Influence of Hall Current and Thermal Radiation on MHD Convective Heat and Mass Transfer in a Rotating Porous Channel with Chemical Reaction

Abstract

A theoretical study is carried out to obtain an analytic solution of heat and mass transfer in a vertical porous channel with rotation and Hall current. A constant suction and injection is applied to the two insulating porous plates. A strong magnetic field is applied in the transverse direction. The entire system rotates with uniform angular velocity Ω about the axis normal to the plates. The governing equations are solved by perturbation technique to obtain the analytical results for velocity, temperature, and concentration fields and shear stresses. The steady and unsteady resultant velocities along with the phase differences for various values of physical parameters are discussed in detail. The effects of rotation, buoyancy force, magnetic field, thermal radiation, and heat generation parameters on resultant velocity, temperature, and concentration fields are analyzed.

1. Introduction

Free convection in channel flow has many important applications in designing ventilating and heating of buildings, cooling of electronic components of a nuclear reactor, bed thermal storage, and heat sink in the turbine blades. Convective flows driven by temperature difference of the bounding walls of channels are important in industrial applications. El-Hakiem [1] studied the unsteady MHD oscillatory flow on free convection radiation through a porous medium with a vertical infinite surface that absorbs the fluid with a constant velocity. Jaiswal and Soundalgekar [2] analyzed the effects of suction with oscillating temperature on a flow past an infinite porous plate. Singh et al. [3] studied the unsteady free convective flow in a porous medium bounded by an infinite vertical porous plate in the presence of rotation. Pal and Shivakumara [4] studied the mixed convection heat transfer from a vertical plate in a porous medium.

Hydromagnetic convection with heat transfer in a rotating medium has important applications in MHD generators and accelerators design, geophysics, and nuclear power reactors. MHD free convection and mass transfer flows in a rotating system have diverse applications. The effects of Hall currents cannot be neglected as the conducting fluid when it is an ionized gas, and applied field strength is strong then the electron cyclotron frequency 𝜔=𝑒𝐵/𝑚 (where 𝑒, 𝐵, and 𝑚 denote the electron charge, the applied magnetic field, and mass of an electron, resp.) exceeds the collision frequency so that the electron makes cyclotron orbit between the collisions which will divert in a direction perpendicular to the magnetic and electric fields directions. Thus, if an electric field is applied perpendicular to the magnetic field then whole current will not pass along the electric field. This phenomena of flow of the electric current across an electric field with magnetic field is known as Hall effect, and accordingly this current is known as Hall current [5]. Thus, it is essential to analyze the effects of Hall currents in many industrial problems. Gupta [6] has studied the influence of Hall current on steady MHD flow in a viscous fluid. Jana et al. [7] analyzed the hall effect in steady flow past an infinite porous flat plate. Makinde and Mhone [8] studied hydromagnetic oscillatory flow through a channel having porous medium. Zhang and Wang [9] analyzed the effect of magnetic field in a power-law fluid over a vertical stretching sheet. Hameed and Nadeem [10] analyzed unsteady hydromagnetic flow of a non-Newtonian fluid over a porous plate. Makinde et al. [11] examined the effect of magnetic field in a rotating porous medium cylindrical annulus. Sibanda and Makinde [12] analyzed effects of magnetic fields on heat transfer on a rotating disk in a porous medium with Ohmic heating and viscous dissipation. Pop and Watanabe [13] analyzed convective flow of a conducting fluid in the presence of magnetic field and Hall current. Saha et al. [14] studied Hall current effect on MHD natural convection flow from vertical flat plate. Recently, Pal et al. [15] examined the influence of Hall current and chemical reaction on oscillatory mixed convection radiation of a micropolar fluid in a rotating system.

Radiation effects on free convection flow have become very important due to its applications in space technology, processes having high temperature, and design of pertinent equipments. Moreover, heat and mass transfer with thermal radiation on convective flows is very important due its significant role in the surface heat transfer. Recent developments in gas cooled nuclear reactors, nuclear power plants, gas turbines, space vehicles, and hypersonic flights have attracted research in this field. The unsteady convective flow in a moving plate with thermal radiation were examined by Cogley et al. [16] and Mansour [17]. The combined effects of radiation and buoyancy force past a vertical plate were analyzed by Hossain and Takhar [18]. Hossain et al. [19] analyzed the influence of thermal radiation on convective flows over a porous vertical plate. Seddeek [20] explained the importance of thermal radiation and variable viscosity on unsteady forced convection with an align magnetic field. Muthucumaraswamy and Senthil [21] studied the effects of thermal radiation on heat and mass transfer over a moving vertical plate. Pal [22] investigated convective heat and mass transfer in a stagnation-point flow towards a stretching sheet with thermal radiation. Aydin and Kaya [23] justified the effects of thermal radiation on mixed convection flow over a permeable vertical plate with magnetic field. Mohamed [24] studied unsteady MHD flow over a vertical moving porous plate with heat generation and Soret effect. Chauhan and Rastogi [25] analyzed the effects of thermal radiation, porosity, and suction on unsteady convective hydromagnetic vertical rotating channel. Ibrahim and Makinde [26] investigated radiation effect on chemically reaction MHD boundary layer flow of heat and mass transfer past a porous vertical flat plate. Pal and Mondal [27] studied the effects of thermal radiation on MHD Darcy-Forchheimer convective flow past a stretching sheet in a porous medium. Palani and Kim [28] analyzed the effect of thermal radiation on convection flow past a vertical cone with surface heat flux. Recently, Mahmoud and Waheed [29] examined thermal radiation on flow over an infinite flat plate with slip velocity.

The study of heat and mass transfer due to chemical reaction is also very importance because of its occurrence in most of the branches of science and technology. The processes involving mass transfer effects are important in chemical processing equipments which are designed to draw high value products from cheaper raw materials with the involvement of chemical reaction. In many industrial processes, the species undergo some kind of chemical reaction with the ambient fluid which may affect the flow behaviour and the production quality of final products. Aboeldahab and Elbarbary [30] examined heat and mass transfer over a vertical plate in the presence of magnetic field and Hall effect. Abo-Eldahab and El Aziz [31] investigated the Hall current and Joule heating effects on electrically conducting fluid past a semi-infinite plate with strong magnetic field and heat generation/absorption. Kandasamy et al. [32] discussed the effects of chemical reaction and magnetic field on heat and mass transfer over a vertical stretching surface. Muthucumaraswamy and Janakiraman [33] analyzed the effects of mass transfer over a vertical oscillating plate with chemical reaction. Sharma and Singh [34] have analyzed the unsteady MHD free convection flow and heat transfer over a vertical porous plate in the presence of internal heat generation and variable suction. Sudheer Babu and Satya Narayan [35] examined chemical reaction and thermal radiation effects on MHD convective flow in a porous medium in the presence of suction. Makinde and Chinyoka [36] studied the effects of magnetic field on MHD Couette flow of a third-grade fluid with chemical reaction. Recently, Pal and Talukdar [37] investigated the influence of chemical reaction and Joule heating on unsteady convective viscous dissipating fluid over a vertical plate in porous media with thermal radiation and magnetic field.

The objective of the present study is to analyze the effects of Hall current, thermal radiation, and first-order chemical reaction on the oscillatory convective flow and mass transfer with suction injection in a rotating vertical porous channel. The present results are compared with those of Singh and Kumar [38], and a very good agreement is found.

2. Problem Formulation

We consider unidirectional oscillatory free convective flow of a viscous incompressible and electrically conducting fluid between two insulating infinite vertical permeable plates separated by a distance 𝑑. A constant injection velocity 𝑤₀ is applied at the stationary plate 𝑧^∗=0. Also, a constant suction velocity 𝑤₀ is applied at the plate 𝑧^∗=𝑑, which oscillates in its own plane with a velocity 𝑈^∗(𝑡^∗) about a nonzero constant mean velocity 𝑈₀. The channel rotates as a rigid body with angular velocity Ω^∗ about the 𝑧^∗-axis perpendicular to the planes of the plates. A strong transverse magnetic field of uniform strength 𝐻₀ is applied along the axis of rotation by neglecting induced electric and magnetic fields. The fluid is assumed to be a gray, emitting, and absorbing, but nonscattering medium. The radiative heat flux term can be simplified by using the Rosseland approximation. It is also assumed that the chemically reactive species undergo first-order irreversible chemical reaction.

The solenoidal relation for the magnetic field ∇⋅𝐻=0, where 𝐻=(𝐻^∗_𝑥,𝐻^∗_𝑦,𝐻^∗_𝑧) gives 𝐻^∗_𝑧=𝐻₀ (constant) everywhere in the flow field, which gives 𝐻=(0,0,𝐻₀). If (𝐽^∗_𝑥,𝐽^∗_𝑦,𝐽^∗_𝑧) are the component of electric current density 𝐽, then the equation of conservation of electric charge ∇⋅𝐽=0 gives 𝐽^∗_𝑧=constant. This constant is zero, that is, 𝐽^∗_𝑧=0 everywhere in the flow since the plate is electrically non-conducting. The generalized Ohm’s law, in the absence of the electric field [39], is of the form 𝜔𝐽+^𝑒𝜏_𝑒𝐻₀𝐽×𝐻=𝜎𝜇_𝑒1𝑉×𝐻+𝑒𝑛_𝑒∇𝑝_𝑒, (1) where 𝑉, 𝜎, 𝜇_𝑒, 𝜔_𝑒, 𝜏_𝑒, 𝑒, 𝑛_𝑒, and 𝑝_𝑒 are the velocity, the electrical conductivity, the magnetic permeability, the cyclotron frequency, the electron collision time, the electric charge, the number density of the electron, and the electron pressure, respectively. Under the usual assumption, the electron pressure (for a weakly ionized gas), the thermoelectric pressure, and ion slip are negligible, so we have from the Ohm’s law 𝐽^∗_𝑥+𝜔_𝑒𝜏_𝑒𝐽^∗_𝑦=𝜎𝜇_𝑒𝐻₀𝑣^∗𝐽^∗_𝑦−𝜔_𝑒𝜏_𝑒𝐽^∗_𝑥=−𝜎𝜇_𝑒𝐻₀𝑢^∗, (2) from which we obtain that 𝐽^∗_𝑥=𝜎𝜇_𝑒𝐻₀𝑚𝑢^∗+𝑣^∗1+𝑚²,𝐽^∗_𝑦=𝜎𝜇_𝑒𝐻₀𝑚𝑣^∗−𝑢^∗1+𝑚². (3) Since the plates are infinite in extent, all the physical quantities except the pressure depend only on 𝑧^∗ and 𝑡^∗. The physical configuration of the problem is shown in Figure 1. A Cartesian coordinate system is assumed, and 𝑧^∗-axis is taken normal to the plates, while 𝑥^∗- and 𝑦^∗-axes are in the upward and perpendicular directions on the plate 𝑧^∗=0 (origin), respectively. The velocity components 𝑢^∗, 𝑣^∗, 𝑤^∗ are in the 𝑥^∗-, 𝑦^∗-, 𝑧^∗-directions, respectively. The governing equations in the rotating system in presence of Hall current, thermal radiation, and chemical reaction are given by the following equations: 𝜕𝑤^∗𝜕𝑧^∗=0⟹𝑤^∗=𝑤₀,𝜕𝑢^∗𝜕𝑡^∗+𝑤₀𝜕𝑢^∗𝜕𝑧^∗−2Ω^∗𝑣^∗1=−𝜌𝜕𝑃^∗𝜕𝑥^∗𝜕+𝜈²𝑢^∗𝜕𝑧^∗2+𝑔₀𝛽𝑇^∗−𝑇_𝑑+𝑔₀𝛽^∗𝐶^∗−𝐶_𝑑+𝐻₀𝜌𝐽^∗_𝑦,𝜕𝑣^∗𝜕𝑡^∗+𝑤₀𝜕𝑣^∗𝜕𝑧^∗+2Ω^∗𝑢^∗1=−𝜌𝜕𝑃^∗𝜕𝑦^∗𝜕+𝜈²𝑣^∗𝜕𝑧^∗2−𝐻₀𝜌𝐽^∗_𝑥,𝜕𝑇^∗𝜕𝑡^∗+𝑤₀𝜕𝑇^∗𝜕𝑧^∗=𝜅𝜌𝑐_𝑝𝜕²𝑇^∗𝜕𝑧^∗2−𝑄₀𝜌𝑐_𝑝𝑇^∗−𝑇_𝑑−1𝜌𝑐_𝑝𝜕𝑞^∗_𝑟𝜕𝑧^∗,𝜕𝐶^∗𝜕𝑡^∗+𝑤₀𝜕𝐶^∗𝜕𝑧^∗=𝐷_𝑚𝜕²𝐶^∗𝜕𝑧^∗2−𝑘₁𝐶^∗−𝐶_𝑑, (4)(5)(6)(7)(8) where 𝑚(=𝜔_𝑒𝜏_𝑒) is the Hall parameter, 𝛽 and 𝛽^∗ are the coefficients of thermal and solutal expansion, 𝑐_𝑝 is the specific heat at constant pressure, 𝜌 is the density of the fluid, 𝜈 is the kinematics viscosity, 𝜅 is the fluid thermal conductivity, 𝑔₀ is the acceleration of gravity, 𝑄₀ is the additional heat source, 𝑞^∗_𝑟 is the radiative heat flux, 𝐷_𝑚 is the molecular diffusivity, 𝑘₁ is the chemical reaction rate constant. The radiative heat flux is given by 𝑞^∗_𝑟=−(4𝜎^∗/3𝑘^∗)(𝜕𝑇^∗4/𝜕𝑧^∗), in which 𝜎^∗ and 𝑘^∗ are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively.

Figure 1 

Physical configuration of the problem.

The initial and boundary conditions as suggested by the physics of the problem are 𝑢^∗=𝑣^∗=0,𝑇^∗=𝑇₀+𝜖𝑇₀−𝑇_𝑑cos𝜔^∗𝑡^∗,𝐶^∗=𝐶₀+𝜖𝐶₀−𝐶_𝑑cos𝜔^∗𝑡^∗at𝑧^∗𝑢=0_∗=𝑈^∗𝑡^∗=𝑈₀1+𝜖cos𝜔^∗𝑡^∗,𝑣^∗𝑇=0,_∗=𝑇_𝑑,𝐶^∗=𝐶_𝑑at𝑧^∗=𝑑, (9) where 𝜖 is a small constant.

We now introduce the dimensionless variables as follows: 𝑧𝜂=^∗𝑑𝑢,𝑢=^∗𝑈₀𝑣,𝑣=^∗𝑈₀𝜔,𝑡=^∗𝑡^∗,𝜔𝜔=^∗𝑑²𝜈Ω,Ω=^∗𝑑²𝜈𝑤,𝜆=⁰𝑑𝜈,𝑇𝜃=^∗−𝑇_𝑑𝑇₀−𝑇_𝑑𝐶,𝜙=^∗−𝐶_𝑑𝐶₀−𝐶_𝑑. (10) After combining (5) and (6) and taking 𝑞=𝑢+𝑖𝑣, then (5)–(8) reduce to 𝜔𝜕𝑞𝜕𝑡+𝜆𝜕𝑞=𝜕𝜕𝜂²𝑞𝜕𝜂²+𝑤𝜕𝑈−𝑀𝜕𝑡−2𝑖Ω𝑞−𝑈₂(1+𝑖𝑚)1+𝑚²𝑞−𝑈+𝜆²,𝜔Gr𝜃+Gm𝜙𝜕𝜃𝜕𝑡+𝜆𝜕𝜃=1𝜕𝜂4Pr1+𝜕3𝑅²𝑞𝜕𝜂²−𝑄_𝐻𝜔Pr𝜃,𝜕𝜙𝜕𝑡+𝜆𝜕𝜙=1𝜕𝜂𝜕Sc²𝜙𝜕𝜂²−𝜉𝜙, (11) where Gr=𝑔₀𝛽𝜈(𝑇0_𝑤−𝑇_𝑑)/𝑈₀𝑤²₀ is the modified thermal Grashof number, Gm=𝑔₀𝛽^∗𝜈(𝐶₀−𝐶_𝑑)/𝑈₀𝑤²₀ is the modified solutal Grashof number, Pr=𝜈𝜌𝑐_𝑝/𝜅 is the Prandtl number, 𝑀=𝐻₀𝑑𝜎/𝜇 is the Hartmann number, 𝑄_𝐻=𝑄₀𝑑²/𝜅 is the heat source parameter, 𝑅=𝜅𝑘^∗/4𝜎^∗𝑇_𝑑 is the radiation parameter, Sc=𝜈/𝐷𝑚 is the Schmidt number, and 𝜉=𝑘₁𝑑²/𝜈 is the reaction parameter.

The boundary conditions (9) can be expressed in complex form as 𝜖𝑞=0,𝜃=1+2𝑒^𝑖𝑡+𝑒^−𝑖𝑡𝜖,𝜙=1+2𝑒^𝑖𝑡+𝑒^−𝑖𝑡(𝑡)𝜖at𝜂=0,𝑞=𝑈=1+2𝑒^𝑖𝑡+𝑒^−𝑖𝑡,𝜃=0,𝜙=0at𝜂=1. (12)

3. Method of Solution

The set of partial differential equations (11) cannot be solved in closed form. So it is solved analytically after these equations are reduced to a set of ordinary differential equations in dimensionless form. We assume that ℜ𝜂,𝑡=ℜ₀𝜂+𝜖2ℜ₁𝜂𝑒^𝑖𝑡+ℜ₂𝜂𝑒^−𝑖𝑡, (13) where ℜ stands for 𝑞 or 𝜃 or 𝜙, and 𝜖≪1 which is a perturbation parameter. The method of solution is applicable for small perturbation.

Substituting (13) into (11) and comparing the harmonic and nonharmonic terms, we obtain the following ordinary differential equations: 𝑞₀^′′−𝜆𝑞^′₀−𝑆𝑞₀=−𝑆−𝜆²Gr𝜃₀+Gm𝜙₀,𝑞₁^′′−𝜆𝑞^′₁−(𝑆+𝑖𝜔)𝑞₁=−(𝑆+𝑖𝜔)−𝜆²Gr𝜃₁+Gm𝜙₁,𝑞₂^′′−𝜆𝑞^′₂−(𝑆−𝑖𝜔)𝑞₁=−(𝑆−𝑖𝜔)−𝜆²Gr𝜃₂+Gm𝜙₂,𝜃₀^′′−3𝜆Pr𝑅𝜃3𝑅+4^′0−3𝑅𝑄_𝐻𝜃3𝑅+4⁰𝜃=0,_1′′−3𝜆Pr𝑅𝜃3𝑅+4^′1−3𝑅3𝑅+4𝑖𝜔Pr+𝑄_𝐻𝜃₁𝜃=0,_2′′−3𝜆Pr𝑅𝜃3𝑅+4^′2+3𝑅3𝑅+4𝑖𝜔Pr−𝑄_𝐻𝜃₂𝜙=0,_0′′−Sc𝜆𝜙^′₀−Sc𝜉𝜙₀𝜙=0,_1′′−Sc𝜆𝜙^′₁−Sc(𝑖𝜔+𝜉)𝜙₁𝜙=0,_2′′−Sc𝜆𝜙^′₂+Sc(𝑖𝜔−𝜉)𝜙₂=0, (14) where 𝑆=(𝑀²(1+𝑖𝑚)/(1+𝑚²))+2𝑖Ω and dashes denote the derivatives w.r.t. 𝜂.

The transformed boundary conditions are 𝑞₀=0,𝑞₁=0,𝑞₂𝜃=0,₀=1,𝜃₁=1,𝜃₂𝜙=1,₀=1,𝜙₁=1,𝜙₂𝑞=1at𝜂=0,₀=1,𝑞₁=1,𝑞₂𝜃=1,₀=0,𝜃₁=0,𝜃₂𝜙=0,₀=0,𝜙₁=0,𝜙₂=0at𝜂=1. (15) The solutions of (14) under the boundary conditions (15) are 𝑞₀=1−𝑒^ℎ14𝜂+𝐴₁𝑒^ℎ14𝜂−𝑒^ℎ8𝜂−𝐴₂𝑒^ℎ14𝜂−𝑒^ℎ7𝜂+𝐴₃𝑒^ℎ14𝜂−𝑒^ℎ2𝜂−𝐴₄𝑒^ℎ14𝜂−𝑒^ℎ1𝜂+𝑒^ℎ13𝜂−𝑒^ℎ14𝜂𝑒^ℎ13−𝑒^ℎ14×𝑒^ℎ14−𝐴₁𝑒^ℎ14−𝑒^ℎ8+𝐴₂𝑒^ℎ14−𝑒^ℎ7−𝐴₃𝑒^ℎ14−𝑒^ℎ2+𝐴₄𝑒^ℎ14−𝑒^ℎ1,𝑞₁=1−𝑒^ℎ16𝜂+𝐴₅𝑒^ℎ16𝜂−𝑒^ℎ10𝜂−𝐴₆𝑒^ℎ16𝜂−𝑒^ℎ9𝜂+𝐴₇𝑒^ℎ16𝜂−𝑒^ℎ4𝜂−𝐴₈𝑒^ℎ16𝜂−𝑒^ℎ3𝜂+𝑒^ℎ15𝜂−𝑒^ℎ16𝜂𝑒^ℎ15−𝑒^ℎ16×𝑒^ℎ16−𝐴₅𝑒^ℎ16−𝑒^ℎ10+𝐴₆𝑒^ℎ16−𝑒^ℎ9−𝐴₇𝑒^ℎ16−𝑒^ℎ4+𝐴₈𝑒^ℎ16−𝑒^ℎ3,𝑞₂=1−𝑒^ℎ18𝜂+𝐴₉𝑒^ℎ18𝜂−𝑒^ℎ12𝜂−𝐴₁₀𝑒^ℎ18𝜂−𝑒^ℎ11𝜂+𝐴₁₁𝑒^ℎ18𝜂−𝑒^ℎ6𝜂−𝐴₁₂𝑒^ℎ18𝜂−𝑒^ℎ5𝜂+𝑒^ℎ17𝜂−𝑒^ℎ18𝜂𝑒^ℎ17−𝑒^ℎ18×𝑒^ℎ18−𝐴₉𝑒^ℎ18−𝑒^ℎ12+𝐴₁₀𝑒^ℎ18−𝑒^ℎ11−𝐴₁₁𝑒^ℎ18−𝑒^ℎ6+𝐴₁₂𝑒^ℎ18−𝑒^ℎ5,𝜃₀=𝑒^{ℎ7+ℎ8𝜂}−𝑒^{ℎ8+ℎ7𝜂}𝑒^ℎ7−𝑒^ℎ8,𝜃₁=𝑒^{ℎ9+ℎ10𝜂}−𝑒^{ℎ10+ℎ9𝜂}𝑒^ℎ9−𝑒^ℎ10,𝜃₂=𝑒^{ℎ11+ℎ12𝜂}−𝑒^{ℎ12+ℎ11𝜂}𝑒^ℎ11−𝑒^ℎ12,𝜙₀=𝑒^{ℎ1+ℎ2𝜂}−𝑒^{ℎ2+ℎ1𝜂}𝑒^ℎ1−𝑒^ℎ2,𝜙₁=𝑒^{ℎ3+ℎ4𝜂}−𝑒^{ℎ4+ℎ3𝜂}𝑒^ℎ3−𝑒^ℎ4,𝜙₂=𝑒^{ℎ5+ℎ6𝜂}−𝑒^{ℎ6+ℎ5𝜂}𝑒^ℎ5−𝑒^ℎ6. (16)(17)(18)(19)(20)(21)(22)(23)(24)

4. Amplitude and Phase Difference due to Steady and Unsteady Flow

Equation (16) corresponds to the steady part, which gives 𝑢₀ as the primary and 𝑣₀ as secondary velocity components. The amplitude (resultant velocity) and phase difference due to these primary and secondary velocities for the steady flow are given by 𝑅₀=𝑢²₀+𝑣²₀,𝛼₀=tan⁻¹𝑣₀𝑢₀, (25) where 𝑢₀(𝜂)+𝑖𝑣₀(𝜂)=𝑞₀(𝜂).

Equations (17) and (18) together give the unsteady part of the flow. Thus, unsteady primary and secondary velocity components 𝑢₁(𝜂) and 𝑣₁(𝜂), respectively, for the fluctuating flow can be obtained from the following: 𝑢₁𝜂,𝑡=Real𝑞₁𝜂+Real𝑞₂𝜂−cos𝑡Im𝑞₁𝜂−Im𝑞₂𝜂𝑣sin𝑡,₁𝜂,𝑡=Real𝑞₁𝜂−Real𝑞₂𝜂+sin𝑡Im𝑞₁𝜂+Im𝑞₂𝜂cos𝑡. (26) The amplitude (resultant velocity) and the phase difference of the unsteady flow are given by 𝑅_𝑣=𝑢²₁+𝑣²₁,𝛼₁=tan⁻¹𝑣₁𝑢₁, (27) where 𝑢₁(𝜂)+𝑖𝑣₁(𝜂)=𝑞₁(𝜂)𝑒^𝑖𝑡+𝑞₂(𝜂)𝑒^−𝑖𝑡.

The amplitude (resultant velocity) and the phase difference 𝑅_𝑛=√𝑢²+𝑣²,𝛼=tan⁻¹𝑣𝑢, (28) where 𝑢=Realpart of 𝑞 and 𝑣=Imaginarypart of 𝑞.

5. Amplitude and Phase Difference of Shear Stresses due to Steady and Unsteady Flow at the Plate

The amplitude and phase difference of shear stresses at the stationary plate (𝜂=0) for the steady flow can be obtained as 𝜏_0𝑟=𝜏²_0𝑥+𝜏²_0𝑦,𝛽₀=tan⁻¹𝜏_0𝑦𝜏_0𝑥, (29) where 𝜕𝑞₀𝜕𝜂_𝜂=0=𝜏_0𝑥+𝑖𝜏_0𝑦=−ℎ₁₄+𝐴₁ℎ₁₄−ℎ₈−𝐴₂ℎ₁₄−ℎ₇+𝐴₃ℎ₁₄−ℎ₂−𝐴₄ℎ₁₄−ℎ₁+ℎ₁₃−ℎ₁₄𝑒^ℎ13−𝑒^ℎ14×𝑒^ℎ14−𝐴₁𝑒^ℎ14−𝑒^ℎ8+𝐴₂𝑒^ℎ14−𝑒^ℎ7−𝐴₃𝑒^ℎ14−𝑒^ℎ2+𝐴₄𝑒^ℎ14−𝑒^ℎ1. (30) For the unsteady part of flow, the amplitude and phase difference of shear stresses at the stationary plate (𝜂=0) can be obtained as 𝜏_1𝑟=𝜏²_1𝑥+𝜏²_1𝑦,𝛽₁=tan⁻¹𝜏_1𝑦𝜏_1𝑥, (31) where 𝜏_1𝑥+𝑖𝜏_1𝑦=𝜕𝑞₁𝜕𝜂_𝜂=0𝑒^𝑖𝑡+𝜕𝑞₂𝜕𝜂_𝜂=0𝑒^−𝑖𝑡,𝜕𝑞₁𝜕𝜂_𝜂=0=−ℎ₁₆+𝐴₅ℎ₁₆−ℎ₁₀−𝐴₆ℎ₁₆−ℎ₉+𝐴₇ℎ₁₆−ℎ₄−𝐴₈ℎ₁₆−ℎ₃+ℎ₁₅−ℎ₁₆𝑒^ℎ15−𝑒^ℎ16×𝑒^ℎ16−𝐴₅𝑒^ℎ16−𝑒^ℎ10+𝐴₆𝑒^ℎ16−𝑒^ℎ9−𝐴₇𝑒^ℎ16−𝑒^ℎ4+𝐴₈𝑒^ℎ16−𝑒^ℎ3,𝜕𝑞₂𝜕𝜂_𝜂=0=−ℎ₁₈+𝐴₉ℎ₁₈−ℎ₁₂−𝐴₁₀ℎ₁₈−ℎ₁₁+𝐴₁₁ℎ₁₆−ℎ₆−𝐴₁₂ℎ₁₈−ℎ₅+ℎ₁₇−ℎ₁₈𝑒^ℎ17−𝑒^ℎ18×𝑒^ℎ18−𝐴₉𝑒^ℎ18−𝑒^ℎ12+𝐴₁₀𝑒^ℎ18−𝑒^ℎ11−𝐴₁₁𝑒^ℎ18−𝑒^ℎ6+𝐴₁₂𝑒^ℎ18−𝑒^ℎ5. (32) The amplitude and phase difference of shear stresses at the stationary plate (𝜂=0) for the flow can be obtained as 𝜏=𝜕𝑞𝜕𝜂_𝜂=0=𝜏²_𝑥+𝜏²_𝑦,𝛽₂=tan⁻¹𝜏_𝑦𝜏_𝑥, (33) where 𝜏_𝑥=Realpart of (𝜕𝑞/𝜕𝜂)_𝜂=0 and 𝜏_𝑦=Imaginarypart of (𝜕𝑞/𝜕𝜂)_𝜂=0.

The Nusselt number 4Nu=−1+3𝑅𝜕𝜃𝜕𝜂_𝜂=0=𝑁_𝑥+𝑖𝑁_𝑦. (34) The rate of heat transfer (i.e., heat flux) at the plate in terms of amplitude and phase is given by Θ=𝑁²_𝑥+𝑁²_𝑦,𝛾=tan⁻¹𝑁_𝑦𝑁_𝑥. (35) The Sherwood number Sh=𝜕𝜙𝜕𝜂_𝜂=0=𝑀_𝑥+𝑖𝑀_𝑦. (36) The rate of mass transfer (i.e., mass flux) at the plate in terms of amplitude and phase is given by Φ=𝑀²_𝑥+𝑀²_𝑦,𝛿=tan⁻¹𝑀_𝑦𝑀_𝑥. (37)

6. Results and Discussion

The system of ordinary differential equations (14) with boundary conditions (15) is solved analytically by employing the perturbation technique. The solutions are obtained for the steady and unsteady velocity fields from (16)–(18), temperature fields from (19)–(21), and concentration fields are given by (22)–(24). The effects of various parameters on the thermal, mass, and hydrodynamic behaviors of buoyancy-induced flow in a rotating vertical channel are studied. The results are presented graphically and in tabular form. Temperature of the heated wall (left wall) at 𝑧^∗=0 is a function of time as given in the boundary conditions, and the cooled wall at 𝑧^∗=𝑑 is maintained at a constant temperature. Further, it is assumed that the temperature difference is small enough so that the density changes of the fluid in the system will be small. When the injection/suction parameter 𝜆 is positive, fluid is injected through the hot wall into the channel and sucked out through the cold wall. The numerical results of the amplitude of the shear stresses and the phase difference of the shear stresses at the stationary plate (𝑧^∗=0) for the steady and unsteady flow are presented in Table 1. The effect of various physical parameters on flow, heat, concentration fields, skin-friction, Nusselt number, and Sherwood number are presented graphically in Figures 2–16.

Table 1 

Comparison results for the resultant velocity or amplitude and the phase difference of the unsteady flow with Singh and Kumar [38] for different values of Gr, 𝑀, 𝑚, 𝜆, Ω, Pr, and Gm=0.0, 𝑅=∞, 𝜉=0.0, Sc→0.0 (in present problem).

Figure 2 

Resultant velocity 𝑅𝑛 due to 𝑢 and 𝑣 versus 𝜂 at 𝑡=𝜋/4.

Figure 3 

Resultant velocity 𝑅𝑛 due to 𝑢 and 𝑣 versus 𝜂 for small values of Ω at 𝑡=𝜋/4.

Figure 4 

Resultant velocity 𝑅𝑛 due to 𝑢 and 𝑣 versus 𝜂 for large values of Ω at 𝑡=𝜋/4.

Figure 5 

Resultant velocity 𝑅𝑛 due to 𝑢 and 𝑣 versus 𝜂 for different values of 𝜖 at 𝑡=𝜋/4.

Figure 6 

Resultant velocity 𝑅𝑛 due to 𝑢 and 𝑣 versus 𝑡 for different values of 𝜂.

Figure 7 

Phase angle 𝛼 due to 𝑢 and 𝑣 versus 𝜂 at 𝑡=𝜋/4.

Figure 8 

Phase angle 𝛼 due to 𝑢 and 𝑣 versus 𝜂 for 𝜆=0.5, Ω=10, Sc=0.3, Pr=0.71, 𝑅=0.2, 𝑄𝐻=5.0, 𝜖=0.01, and 𝜉=0.5 at 𝑡=𝜋/4.

Figure 9 

Phase angle 𝛼 due to 𝑢 and 𝑣 versus 𝜂 for 𝜔=5.0 at 𝑡=𝜋/4.

Figure 10 

Concentration profiles against 𝜂 for different values of 𝜉 at 𝑡=𝜋/4.

Figure 11 

Concentration profiles against 𝜂 for different values of Sc at 𝑡=𝜋/4.

Figure 12 

Temperature profiles against 𝜂 for different values of 𝑄𝐻 at 𝑡=𝜋/4.

Figure 13 

Temperature profiles against 𝜂 for different values of 𝑅 at 𝑡=𝜋/4.

Figure 14 

Skin friction coefficient against 𝜔 for different values of Gr with Gm=5.0, 𝑀=2.0, 𝑚=1.0, Ω=10.0, Pr=0.71, 𝑅=0.2, 𝑄𝐻=5.0, Sc=0.3, 𝜉=0.5, 𝜖=0.01 at 𝑡=𝜋/4.

Figure 15 

Nusselt number against 𝜔 for different values of 𝑄𝐻 with 𝜆=0.5, Pr=0.71, 𝑅=2.0, 𝜖=0.01 at 𝑡=𝜋/4.

Figure 16 

Sherwood number against 𝜔 for different values of 𝜉 with 𝜆=0.5, 𝜖=0.01, Sc=0.3 at 𝑡=𝜋/4.

Table 1 shows a comparative study of the present results of amplitude and phase difference of shear stresses for the steady flow with those of Singh and Kumar [38]. It is seen from this table that the present results coincide very well with those of Singh and Kumar [38]. This confirms that the present analytical solutions are correct and accurate. Further, it is observed from this table that the effects of increasing the value of thermal Grashof number Gr, magnetic field 𝑀, and injection/suction parameter 𝜆 are to increase amplitude and decrease the phase difference of shear stresses for the steady flow, whereas reverse effect is found by increasing the Hall parameter, 𝑚. The effects of increasing the angular velocity Ω are to increase both amplitude and phase difference of shear stresses, whereas reverse effects are seen by increasing the values of the Prandtl number. The computed results 𝜏_0𝑟, 𝜏_1𝑟, 𝛽₀, 𝛽₁ for the present problem are provided in Table 2 for various values of Gm, 𝑅, 𝑄_𝐻, Sc, 𝜉, and 𝑚. It is seen from this table that the values of 𝜏_0𝑟 and 𝜏_1𝑟 increase whereas the values of 𝛽₀ and 𝛽₁ decrease with the increase of solutal Grashof number Gm, but the effects are reversed with an increase in the Hall parameter 𝑚, that is, the value of 𝛽₀ and 𝛽₁ are increased whereas there is decrease in the values of 𝜏_0𝑟 and 𝜏_1𝑟. Also, it is found that the values of 𝜏_0𝑟, 𝛽₀𝛽₁ increase with an increase in the radiation parameter 𝑅. It is noted that the values of both 𝜏_0𝑟 and 𝛽₀ increase due to an increase in the heat source parameter 𝑄_𝐻 and chemical reaction parameter 𝜉, whereas the effects are reversed with the increase in the Schmidt number, that is, the values of 𝜏_0𝑟, 𝜏_1𝑟𝛽₁ decrease with an increase in the Schmidt number. Also, it is found that the value of 𝛽₁ decreases with an increase in the heat source parameter 𝑄_𝐻, chemical reaction parameter 𝜉 and Schmidt number Sc.

Table 2 

Values of 𝜏_0𝑟, 𝛽₀, 𝜏_1𝑟, and 𝛽₁ for the reference values of Gr=5.0, 𝑀=2.0, 𝜆=0.5, Ω=10.0, Pr=0.71, and 𝜔=5.0 at 𝑡=𝜋/4.

The profiles for resultant velocity 𝑅_𝑛 for the flow are shown in Figures 2–6 for suction/injection parameter 𝜆 and for small and large values of rotation parameter Ω, 𝜖, and 𝜂, respectively. From Figure 2, it is observed that the increase in the suction parameter 𝜆 leads to an increase of 𝑅_𝑛 within the stationary plates. Similar trend of 𝑅_𝑛 profiles is seen by increasing the rotation parameter Ω, that is, resultant velocity profiles increase with increase in the rotation parameter Ω (small values) as shown in Figure 3. However, the opposite effect occurs near the right wall for large values of Ω as shown in Figure 4. This effect is due to the rotation effects being more dominant near the walls, so when Ω reaches high values, the secondary velocity component 𝑣 decreases with increase in Ω while approaching to the right plate. From Figure 5, it is observed that the increase in the 𝜖 leads to an increase of 𝑅_𝑛 within the stationary plates. From Figure 6, it is seen that the resultant velocity profiles increases with increase in 𝜂; also it is observed that the velocity oscillates with increasing time. The phase difference 𝛼 for the flow is shown graphically in Figure 7 for various values of rotation parameter Ω. From this figure, it is observed that the phase angle 𝛼 decreases with an increase in rotation parameter. Figure 8 shows the variation of 𝛼 against 𝜂 for different values of thermal Grashof number Gr, solutal Grashof number Gm, Hartmann number 𝑀, and Hall parameter 𝑚. From this figure it is found that the values of 𝛼 decrease with an increase in the value of Gr, Gm and 𝑀, whereas reverse trend is seen on the values of 𝛼 by increasing the value of the Hall parameter 𝑚. The phase difference 𝛼 for the flow is shown graphically in Figure 9 for various positive values of suction/injection parameter 𝜆. The figure shows that the phase angle 𝛼 decreases with the increase of suction parameter.

The effect of reaction rate parameter 𝜉 on the species concentration profiles for generative chemical reaction is shown in Figure 10. It is noticed for the graph that there is a marked effect of increasing the value of the chemical reaction rate parameter 𝜉 on concentration distribution 𝜙 in the boundary layer. It is observed that increasing the value of the chemical reaction parameter 𝜉 decreases the concentration of species in the boundary layer; this is due to the fact that destructive chemical reduces the solutal boundary layer thickness and increases the mass transfer. Opposite trend is seen in the case when Schmidt number is increased as noted in Figure 11. It may also be observed from this figure that the effect of Schmidt number Sc is to increase the concentration distribution in the solutal boundary layer.

Figure 12 has been plotted to depict the variation of temperature profiles against 𝜂 for different values of heat absorption parameter 𝑄_𝐻 by fixing other physical parameters. From this graph, we observe that temperature 𝜃 decreases with increase in the heat absorption parameter 𝑄_𝐻 because when heat is absorbed, the buoyancy force decreases the temperature profile. Figure 13 represents graph of temperature distribution with 𝜂 for different values of radiation parameter. From this figure, we note that the initial temperature 𝜃=1.0 decreases to zero satisfying the boundary condition at 𝜂=1.0. Further, it is observed from this figure that increase in the radiation parameter decreases the temperature distribution in the thermal boundary layer due to decrease in the thickness of the thermal boundary layer with thermal radiation parameter 𝑅. This is because large values of radiation parameter correspond to an increase in dominance of conduction over radiation, thereby decreasing the buoyancy force and the temperature in the thermal boundary layer.

Figures 14–16 show the amplitude of skin-friction, Nusselt number, and Sherwood number against frequency parameter 𝜔 for different values of Gr, 𝑄_𝐻 and 𝜉, respectively. From Figure 14, it is observed that the skin friction increases with increasing the values of Gr. Also, the skin friction decreases slowly with increasing the value of 𝜔. The amplitude of Nusselt number decreases with increasing the value of heat source parameter 𝑄_𝐻 which is shown in Figure 15. Figure 16 shows the variation of Sherwood number with 𝜉 and 𝜔. From this figure, it is seen that the Sherwood number decreases with increasing the values of chemical reaction parameter 𝜉, and the opposite trend is seen with increasing the values of 𝜔.

7. Conclusions

The influence of hall current and chemical reaction on unsteady MHD heat and mass transfer of an oscillatory convective flow in a rotating vertical porous channel with thermal radiation and injection is studied analytically. Computed results are presented to exhibit their dependence on the important physical parameters. We conclude the following from the numerical results.(i)An increase in 𝑄_𝐻 leads to an increase in 𝜏_0𝑟, 𝛽₀, 𝜏_1𝑟 and decrease in 𝛽₁.(ii)An increase in radiation parameter 𝑅 and chemical reaction parameter 𝜉 leads to increase in 𝜏_0𝑟, 𝛽₀, and 𝛽₁ but decrease in 𝜏_1𝑟.(iii)An increase in Gr, Gm, 𝑀, Ω, and 𝜆 leads to decrease in 𝛼₀ and 𝛼₁, whereas reverse effect is seen by increasing Hall parameter 𝑚.(iv)The amplitude 𝑅_𝑛 increases with the increase of 𝜆 and Ω.(v)The value of 𝑅_𝑛 decreases with the increase in chemical reaction parameter 𝜉 and oscillation parameter 𝜔.(vi)The skin friction increases with increase in thermal Grashof number Gr.

Appendix

Consider the following: 𝑅₁=3𝑅3𝑅+4,𝑅₂=𝑅₁ℎPr,₁=Sc𝜆+(Sc𝜆)²+4𝜉Sc2,ℎ₂=Sc𝜆−(Sc𝜆)²+4𝜉Sc2,ℎ₃=Sc𝜆+(Sc𝜆)²+4(𝑖𝜔+𝜉)Sc2,ℎ₄=Sc𝜆−(Sc𝜆)²+4(𝑖𝜔+𝜉)Sc2,ℎ₅=Sc𝜆+(Sc𝜆)²−4(𝑖𝜔−𝜉)Sc2,ℎ₆=Sc𝜆−(Sc𝜆)²−4(𝑖𝜔−𝜉)Sc2,ℎ₇=𝑅₂𝜆+𝑅₂𝜆²+4𝑅₁𝑄_𝐻2,ℎ₈=𝑅₂𝜆−𝑅₂𝜆²+4𝑅₁𝑄_𝐻2,ℎ₉=𝑅₂𝜆+𝑅₂𝜆²+4𝑅₁𝑖𝜔Pr+𝑄_𝐻2,ℎ₁₀=𝑅₂𝜆−𝑅₂𝜆²+4𝑅₁𝑖𝜔Pr+𝑄_𝐻2,ℎ₁₁=𝑅₂𝜆+𝑅₂𝜆²−4𝑅₁𝑖𝜔Pr−𝑄_𝐻2,ℎ₁₂=𝑅₂𝜆−𝑅₂𝜆²−4𝑅₁𝑖𝑤Pr+𝑄_𝐻2,ℎ₁₃=𝜆+𝜆²+4𝑆2,ℎ₁₄=𝜆−𝜆²+4𝑆2,ℎ₁₅=𝜆+𝜆²+4(𝑆+𝑖𝜔)2,ℎ₁₆=𝜆−𝜆²+4(𝑆+𝑖𝜔)2,ℎ₁₇=𝜆+𝜆²+4(𝑆−𝑖𝜔)2,ℎ₁₈=𝜆−𝜆²+4(𝑆−𝑖𝜔)2,𝐴₁=𝜆²Gr𝑒^ℎ7𝑒^ℎ7−𝑒^ℎ8ℎ²₈−𝜆ℎ₈,𝐴−𝑆₂=𝜆²Gr𝑒^ℎ8𝑒^ℎ7−𝑒^ℎ8ℎ²₇−𝜆ℎ₇,𝐴−𝑆₃=𝜆²Gr𝑒^ℎ1𝑒^ℎ1−𝑒^ℎ2ℎ²₂−𝜆ℎ₂,𝐴−𝑆₄=𝜆²Gr𝑒^ℎ2𝑒^ℎ1−𝑒^ℎ2ℎ²₁−𝜆ℎ₁,𝐴−𝑆₅=𝜆²Gr𝑒^ℎ9𝑒^ℎ9−𝑒^ℎ10ℎ²₁₀−𝜆ℎ₁₀−(,𝐴𝑆+𝑖𝜔)₆=𝜆²Gr𝑒^ℎ10𝑒^ℎ9−𝑒^ℎ10ℎ²₉−𝜆ℎ₉−(,𝐴𝑆+𝑖𝜔)₇=𝜆²Gm𝑒^ℎ3𝑒^ℎ3−𝑒^ℎ4ℎ²₄−𝜆ℎ₄−(,𝐴𝑆+𝑖𝜔)₈=𝜆²Gm𝑒^ℎ4𝑒^ℎ3−𝑒^ℎ4ℎ²₃−𝜆ℎ₃−(,𝐴𝑆+𝑖𝜔)₉=𝜆²Gr𝑒^ℎ11𝑒^ℎ11−𝑒^ℎ12ℎ²₁₂−𝜆ℎ₁₂−(,𝐴𝑆−𝑖𝜔)₁₀=𝜆²Gr𝑒^ℎ12𝑒^ℎ11−𝑒^ℎ12ℎ²₁₁−𝜆ℎ₁₁−(,𝐴𝑆−𝑖𝜔)₁₁=𝜆²Gm𝑒^ℎ5𝑒^ℎ5−𝑒^ℎ6ℎ²₆−𝜆ℎ₆−(,𝐴𝑆−𝑖𝜔)₁₂=𝜆²Gm𝑒^ℎ6𝑒^ℎ5−𝑒^ℎ6ℎ²₅−𝜆ℎ₅−(.𝑆−𝑖𝜔) (A.1)

Nomenclature

𝐶∗:	Dimensional concentration
𝐶0:	Concentration at the left plate
𝐶𝑑:	Concentration at the right plate
𝑐𝑝:	Specific heat at constant pressure
𝑑:	Distance of the plates
𝐷𝑚:	Chemical molecular diffusivity
𝑒:	Electric charge
𝑔0:	Acceleration due to gravity
Gm:	Modified Grashof number for mass transfer
Gr:	Modified Grashof number for heat transfer
𝐻:	Magnetic field
𝐻0:	Magnetic field of uniform strength
𝐻𝑥:	𝑥-component of magnetic field
𝐽:	Current density
𝐽𝑥:	𝑥-component of current density
𝑘∗:	Mean absorption coefficient
𝑘1:	Chemical reaction rate constant
𝑚:	Hall parameter
𝑀:	Hartmann number
Nu:	Nusselt number
𝑛𝑒:	Number density of the electron
𝑃∗:	Dimensional pressure
𝑝𝑒:	Electron pressure
Pr:	Prandtl number
𝑞∗𝑟:	Radiative heat flux
𝑄0:	Dimensional heat source
𝑄𝐻:	Heat source parameter
𝑅:	Radiation parameter
𝑅0:	Amplitude for steady flow
𝑅𝑛:	Resultant velocity
𝑅𝑣:	Amplitude for unsteady flow
Sc:	Schmidt number
Sh:	Sherwood number
𝑡∗:	Dimensional time
𝑇∗:	Dimensional temperature
𝑇0:	Temperature at the left wall
𝑇𝑑:	Temperature at the right wall
𝑈0:	Nonzero constant mean velocity
𝑢0:	Primary velocity component for steady flow
𝑢1:	Primary velocity component for unsteady flow
𝑉:	Electron velocity
𝑣0:	Secondary velocity component for steady flow
𝑣1:	Secondary velocity component for unsteady flow
𝑢∗,𝑣∗,𝑤∗:	Velocity components are in the 𝑥∗-, 𝑦∗-, 𝑧∗-directions, respectively
𝑤0:	Dimensional injection/suction velocity.

Greek Symbols

𝛼0:	Phase difference for steady flow
𝛼1:	Phase difference for unsteady flow
𝛼:	Phase difference of the flow
𝛽:	Coefficient of thermal expansion
𝛽∗:	Coefficient of solutal expansion
𝛽0:	Phase difference of shear stresses for the steady flow
𝛽1:	Phase difference of shear stresses for the unsteady flow
𝛽2:	Phase difference of shear stresses for the flow
𝛿:	Phase difference of mass flux
𝜖:	Small positive constant
𝜂:	Dimensionless distance
𝛾:	Phase difference of heat flux
𝜅:	Fluid thermal conductivity
𝜆:	Injection/suction parameter
𝜇:	Dynamic viscosity
𝜇𝑒:	Magnetic permeability
𝜈:	Kinematic viscosity
𝜔:	Oscillation parameter
Ω∗:	Dimensional angular velocity
Ω:	Angular velocity
𝜔𝑒:	Cyclotron frequency
Φ:	Amplitude of mass flux
𝜙:	Nondimensional concentration
𝜌:	Density
𝜎:	Electric conductivity
𝜎∗:	Stefan-Boltzmann constant
𝜏:	Amplitude of shear stresses for the flow
𝜏0𝑟:	Amplitude of shear stresses for the steady flow
𝜏1𝑟:	Amplitude of shear stresses for the unsteady flow
𝜏𝑒:	Electron collision time
𝜃:	Non-dimensional temperature
Θ:	Amplitude of heat flux.

Acknowledgment

One of the authors (Dulal Pal) is grateful to the University Grants Commission (UGC), New Delhi, for providing financial support under SAP-DRS (Phase-II) Grant.