Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium

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ORIGINAL ARTICLE

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Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium Dulal Pala,n, Sukanta Biswasb

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a

Department of Mathematics, Visva-Bharati University, Siksha-Bhavana (Institute of Science), Santiniketan, West Bengal 731235, India Mathematics Department, Ananda Mohan College, Kolkata, West Bengal 700009, India

b

Received 23 March 2016; accepted 14 December 2017

KEYWORDS Chemical reaction; Convective boundary condition; Heat and mass transfer; Micropolar fluid; Magnetohydrodynamics; Porous medium; Thermal radiation

Abstract This paper deals with the investigation of double-diffusive heat and mass transfer characteristics of an oscillatory viscous electrically conducting micropolar fluid over a moving plate with convective boundary condition and chemical reaction. The non-linear partial differential equations are first converted into nonlinear ordinary differential equations by means of perturbation analysis and the governing equations are solved analytically. The effects of magnetic field, chemical reaction, permeability parameter, Prandtl number, Schmidt number, thermal radiation and viscosity parameter are analyzed on skin friction, Nusselt number, velocity, and temperature & concentration distributions. It is observed that the concentration profiles decrease with increase in the dimensionless time and increase with increase in the chemical reaction parameter. It is also observed that the velocity profile increases with increase in time but reverse effects are found by increasing the value of the viscosity ratio parameter. Further, it is seen that the effect of magnetic field parameter is to increase the micro-rotational velocity profiles but reverse effect is observed by increasing time. & 2018 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

n

Corresponding author. Tel./Fax: þ91 3463 261029. E-mail addresses: [email protected] (Dulal Pal), [email protected] (Sukanta Biswas).

Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China. https://doi.org/10.1016/j.jppr.2018.05.004 2212-540X & 2018 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Dulal Pal, Sukanta Biswas

Nomenclature

up

A B0

Up V0 v* v x, y

C C∞ Cf 0 Cω cp D Gr g GrT GrC j* j K K 1 K2 M n n1 Nu Pr qr Rex r Sc Shx T T∞ Tω t* t u* u U0

small real positive constant externally imposed transverse magnetic field strength, gauss meter concentration of the fluid (uint: mol  m−3) free stream concentration (uint: mol  m−3) frictional coefficient local couple stress coefficient at the wall specific heat at constant pressure (uint: J  kg−1  K−1) molecular diffusivity (uint: m2  s−1) local Grashof number acceleration due to gravity (uint: m  s−1) thermal Grashof number mass Grashof number microinertia per unit mass (uint: m2) dimensionless microinertia per unit mass permeability parameter (uint: m2) mass absorption coefficient (uint: m−1) dimensionless permeability parameter magnetic field parameter frequency parameter (uint: Hertz) parameter related to micro-gyration vector and shear stress local Nusselt number Prandtl number thermal radiative heat flux (uint: W  m−2) local Reynolds number radiation parameter Schmidt number Sherwood number temperature of the fluid (uint: K) free stream temperature (uint: K) temperature at the wall (uint: K) dimensional time (uint: s) dimensionless time velocity component in x-direction (uint: m  s−1) dimensionless velocity component in x-direction free stream velocity (uint: m  s−1)

1. Introduction In recent years. the study of heat and mass transfer in micropolar fluid has been considered extensively due to their occurrences in several industrial applications. The investigation of convective heat and mass transfer in Newtonian and non-Newtonian fluids can be extensively used in polymer production and in a number of industrial applications such as fiber and granular insulation, geothermal systems, glass-fiber and paper production, cooling of metallic sheets, geothermal reservoirs, thermal insulation, enhanced oil recovery, packed bed catalytic reactors etc. The theory of micropolar fluid given by Eringen [1] described the characteristics of polymeric fluids, liquid crystals, animal blood etc. Extensive research works have been carried out to study heat and mass transfer in a

uniform velocity of the fluid in its own plane (uint: m  s−1) dimensionless velocity of the plate scale of suction velocity at the plate (uint: m  s−1) velocity component in y-direction (uint: m  s−1) dimensionless velocity component in y-direction distance along and perpendicular to the plate, respectively (uint: m)

Greek symbols α β βC βT γ γ*1 γ1 ε θ η μ υ υr ρ σ σ* ω* ω Λ κ λ ϕ

thermal diffusivity (uint: m2  s−1) dimensionless viscosity ratio coefficient of mass expansion of the fluid (uint: K−1) coefficient of thermal expansion of the fluid (uint: K−1) spin gradient viscosity (uint: kg  m  s−1) chemical reaction parameter dimensionless chemical reaction parameter small positive quantity dimensionless temperature of the fluid similarity variable coefficient of viscosity kinematic viscosity (uint: m2  s−1) kinematic rotational viscosity (uint: m2  s−1) density of the fluid (uint: kg  m−3) electrical conductivity of the fluid (uint: S  m−1) Stephan-Boltzmann constant (uint: W  m−2  K−4) component of angular velocity (uint: m2  s−2) dimensionless component of angular velocity coefficient of vortex viscosity (uint: kg  m−1  s−1) thermal conductivity of the fluid (uint: W  m−1  K−1) heat generation/absorption parameter concentration distribution

Superscript ′

differentiation with respect to y

micropolar fluid past a semi-infinite plate under different boundary conditions. Motsa et al. [2] examined numerical analysis of mixed convection magnetohydrodynamic heat and mass transfer past a stretching surface in a micropolar fluid-saturated porous medium in the presence of Ohmic heating and they noticed that the fluid velocity increases with increase in the Grashof numbers and the fluid temperature increases with increasing values of magnetic field. Ferdows et al. [3] examined hydromagnetic convection heat transfer in a micropolar fluid over a vertical plate and they conclude that the local skin-friction coefficient and the local Nusselt number decreases with an increase in the value of the magnetic parameter. Gupta et al. [4] studied unsteady mixed convection flow of micropolar fluid over a porous shrinking sheet and they found that a fast rate of cooling can be achieved by Eckert number. Pal [5]

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Magnetohydrodynamic convective-radiative oscillatory flow

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investigated magnetohydrodynamic flow and heat transfer past a semi-infinite vertical plate embedded in a variable permeability type of porous medium. Modather et al. [6] studied analytically magnetohydrodynamic (MHD) heat and mass transfer in an oscillatory flow of a micropolar fluid over a vertical permeable plate in a porous medium. It is found that the fluid flow along the wall of the plate accelerated as the chemical reaction parameter is increased. Krishna Reddy et al. [7] investigated heat and mass transfer on an unsteady MHD free convection flow past a vertical permeable moving plate in the presence of thermal radiation. It was found from their investigation that the velocity profiles increased due to decrease in the chemical reaction parameter, Schmidt number and magnetic field parameter. Karunakar Reddy et al. [8] studied MHD heat and mass transfer flow of a viscoelastic fluid past an impulsively started infinite vertical plate with chemical reaction. They observed that the velocity profile increases with increase in the Grashof number and permeability of porous medium. Uddin and Kumar [9] investigated thermal radiation effect on an unsteady MHD heat and mass transfer flow on a moving inclined porous heated plate with chemical reaction. They concluded that the velocity of the fluid increases with an increase in the value of magnetic field parameter. Tiwari et al. [10] examined free convection heat and mass transfer in the presence of sinusoidal suction with time-dependent permeability. It was found that when thermal and solutal Grashof numbers were increased, the thermal and concentration buoyancy effects were enhanced then the fluid velocity increased. Hayat et al. [11] studied flow of a second grade fluid with convective boundary conditions and conclude that the local Nusselt number increases with increase in the Prandtl number and decreases when Eckert number is increased. In some industrial applications, thermal radiation plays a major role since many tasks in engineering level are operated through high temperature like as in nuclear power plants, the various propulsion devices for aircraft, missiles, gas turbines, satellites, and space vehicles etc. Samad et al. [12] examined effects of thermal radiation, heat generation and viscous dissipation on MHD free convection flow along a stretching sheet and it shows that the velocity and temperature profiles decrease with the increase of Prandtl number. Kishore et al. [13] investigated the effects of radiation and chemical reaction on unsteady MHD free convection flow of viscous fluid past an exponentially accelerated vertical plate. They found that the velocity decreases with increase in the magnetic parameter. Pal and Mandal [14] examined the importance of magnetic field and thermal radiation on boundary layer flow of micropolar nanofluid over a stretching sheet with non-uniform heat source/sink. Shit et al. [15] investigated thermal radiation and Hall effect on MHD flow, heat and mass transfer over an inclined permeable stretching sheet. They found that

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increasing the values of the thermal radiation parameter leads to the decreasing of thermal boundary layer thickness. Pal et al. [16] examined the influence of Hall current and thermal radiation on MHD convective heat and mass transfer in a rotating porous channel with chemical reaction. They concluded that with increase in thermal Grashof number, the skin friction increases. Rao et al. [17] studied MHD transient free convection and chemically reactive flow past a porous vertical plate with radiation and temperature gradient dependent heat source in slip flow regime. It shows from their investigation that the concentration decreases with an increase in the Schmidt number and chemical reaction parameter. Yazdi et al. [18] investigated thermal radiation effects on MHD stagnation-point flow in a nanofluid. Pal et al. [19] used perturbation analysis to study unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. They found that the velocity as well as concentration decrease with increasing the chemical reaction, heat source and thermal radiation parameters. Pal et al. [20] investigated the effects of thermal radiation and Ohmic dissipation on hydromagnetic Casson nanofluid flow over a vertical non-linear stretching surface using scaling group transformation. Singh [21] studied viscous dissipation and chemical reaction effects on flow past a stretching porous surface in a porous medium. Jat et al. [22] analyzed MHD heat and mass transfer of viscous flow over nonlinearly stretching sheet in a porous medium. It is observed that with increasing the values of magnetic field parameter, velocity boundary layer thickness increase. Pal and Mondal [23] examined the presence of Soret-Dufour effects on hydromagnetic non-Darcy convective-radiative heat and mass transfer over a stretching sheet in porous medium with viscous dissipation. Baoku et al. [24] investigated magnetic field and thermal radiation effects on steady hydromagnetic Couette flow through a porous channel. Sugunamma et al. [25] investigated the effects of inclined magnetic field and chemical reaction on flow over a semi infinite vertical porous plate through porous medium. It is evident from their investigation that the temperature decreases by an increase in the thermal radiation and Prandtl number. Pal et al. [26] presented influence of magnetic field, Ohmic dissipation and thermal radiation in Casson nanofluid flow over a vertical non-linear stretching surface using scaling group transformation. Perturbation techniques is applied to the present problem for finding the approximate solutions of the momentum, energy and mass diffusion equations in the boundary layer flow of electrically conducting micropolar fluid over a moving plate. Karthikeyan et al. [27] studied thermal radiation effects on MHD convective flow over a plate in a porous medium by perturbation technique. It is found that

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Dulal Pal, Sukanta Biswas

the velocity is reduced by the effect of magnetic parameter and radiation parameter while the effect of porous permeability enhances the velocity profile. Mamatha et al. [28] analyzed thermal diffusion effect on MHD mixed convection unsteady flow of a micropolar fluid past a semi-infinite vertical porous plate with radiation and mass transfer. Mamatha et al. [29] investigated unsteady MHD mixed convection, radiative boundary layer flow of a micropolar fluid past a semi-infinite vertical porous plate with suction. They used the regular perturbation technique to solve the non linear differential equations. Owing to the above mentioned studies, we propose to investigate the influence of chemical reaction and thermal radiation on doublediffusive convective heat and mass transfer of MHD oscillatory flow of a micropolar fluid over a vertical plate embedded in a porous medium with convective boundary conditions.

2. Formulation of the problem Mixed convection and diffusion mass transfer of a viscous incompressible electrically conducting micropolar fluid over an infinite vertical porous mov ing permeable plate in a porous medium in the presence of thermal radiation and chemical reaction with convective boundary condition have been considered in his paper. In the Cartesian coordinate system, x* is measured along the plate and y*-axis normal to the plate. A uniform magnetic field of strength B0 is applied in the y*-direction. Induced magnetic field is negligible in comparison to the applied magnetic field. It is assumed that the magnetic field is of small intensity. Since electric dissipation is neglected so that Joule heating is negligible. The fluid is considered to be gray absorbing emitting or radiation but not scatting medium. The Rosseland approximation is used to describe the radiative heat flux in the y*-direction. The plate moves continuously with uniform velocity up* in its own plane. It is assumed that the temperature of the surface is held uniform at Tω while those of the ambient temperature takes the constant value T∞ so that Tω 4 T∞. Concentration of the uniformly at Cω and that of the ambient fluid is taken as C∞. Under these assumptions,the boundary layer equation of motion, energy and mass-diffusion equations under the influence of uniform transverse magnetic field in the presence of thermal radiation are as follows: ∂v ¼ 0; ð1Þ ∂y
 ∂u ∂2 u  ∂ω  ∂u þ v Þ þ 2ν ¼ ðν þ ν r r ∂t  ∂y ∂y2 ∂y þ gβT ðT−T ∞ Þ þ gβC ðC−C ∞ Þ− 

ρj




  ∂ω ∂ 2 ω  ∂ω þ v ; ¼ γ ∂t  ∂y ∂y2

σB20 ρ

u −

ν þ νr  u K

ð2Þ ð3Þ

∂T ∂2 T 1 ∂qr  ∂T þ v ¼ α − ; ∂t  ∂y ∂y2 ρcp ∂y

ð4Þ

∂C ∂2 C  ∂C þ v ¼ D þ γ 1 ðC−C ∞ Þ ∂t  ∂y ∂y2

ð5Þ

where (u*,v*) are the component of the velocity at any point (x*,y*), ω* is the component of the angular velocity normal to the x*y* plane and T is temperature of the fluid and C is the mass concentration of the species in the flow. The appropriate boundary conditions of the problem are:    u ¼ up ; ω ¼ −n1 ∂u −k ∂T ∂y ; ∂y ¼ qω cos ωt; −D

∂C  ¼ qm cos ωt ∂y

at

u →0; ω →0; T→T ∞ ; C→C ∞

y ¼ 0; as y →∞

ð6Þ

It is well established that when n1 ¼ 0, the boundary condition used for the microrotation as stated in Eq. (6), we have microrotation, ω* ¼ 0. This represents the case of concentrated particle flows in which the microelements close to the wall are not able to rotate (Jenaand Mathur [30]). The case corresponding to n1 ¼ 0.5 in Eq. (6), results in the vanishing of the antisymmetric part of the stress tensor and represents weak concentrations. Ahmadi [31] suggested that the particle spin is equal to the fluid vorticity at the boundary for fine particle suspensions. According to Peddieson [32], when n1 ¼ 1, it relates to then this is the case of turbulent boundary layer flows. Further, when n1 ¼ 0, the particles are not free to rotate near the surface. However, when n1 ¼ 0.5 or n1 ¼ 1.0, the microrotation term gets augmented and induces flow enhancement. From the continuity Eq. (1), it is clear that the suction velocity normal to the plate is a function of time only, so we can write v* in the following form: v ¼ −ð1 þ εAent ÞV 0 ;

ð7Þ

where A is a real positive constant, εA is a small quantity less than unity and V0 is a scale of suction velocity which has a non-zero positive constant. The radiative heat flux term by using the Rosseland approximation is given by qr ¼ −

4σ  ∂T 4 : 3K 1 ∂y

We assume that the temperature difference within the flow are sufficiently small such that T*4 may be expressed as a linear function of the temperature. This is accomplished by expanding T*4 in a Taylor series about T*∞ and neglecting the higher order terms, we get  4 T 4  4T 3 ∞ T −3T ∞ :

We now employ the following dimensionless variables:   u ¼ U 0 u; v ¼ V 0 v; y ¼ v=V 0 y;     up ¼ U 0 U p ; ω ¼ U 0 V 0 =v ω; t  ¼ v=V 20 t;     n ¼ V 20 =v n; j ¼ v2 =V 20 j; Pr ¼ v=α;

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Magnetohydrodynamic convective-radiative oscillatory flow

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Sc ¼ v=D; M ¼ ðσB20 vÞ=ρV 20 ; GrT ¼ νgβT ðT ω −T ∞ Þ=U 0 V 20 ; GrC ¼ vgβC ðC ω −C∞ Þ=U 0 V 20 ;     Γ ¼ μ þ Λ=2 j ¼ μj 1 þ β=2 ; β ¼ Λ=μ ¼ νr =ν;

5

0

ð3 þ 4rÞθ″0 þ 3Prθ0 ¼ 0; 0

ð8Þ

∂ϕ ∂ϕ 1∂ϕ −ð1 þ εAent Þ ¼ þ γ1ϕ ∂t ∂y Sc ∂y2 2

0

ð20Þ ð21Þ

0

ð22Þ

with the following boundary conditions: u0 ¼ U p ; u1 ¼ 0; ω0 ¼ −n1 u00 ; ω1 ¼ −n1 u01 ; 1 0 1 0 θ0 ¼ − ; θ1 ¼ 0; ϕ00 ¼ − ; ϕ01 ¼ 0 at y ¼ 0 2 2 u0 ¼ u1 ¼ ω0 ¼ ω1 ¼ θ0 ¼ θ1 ¼ f0 ¼ f1 ¼ 0 as y → ∞: ð23Þ The solution of the linear coupled Eqs.(15)–(22) with the help of boundary conditions (23) are as follows: ð9Þ

uðy; tÞ ¼ a6 e−h4 y þ a3 e−h1 y þ a4 e−h3 y þ a5 e−ηy þ εent ða20 e−h7 y þ a10 e−h4 y þ a11 e−h1 y þ a12 e−h3 y

ð10Þ ð11Þ ð12Þ

with the following dimensionless boundary conditions u ¼ U p ; ω ¼ −n1

0

ϕ″0 þ Scϕ0 þ γ 1 Scϕ0 ¼ 0; ϕ″1 þ Scϕ1 þ ðγ 1 −nÞScϕ1 ¼ −AScϕ0

With the help of dimensionless variables (8) , Eqs. (1)–(7) reduced to the following initial value problem:

∂ω ∂ω 1 ∂2 ω −ð1 þ εAent Þ ¼ ; ∂t ∂y η ∂y2
 ∂θ ∂θ 1 4r ∂2 θ −ð1 þ εAent Þ ¼ 1þ ; ∂t ∂y Pr 3 ∂y2

0

ð3 þ 4rÞθ″1 þ 3Prθ1 −3nPrθ1 ¼ −3APrθ0 ;

K 0 ¼ KU 0 V 20 =ν2 ; η ¼ μj =γ ¼ 2=ð2 þ βÞ; γ 1 ¼ vγ 1 =VV 20 ; Pr ¼ μcp =K; ν ¼ μ=ρ;   r ¼ 4σ  T 3 ∞ =kK 1 ; θ ¼ ðT−T ∞ ÞkV 0 =qω ν;  ϕ ¼ ðC−C ∞ ÞDV 0 =qm ν:

∂u ∂u ∂2 u ∂ω −ð1 þ εAent Þ ¼ ð1 þ βÞ 2 þ 2β þ GrT θ ∂t ∂y ∂y ∂y 1þβ u; þ GrC ϕ−Mu− K0

ð19Þ

∂u ∂θ 1 ∂ϕ 1 ; ¼ − ðeint þ e−int Þ; ¼ − ðeint þ e−int Þ at y ¼ 0 ∂y ∂y 2 ∂y 2

þ a13 e−ηy þ a14 e−h2 y þ a15 e−h1 y þ a16 e−h5 y þ a17 e−h3 y þ a18 e−h6 y þ a19 e−ηy Þ; ωðy; tÞ ¼ a8 e−ηy þ εent ða21 e−h6 y þ a9 e−ηy Þ;
 −h1 y nt −a2 h1 −h2 y −h1 y þ εe e þ a2 e θðy; tÞ ¼ a1 e ; h2
 1 −h3 y −a7 h3 −h5 y e þ εent e þ a7 e−h3 y : ϕðy; tÞ ¼ 2h3 h5

ð24Þ ð25Þ ð26Þ ð27Þ

ð13Þ

The velocity gradient, microrotational velocity gradient, temperature and concentration gradients are given as follows:

To solve Eqs. (9)–(12) subject to the boundary conditions (13) we may use the following linear transformations for low value of ε, so we have:

u0 ðy; tÞ ¼ −a6 h4 e−h4 y −a3 h1 e−h1 y −a4 h3 e−h3 y −a5 ηe−ηy  þ εent −a20 h7 e−h7 y −a10 h4 e−h4 y −a11 h1 e−h1 y −a12 h3 e−h3 y −a13 ηe−ηy −a14 h2 e−h2 y −a15 h1 e−h1 y −a16 h5 e−h5 y  −a17 h3 e−h3 y −a18 h6 e−h6 y −a19 ηe−ηy ; ð28Þ

u→0;

ω→0;

θ→0;

ϕ→0

as y→∞

uðy; t Þ ¼ u0 ðyÞ þ εent u1 ðyÞ þ Oðε2 Þ; ωðy; t Þ ¼ ω0 ðyÞ þ εent ω1 ðyÞ þ Oðε2 Þ; θðy; t Þ ¼ θ0 ðyÞ þ εent θ1 ðyÞ þ Oðε2 Þ; ϕðy; t Þ ¼ ϕ0 ðyÞ þ εent ϕ1 ðyÞ þ Oðε2 Þ:

ð14Þ

After substituting the expression (14) into Eqs. (9)–(13), we have


 1þβ ð1 þ βÞu000 þ u0 − M þ u0 ¼ −GrT θ0 −GrC ϕ0 −2βω00 ; ð15Þ K0
 1þβ 0 0 ð1 þ βÞu001 þ u01 − M þ 0 −n u1 ¼ −Au0 −GrT θ1 −GrC ϕ1 −2βω1 ; K 0

ð16Þ ω000 þ ηω00 ¼ 0; 0

ω001 þ ηω1 −nηω1 ¼ −Aηω00 ;

ð17Þ ð18Þ

ω0 ðy; tÞ ¼ −a8 ηe−ηy þ εent ð−a21 h6 e−h6 y −a9 ηe−ηy Þ;

ð29Þ

θ0 ðy; tÞ ¼ −a1 h1 e−h1 y þ εent ða2 h1 e−h2 y −a2 h1 e−h1 y Þ;

ð30Þ

1 ϕ0 ðy; tÞ ¼ − e−h3 y þ εent ða7 h3 e−h5 y −a7 h3 e−h3 y Þ ð31Þ 2 where, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3Pr 3Pr þ 9Pr2 þ 12nPrð3 þ 4rÞ ; h2 ¼ ; h1 ¼ 3 þ 4r 2ð3 þ 4rÞ rffiffiffiffiffiffiffiffiffiffiffiffiffi! Sc 4γ 1 þ 1− 1 ; h3 ¼ 2 Sc sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #
 1 1þβ 1þ 1þ4 Mþ h4 ¼ ð1 þ βÞ ; 2ð1 þ βÞ K′

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Dulal Pal, Sukanta Biswas

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Sc 4ðγ −nÞ 1 þ 1− 1 h5 ¼ ; 2 Sc sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! η 4n 1þ 1þ h6 ¼ ; 2 η sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #
 1 1þβ 1þ 1þ4 Mþ −n ð1 þ βÞ ; h7 ¼ 2ð1 þ βÞ K′ −A ; 2n −GrT a3 ¼    ; 2h1 ð1 þ βÞh21 −h1 − M þ 1þβ K0 −GrC a4 ¼    ; 2h3 ð1 þ βÞh23 −h3 − M þ 1þβ K0 2βηc2 a5 ¼   ¼ λ 1 c2 ; 2 ð1 þ βÞη −η− M þ 1þβ K0 2βη λ1 ¼  ; 2 ð1 þ βÞη −η− M þ 1þβ K0 a6 ¼ U p −a3 −a4 −a5 ; 1 ; 2h1

a1 ¼

a2 ¼

ASc ; 2½h23 −Sch3 þ ðγ 1 −nÞSc a8 ¼ n1 ða6 h4 þ a3 h1 þ a1 h3 þ a5 ηÞ; −Aa8 η ; a9 ¼ n Aa6 h4 a10 ¼ ; ½ð1 þ βÞh24 −h4 −l1  Aa3 h1 ð1 þ βÞ a11 ¼ ; l1 ¼ M þ −n; 2 K0 ð1 þ βÞh1 −h1 −l1 Aa4 h3 a12 ¼ ð1 þ βÞh23 −h3 −l1 Aa5 η a13 ¼ ; ð1 þ βÞη2 −η−l1 a7 ¼

GrT a2 h1 ; h2 ½ð1 þ βÞh22 −h2 −l1  −GrT a2 ; a15 ¼ ð1 þ βÞh21 −h1 −l1

a14 ¼

GrC a7 h3 ; h5 ½ð1 þ βÞh25 −h5 −l1  −GrC a7 a17 ¼ ; ð1 þ βÞh23 −h3 −l1

a16 ¼

a18 ¼ λ2 ¼

2βh6 c10 ¼ λ2 c10 ; ð1 þ βÞh26 −h6 −l1

2βh6 ; ð1 þ βÞh26 −h6 −l1

a19 ¼

2βa9 η ; ð1 þ βÞη2 −η−l1

a20 ¼ −a10 −a11 −a12 −a13 −a14 −a15 −a16 −a17 −a18 −a19 ;

a21 ¼ a9 þ n1 ða20 h7 þ a10 h4 þ a11 h1 þ a12 h3 þ a13 η þa14 h2 þ a15 h1 þ a16 h5 þ a17 h3 þ a18 h6 þ a19 ηÞ; n1 ½h4 ðU p −a3 −a4 Þ þ a3 h1 þ a1 h3 ; c2 ¼ 1 þ nλ1 ðh4 −ηÞ R c10 ¼ ; 1 þ n1 λ2 ðh7 −h6 Þ R ¼ a9 þ n1 ½h7 ð−a10 −a11 −a12 −a13 −a14 −a15 −a16 −a17 −a18 −a19 Þ þ a10 h4 þ a11 h1 þ a12 h3 þ a13 η þ a14 h2 þ a15 h1 þa16 h5 þ a17 h3 þ a19 η: The local skin friction coefficient, local wall couple stress coefficient, local Nusselt number, and local Sherwood number are important physical quantities for this type of heat and mass transfer problem, which are defined below. The wall shear stress may be written as: ∂u τω ¼ ðμ þ ΛÞ  y ¼ 0 ¼ ρU 0 V 0 ½1 þ ð1−n1 Þβu0 ð0; tÞ; ð32Þ ∂y where, u0 ð0; tÞ ¼ −a6 h4 −a3 h1 −a4 h3 −a5 η þ εent ð−a20 h7 −a10 h4 −a11 h1 −a12 h3 −a13 η−a14 h2 −a15 h1 −a16 h5 −a17 h3 −a18 h6 −a19 ηÞ is obtained from Eq. (28). Therefore, the local skin-friction factor is given by: Cf ¼

2τω ¼ 2½1 þ ð1−n1 Þβu0 ð0; tÞ: ρU 0 V 0

The wall couple stress may be written as: ∂ω M ω ¼ γ  y ¼ 0 ∂y

ð33Þ

ð34Þ

Therefore, the local couple stress coefficient is given by: 0

Cω ¼ γ

M ω v2 ¼ ω0 ðy; tÞ γU 0 V 20

ð35Þ

where, ω′(0,t) ¼ −a8ηþεent[−a21h6−a9η] is obtained from Eq. (29). The rate of heat transfer at the surface in terms of the local Nusselt number can be written as: Nu ¼ x

ð∂T=∂y Þy ¼ 0 ; T ∞ −T ω

ð36Þ

Using Eqs. (8) and (34) in Eq. (36), we get 0 NuRe−1 x ¼ −θ ð0; t Þ;

ð37Þ

where, Rex ¼ xV0/η is the local Reynolds number, and θ′(0,t) ¼ −a1h1 is obtained from Eq. (30). The rate of mass transfer at the surface in terms of the local Sherwood number is given by: Sh ¼ x

ð∂C=∂y Þy ¼ 0 : C∞ −C ω

ð38Þ

Using Eqs. (8) and (36) in Eq. (38), we get 0 ShRe−1 x ¼ −ϕ ð0; t Þ;

ð39Þ

where ϕ′(0,t) ¼ −1/2 is obtained from Eq. (31).

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Magnetohydrodynamic convective-radiative oscillatory flow

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Figure 1 Velocity profile for various values of time with y.

Figure 2 Velocity profile for various values of the viscosity ratio with y.

Figure 3 Unsteady microrotational velocity profile for different values of dimensionless time with y.

3. Results and discussions The analytical solutions are obtained by solving the basic equations with perturbation method. The computed results of the analytical solutions are presented in

7

Figure 4 Unsteady microrotational velocity profile for different values of the local magnetic field parameter with y.

Figure 5 Temperature profile for various values of the Prandtl number with y.

Figure 6 Variation of temperature profile with different value of the radiation parameter with y.

graphical and tabular forms. The influence of various physical properties is analyzed on the horizontal velocity field, temperature distribution, local skin friction factor and Nusselt number. The velocity profiles are drawn for different values of dimensionless time against y as

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Dulal Pal, Sukanta Biswas

Figure 7 Concentration distribution for various values of the chemical reaction parameter with y.

Figure 8

Figure 10 Microrotational velocity gradient for various values of the magnetic field parameter with y.

Concentration distribution for various values of t with y. Figure 11 Temperature gradient for various values of the Prandtl number with y.

Figure 9 Velocity gradient for various values of the magnetic field parameter with y.

Figure 12 Concentration gradient for various values of the chemical reaction parameter with y.

depicted in Figure 1. From this figure we observe that the velocity profiles increase by increasing the values of the dimensionless time. Also, it is seen that the velocity profile decreases rapidly by increasing the value of y. It is observed that the initial value of the horizontal velocity 0.5 at y ¼ 0 reaches gradually to zero and the value becomes zero as y tends to ∞, thereby matching the boundary condition. Figure 2 depicts the plot of velocity

profiles against y for different values of the viscosity ratio parameter. From this figure it is seen in the velocity profile decreases with increasing the values of the viscosity ratio parameter. It is found that the minimum value of the horizontal velocity is observed around y ¼ 0.25. The effects of micro rotational velocity distribution ω against y for various value of time t are seen in Figure 3.

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Magnetohydrodynamic convective-radiative oscillatory flow

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Figure 13

Cf for different values of the Schmidt number with t.

Figure 14 Frictional coefficient for various values of the viscosity ratio with time.

Figure 15 Frictional coefficient for various values of the chemical reaction parameter with time.

We can observe from this figure that the microrotational velocity distribution decreases with increase in time. Further, the value of microrotational velocity distribution decreases sharply as y is increased. Figure 4 depicts the variation of micro-rotational velocity distribution against y for different value of magnetic field parameter. We observe from this figure that the microrotational velocity distribution has an increasing effect with

9

Figure 16 Frictional coefficient for various values of the permeability parameter with time.

Figure 17 Local couple stress coefficient at wall for various values of the permeability parameter with time.

Figure 18 Local couple stress coefficient at wall for various values of the Prandtl number with time.

increase in the magnetic field parameter and reverse effect of magnetic field is observed as y is increased which ultimately tends to zero as y tends to ∞. This is due to the fact that the application of the transverse magnetic field, which acts as a drag force, called Lorentz force decrease the velocity profile so there is an increase in the microrotational velocity profile.

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Dulal Pal, Sukanta Biswas

Figure 5 depicts the variation of the temperature profile for different values of the Prandtl number with y. It is seen from this figure that the temperature profile decreases by increasing the values of the Prandtl number. It is also observed that the temperature profiles decreased with increasing y which go to zero as y tends to ∞, thereby matching the boundary condition (13) on temperature field. This is due to the fact that smaller values of Prandtl number increases the thermal conductivity of the fluid and therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of Prandtl number. So, the thermal boundary layer is thicker for low values of Pr and the rate of heat transfer is reduced. The effect of the thermal radiation on the temperature distribution with y is illustrated in Figure 6. It is seen from this figure that the temperature distribution increases with the increase in the thermal radiation parameter due to increase in the thermal boundary layer thickness. The result agrees due to the physical fact that the rate of transport to the fluid increase by the effect of thermal radiation, which causes increase in the temperature of the fluid.

Figure 19 Local couple stress coefficient at wall for various values of magnetic field parameter with time.

Table 1

Figure 7 depicts the variation of the concentration profile for different values of the chemical reaction parameter with y. It is observed from this figure that by increasing the value of the chemical reaction parameter there is increasing effect in the concentration distribution for all the values of y. Further, it is observed that the concentration distribution decreases with increasing the value of y which ultimately becomes zero as y tends to ∞. Figure 8 is drawn to study the variation of the concentration distribution against y for different values of time. It is observed from this figure that the concentration distribution decreases with increase in time. Further, it is observed that there is steep fall in the concentration distribution as the value of y is increased for all time. Figure 9 depicts the variation of the velocity gradient for different values of the magnetic field parameter. It is seen from this figure that the velocity gradient decreases by increasing the value of the magnetic field parameter but the effect of the magnetic field gets reversed as y reaches closer to 1. It is observed that the velocity gradient profiles crosses around y ¼ 0.8 and all the profiles meets at y ¼ 3.5. The effect of magnetic field parameter on microrotational velocity gradient is seen in Figure 10. It is seen from this figure that the microrotational velocity gradient decreases by increasing the value of the magnetic field parameter. Further, it is observed that the value of ω' increases as y is increased and reaches the maximum value at y ¼ 8 i.e. ω′ ¼ 0 as y→∞ which matches the boundary condition as y→∞. Variation of the temperature gradient for different values of the Prandtl number is seen in Figure 11. From this figure we can observe that by increasing the value of y temperature gradient profiles initially begin to increase from a fixed value of the Prandtl number and after reaching a peak value it starts decreasing and ultimately reaches to a minimum value. Further, it is noticed that the temperature gradient decreases by increasing the value of the Prandtl number when y41.0. It is interesting to note that the peak value

Effects of variations of flow conditions and fluid properties for various values of few physical parameters.

Sc

Pr

M

r

γ1

β

K′

Cf

C'ω

−θ′(1,1)

−ϕ′(1,1)

2 1 0.5 2

1

2

0.1

0.1

1

5

−2.0229123 −0.8999443 0.7085748 −2.5227678 −2.5431981 −2.4265382 −2.8752875 −1.7611594 −1.4752831 −1.9986191 −2.0171862 −2.5083320 −2.9472053 −2.0652654 −2.1344273 −2.2687206

−0.6407067 −0.1703660 0.0770802 −0.4576026 −0.4121400 −0.7163391 −0.8665182 −0.6958460 −0.7483035 −0.6360424 −0.6612576 −0.5428798 −0.4726496 −0.6474990 −0.6590133 −0.6832638

0.2067226 0.2067226 0.2067226 0.0353358 0.0060400 0.2067226 0.2067226 0.2445975 0.2742430 0.2067226 0.2067226 0.2067226 0.2067226 0.2067226 0.2067226 0.2067226

0.0750432 0.2056505 0.3479516 0.0750432 0.0750432 0.0750432 0.0750432 0.0750432 0.0750432 0.0845826 0.0974748 0.0750432 0.0750432 0.0750432 0.0750432 0.0750432

3 5 1

3 4 2

0.3 0.2 0.1

0.2 0.3 0.1

1.5 2 1

4 3 2

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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Magnetohydrodynamic convective-radiative oscillatory flow

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gradually shift towards right side as y is increased from y ¼ 0 to y ¼ 12. Figure 12 depicts the variation of the concentration gradient for different values of the chemical reaction parameter. From this figure we can observed that by increasing the value of the chemical reaction parameter, concentration gradient decreases. Also, it is noticed that the concentration gradient increases by increasing the value of y and the values of the concentration gradient become zero as y tends to ∞. Further, it is observed that the effect of the chemical reaction parameter is more significant in the region 0.5 r y r 2.5. The effects of the Schmidt number on frictional coefficient against time are shown in Figure 13. It is seen from this figure that the frictional coefficient increases by decreasing the values of Schmidt number closer to the plate. This is due to the fact that in the presence of heavier diffusing species a retarding effect in the velocity of the flow field is seen. It is also observed that the curves for Sc ¼ 0.4, 0.5 bend towards the lower values of frictional coefficient as time t 4 300, while the curves for Sc ¼ 1.0, 2.0 bend towards the higher values of frictional coefficient as time t 4 220, i.e. when the value of Sc ¼ 0.4, 0.5 then frictional coefficient decreases with time (t 4 300) and reverse effect of time (t 4 220) is seen on frictional coefficient when Sc ¼ 1.0, 2.0. Figure 14 depicts the effect of the frictional coefficient against time t for different values of the viscosity ratio. From this figure, we can observe that the value of the frictional coefficient decreases with increase in the viscosity ratio. Also, frictional coefficient increases with increase in time for any value of the viscosity ratio β. Figure 15 has been drawn to depict the effects of the frictional coefficient against t for different values of the dimensionless chemical reaction parameter. This figure shows that the skin frictional coefficient increases with increase in the dimensionless chemical reaction parameter but when chemical reaction parameter is greater than 0.2, frictional coefficient starts decreasing due to reduction in the boundary layer thickness which results in increase mass transfer by destructive chemical. From this figure we also observe that the frictional coefficient increases with time for all values of the chemical reaction parameter. The variation of the frictional coefficient against t for different value of permeability parameter is shown in Figure 16. Here, we observed that the frictional coefficient increases with increase in the porous permeability parameter. This is because the frictional force increases for the presence of porous medium in which results in increase in the skin friction coefficient. Further, from Figures 15, 16, it is observed that the skin friction coefficient increases with time. Figure 17 depicts the effects for different values of porous permeability parameter on the local couple stress coefficient at the wall against time. It is seen from this figure that the local couple stress coefficient at the wall increases by increasing the value of the porous permeability parameter. Also, the value of the local couple stress

11

coefficient at the wall increases with time. Figure 18 is drawn to analyze the variation of the local couple stress coefficient at the wall against t for different values of the Prandtl number. It is observed from this figure that there is an increasing effect in the local couple stress coefficient at the wall by increasing the value of the Prandtl number. Further, it is observed that the values of the local couple stress coefficient at the wall increases with increase in time. Figure 19 depicts the effects of magnetic field on the local couple stress coefficient at the wall against time. It is seen from this figure that the local couple stress coefficient at the wall decreases with increase in the magnetic field parameter. Physically, a retarding force called Lorentz force is seen for larger value of magnetic parameter which provide more heat to fluid and then the thicker thermal boundary layer thickness occur which decreases the values of the local couple stress coefficient at the wall. Table 1 depicts the effect of the local skin friction coefficient, local couple stress coefficient at the wall, temperature gradient and concentration gradient for the different value of the parameters Sc, Pr, M, r, γ1, β and K' by keeping the other physical parameters constant. In all the figures and tables, the constant values which are taken are ε ¼ 0.01, n ¼ 0.1, n1 ¼ 0.5, GrT ¼ 2, GrC ¼ 1, Up ¼ 0.5. Also the values in Table 1 has been calculated for y ¼ 1 and t ¼ 1. Here, we can easily see that the frictional coefficient increases with the increase in Pr, M, r, and decreases with the increase in Sc, γ1, β and K′. The local couple stress coefficient at the wall increases with increase in Pr, β, K′ and decreases with increase in Sc, M and r but it increases with increase in γ1 upto 0.2 and decreases for γ1 ¼ 0.3. Temperature gradient increases with increase in the radiation parameter and decreases rapidly with increase of Pr, but it remains unchanged for different values of other flow parameters. Also, we can see that the concentration gradient also changes only for different values of Schmidt number and chemical reaction parameter i.e., it is found that the influence of increasing the value of the Schmidt number and chemical reaction parameter is to increase the values of the concentration gradient.

4. Conclusions In this study, we have analyzed the effects of various physical parameters on thermal radiation, magnetic field, chemical reaction, porous permeability, viscosity ratio, Prandtl number, Schmidt number on double diffusive magnetohydrodynamic radiative heat and mass transfer of an oscillatory flow of a micropolar fluid over a vertical permeable plate in a porous medium with convective boundary condition and chemical reaction. The governing differential equations are transformed into a linear form by perturbation method and then solved analytically. From the present investigation, following conclusions are drawn:

Please cite this article as: Dulal Pal, Sukanta Biswas, Magnetohydrodynamic convective-radiative oscillatory flow of a chemically reactive micropolar fluid in a porous medium, Propulsion and Power Research (2018), https://doi.org/10.1016/j.jppr.2018.05.004

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(i) Velocity profiles increase with increase in time but it decrease by increasing the value of the viscosity ratio parameter. (ii) The micro rotational velocity distribution decreases with increase in time but it increases with increase in the magnetic field parameter. (iii) The temperature profile increases with the increase in the radiation parameter and it decreases with increase in the Prandtl number and in both the cases temperature profiles become 0 as y→∞. (iv) Concentration distribution decreases with increase in the dimensionless time and increases with increase in the chemical reaction parameter, (v) The effect of increase in the porous permeability parameter is to increase the skin frictional. Also, skin frictional coefficient increases with increase in the chemical reaction parameter. (vi) The local couple stress coefficient at the wall increases with increase in the permeability parameter and Prandtl number but it decreases by increasing the magnetic field parameter.

Dulal Pal, Sukanta Biswas

[10]

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