Ram Prakash Sharma ,S.R.Mishra

a Department of Mechanical Engineering,National Institute of Technology,Arunachal Pradesh,Yupia 791112,Papum Pare District,Arunachal Pradesh

b Department of Mathematics,Siksha ‘O’ Anusandhan Deemed to be University,Bhubaneswar 751030,Odisha,India

Keywords:Magnetohydrodynamic Thermal radiation Heat source Thermo-diffusion Chemical reaction Runge-Kutta-Felhberg method

ABSTRACT The current paper explores the behavior of the thermal radiation on the time-independent flow of micropolar fluid past a vertical stretching surface with the interaction of a transverse magnetic field.The effect of thermo-diffusion (Soret) along with the heat source is incorporated to enhance the thermal properties.Also,the convective solutal condition is considered that affects the mass transfer phenomenon.The transformed equations are modeled using suitable similarity transformation.However,the complex coupled equations are handled mathematically employing the Runge-Kutta-Felhberg method.The behavior of characterizing parameters on the flow phenomena as well as the engineering coefficients are displayed via graphs and the validation of the current outcome is reported with the previously published results in particular cases.

1.Introduction

Micropolar liquids are liquids with microstructures.Micropolar liquids fit into a class of liquids with non-symmetrical strain tensor that we intend to name polar liquids,and contain,as a different instance,the closely conventional Navier-Stokes model of standard liquids that we shall name ordinary liquids.Substantially,micropolar liquids can signify liquids containing inflexible,arbitrarily focused on (or spherical) elements put off in a viscous media,where the distortion of liquid elements is overlooked.In current centuries examination of flow and heat transport of micropolar fluids in holes has been of countless attention due to the Newtonian liquids cannot positively define the properties of liquid with deferred particle.Dissimilar to the other liquids,micropolar liquids can be defined as non-Newtonian liquids containing dumbbell particles or petite inflexible cylindrical components,polymer liquids,liquid interruption,etc.Furthermore,by the standard momentum description,a micro-revolution path and a rotation parameter are presented in the micropolar liquid structure to explore the kinematics of micro-revolution.Since the concept of micropolar liquid,it is similarly predictable to effectively designate the non-Newtonian performance of convinced liquids,for example,fluid crystals,ferrofluids,colloidal liquids,fluids with polymer extracts,animal plasma booming deformable elements,clouds with burn,interruptions,slurries,and fluid crystals.The concept of micropolar liquids given by Eringen [1,2].Sharma and Mishra [3]have reported the 2-D steady magnetohydrodynamic laminar flow of viscid micropolar fluid past a stretching sheet with heat source and nth-order chemical reaction.The squeezing motion of an unsteady magnetohydrodynamic micropolar liquid on radial and angular momentum with a transverse magnetic field with the permeable media has been examined by Pradhan et al.[4].Waqas et al.[5]have scrutinized the effect of viscid dissipation,Joule energy,and convective boundary condition on magnetohydrodynamic motion of micropolar liquid past a nonlinear extended surface.Anwar et al.[6]have reported the numerical investigation for the magnetohydrodynamic inertiapoint motion of a micropolar nanoliquid over an extending surface in the impact of a uniform magnetic field.Hosseinzadeh et al.[7]have inspected the impact of free convection energy transport,magnetic parameter,micropolar parameter,and nanoparticle volume fraction on micropolar MHD liquid motion transitory past a perpendicular sheet.Talarposhti et al.[8]examined the influence of Hall current on micropolar nanoliquid between 2-horizontal sheets in a revolving structure through a novel and advanced semi-analytical technique named Akbari-Ganji’s method (AGM).Zaib et al.[9]have observed the impact of Prandtl number on the assorted convective motion of Al2O3-H2O nanofluids in the influence of micropolar fluids driven over the wedge.Zaib et al.[10]have examined the influence of entropy generation on assorted convection motion of micropolar liquid holding water-based TiO2 nanomaterial over a perpendicular plate with a non-Darcy permeable media.Sheremet et al.[11]have investigated the impact of the non-dimensional time,Prandtl number,vortex viscosity parameter,and undulation number on free convective of micropolar liquid in a right-angled wavy triangular cavity.Gibanov et al.[12]have analyzed the free convective of micropolar liquid in a curvy differentially animated cavity.Mandal and Mukhopadhyay[13]have studied the impact of nonlinear convective on the laminar flow of micropolar fluid over an exponentially extending surface with radiative heat transfer and exponentially moving natural stream.Bhattacharyya et al.[14]surveyed the impact of thermal radiation on the transfer of energy and micropolar liquid motion past a permeable reduction surface.Pal and Chatterjee [15]have scrutinized the study of MHD laminar motion,energy,and concentration transport property on a steady two-dimensional flow of a micropolar fluid over a stretching sheet surrounded in a non-Darcian permeable media in the impact of Ohmic dissipation and thermal radiation.Pal and G Mandal [16]have inspected the influence of magnetohydrodynamic and thermal radiation on the laminar motion of micropolar nanoliquid over an extending plate in the effect of non-uniform heat source/sink.Sandeep and Sulochana[17]have reviewed the impact of the chemical reaction and nonuniform heat source/sink on an unsteady assorted convection motion of magnetohydrodynamic micropolar liquid over an extending/shrinking surface.Kumar et al.[18]surveyed the influence of variable heat sink/source,non-linear Rosseland approximation,and Biot number on magnetohydrodynamic radiative non-aligned stagnation point flow of non-Newtonian fluid past an extended sheet.Very recently,Mohanty et al.[19]demonstrated theoretically the convective heat transport properties of a radiating micropolar fluid and used an analytical approach for the governing equations.Further,Pattnaik et al.[20]prepared a note on the thermophysical properties of micropolar nanofluid combined with exponential heat source and numerical treatment is employed for the simulation purpose.Statistical analysis on the flow of time-dependent hybrid nanofluids with the inclusion of radiative heat transfer phenomenon is carried out by Mackolil and Mahanthesh [21].Earlier that,Noor et al.[22]worked on the mixed convection of micropolar nanofluid for the impact of slip.

Silambarasan et al.[23]have surveyed in the cylindrical rod,the collective substantial and operational constants are stated as general incompressible materials,the waves that travelling via such rods are the longitudinal strain waves.Basha et al.[24]have investigated the fluid transfer characteristics of ferromagnetic Carreau nanoliquid past a permeable block,sheet,and stagnation point in the presence of magnetic dipole impact for shear thinning/shear thickening instances.

Alp Ilhan et al.[25]have inspected using tan(φ/2)-expansion method and find the exact solutions for Nematicons in liquid crystals arising in fluid mechanics.Baskonus [26]has examined the novel acoustic wave performances to the Davey-Stewartson equation with power-law nonlinearity rising in liquid dynamics.

Ramly et al.[27]have examined the passive control and active control of thermal radiation boundary layer motion of nanoliquid past a radially stretched sheet.Noor et al.[28]have analysed the influence of internal heat generation on energy transport examination of magnetohydrodynamic motion due to a porous shrinking surface surrounded in a porous media.Noor and Hashim[29]have studied the effect of transverse magnetic field on MHD viscous motion past a linearly stretching surface surrounded in a non-Darcian permeable media.

Nadhirah and Noor [30]have investigated assorted convective motion of Powell-Eyring nanoliquid near a stagnation point along a vertical stretching surface.

Keeping in mind from the aforesaid literature review,this study aims to investigate the natural convection stagnation point flow past a stretching surface with the impact of thermo-diffusion(Soret) along with the thermal radiation and heat source on the electrically conducting micropolar fluid.Additionally,the solutal convective condition is assured that provides a boot in the concentration.Therefore,due to its practical interest in the engineering fields such as building design,cooling of electronic components,solar energy systems,solar collectors,liquid crystals,animal blood,colloidal fluids,and polymeric fluids [31,32],the topic needs to be further explored.However,the present result validates with earlier studies in particular cases with good agreement.

2.Flow analysis

The steady,2D viscous incompressible electrically conducting micropolar fluid near a stagnation point on a vertical hot plane is considered.Two equal and opposite forces are applied alongx-axis so that the surface is stretched retaining the origin fixed andy-axis is normal to the flow direction.The plate temperature and concentration are assumed to beTwandCswhereas the ambient conditions areT∞andC∞respectively.

Fig.1.Physical model and coordinate system.

The magnetic Reynolds number of the conducting fluid is supposed to be very small so that the Hall effect and induced magnetic field may be neglected.Therefore,the magnetic field effect in momentum is taken into account in the present study.Moreover,a uniform magnetic field of strengthB0is imposed in the transverse direction of the flow (Fig.1).Following Bhattacharyya[14]the equations govern the flow phenomena of the polar fluid with boundary conditions that are

The governing equations are

The applicable boundary conditions are given by

Following Ishaketal.[33],it is assumed thatγ=(μ+κ/2)j=μ(1+K/2)j,whereK=κ/μis the material parameter.The aforesaid hypothesis is implemented due to the restrictive situation once the microstructure properties convert insignificant and the entire spinΓdiminutions to the angular momentum.The ‘ ±’ sign that appeared in the last term of the Eq.(2) denotes the impact of the thermal and solutal buoyancy for together assisting and opposing the motion regions,correspondingly.

Symbolizeqras the radiative heat flux.Rendering to Rosseland’s approximation [34,35],we takeqr=-4σ/3k*?yT4,whereσandk*are the Stefan-Boltzmann constant and absorption coeffi-cient.We undertake that the energy variationT4can be expended in Taylor’s series.Neglecting higher-order relations and expendingT4aboutT∞yieldsEq.(4) then reduces to

Letψ(x,y)be the stream function satisfying Eq.(1) with

Let us considerX=hence we can define the stream function asψ(X,η)=υX f(η).

Similarity transformation and non-dimensional variables are introduced

In assessment of Eqs.(7)-(9) and Eqs.(2)-(5) can be written as,

The boundary conditions (6) reduce to the following forms:

where prime symbolize diff.w.r toη.

The definitions of Soret and Biot parameters are

3.Quantity of main engineering interest

The main engineering quantities of attention are defined as

where surface shear stress,surface heat,and mass flux are defined as

Using the dimensionless variables (9),we attain from Eqs.(16) and (17) as

4.Numerical solution

The methodology adopted for the solution of governing equations (10)-(13) with their boundary conditions (14) is Runge-Kutta Fehlberg 5thorder method and the iterative scheme as follows;

The whole system is discretized into a set of first-order differential equations and due to the unavailability of the required initial conditions,the suitable guess values are considered and further evaluated using the shooting technique.

However,with suitable step sizeh,RKF 5thorder technique is used for the function asf(t,y),and the procedure as follows;

Hence,Runge-Kutta fifth-order iterative process is

In general,the methodology is known as RKF45.Here,‘4’ stands for the local error ofO(h4)and ‘5’ represents the local order ofO(h5).The amount of computation in both RK45 and RKF45 are almost proportional and the only advantage is that the error estimation is accurate if the step size is not known to us that will useful for the problem.

5.Results and discussion

The stagnation point motion of conducting micropolar liquid flow past a stretching surface is considered in the current investigation.The inclusion of the thermo-diffusion (Soret) impact together with the thermal radiation will enhance the motion phenomena of micropolar fluid.The crux of the investigation is the influence of convective solutal boundary conditions affects the mass transfer phenomenon significantly.However,the main attraction of the literature is to observe the behavior of the assisting and opposing cases due to the buoyant forces.The transformed governing equations are coupled in nature and due to the complexity of the nonlinear problem,the numerical method i.e.Runge-Kutta-Felhberg method is employed to tackle the situation.The performance of the contributing physical parameters is accessible via graphs and the validation of the current result with the works of Ishak et al.[33]and Lok et al.[36]is made in particular cases.In the entire discussion,the physical significance of the particular parameter is displayed in the respective graph keeping other parameters as fixed.

Here,Table 1 is displayed for the validation of the present result with the work of Lok et a.[36]and Ishak et al.[33]for the particular values ofM=K=Ra=0,andλ=1 the other fixed values.These computational results confirm the suitability of the methodology employed to great accuracy.

Table 1 Comparison of f ′′ (0) and-θ′ (0) for M=K=Ra=0,and λ=1.

Fig.2.Streamlines for the stagnation point flow.

Fig.3.Variation of material parameter on velocity description for both assisting and opposing motion.

Fig.2 shows the streamlines for different physical parameters.The intensity of the fluid on the stretching surface increases near the stagnation pointX=0.The sharpness of the streamlines towards the free stream shows the increasing impact of the fluid on the surface near the stagnation point.The appearance of the material parameter,Kdue to the vortex viscosity plays a significant part in the motion phenomena.It reflects (K/=0) the non-Newtonian characteristics of the polar liquid.However,withdrawingKi.e.K=0 exhibits the Newtonian fluid characteristics.At present,the assisting case (λ ＞0,δ ＞0) and opposing case (λ＜0,δ＜0) situations are described for the variation of the other parameters.Fig.3 portrays the impact of the material constraint for both the non-Newtonian as well as Newtonian circumstances on the momentum profile of the micropolar fluid.The dotted lines in the figure represent the opposing case and here it is considered asλ=-1,δ=-1 and the bold stands for assisting case i.e.,assumed to beλ=1,δ=1.The overview is very clear that the Newtonian fluid overrides the case of non-Newtonian fluid and exhibits maximum velocity within the flow domain.Moreover,increasing the material parameter retards the profile throughout.From the definition ofK(=κ/μ),it is clear to note that,as increasing material parameter i.e.the dynamic viscosity falls and resulted in the velocity regards.It is stimulating to detect that,assisting case overrides the outline in comparison to the opposing case since buoyancy overshoots the velocity in the entire domain.Difference of the material parameter on the angular momentum for both the cases of assisting as well as opposing is displayed in Fig.4.The reverse trend in the angular momentum is rendered comparing to that of momentum distribution.The assisting case lowers down the angular velocity due to the microrotation of the fluid particle and it is also reflected that the enhanced values of the material parameter enhance the angular velocity within the domainη＜1.25 approximately and afterward the effect is opposite.Fig.5 exhibits the influence of buoyant forces in conjunction with the numerous values of the magnetic parameter on the momentum distribution.The discussion is based upon the opposing,assisting as well as the absence of buoyancy parameter for the absence/presence of the magnetic parameter.The inclusion of a magnetic parameter leads to the appearance of a resistive force that resists the liquid flow.Thus,it is perceived that a rise in M shows the description hikes but in other words,it is visible that the width of the boundary layer retards.Moreover,enhancing buoyant forces i.e.from opposing to assisting overrides the velocity profile significantly.Comparing with Fig.5,from Fig.6 the dual characteristics in the profiles of angular momentum are observed for the variation of buoyancy parameter and the magnetic parameter.The entire domain is partitioned into two distinct regions from the point of inflection marked nearη=1.5.The profile retards the increasing buoyancy parameter within the first region however in the second region the insignificant enhancement is marked.An enhance in the magnetic parameter also retards the profile as well.Observation of the heat source/sink parameter on the liquid energy for various Prandtl numbers is shown in Fig.7.The fluid heat enhances to rise in the additional heat source and the opposite effect is rendered for the sink i.e.sink favors into retards the liquid energy in its boundary layer.Therefore,the thinning in the layer is appended in the figure.It is a clear picture of the Prandtl number on the fluid energy that the enhancing Pr also lowers down the profiles.From the scientific point of opinion,it is clear that as Pr enhances the decrease in the thermal diffusivity reduces the liquid temperature in its domain.The electromagnetic radiation because of the thermal flow in the particle generates the radiation and the behavior of the thermal radiation of the different Prandtl numbers on the fluid energy is observed in Fig.8.The emission of radiative heat energy boosts up the profile significantly irrespective of the values of Pr.The stored energy near the surface grows up to hike the fluid temperature in its boundary layer.The description shows its asymptotic nature throughout the domain.The variation of chemical reaction on the liquid concentration for the supporting besides contrasting case is displayed in Fig.9.The profile becomes disturbed in the opposing cases for the negative chemical reaction parameter.However,the increasing chemical reaction from destructive to constructive retards the profile.Assisting the case also favors lowering fluid concentration.Fig.10 exhibits the variance of the Soret number on the fluid mass.It is detected that the higher values of Soret or the increasing Soret number enhance the solutal concentration in the entire domain.

Fig.4.Variation of material parameter on angular velocity description for both assisting and opposing motion.

Fig.5.Variation of buoyancy parameters and magnetic parameter on velocity description for assisting motion.

Fig.6.Variation of buoyancy parameters and magnetic parameter on angular velocity description for assisting motion.

Fig.7.Variation of heat source/sink and Prandtl number on energy profile for assisting motion.

Fig.8.Variation of thermal radiation and Prandtl number on a temperature profile for assisting the flow.

Fig.9.Variation of destructive and generative chemical reaction on the mass description.

Fig.10.Variation of Soret parameter on the mass description.

6.Conclusive remarks

The present study reveals the flow of a conducting micropolar fluid for the influence of thermal radiation past a vertical stretching surface.The crux of the investigation is the behavior of the Soret and heat source that affect the heat transfer properties of the polar fluid.Further,the convective boundary condition for the solutal profile at the surface is assumed.The numerical simulation for the transformed governing equations is carried out employing the Runge-Kutta-Felhberg method and the characteristics of the pertinent parameters are presented and described.However,the conclusive remarks are laid down as;

·The validation of the present result with that of earlier work in the particular case shows the convergence criteria of the methodology adopted and gives a road ap to conduct further investigation for the flow phenomena using various contributing parameters.

·In both the assisting and opposing cases the higher material parameter retards the velocity profile significantly due to the lower down of the dynamic viscosity however,the opposite impact is rendered for the case of angular velocity distribution.

·The velocity distribution enriches with an increasing buoyancy i.e.,assisting flow favors enhancing the velocity profiles.

·Thinning in the thermal bonding surface is observed due to the involvement of the heat sink whereas the external heat source boosts up the fluid temperature throughout the domain.

·The augmentation in the thermo-diffusion i.e.the Soret number augments the solutal concentration profile.

Declaration of Competing Interest

The authors declare no conflict of interest.