, this leads to an increase in the concentration of homogeneous bulk fluid and consequently leads to decrease in the concentration gradient of homogeneous bulk fluid within the fluid domain as shown in Fig. 2b. Fig. 2c reveals that concentration of reactant B (Heterogeneous) of the nanofluid decreases with the peak at ς = 0.5 when Nb = 0.1. For smaller value of Nb, a kind of overshoot is found in H′(ς) near the upper horizontal surface of paraboloid of revolution 0 b ς ≤ 0.70; see Fig. 2d.

4.2. Effects of thermophoresis parameter The influence of thermophoresis parameter Nt on G(ς), G′(ς), H(ς) and H′(ς) are depicted in Fig. 3a–d. The increment of Nt results in decrease of G(ς). On the other hand, H(ς) is an increasing function of Nt. Physically, increase in the magnitude of thermophoresis parameter Nt corresponds to an increase in thermophoretic force and this often move nanoparticles from region of higher to lower temperature. Consequently, this leads to decrease in the concentration of reactant A (homogeneous) and increase in the concentration of reactant B (heterogeneous); see Fig. 3a and c. It is also observed in Fig. 3b and d that for small values of thermophoresis parameter Nt, concentration gradient of reactant A is small while concentration gradient of reactant B is large within the domain. 4.3. Effects of modified local thermal Grashof parameter, bioconvection Rayleigh number and magnetic field parameter Fig. 4a–d depicts the influence of local thermal Grashof parameter Grm, bioconvection Rayleigh number Rb and magnetic field parameter ζ on the nanofluid flow when Pr = 6.8, Smm = 1, Sca = 0.62, Scb = 1.3, K = 0.4, γ = 0.07, n = 0.3, m = 0.25, Nb = 0.4, Nt = 0.4, χ = 0.3, Λ = 0.4, δ = 1.2, R = 0.7, θw = 1.2 and Pe = 0.3. At a constant value of bioconvection Rayleigh number and local thermal Grashof parameter, it is observed in Fig. 4a that Re1/2 x Cf decreases with an increase in the magnitude of magnetic field parameter ζ. Moreover, at a fixed value of

Fig. 3. Effect of Nt on G(ς). b: Effect of Nt on G'(ς). c: Effect of Nt on H(ς). d: Effect of Nt on H'(ς).


741

−1/2 Fig. 4. a: Variations in Re1/2 with Rb and ζ. c: Effect of Grm on F'(ς). d: Effect of Grm on Mm(ς). e: Effect of Rb on F'(ς). f: Effect of Rb on Mm(ς). x Cf with Grm and ζ. b: Variations in JnxRex

magnetic field parameter ζ and bioconvection Rayleigh number Rb it is observed that Re1/2 x Cf increases with Grm. At constant value of buoyancy with Rb and ζ are illustrated in Fig. parameter, the variations in JnxRe−1/2 x −1/2 4b. We notice similar trend for the variation of Re1/2 x Cf and JnxRex with respect to these pertinent parameters. Furthermore, decrease in with ζ is more significant than that of Re1/ the coefficient of JnxRe−1/2 x x 2 Cf with ζ. Physically, this can be traced to the fact that the effects of

Lorentz force on flow velocity and consequently on shear stress are more substantial. The variations of F′(ς) and Mm(ς) along ς with different values of buoyancy parameter Grm are plotted in Fig. 4c and d. It is seen that the increase of Grm leads to the enhancement of velocity profiles and decrease of diffusion of motile microorganism. We further notice that as Grm becomes large its effects on Mm(ς) near an upper horizontal surface

742


of paraboloid of revolution is negligible. Fig. 4e and f presents the effect of bioconvection Rayleigh number Rb on F′(ς) and Mm(ς). It is observed that F′(ς) enlarges continuously and significantly near the surface as Rb grows. However, diffusion of motile microorganism Mm(ς) decreases with Rb. 5. Conclusion This paper presents quartic kind of homogeneous-heterogeneous chemical reaction in nanofluid flow past upper surface of horizontal paraboloid of revolution. Buoyancy induced model for flow along upper surface of horizontal paraboloid of revolution in which there exist changes in the volume due to changes in temperature and motile microorganisms density difference is presented. Similarity transformation of the governing mathematical equations are achieved and the corresponding ordinary differential equations are solved numerically. The results show that • the proposed buoyancy-induced model for the case of convective flow along variable thickness when (m b 1) is suitable enough to investigate boundary layer formed on upper half surface of paraboloid of revolution (where Tw N T∞), • Brownian motion boosts concentration of bulk fluid while thermophoresis reduces it, • in the presence of Brownian motion and thermophoresis decreases and increases concentration of reactant B (heterogeneous) respectively. Diffusion of motile microorganism Mm(ς) is a decreasing function of bioconvection Rayleigh number and local thermal Grashof parameter.

References [1] R. Tsai, Y.P. Chang, T.Y. Lin, Combined effects of thermophoresis and electrophoresis on particle deposition onto a wafer, J. Aerosol Sci. 29 (1998) 811–825. [2] T. Kowalkowski, B. Buszewski, C. Cantado, F. Dondi, Field flow fractionation: theory, techniques, applications and the challenges, Crit. Rev. Anal. Chem. 36 (2006) 129–135. [3] E.J. Davis, Thermophoresis of particles, Encyclopedia of Surface and Colloid Science, Online version of Taylor & Francis 2006, pp. 6274–6282, http://dx.doi.org/10.1081/ E-ESCS-120000614. [4] L. Talbot, R.K. Cheng, R.W. Scheffer, D.P. Wills, Thermophoresis of particles in a heated boundary layer, J. Fluid Mech. 101 (1980) 737–758. [5] I.L. Animasaun, Effects of thermophoresis, variable viscosity and thermal conductivity on free convective heat and mass transfer of non-darcian MHD dissipative Casson fluid flow with suction and nth order of chemical reaction, J. Niger. Math. Soc. 34 (2015) 11–31. [6] N. Sandeep, C. Sulochana, Dual solutions of radiative MHD nanofluid flow over an exponentially stretching sheet with heat generation/absorption, Appl. Nanosci. (2015), http://dx.doi.org/10.1007/s13204-015-0420-z. [7] R. Tsai, A simple approach for evaluating the effect of wall suction and thermophoresis on aerosol particle deposition from a laminar flow over a flat plate, Int. Commun. Heat Mass Transfer 26 (1999) 249–257. [8] A. Einstein, A.D. Cowper, Investigation on the theory of the Brownian movement, Sci. Mon. 83 (6) (1956) 314. [9] S.P. Jang, S.U.S. Choi, Role of Brownian motion in the enhanced thermal conductivity of nanofluids, Appl. Phys. Lett. 84 (21) (2004) 4316–4318, http://dx.doi.org/10. 1063/1.1756684. [10] J. Buongiorno, Convective transport in nanofluids, J. Heat Transf. ASME 128 (3) (2006) 240–250, http://dx.doi.org/10.1115/1.2150834 (2006). [11] K. Zaimi, A. Ishak, I. Pop, Boundary layer flow and heat transfer over a nonlinearly permeable stretching/shrinking sheet in a nanofluid, Sci. Rep. 4 (2014) 4404, http://dx.doi.org/10.1038/srep04404. [12] M.A. Chaudhary, J.H. Merkin, A simple isothermal model for homogeneous-heterogeneous reactions in boundary-layer flow. I equal diffusivities, Fluid Dyn. Res. 16 (1995) 311–333. [13] S.K. Scott, Isolas, mushrooms and oscillations in isothermal, autocatalytic reactiondiffusion equations, Chem. Eng. Sci. 42 (1987) 307–315. [14] R. Nandkeolyar, S.S. Motsa, P. Sibanda, Viscous and joule heating in the stagnation point nanofluid flow through a stretching sheet with homogenous-heterogeneous reactions and nonlinear convection, J. Nanotechnol. Eng. Med. 4 (2013) 041001, http://dx.doi.org/10.1115/1.4027435. [15] I.L. Animasaun, C.S.K. Raju, N. Sandeep, Unequal diffusivities case of homogeneousheterogeneous reactions within viscoelastic fluid flow in the presence of induced magnetic-field and nonlinear thermal radiation, Alex. Eng. J. (2016), http://dx.doi. org/10.1016/j.aej.2016.01.018 (in press).

[16] S.U.S. Choi, Enhancing Thermal Conductivity of Fluids with Nanoparticles, Proceedings of the ASME International Mechanical Engineering Congress and Exposition, San Francisco, Calif, USA, Vol. 661995 99–105. [17] H.A. Mintsa, C.T. Nguyen, G. Roy, New Temperature Dependent Thermal Conductivity Data of Water Based Nanofluids, Proceedings of the 5th IASME/WSEAS Int. Conference on Heat Transfer, Thermal Engineering and Environment, Athens, Greece, Vol. 2902007 25–27. [18] J.R. Platt, “Bioconvection patterns” in cultures of free-swimming organisms, Science 133 (1961) 1766–1767. [19] S. Ghorai, N.A. Hill, Wavelengths of gyrotactic plumes in bioconvection, Bull. Math. Biol. 62 (2000) 429–450. [20] A.V. Kuznetsov, A.A. Avramenko, Effect of small particles on the stability of bioconvection in a suspension of gyrotactic microorganisms in a layer of finite depth, Int. Commun. Heat Mass Transfer 31 (2004) 1–10. [21] W.A. Khan, O.D. Makinde, MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet, Int. J. Therm. Sci. 81 (2014) 118–124, http://dx.doi.org/10.1016/j.ijthermalsci.2014.03.009. [22] W.A. Khan, O.D. Makinde, Z.H. Khan, MHD boundary layer flow of a nanofluid containing gyrotactic microorganisms past a vertical plate with Navier slip, Int. J. Heat Mass Transf. 74 (2014) 285–291. [23] W.N. Mutuku, O.D. Makinde, Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms, Comput. Fluids 95 (2014) 88–97. [24] O.D. Makinde, W.A. Khan, J.R. Culham, MHD variable viscosity reacting flow over a convectively heated plate in a porous medium with thermophoresis and radiative heat transfer, Int. J. Heat Mass Transf. 93 (2016) 595–604. [25] S. Das, R.N. Jana, O.D. Makinde, Magnetohydrodynamic free convective flow of nanofluid past an oscillating porous flat plate in a rotating system with thermal radiation and Hall effects, J. Mech. 32 (2) (2016) 197–210. [26] W.A. Khan, O.D. Makinde, Z.H. Khan, Non-aligned MHD stagnation point flow of variable viscosity nanofluids past a stretching sheet with radiative heat, Int. J. Heat Mass Transf. 96 (2016) 525–534. [27] S.S. Motsa, I.L. Animasaun, A new numerical investigation of some thermo-physical properties on unsteady MHD non-Darcian flow past an impulsively started vertical surface, Therm. Sci. 19 (Suppl. 1) (2015) S249–S258. [28] O.D. Makinde, F. Mabood, W.A. Khan, M.S. Tshehla, MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat, J. Mol. Liq. 219 (2016) 624–630. [29] N. Sandeep, C. Sulochana, I.L. Animasaun, Stagnation-point flow of a Jeffrey nano fluid over a stretching surface with induced magnetic field and chemical reaction, Int. J. Eng. Res. Afr. 20 (2016) 93–111, http://dx.doi.org/10.4028/www.scientific. net/JERA.20.93. [30] A. Raees, X.U. Hang, S.U.N. Qiang, I. Pop, Mixed convection in gravity-driven nanoliquid film containing both nanoparticles and gyrotactic microorganisms, Appl. Math. Mech. (Eng. Ed.) 36 (2015) 163–178, http://dx.doi.org/10.1007/s10483015-1901-7. [31] L.L. Lee, Boundary layer over a thin needle, Phys. Fluids 10 (1967) 820, http://dx.doi. org/10.1063/1.1762194. [32] R.T. Davis, M.J. Werle, Numerical solutions for laminar incompressible flow past a paraboloid of revolution, Am. Inst. Aeronaut. Astronaut. J. 10 (1972) 1224–1230. [33] S.P. Anjali Devi, M. Prakash, Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet, J. Niger. Math. Soc. (2015), http://dx.doi.org/10.1016/j.jnnms.2015.07.002 (in press). [34] T. Fang, J. Zhang, Y. Zhong, Boundary layer flow over a stretching sheet with variable thickness, Appl. Math. Comput. 218 (2015) 7241–7252. [35] N. Liron, H.E. Wilhelm, Integration of the magnetohydrodynamic boundary-layer equations by Meksin's method, J. Appl. Math. Mech. (ZAMM) 54 (1974) 27–37. [36] A.V. Kuznetsov, D.A. Nield, Natural convective boundary layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243–247. [37] A.V. Kuznetsov, D.A. Nield, Double-diffusive natural convective boundary layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 50 (2011) 712e717, http://dx. doi.org/10.1016/j.ijthermalsci.2011.01.003. [38] D.T. Lynch, Chaotic behavior of reaction systems: mixed cubic and quadratic autocatalysis, Chem. Eng. Sci. 47 (1992) 4435–4444. [39] T.J. Pedle, N.A. Hill, J.O. Kessler, The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms, J. Fluid Mech. 195 (1988) 223–338. [40] A.V. Kuznetsov, The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms, Int. Commun. Heat Mass Transfer 37 (2010) 1421–1425. [41] J. Boussinesq, Théorie Analytique de la Chaleur, Gauthier-Villars, 1903. [42] J. Boussinesq, Theorie de l'ecoulement Tourbillonnant et Tumultueux Des Liquides Dans Les Lits Rectilignes a Grande Section, vol. 1, Gauthier-Villars, Paris, 1897 (Open Library OL7070543M). [43] A. Oberbeck, Uber die wärmleitung der flüssigkeiten bei berücksichtigung der strömungen infolge von temperatur differenzen, Ann. Phys. Chem. 7 (1879) 271–292. [44] A. Oberbeck, Uber die bewegungsercheinungen der atmosphere, Sitz. Ber. K. Preuss. Akad. Miss. 383 (1888) 1120. [45] F. Shahraki, Modeling of buoyancy-driven flow and heat transfer for air in a horizontal annulus: effects of vertical eccentricity and temperature-dependent properties, Numer. Heat Transfer, Part A 42 (2002) 603–621, http://dx.doi.org/10.1080/ 10407780290059729. [46] J.I. Diaz, J.M. Rakotoson, P.G. Schmidt, A parabolic system involving a quadratic gradient term related to the Boussinesq approximation, Rev. Real Acad. Cien. Ser. A Matematicas (RACSAM) 101 (2007) 113–118.

O.D. Makinde, I.L. Animasaun / Journal of Molecular Liquids 221 (2016) 733–743 [47] I.L. Animasaun, Double diffusive unsteady convective micropolar flow past a vertical porous plate moving through binary mixture using modified Boussinesq approximation, Ain Shams Eng. J. (2016), http://dx.doi.org/10.1016/j.asej.2015.06.010 (in press). [48] S. Rosseland, Astrophysik aud Atom-Theoretische Grundlagen, Springer, Berlin, 1931 41–44. [49] T.Y. Na, Computational Methods in Engineering Boundary Value Problems, Academic Press, New York, 1979.

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[50] A. Pantokratoras, A common error made in investigation of boundary layer flows, Appl. Math. Model. 33 (2009) 413–422. [51] F.S. Gökhan, in: C.M. Ionescu (Ed.), Effect of the Guess Function & Continuation Method on the Run Time of MATLAB BVP Solvers 2011, p. 1. [52] J. Kierzenka, L.F. Shampine, A BVP solver based on residual control and the MATLAB PSE, ACM TOMS 27 (3) (2001) 299–316.