Zaheer Abbas*,Mariam Sheikh

Department of Mathematics,The Is lamia University of Bahalwalpur,Bahawalpur 63100,Pakistan

1.Introduction

Nanotechnology affects the major industrial revolution of the present century therefore the study of nano fluid has acquired enough attention among the researchers.Choi[1]initially studied the thermal conductivity of the fluids with nanoparticles.Nano fluid is the type of fluid that contains a suspension of solid particles(100-nm diameter).Due to tiny size and low volume fraction of nanoelements,these are extremely stable and free of extra challenges like erosion,sedimentation,non-Newtonian behavior and additional pressure drop.Nano fluids are particularly important in applications such as in power generators,hybrid-powered engines nuclear reactors,micro-manufacturing,thermal therapy for cancer treatment,magnetic drug targeting and magnetic cell separation.Khan and Pop[2]discussed the boundary layer flow of a nano fluid over a linear stretching sheet.Ahmad and Pop[3]analyzed the effects of four different nano particles(Cu,Ag,Al2O3,TiO2)on mixed convection boundary layer flow over a vertical plate embedded in a porous medium.In the presence of magnetic field,Hamad[4]proposed an analytical soluti on of natural convection flow of anano fluid past a stretching sheet.Sharmaet al.[5]studied the boundary layer flow of a nano fluid with heat transfer over a stretching surface with velocity slip condition.Vajravelu[6]carried out a numerical study to see the influence of temperature dependent viscosity on the flow of nano fluid over a flat surface with heat transfer.Bachoket al.[7]discussed the steady two-dimensional stagnation point flow of Cu(copper)–water nano fluid over a permeable stretching/shrinking sheet with heat transfer characteristics.Two-dimensional flow of water based nano fluid towards a linearly semi-in finite stretching sheet was investigated by Ganeshet al.[8]in the presence of magnetic field.

On the other side,magnetic or ferro fluids are special types of nano fluids which are based on very small magnetic particles such that they consist of only one single magnetic domain.In such fluids nanoparticles like magnetite,hematite,cobalt ferrite and also some other substances that contain iron are suspended in a convectional base fluids(water,mineral oil and ethylene glycol).Such fluids have the fluid properties of liquid as well as the magnetic properties of solid.The availability of such magnetic fluids has resulted in a number of unique applications including energy conversion devices,viscous dampers for gravity gradient satellites,accelerometers,rotating anode X-ray generators,vacuum chambers in semi-conductor industry,in high speed computer disk devices to eliminate harmful dust particles and other impurities and in biomedical applications.Tang thienget al.[9]studied the finite element simulations of heat transfer to a ferro fluid for flow between flat plates and in a box with magnetic field.Li and Xuan[10]carried out experiments to analyze the effects of convection heat transfer of ferro fluid over a fine wire under the impact of external magnetic field.By applying external magnetic field,Yamaguchiet al.[11]discussed the natural convection in a partitioned square cavity.Abraham and Titus[12]considered boundary layer flow of ferro fluid over a stretching sheet in the presence of heat source and found a numerical solution.Khanet al.[13]investigated the stagnation point flow of ferro fluid and heat transfer over a stretching sheet in the presence of viscous dissipation.Sheikholeslamiet al.[14]studied the force convection heat transfer in semi annulus enclosure filled with Fe3O4–water nano fluid in the presence of non-uniform magnetic field.Sheikholeslami and Ganji[15]discussed the effect of an external magnetic field on ferro fluid flow in a semi-annulus enclosure with heat transfer.

Whenever the flow velocity at fluid–solid interface approaches to zerothen insuchcase the flow is termed asno-slip flow but if the velocity is non-zero at the solid wall then slip flow occurs.Generally,slip flow phenomena take place in the fluids like emulsion suspension and form and polymer solutions.The linear slip boundary condition was discussed by Navier[16]and Maxwell[17].They stated that in slip flow model the tangential components of the fluid velocity at the solid surface have a proportional relationship with the shear stress on the fluid–solid interface,where the constant of proportionality is referred to slip length that describes the slipperiness of the surface. An exact similarity solution of the Navier–Stokes equations towards a stretching sheet with constant slip length(partial slip)was discussed by Wang[18].Influenced by molecular dynamical simulation,Thompson and Troian[19]presented generalized slip boundary condition.In generalized slip boundary condition the slip length is a function of shear stress instead of constant.Using non-linear Navier boundary conditions,Mathews and Hill[20]studied the flow of Newtonian fluid.Wang[21]investigated the viscous flow with suction and slip boundary condition towards a stretching sheet.Fanget al.[22]presented exact solution of viscous fluid over a stretching sheet with slip effects in the presence of magnetic field.Sajidet al.[23]studied the effects of generalized slip condition on both plannar and axisymmetric flows.The boundary layer flow with slip and constant heat flux surface conditions over a flat plate was studied by Aziz[24].Rashidiet al.[25]analyzed the effect of thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk with partial slip.Mutuku-Njane and Makinde[26]examined the effect of magnetic field on boundary layer flow of water-based nano fluids towards a convectively heated vertical porous surface with Navier slip boundary condition.Mansur and Ishak[27]investigated the MHD boundary layer flow of a nano fluid over a stretching/shrinking sheet with velocity,thermal and solutal slip effects.Abbaset al.[28]discussed the MHD viscous fluid and heat transfer near a stagnation point with generalized slip condition for both planner and axisymmetric sheets.Recently,Abbaset al.[29]performed a study to analyze the effects of homogeneous–heterogeneous reactions on MHD viscous fluid in the vicinity of stagnation point with uniform suction and generalized slip boundary condition past a stretching/shrinking sheet.

Both homogeneous–heterogeneous reactions involve in many chemicallyreactive systems for example in combustion, catalysis, aircraft androcket engines and in biochemical systems. Several reactions occur veryslowly or not even happenwithout the presence of catalyst, however theassociation between homogeneous–heterogeneous reactions that takeplace on some catalytic surface is quite complicated which can becontained in the generation as well as in consumption of reactant speciesat distinct rate both within the fluid and on catalytic surfaces. Chaudaryand Merkin [30,31] constructed a simple model of homogeneous–heterogeneous reactions in a boundary layer stagnation point flowfor both equal and unequal diffusivities. In another study, Chaudaryand Merkin [32] assumed effects of loss of autocatalyst in theirprevious study. Some important studies relevant to homogeneous–heterogeneous reactions have been listed in Refs. [33–40].

Ferro fluids have unique physical and chemical properties and from these fluids it is easy to stimulate significant magnetic forces which lead to fluid motion.Ferro fluids in the presence of magnetic field have distinct applications like development of stable liquid spikes,specific gravity analyzer,stable levitation of an object and treatment of cranial aneurysms.Therefore,the objective of the present investigation is to examine the two-dimensional stagnation point flow of ferro fluid over a flat plate with non-linear slip condition in the presence of magnetic field.Both homogeneous and heterogeneous reactions are considered with equal and unequal diffusivities of reactant and autocatalys is.The resultant non-linear ordinary differential equations with non-linear boundary conditions are solved numerically with shooting method and Runge–Kutta integrating scheme. To the best of author's knowledgesuch type of study for fer rofluid with non-linear slip boundary conditionin the vicinity of stagnation point and homogeneous–heterogeneous reactionshas not been done before.

2.Formulation of the Problem

Consider the steady two-dimensional incompressible flow of ferro fluid near a stagnation point over a flat horizontal plate.The Cartesian co-ordinates system is employed with thex-axis parallel to the plate andy-axis perpendicular to it.We assumed that the linear free stream velocity isue=cx,herec＞0 represents the strength of stagnation point flow.An external magnetic field of constant strengthB0is applied normal to the sheet.The induced magnetic field is considered to be negligible and external electric field is zero due to the assumption of small magnetic Reynolds number.We also considered a simplemodel of homogeneous–heterogeneous reactions in which two chemical species A and B are associated in a boundary layer flow as mentioned by Chaudary and Merkin[30]and Merkin[33]:

in above Eqs.(1)and(2),aandbdenote the concentrations of the chemical species A and B,whereaski(i=c,s)are the rate constants.We considered both reactions as isothermal processes and at the ambient fluid a uniform concentrationa0of reactant A is present and there is no auto catalyst B.According to these assumptions and after applying boundary layer approximation,the basic equations of the present problem are

hereuandvrepresent the velocity components inxandydirections,μnf,ρnfand σnfare the dynamic viscosity,density and electrical conductivity of the ferro fluid,DAandDBare the respective coefficients of A and B.The dynamic viscosity and effective density of the ferro fluid are given as[41]

where ? is the solid volume fraction of ferrofluid,whilenf,fandsin subscript stand for thermophysical properties of the ferro fluid,base fluid and ferrosolid particles,respectively.Table 1 shows the thermophysical properties of base fluids and magnetic and non-magnetic nanoparticles.

Table 1Thermophysical properties of base fluids,magnetic and non-magnetic nanoparticles

As suggested by Thompson and Troian[19],we take generalized slip boundary condition as

utindicates the tangential velocity,α?indicates the Navier's slip constant,β?indicates the reciprocal of some critical shear rate and τwindicates the shear rate at the wall.

The boundary conditions for the present analysis are:

By applying the similarity transformations

The continuity Eq.(3)is satisfied and Eqs.(4)–(6)take the form

with the boundary conditions

here prime symbolizes the differentiation with respect toη,M2the magnetic parameter,Scthe Schmidt number,KandKsshow the measure of the strength of homogeneous and heterogeneous reactions,δis the ratio of diffusion coefficient,α and β are the dimensionless velocity slip parameter and the dimensionless critical shear rate,respectively and all these quantities are de fined as

Following Aziz[42],for similarity solution of Eqs.(10)–(12)the quantities α and β(x)must be constant instead of function of variablex,for this β(x)needs to be proportional tox-1,therefore we assume

herea?andb?are constants.By using Eq.(17)into Eq.(16)we have

It is worth mentioning here that Eqs.(10)–(12)give the similarity solutions with the values of α and β de fined in Eq.(18)but with α and β described in Eq.(16),the solutions developed are the local similarity solutions.

In most applications,diffusion coefficients of chemical species A and B are of a comparable size due to this we can further assume that the diffusion coefficientsDAandDBare equal(δ=1,see[32]).Because of this assumption,we have

Now Eqs.(11)and(12)take the form

subject to the boundary conditions

The skin friction coefficientCfalongxdirection is given by

where τw=μ(?u/?y)y=0is the shear stress at the surface of the wall.Using Eq.(9)we get dimensionless skin friction coefficient as

3.Numerical Solution

To find the numerical solution of non-linear ordinary differential Eqs.(10)–(12)along with boundary conditions(13)–(15)for unequal diffusivities and Eqs.(10)and(20)with boundary conditions(13)and(21)for equal diffusivities we used shooting method with Runge–Kutta iterative method of fourth order.For this method,we need to transform the system of equations into first order initial value problem.

The system of first order initial value problems for unequal diffusivities is given as:

with boundary conditions

The first order initial value problem of concentration equation for equal diffusivities is given as:

with boundary conditions

The initial value problems(IVPs)are solved using Runge–Kutta iterative scheme.The missing conditions or initial guessesu1,u2,u3are adjusted to find the better approximation.

4.Results and Discussion

Numerical calculations are carried out for Eqs.(10)and(20)with the boundary conditions(13)and(21)by using shooting method with Runge–Kutta algorithm.Graphical representations of kerosene based ferroparticles are given in Figs.(1)–(10)for various parameters namely magnetic parameterM,velocity slip parameter α,critical shear rate parameter β,K(strength of homogeneous reaction),Ks(strength of heterogeneous reaction),Schmidt numberScand the type of nanoparticles on velocity and concentration pro files.Tables 2–6 are made to show the numerical values off″(0)andg′(0)for different involving parameters for both water based and kerosene based magnetic and non-magnetic nanoparticles.

Fig.1.Effects of α on velocity pro files for ? =0.4,M=1.5 and β =0.3.

Fig.2.Effects of β on velocity pro files for ? =0.5,M=1.5 and α =0.2.

Fig.3.Effects of α on concentration pro files for ?=0.4,M=1.5,β=0.3,K=10,Ks=0.5 and Sc=0.5.

Fig.4.Effects of β on concentration pro files for ?=0.5,M=1.5,α=0.2,K=10,Ks=0.5 and Sc=0.5.

Fig.5.Effects of Sc on concentration pro files for ?=0.4,M=1.5,α=0.2,β=0.3,K=10,and Ks=0.5.

Fig.1 depicts the variation of velocity slip parameter α on velocity pro filesf′(η).From this figure it can be seen that velocity pro filesf′(η)increase with increasing values of α,whereas boundary layer thickness decreases for both ferroparticles,however the decrement is faster for kerosene based cobalt ferrite.Fig.2 gives the influence of critical shear rate β on the velocity of the fluidf′(η).It is noted from the figure that boundary layer thickness decreases more rapidly for large values of β.

Fig.6.Effects of K on concentration pro files for ?=0.4,M=1.5,α=0.2,β=0.3,K=10,and Ks=0.5.

Fig.7.Effects of Kson concentration pro files for ?=0.4,M=1.5,α=0.2,β=0.3,K=10,and Sc=0.5.

Figs.3 and 4 describe the variation of species concentrationg(η)for different values of velocity slip parameter α and critical shear rate β.It can be seen from these figures thatg(η)is an increasing function for both parameters α and β and concentration boundary layer thickness decreases with increasing values of α and β.Fig.5 gives the influence of Schmidt numberScon fluid concentrationg(η).Since Schmidt numberScis the ratio of viscous diffusion rate and molecular diffusion rate so,for large values ofSc,species concentration increases while in contrast concentration boundary layer thickness suppresses with increasing values ofSc.The intensity of homogeneous and heterogeneous reactions on concentration pro filesg(η)is observed from Figs.6 and 7,respectively.These figures indicate that for bothKandKs,the concentration boundary layer increases with η,moreover they all coincide after a certain value of η.In other words,after a certain value of η,homogeneous and heterogeneous reaction shave not impact any on the concentration of reactant and this critical value of η (η∞)is influenced by the strength of homogeneous reaction.Fig.8 shows the behavior of dimensionless concentrationg(0)versus Kfor various values ofKs.It can be seen from this figure that the concentration decreases for both the increasing values ofKandKs.Fig.9 shows the variation of ferroparticles volume fraction parameter ? and magnetic parameterMonf″(0)for three different types of kerosene based ferro particles.It is evident from the figure thatf″(0)decreases for large values of solid volume fraction parameter ? while increases with increasing values ofM.The variation of magnetic parameterMand solid volume fraction parameter ? ong′(0)can be seen from Fig.10 andg′(0)increases withMsince it decreases with ?.It is further noticed from Figs.1–8 that both lines(solid and dashed)are close to each other.This is due to the fact that the values of densities of Fe3O4and CoFe2O4are 5180 and 4907 respectively.Similar situation occurs in Figs.9 and 10 for CoFe2O4and Mn–ZnFe2O4.Fig.11 shows the comparison of concentration pro filesg(η)andh(η)with equal and unequal diffusivities of speciesAandBfor Fe3O4–kerosene ferro fluid.

Fig.8.Variation ofg(0)withKfor somevalues ofKsfor kerosene based Fe3O4,CoFe2O4and Mn–ZnFe3O4when ?=0.4,M=1.5,α=0.2,β=0.3 and Sc=0.5.

Fig.9.Variation of f″(0)with ? for some values of M for kerosene based Fe3O4,CoFe2O4 and Mn–ZnFe3O4when α=0.2 and β=0.3.

Fig.10.Variation of g′(0)with ? for some values of Ksfor kerosene based Fe3O4,CoFe2O4 and Mn–ZnFe3O4when α=0.2,β=0.3,K=1,Ks=0.5 and Sc=0.5.

Table 2Comparison of f″(0)at M=?=α=β=0

Table 3Comparison of g(0)for various values of Ks,K and Sc at M=?=α=β=0

Table 4Numerical values of f″(0)for water based magnetic and non-magnetic nano fluids

Table 5Numerical values of g′(0)for water based magnetic and non-magnetic nano fluids

Fig.11.Comparison of concentration pro files by taking diffusion coefficients DAand DB equal(solid line)and unequal(dashed line)for kerosene based Fe3O4when ?=0.4,M=1.5,α=0.2,β=0.3,K=1,Ks=1 and Sc=1.

Tables 2 and 3 are made to show the comparison for numerical values off″(0)andg'(0)with the existing literature and found in good agreement.Table 4 represents the numerical values off″(0)for different values ofM,α,β and ?.In fact this table shows the comparison between pure fluid,magnetic and non-magnetic nano fluids.From this table we can observe that values off″(0)increase with the addition of nanoparticles due to the fact that resistance is lower for the higher values of ?.It is also noticed from this table that for large values ofM,f″(0)increases while decreases with large values of α and β.The increment in non-magnetic nanoparticle Al2O3is higher than magnetic nanoparticle Fe3O4because the thermal conductivity is higher in nonmagnetic nanoparticle Al2O3.Table 5 is made to show the numerical values ofg′(0)for various values ofM,α,β,?,K,KsandSc.Same asf″(0),numerical values ofg′(0)also increase with an addition of nanoparticles.However,g′(0)increases with increasing values ofM,α,β,?,KsandSc,whereas decreases with large values ofK.The numerical values off″(0)andg′(0)for water based ferro fluid and kerosene based ferro fluid are displayed in Table 6.The solid volume fraction of ferro fluid ? and viscosity are the main factors that affect the velocity of the ferro particles in the fluid.Therefore,it is evident from this table that in the presence and absence of magnetic field the value off″(0)for magnetic ferro fluids in kerosene is decreased by increasing the values of ? from 0.1 to 0.2.This is due to high resistance developed among the ferroparticles.The viscosity of kerosene and low solid volume fraction of ferro particles ? are taken into account for the high resistance.Further,increase in ? from 0.2 to 0.3 the values off″(0)in kerosene are increased because higher concentration of ferroparticles drastically enhanced the velocity of the fluid.In comparison with the water,the value off″(0)in kerosene is low in the absence of magnetic field.This is due to the low viscosity of water as kerosene.Low viscosity reduced the resistance among the ferro particles thereforef″(0)increases in water by increasing ?.Similar behavior is shown forg'(0).

5.Concluding Remarks

In the present study,the stagnation point flow of ferro fluid over a flat plate with homogeneous–heterogeneous reactions and non-linear slip boundary condition is studied.The governing equations of momentum and concentration are converted into system of non-linearcoupled ordinary differential equations using similarity transformations.Shooting method with Runge–Kutta scheme is applied to solve the problem under consideration.The outcomes of the present investigation are as follows:

Table 6Numerical values of f″(0)and g′(0)for water based and kerosene based magnetic nano fluids with α=0.3,β=0.5,K=1,Ks=1 and Sc=0.5,where parenthesis gives the values of g′(0)

·The dimensionless velocity increases with increasing values of velocity slip parameter α and critical shear rate β for both kerosene-based magnetite and cobalt ferrite.

·The dimensionless concentration increases with large values of α,β andScbut in contrast decreases with higher values ofKandKsfor both types of kerosene-based ferroparticles.

·Numerical calculations show that the values off″(0)andg′(0)are higher for magnetic(Fe3O4)and non-magnetic(Al2O3)nanoparticles as compare to pure fluid,and this increment is higher for nonmagnetic nano fluid.

·Numerical values off″(0)andg′(0)for water based ferro fluid are higher than the kerosene based ferro fluid.

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