M.Veera Krishna

Department of Mathematics,Rayalaseema University,Kurnool,Andhra Pradesh 518007,India

Keywords: Chemical reaction Free convection Heat source Heat transfer Mass diffusion Radiation absorption

ABSTRACT The investigation of radiation-absorption,chemical reaction,Hall and ion-slip impacts on unsteady MHD free convective laminar flow of an incompressible viscous,electrically conducting and heat generation/absorbing fluid enclosed with a semi-infinite porous plate within a rotating frame has been premeditated.The plate is assumed to be moving with a constant velocity in the direction of fluid movement.A uniform transverse magnetic field is applied at right angles to the porous surface,which is absorbing the fluid with a suction velocity changing with time.The non-dimensional governing equations for present investigation are solved analytically making use of two term harmonic and non-harmonic functions.The graphical results of velocity,temperature and concentration distributions on the analytical solutions are displayed and discussed with reference to pertinent parameters.It is found that the velocity profiles decreased with an increasing in Hartmann number,rotation parameter,the Schmidt number,heat source parameter,while it increased due to an increase in permeability parameter,radiation-absorption parameter,Hall and ion slip parameters.However,the temperature profile is an increasing function of radiation-absorption parameter,whereas an increase in chemical reaction parameter,the Schmidt number Sc or frequency of oscillations decrease the temperature profile on cooling.Also,it is found that the concentration profile is decreased with an escalating in the Schmidt number or the chemical reaction parameter.

1.Introduction

The combined heat and mass transport problems through the chemical reaction are of significance in a lot of processes and have obtained an extensive value of concentration in current years.In developments such as drying,disappearance at the external of a fluid body,energy transportation in a drenched cooling increase and the flow in a desert cooler,heat and mass transport happen simultaneously.Possible applications of that category of flow can be established in numerous industries.Some examples,in the power industries,between the techniques of generating electric energy is solitary in this electrical energy are extracted directly exciting from a conducting fluid.It is predominantly attracted in cases of diffusion and chemical reaction occurs at approximately the identical speediness.Once diffusion is to a great extent faster than chemical reaction,then merely chemical reaction influences the rate of chemical reaction; when diffusion is not much quicker than chemical reaction,the diffusion as well as kinetics interacts to construct very dissimilar consequences.The investigation of heat generation or absorption consequences in moving fluids is significant in sight of quite a few substantial problems,they are,and fluids undergo exothermic or else endothermic chemical reaction.Outstanding to the quick development of electronic technology,effectual freezing of electronic apparatus has became certified and freezing of electronic apparatus ranges from own transistors to foremost structure computers and from energy providers to telephone switch panels and thermal diffusion impacts has been exploited for isotopes separation in the combination among gases with extremely low molecular weight (H2and He) and average molecular weight.

Magnetic field plays an important role in numerous fields such as biological,chemical,mechanical and medical research.In clinical and medical research the magnets are extremely important to create three dimensional images of anatomical and diagnostic importance from nuclear magnetic resonance signals.In view of these applications,the present research is to the effects of radiation on magnetohydrodynamic (MHD) flow and heat transfer problem has become innovative and significant industrially.At high functioning of heat and radiation impact might be moderately momentous.Numerous processes in manufacturing regions occur at huge temperature and consideration of radiation heat transport becomes extremely significant for the construction of the significant apparatus.Nuclear energy plants,some gas turbines and the different propulsion machines for satellites,missiles,aircraft,and space vehicles are illustrations of several engineering regions.

Bestman [1]investigated the free convection boundary layer flow with simultaneous heat and mass transfer in a porous medium when the boundary wall moves in its own plane with suction.Abdus Sattar and Hamid Kalim [2]studied the unsteady free convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate.Makinde [3]explored the combined free convection boundary layer flow with thermal radiation and mass transfer past a permeable vertical plate.Makindeet al.[4]investigated the problem of unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture in an optically thin environment.Muthucumaraswamy and Ganesan [5]explored the impact of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate.Dekaet al.[6]derived an exact solution to the flow due to impulsive motion of an infinite vertical plate in its own plane in the presence of species concentration,constant heat flux at the plate and chemical reaction of first order by the Laplace-transform technique.Analytical solutions for heat and mass transfer by laminar flow of a Newtonian,viscous,electrically conducting and heat generating/absorbing fluid on a continuously moving vertical permeable surface in the presence of a magnetic field and a first-order chemical reaction have been reported by Chamkha[7].An exact solution for the Stokes problem for an isothermal infinite vertical plate on taking into account the constant heat flux at the plate has been derived by Soundalgekar and Patti[8].Gebhart and Pera[9]studied the laminar flows which arise in fluids due to the interaction of the force of gravity and density differences caused by the simultaneous diffusion of thermal energy and of chemical species.The problem of unsteady MHD two-dimensional flow of a viscous,incompressible,electrically conducting and heat-absorbing fluid along a semi-infinite vertical permeable moving plate has been discussed by Chamkha[10].Raptis [11]investigated the steady two dimensional free convection flow through a porous medium bounded by a vertical infinite porous plate by the presence of thermal radiation and constant suction velocity.

The inspect of thermal radiation in heterogeneous areas in engineering such as nuclear power plants,multifarious propulsion devices for missiles,liquid metal fluids,gas turbines and so forth,due to small coefficient of convective heat transfer,it demonstrates the surface heat transfer.Thermal radiation is distinguished in numerous applications because of the perspective in which radiant emission relies on temperature.Biswaset al.[12]investigated the unsteady MHD heat and mass transfer micropolar fluid flow in the presence of radiation and chemical reaction through a vertical porous plate.The MHD pulsating flow of Casson fluid in a porous channel with thermal radiation,chemical reaction has been explored by Srinivaset al.[13].Sobamowoet al.[14]studied the heat transfer effects of thermal radiation on free convection flow of Casson nanofluid over a vertical plate.Effects of chemical reaction and heat generation/absorption on MHD Casson fluid flow over an exponentially accelerated vertical plate embedded in porous medium with ramped wall temperature and ramped surface concentration have been investigated by Hari and Harshad [15].Ahmmedet al.[16]have been discussed the unsteady MHD free convection flow of nanofluid through an exponentially accelerated inclined plate embedded in a porous medium with variable thermal conductivity in the presence of radiation.

The liquefied flow problems in rotating medium have drawn attention of many researchers who investigated hydrodynamic flow of a viscous,incompressible fluid in rotating medium considering different aspects of the problem.There has been considerable interest in the problems of MHD flow in a rotating medium during past few decades due to their geophysical and astrophysical significance and their application in fluid engineering.Many imperative problems,like,maintenance and secular variations of terrestrial magnetic field due to motion in Earth’s liquid core,internal rotation rate of the sun,structure of rotating magnetic stars,planetary and solar dynamo problems,MHD Ekman pumping,turbo machines,rotating MHD generators,rotating drum type separator in closed cycle two phase MHD generator flowetc.are directly governed by the action of Coriolis and magnetic forces.The effects of Coriolis force are more significant than that of inertial and viscous forces.In addition to it,Coriolis and magnetic forces are comparable in magnitude.In many realistic applications requiring strong magnetic field there is a need to think both the Hall and ion slip currents because of the significant effect they have on the vector of the current density and transitively on the magnetic force idiom.Ellahiet al.[17]discussed the Hall and ion-slip impacts on the peristalsis Jeffreys fluid flow through a rectangular channel everywhere it is inhomogeneous.Bhattiet al.[18]carried out research on the effects of Hall and ion-slip on non-Newtonian fluid flow with the Reynolds number as zero.Srinivasacharya and Shafeeurrahman [19]studied amid two analogous concentric cylinders for the Hall and ion slip effects on MHD mixed convective nanofluid.Jitendra and Srinivasa [20]discussed Hall and ion slip effects on convective flow of rotating fluid past an exponential accelerated vertical plate.Krishna and Chamkha [21]explored the thermal radiation and heat absorption,diffusion-thermo and Hall and ion slip effects on MHD free convection rotating flow by nanofluids past a moving porous plate with constant heat source.Sara and Bhatti [22]investigated the MHD peristaltic induced flow of a nanofluid in a non-uniform channel with chemical reaction,Hall and ion slip effects.Srinivasacharya and Kaladhar[23]investigated the Hall and ion slip effects on fully developed electrically conducting couple stress fluid flow between parallel disks using Homotopy analysis method.Ibrahim and Anbessa[24]discussed the problem of Hall and ion-slip effects on a mixed convection flow of an electrically conducting nanofluid over a stretching sheet in a permeable medium and is solved numerically using a spectral relaxation method.Nawaz and Zubair [25]explored the three dimensional flow of nano-plasma in the presence of uniform applied magnetic field perpendicular to two dimensional nonlinear stretching surfaces with Hall and ion slip currents when Joule heating and viscous dissipation are of considerable order of magnitude.Krishnaet al.[26]investigated the Hall and ion slip effects on the unsteady MHD free convective rotating flow through porous medium past an exponentially accelerated inclined plate.The combined effects of Hall and ion slip on MHD rotating flow of ciliary propulsion of microscopic organism through porous medium have been studied by Krishnaet al.[27].Krishna and Chamkha [28]investigated the Hall and ion slip effects on the MHD convective flow of elastico-viscous fluid through porous medium between two rigidly rotating parallel plates with time fluctuating sinusoidal pressure gradient.Krishna [29]reported that the Hall and ion slip effects on MHD free convective rotating flow bounded by the semiinfinite vertical porous surface.Krishna [30]discussed the MHD laminar flow of an elastico-viscous electrically conducting Walter’s fluid through a circular cylinder or a pipe.Hall and ion slip effects on unsteady MHD Convective Rotating flow of Nanofluids have been discussed by Krishna and Chamkha[31].Ibrahim and Anbessa[32]explored the mixed convection flow of Eyring-Powell nanofluid over a linearly stretching sheet through a porous medium with Cattaneo–Christov heat and mass flux model in the presence of Joule heating,Hall and ion slip effects.Ibrahimet al.[33]studied the effects of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate by the influence of heat source and suction.Ajibade and Umar [34]investigated the effects of chemical reaction and radiation absorption on the unsteady MHD natural convection flow in a vertical channel filled with porous materials,one of the plate moves with a constant velocity in the direction of fluid flow while the other plate is stationary and which absorbs the fluid with a suction velocity varying with time.Krishna [35]investigated the influence of thermal radiation,Hall and ion-slip impacts on the unsteady MHD free convective rotating flow of Jeffreys fluid past an infinite vertical porous plate with the ramped wall temperature.

With the motivation from all aforementioned above,the radiation-absorption,chemical reaction,Hall and ion-slip impacts on unsteady MHD free convective laminar flow of an incompressible viscous,electrically conducting and heat generation/absorbing fluid enclosed with a semi-infinite porous plate within a rotating frame have not been discussed yet.Therefore the radiationabsorption,chemical reaction,Hall and ion-slip impacts on unsteady MHD free convective laminar flow of an incompressible viscous,electrically conducting and heat generation/absorbing fluid enclosed with a semi-infinite porous plate within a rotating frame has been premeditated.This investigation organized in six sections.Section two contains problem statement,its modelling and solution procedure is given.Results and discussion are described in detail in section three.It is also discussed the relation among the momentum,thermal and concentration boundary layers in section four,and code validation in section five.Finally the conclusions are specified in section six.

2.Formulation and Solution of the Problem

We have deemed the radiation-absorption,chemical reaction,Hall and ion-slip impacts on the unsteady two-dimensional flow of a laminar,viscous,electrically conducting fluid over a semiinfinite vertical moving porous plate embedded in a uniform porous medium with a uniform transverse magnetic field in the existence of thermal and concentration buoyancy effects.The physical configuration of the problem is as shown in Fig.1.It is assumed that there is no applied voltage which implies that,the absence of an electrical field.The fluid properties are assumed to be constant except that the influence of density variation with temperature has been considered only in the body-force term.The concentration of diffusing species is very small in comparison to other chemical species,the concentration of species far from the wall,C∞,is infinitesimally small and hence the Soret and Dufour effects are neglected.The chemical reactions are taking place in the flow and all thermophysical properties are assumed to be constant of the linear momentum equation which is approximated according to the Boussinesq approximation.Due to the semiinfinite plane surface assumption,the flow variables are functions ofzand the timetonly.

Under these assumptions,the governing equations that describe the physical situation in a rotating frame are specified by,

The magnetic and viscous dissipations are neglected in this study.The third and fourth terms on the right hand side of the momentum Eq.(2) denote the thermal and concentration buoyancy effects,respectively.Also,the second and third terms on the right hand side of the energy Eq.(4) represents the heat and radiation absorption effects,respectively.It is assumed that the permeable plate moves with a variable velocity in the direction of fluid flow.In addition,it is assumed that the temperature and the concentration at the wall as well as the suction velocity are exponentially varying with time.Under these assumptions,the suitable boundary conditions for the velocity,temperature and concentration fields are

Fig.1.Physical model.

It is clear from Eq.(1)that the suction velocity at the plate surface is not a function of z but a function of time only.Assuming that it takes the following exponential form as,

whereAis a real positive constant,ε and εAare small less than unity,andw0is a suction velocity scale,which has non-zero positive constant.The negative sign indicates that suction is towards porous materials.

The electron-atom collision frequency is assumed to be very high,so that Hall and ion slip currents cannot be neglected.Hence,the Hall and ion slip currents give rise to the velocity iny-direction.When the strength of the magnetic field is very large,the generalized Ohm’s law is modified to include the Hall and ion slip effect(Sutton and Sherman [22]),

Additionally,it is assumed that the Hall parameter βe=ωeτe～O（1） and the ion slip parameter βi=ωiτi<<1，in the Eq.(9),the electron pressure gradient and thermo-electric effects are abandoned,i.e.,the electric fieldE=0 under these assumptions,the Eq.(9) condensed to,

Making use of the above non-dimensional variables,the governing Eqs.(4),(5)and(16)can be expressed in non dimensional form as,

The boundary conditions are

To determine the solutions of the Eqs.(17)–(19) subjected to the boundary conditions (20) and (21).The Eqs.(17)–(19) represent a set of partial differential equations that cannot be solved in closed form.However,it can be reduced to a set of ordinary differential equations in dimensionless form that can be solved analytically.This can be done by representing the velocity,temperature and the concentration as,

Substituting the Eqs.(22)–(24)into the Eqs.(17)–(19),equating the harmonic and non-harmonic terms,and neglecting the higher order powers of ε,and simplifying,it is obtained the following pairs of equations of zeroth and first orders be,

Subjected to the boundary conditions

Through the boundary conditions

Without going into detail,the solutions of Eqs.(25)–(27) and(30)–(32) subject to boundary conditions (28) and (29) and (33)and (34) can be shown to be,

The physical quantities of interest are the wall shear stress τwand the local surface heat transfer rateqw.These are defined by

Consequently,the local friction factorCfis given by

The local surface heat flux is given by,

where,k2is the effective thermal conductivity,together with the definition of the local Nusselt number,

It is written be,

where,Rex=w0x/ν is the local Reynolds number.

The local surface mass flux is given by,

whereDmis the effective molecular diffusivity,together with the definition of the local Sherwood number,

It is written be,

3.Results and Discussion

The present problem is explored the radiation-absorption,chemical reaction,Hall and ion slip impacts on unsteady MHD free convective laminar flow of an incompressible viscous,electrically conducting and heat generation/absorbing fluid enclosed with a semi-infinite porous plate within a rotating frame.The plate is assumed to be moving with a constant velocity in the direction of flow.A uniform transverse magnetic field is applied at perpendicular to the porous surface,which is absorbing the fluid with a suction velocity changing with time.The non-dimensional governing equations for present investigation are solved analytically making use of perturbation method.The governing flow presides over the non-dimensional parameters namelyviz.Hartmann numberM,permeability parameterK,rotation parameterR,thermal Grashof numberGr,mass Grashof numberGm,Schmidt numberSc,chemical reaction parameterKc,heat source parameterH,radiation-absorption parameterQl,Prandtl numberPr,Hall parameter βeand ion slip parameter βi.The velocity,temperature and concentration profiles are displayed in Figs.2–4,Fig.5 and Fig.6 respectively.The Tables 1–3 symbolize the skin friction,Nusselt number and Sherwood number for dissimilar deviations in the pertinent parameters.When the magnetic field is large,then the Hall current will be developed in the flow field.Therefore,it is considered that the values of Hartmann number 0

Table 1The shear stresses （A=0˙5，n=0˙5，t=0˙5，ε=0˙001，u0=0˙5）

Table 2The Nusselt number (Nu)（A=0˙5，ε=0˙001）

Table 3The Sherwood number (Sh) （A=0˙5，ε=0˙001）

The significance of Hartmann numberMon both velocity components is interpreted from Fig.2(a).The primary velocity componentulessens and secondary velocity component v augments with an increasing in the Hartmann numberM.As likely,the enlargement inMdiminishes the resultant velocity.It is seeing as of the classical result,the Lorentz force,this emerges due towards an application of magnetic field just before an electrical conducting fluid and provide to a resistive type of force.This shows a tendency to refuse to go along with the flow throughout the fluid medium.As a result,it is observed as the outcome of magnetic field in the appearance of porous material,endorses a decelerating effect on the velocity distribution and thickness of the momentum boundary layer.It is looked from the Fig.2(b)that,the primary velocity componentuheightens and secondary velocity component v condenses with ever increasing permeability parameterK.The larger values ofK,enhances the resultant velocity and consequently enlarge the thickness of the momentum boundary layer.Lower the permeability causes lesser the fluid speed is observed in the flow region occupied by the fluid.Physically,the porous medium impact on the boundary layer growth is significant due to the increase in the thickness of the thermal boundary layer.It is expected that,an increase in the permeability of porous medium lead to a rise in the flow of the fluid through it,since when the holes of the porous medium become large,the resistance of the medium may be neglected.The Fig.2(c)depicted that the behaviour of the velocity distributions ofuand v with rotation parameterR.It is observed that,as reinforcement in rotation parameter,the primary velocity componentureduces throughout the fluid region.The secondary velocity component v is lessens forz≤1 and then enhances with a growing in rotation parameter throughout the fluid region.This results,diminishes the resultant velocity and the thickness of the momentum boundary layer in the fluid region when an increase in rotation parameter.Even though rotation is identified to persuade fluid velocity its decelerated significance is extensive only through the fluid region nearby the plate while it has overturned achievement on other area gone as of the plate.It is owing to the motivation that Coriolis force is overriding through the region in close proximity to the axis of rotation.The improvements of velocity profiles foruand v with the Prandtl numberPrare designed in Fig.2(d).This is perceived that the enlargement of Prandtl number made the fluid movement slowing down for primary velocity,where as it enhances for secondary velocity throughout the fluid region.Therefore,the resultant velocity is reduced with an increase in Prandtl number throughout the region occupied by the fluid.The Prandtl number controls the relative thickness of the momentum and thermal boundary layers.WhenPris small,it means that the heat diffuses quickly compared to the momentum.This means that for liquid metals the thermal boundary layer is much thicker than the velocity boundary layer.This is perceived that,the enlargement ofPrmade then the fluid flow is slowing down.Actually,it is justified owing to the fact that so as to the fluid with the highest Prandtl number has high viscosity,this made as the fluid has substantial thickness.

Fig.2.(a–d) The velocity profiles against M, K, R and Pr.

Fig.3.(a–d) The velocity profiles against Gr, Gm, Sc and Kc.

Fig.4.(a–d) The velocity profiles against H， Ql，βe and βi.

The Fig.3(a) and (b) displayed the effects of thermal and mass Grashof number on the fluid velocity components foruand v.The primary velocityuimproves and secondary velocity v lessens with growing in thermal Grashof numberGrand solutal Grashof numberGmthroughout the fluid region.The thermal Grashof numberGrimplies the experienced outcome of the thermal buoyancy force to the viscous hydrodynamic force through the boundary layer,at the same time as the mass Grashof numberGmestablishes the ratio for the solutal buoyancy force to the viscous hydrodynamic force.As accepted the fluid velocity boosts by virtue of the strengthening of heat and solute buoyancy force strengths.The velocity distribution enlarges rapidly next to the porous surface and later this is decay smoothly to the initial velocity zero.This is due to the fact that,both buoyancy forces enhances velocity and increases thickness of momentum boundary layer through a heightening in thermal Grashof numberGrand mass Grashof numberGm.It is noticed from the Fig.3(c)for the velocity through the dissimilar values of Schmidt numberSc.On enlarges in values of Schmidt number be inclined to decreasing of the both velocity componentsuand v and hence reduce the resultant velocity and the thickness of momentum boundary layer.That is,an increase in the Schmidt number produces a decrease in the momentum boundary layer thickness,associated with the reduction in the velocity profiles.Physically,the increase in the value ofScmeans the decrease of molecular diffusion.Reversal performance is scrutinized for both velocity components with ever-increasing chemical reacting parameterKc(Fig.3(d)).Heavier diffusing species,thus,for the highest value ofKc,the velocity is increasing throughout the fluid region.Furthermore,heavier species through disparaging reaction reasons acceleration within the velocity distribution.Hence,it is to mounting by rate of chemical reaction sources.

The Fig.4(a) portrayed the behaviour of velocity componentsuand v across the boundary layer with heat source parameterH.The primary velocity componentulessens and v augments with escalating heat source parameter throughout the fluid region.Raising the values of heat source parameter trend to decreases the primary velocity component and therefore it dominate to decrease the rate of the momentum boundary layer thickness.The Fig.4(b) demonstrated the effects of radiation absorption parameter for the primary and secondary velocity components.This may be observed from the figures that primary velocity profiles are increasing with increasing radiation absorption parameter,where as secondary velocity profiles reduced with an enlarge in radiation absorption throughout the fluid region.This is fascinating to note that,the resultant velocity and the momentum boundary layer thickness enhances with an escalating in radiation absorption parameter.The Fig.4(c)and(d)depicted that the behaviour of the velocity distribution with Hall and ion-slip parameters.The primary velocityuget better and secondary velocity v minimizes with growing in Hall and ion slip parameters βeand βithroughout the fluid medium.It is marked that,as a strengthening in Hall and ion-slip parameters,which results,enhances the resultant velocity and the momentum boundary layer thickness all over the fluid region.The incorporation of Hall parameter lessens the effectual conductibility and therefore descends the magnetic renitent fierceness.Also,the efficient conductivity augments as enlarge in ion-slip parameter,for this reason the attenuation force are lessening consequently velocity heightens.

Fig.5.(a–f) The temperature profiles against Kc, Ql, Sc, H,Pr and n.

Fig.6.(a and b) The concentration profiles against Kc and Sc.

The Fig.5(a)–(f) exhibits the disparity of temperature profiles through the dissimilar quantities of chemical reaction parameterKc,radiation-absorption parameterQl,Schmidt numberSc,heat resource parameterH,Prandtl number Pr,and frequency of oscillationsn.It is noticed from the Fig.5(a) that,the temperature reduces with increasing chemical reaction parameterKc.The reduction in temperature by chemical reaction may be due to some amount of heat energy is used to augment the chemical reaction.The effect of radiation-absorption parameterQlon temperature distribution is delineated within the boundary layer (Fig.5(b)).The temperature profiles increases with increase of radiationabsorption parameter.This is due to the fact that when heat is absorbed,the buoyancy forces accelerate the flow temperaturethroughout the fluid region.This has been obvious that temperature is a growing function of radiation-absorption.From the Figure 5(c),the temperature reduces with increasing Schmidt numberSc.The mass diffusion in liquids grows with temperature,roughly inversely proportional viscosity variation with temperature,so that the Schmidt number quickly decreases with temperature.The similar tendency is notified with an increasing in heat source parameterH(Figure 5(d)).As intended that,an enlargement in heat source parameter reduce the temperature distribution.The heat resource parameter indicates the experienced role of conducting heat transport to heat absorption transport.Hence the thickness of the thermal boundary layer is decreased.The temperature distribution lessens with an escalating in Prandtl number intact the fluid region (Fig.5(e)).The elucidation of that performance conceals through the information to Prandtl number was described as the proportion of momentum diffusivity to thermal diffusivity.Hence,at less important quantities of Prandtl fluids acquired large thermal conductivity this elicits the thermal diffusion gone from the heating surface additionally accelerate and quicker contrasted to bigger quantities of Prandtl number.Consequently the thickness of the thermal boundary layer is slow down by an increasing in Prandtl number.The temperature and thickness of thermal boundary layer lessens with increasing in frequency of oscillation in entire fluid region (Fig.5(f)).

The concentration distributions with esteem to the parameters the Schmidt numberSc,chemical reaction parameterKcare displayed in Fig.6(a) and (b).We perceived that,from Fig.6(a) the concentration diminishes through an increasing in Schmidt numberScduring the fluid region.Schmidt number is a dimensionless number defined as the ratio of momentum diffusivity(kinematic viscosity) and mass diffusivity,and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes.TheScis quantizing the virtual efficiency of momentum and mass transport through diffusion in the momentum and concentration boundary layers.This was acquired that an increasing in the value of Schmidt number induces the absorption of particles and these boundary layer thickness to diminish extensively.That is,physically,the increase in the value ofScmeans the decrease of molecular diffusion.Hence,the concentration of the species is higher for smaller values ofScand lower for larger values ofSc.Analogous behaviour is identified with enlarging chemical reaction parameterKc(Fig.6(b)).An enlargement in the chemical reaction parameter Kc declines the concentration distribution quickly.Because,the quantity of solute particles experiencing chemical reaction effect gets increased as chemical reaction parameter enlarges,this leads to diminish in concentration distribution.Therefore,the chemical reaction lessens the solutal boundary layer thickness substantially.

Table 1 is characterized the magnitudes of skin friction.An enlargement of Hartmann number goes ahead to diminishes in Skin friction.Because the Lorentz force on viscous fluid reduced the frictional drag.The similar behaviour is scrutinized through an augment in rotation parameter,Prandtl number and heat source parameter.Additionally,an enlargement in permeability parameterKgoes ahead to bring to bear higher skin friction in enormity on the boundary of the surface,while the similar nature is scrutinized for the same when an augment in radiation-absorption parameter,Schmidt number,chemical reaction parameter,thermal Grashof number,mass Grashof number,Hall and ion slip parameters near the boundary of the surface.Since from the Table 2,an increment in chemical reaction parameter,Schmidt number,Prandtl number,heat source parameter,frequency of oscillation and time goes ahead to an enhancement in Nusselt number.It is reduced with an increasing in radiation-absorption parameter.From the Table 3,an increase in Schmidt number,chemical reaction parameter,frequency of oscillation or time goes ahead to a strengthen in Sherwood number.

4.Relationship among the Three Boundary Layers

It is important to note that the thickness of the boundary layer depends on the flow,but also on how viscous the fluid,how sticky it is and how quickly the velocity can change in it,is called the momentum diffusivity.The momentum boundary layer,the thermal boundary layer and the concentration boundary layer are having a decisive influence on the entire heat and mass transport in a flow.The momentum boundary layer develops whenever there is flow over a surface.It is associated with shear stresses parallel to the surface and results in an increase in velocity through the boundary layer from nearly zero right at the surface to the free stream velocity far from the surface.The boundary layer thickness is by convention defined as the distance from the surface at which the velocity is 99%of the free stream velocity.The thermal boundary layer is associated with temperature gradients near the surface,and develops when there is temperature difference between the fluid free stream and the surface.Right at the fluid-surface interface,heat transfer occurs only through conduction.The thickness of the thermal boundary layer is defined as that point at which the temperature difference between the fluid and surface is 99%of the temperature difference between the free stream fluid and the surface.The concentration boundary layer develops when there is a difference in concentration of a component between the free stream and the surface.A concentration profile develops,and the thickness of the concentration boundary layer is defined as that point at which the difference in concentration between the fluid and the surface is 99% of the difference in concentration between the free stream fluid and the surface.All three boundary layers always influence each other and cannot be considered independently of each other.This leads directly to the dimensionless numbers,which relate two boundary layers to each other.The shear stresses are ultimately responsible for the deceleration of the fluid layers and the subsequent formation of equilibrium between the external pressure gradient and the shear stresses.Since there are no velocity gradients outside the hydrodynamic boundary layer,the shear stresses are negligible.In this undisturbed flow,the viscosity of the fluid plays no role.But within the boundary layer,the shear stresses are usually no longer negligible and have a decisive influence on the flow.Even if the boundary layer is defined very precisely,the change of the properties of the flow inside and outside of this sharp boundary is smooth.

The thermal boundary layer also gradually grows to an almost constant thickness,as the heat penetrates more and more into the fluid and thus heats the fluid layers over time.The thermal boundary layer is thus defined by the fact that temperature gradients exists and thus transport of heat takes place.At the same time,however,the heat transport has an effect on the flow,since the temperature decisively determines the viscosity of the fluid.For example,the fluid starts to flow faster due to the decreasing viscosity caused by an increase in temperature.This,in turn,has an effect on the heat transfer and thus on the flow itself.The thermal and momentum boundary layer therefore influence each other.The thickness of both boundary layers are generally differ from each other.If the Prandtl numberPris large (?1) then momentum flows more readily than temperature,so the thermal boundary layer will be thinner than the momentum one (because momentum effects spread more).IfPris small (?1),then temperature is what flows more readily,so the momentum boundary layer will be thinner than the thermal one.IfPr=1,then the two boundary layers will be exactly the same.

It is noted that the concentration boundary layer does not exist independently of the hydrodynamic or thermal boundary layer.On the one hand,especially for gases,the diffusion coefficient is very strongly dependent on the temperature,so that the thermal boundary layer has a direct influence on mass transport.On the other hand,the flow in the momentum boundary layer takes away the diffused particles.If this removal takes place relatively quickly,a large concentration gradient is formed.This in turn results in increased mass transport.Thus,the momentum boundary layer also directly influences the concentration boundary layer.

5.Code Validation

In applied mathematics and all branches of engineering,perturbation theory comprises mathematical methods for finding an approximate solution to a problem,by starting from the exact solution of a related,simpler problem.A critical feature of the technique is a middle step that breaks the problem into ‘‘solvable” and ‘‘perturbative” parts.Perturbation theory is widely used when the problem at hand does not have a known exact solution,but can be expressed as a ‘‘small” change to a known solvable problem.Perturbation theory is used in a wide range of fields,and reaches its most sophisticated and advanced forms in chemical engineering.Perturbation theory for chemical kinematics imparts the first step on this path.The field in general remains actively and heavily researched across multiple disciplines.The exactitude of numerical code is simulated for propriety,by MATHEMATICA 10.4 software through the perturbation mechanism.Using MATHEMTICA code,it is obtained the primary velocity distributions are shown in Table 4 for quite a few values of pertinent parameters namely Hartmann number,permeability parameter,thermal Grashof number and mass Grashof number.Again,by using same codefor the previous work(Ibrahim[33]),the same results are obtained nearly described above and are shown in Table 4 respectively for numerous values of pertinent parameters.The rigorously identical results are distinguished for both the research problems.Thus,the sensitivity of coding achieved accuracy (Fig.7).

Table 4Comparison of results for primary velocity （A=0˙5， n=0˙5， t=0˙5，ε=0˙001， u0=0˙5， Sc=0˙22， Kc=H=Ql=1， z=0˙5）

Fig.7.Comparison of primary velocity u with A=0˙5，ε=0˙001， u0=0˙5，R=βe=βi=0， M=2，K=0˙5， Gr=5， Gm=3， Pr=0˙71， H=1， Ql=1，Sc=0˙22， Kc=1， n=0˙5， t=0˙5.

6.Conclusions

It is investigated that the radiation-absorption,chemical reaction,Hall and ion-slip impacts on unsteady MHD free convective laminar flow of an incompressible viscous,electrically conducting and heat generation/absorbing fluid enclosed with a semi-infinite porous plate within a rotating frame.The conclusions are made as the following.The resultant velocity is reduced with an increasing in the strength of magnetic field,rotation and Prandtl number.It is also found that,the resultant velocity is improving with an increasing in the permeability of porous medium.If the pore size of the porous medium diminishes,then the velocity is obtained to be declining.The resultant velocity is increased with an enlargement in Hall and ion slip parameters throughout the fluid.The thermal and solutal buoyancy forces contribute to the resultant velocity ever-increasing to mountain.The temperature distribution is trim downs with an escalating in heat source parameter and frequency of oscillation.It is important to note that the temperature distribution increases significantly with an increasing in the radiation-absorption parameter.It is also concluded that,the concentration distribution is reduced with an increasing in chemical reaction parameter and Schmidt number in the entire fluid medium.The heat source and rotation parameters are to diminish the skin friction,whereas it is augmented through an increasing in Hall and ion slip parameters.The nature of viscosity of a fluid controls over conduction after that the rate of heat transfer augments comprehensively.The Schmidt number and chemical reaction parameter enhance the rate of mass transport extensively.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature

Areal positive constant

Bmagnetic induction vector

B0applied magnetic field,A·m-1

Cdimensional concentration,kg·m-3

Cwthe uniform concentration of the fluid at the plate,kg·m-3

C∞the concentration of the fluid far away from the plate,kg·m-3

cpthe specific heat at constant pressure,J·K-1·kg-1

Dcoefficient of mass diffusivity,m2·s-1

Dmeffective molecular diffusivity,m2·s-1

Eelectric field vector,C

Gmmass Grashof number

Grthermal Grashof number

gacceleration due to gravity,m·s-2

Hheat source parameter

Jcurrent density vector,A·m-2

Jx，Jycurrent densities alongxandydirections

Kpermeability parameter

Kcchemical reaction parameter

Klchemical reaction rate constant

kpermeability of porous medium,m2

k1thermal conductivity,W·m-1·K-1

k2effective thermal conductivity,W·m-1·K-1

MHartmann number

Nulocal Nusselt number

nconstant

Peelectron pressure,Pa

PrPrandtl number

Qlradiation absorption parameter

the coefficient of proportionality for the absorption of radiation

Q0the dimensional heat absorption coefficient

qmlocal surface mass flux,kg·s-1·m-2

qwlocal surface heat flux,W·m-2

Rrotation parameter

ScSchmidt number

Shlocal Sherwood number

Twthe uniform temperature of the fluid at the plate,K

T∞the temperature of the fluid far away from the plate,K

ttime,s

u，v the velocity components alongxandydirections respectively,m·s-1

u0plate velocity,m·s-1

wslip velocity,m·s-1

w0scale of suction velocity

β coefficient of thermal expansion of the fluid

β*coefficient of mass expansion of the solid

βeHall parameter

βiion slip parameter

θ non-dimensional temperature

ν kinematic viscosity of the fluid,m2·s-1

Ω angular velocity,s-1

φ non-dimensional concentration,

ρ fluid density,kg·m-3

σ electrical conductivity,s·m-1

τ local skin friction coefficient

τeelectron collision time,s

τwlocal wall shear stress,Pa

ωecyclotron frequency,e·mB-1

Appendix A