Vol 4, No 12 (2013)

Available online throug

ISSN 2229 – 5046

CHEMICAL REACTION AND SORET EFFECT

ON HYDRO MAGNETIC OSCILLATORY FLOW THROUGH A POROUS MEDIUM BOUNDED

BY TWO VERTICAL POROUS PLATES WITH HEAT SOURCE

N. Senapati

_{, B. K. Swain*}

_{and R. K. Dhal}

1

_{Department of mathematics, Ravenshaw University, Cuttack, Odisha-753003, India.}

2

_{Department of mathematics, Ravenshaw University, Cuttack, Odisha-753003, India.}

3

_{J. N. V., Paralakhemundi, Gajapati-761203, India.}

(Received on: 08-11-13; Revised & Accepted on: 24-12-13)

ABSTRACT

H

Keywords: Hydro magnetic, Oscillatory flow, porous medium, chemical reaction, Heat source, soret effect.

INTRODUCTION

Magneto hydrodynamics free convection flows bounded by two vertical plates are studied because of their wide application and hence it has attracted the attention of many research scholars, investigators and scientists. MHD is applied to study the stellar and solar structure, interstellar matter, radio propagation through the ionosphere etc. The ionised gas or plasma can be made to interact with the magnetic field and can frequently alter heat transfer on the bounding surface. Heat transfer by thermal radiation is becoming of great importance when we are concerned with space applications, higher operating temperatures and also power engineering. In processes such as drying, evaporation at the surface of water body, energy transfer in a wet cooling tower and the flow in a desert, cooler, heat and mass transfer occurs simultaneously. The various applications of MHD flows in technological fields have been compiled by Moreau [4]. P. Mohapatra and N. Senapati [5] have been analyzed magneto hydrodynamic free convection flow with mass transfer past a vertical plate.

Oscillatory flows are known to result in higher rates of heat and mass transfer. Many studies have been done to understand its characteristics in different systems such as reciprocating engines, pulse combustors and chemical reactors etc. Stokes first found the exact solution of Navier-Stokes equation which is concerned with flow of viscous incompressible fluid past a horizontal plate oscillating in its own plane. Then Soundalgekar [2] has studied the effect of natural convection on stokes problem. The same problem was considered by Revankar [3] for an impulsive started or oscillating plate. Cooper et al. [6] have made a detailed study on fluid mechanics of oscillatory and modulated flows and associated applications in heat and mass transfer. Muthucumaraswamy [10] has studied the effect of heat and mass transfer on flow past an oscillatory vertical plate with variable temperature.

To study the underground water resources, seepage of water in river beds, filtration and water purification processes in chemical engineering, one need the knowledge of the fluid flow through porous medium. The porous medium is in fact a non homogeneous medium but for the sake of analysis, it may be possible to replace it with a homogeneous fluid which has dynamical properties equal to those of non homogeneous continuum. Model studies of the above phenomena of MHD convection have been made by many researchers. Ram and Mishra [1] studied MHD flow of conducting fluid

through porous media. Ahmed et al. [7] discussed three dimensional free convective flow and heat transfer through a

porous medium. Geindreau et al. [9] studied the effect of magnetic field in flow through porous medium.

Corresponding author: B. K. Swain*

The soret effect or thermophoresis is a phenomenon observed in mixtures of mobile particles where the different particles exhibit different responses to the force of a temperature gradient. The term soret effect is most often applied to aerosol mixtures, but may also commonly refer to the phenomenon in all phases of matter. It has been used in commercial precipitators for applications similar to electro static precipitators, manufacturing of optical fibre in vapour deposition processes, facilitating drug discovery by allowing the detection of aptamer binding by comparison of the bound versus unbound motion of the target molecule. It is also used to separate different polymer particles in field flow fractionation. M. Anghel et al. [8] studied Dufour effect and Soret effect on free convection boundary layer over a vertical surface embedded in porous medium. N. Ahmed [12] studied MHD convection with soret and Dufour effect in a three dimensional flow past an infinite vertical porous plate. Recently Chand et al. [14] have studied hydro magnetic oscillatory flow through a porous medium bounded by two vertical porous plates with heat source and soret effect.

Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have therefore received a considerable amount of attention in recent years. Possible applications of this type of flow can be found in many industries. The effect of chemical reaction depends whether the reaction is homogeneous or heterogeneous, the rate of reaction depends on the concentration of species itself. Muthucumaraswamy et al. [11] have studied the mass transfer effect on isothermal vertical oscillating plate in presence of chemical reaction. Senapati et al. [13] have studied magnetic effect on mass and heat transfer of hydrodynamic flow past a vertical oscillating plate in presence of chemical reaction.

The objective of this paper is to analyze hydro magnetic oscillatory flow through a porous medium bounded by vertical porous plates with heat source and soret effect in presence of chemical reaction.

FORMULATION OF PROBLEM

Consider the flow of an electrically conducting viscous incompressible fluid through saturated porous medium bounded by two insulated vertical porous plates distance‘d’ apart in the presence of the heat source. A coordinate system is chosen with origin at the stationary plate which is subjected to a constant injection velocity 𝑉𝑉₀ .The other plate is oscillating in its own plane with a velocity 𝑈𝑈′(𝑡𝑡′) about a nonzero constant mean velocity 𝑈𝑈₀ and is subjected to same constant suction velocity 𝑉𝑉₀ . A homogeneous magnetic field of strength 𝐵𝐵₀ is applied normal to the plane of the plates. The plates of the channel are assumed infinite in extent. Hence all the physical properties of the fluid will be function of 𝑦𝑦′ and 𝑡𝑡′ except the pressure.

Under the usual Boussinesq approximations the flow is governed by the following equations

𝜕𝜕𝑢𝑢′

𝜕𝜕𝑦𝑦′ = 0 ⇒ 𝑢𝑢′= 𝑉𝑉0 (1)

𝜕𝜕𝑢𝑢′ 𝜕𝜕𝑡𝑡′ + 𝑉𝑉0𝜕𝜕𝑢𝑢

′

𝜕𝜕𝑦𝑦′ = −_𝜌𝜌1𝜕𝜕𝑝𝑝 ′

𝜕𝜕𝑥𝑥′ + 𝜈𝜈𝜕𝜕 2_𝑢𝑢′

𝜕𝜕𝑦𝑦′ 2−

𝜎𝜎𝐵𝐵02𝑢𝑢′

𝜌𝜌 −

𝜈𝜈𝑢𝑢′

𝐾𝐾′ + 𝑔𝑔𝑔𝑔(𝑇𝑇′− 𝑇𝑇𝑑𝑑) + 𝑔𝑔𝑔𝑔𝑐𝑐(𝐶𝐶′− 𝐶𝐶𝑑𝑑) (2)

𝜕𝜕𝑇𝑇′ 𝜕𝜕𝑡𝑡′ + 𝑉𝑉0

𝜕𝜕𝑇𝑇′ 𝜕𝜕𝑦𝑦′ =

𝐾𝐾 𝜌𝜌𝐶𝐶𝑝𝑝

𝜕𝜕2𝑇𝑇′ 𝜕𝜕𝑦𝑦′ 2+

𝑄𝑄′ 𝜌𝜌𝐶𝐶𝑝𝑝(𝑇𝑇

′_{− 𝑇𝑇}

𝑑𝑑) (3)

𝜕𝜕𝐶𝐶′

𝜕𝜕𝑡𝑡′ + 𝑉𝑉0

𝜕𝜕𝐶𝐶′

𝜕𝜕𝑦𝑦′ = 𝐷𝐷

𝜕𝜕2_𝑇𝑇′

𝜕𝜕𝑦𝑦′ 2+ 𝐷𝐷1

𝜕𝜕2_𝑇𝑇′

𝜕𝜕𝑦𝑦′ 2− 𝑅𝑅′(𝐶𝐶′− 𝐶𝐶𝑑𝑑) (4)

With the following boundary conditions

𝑢𝑢′_{= 0, 𝑇𝑇}′ _{= 𝑇𝑇}

0+ 𝜀𝜀(𝑇𝑇0− 𝑇𝑇𝑑𝑑)𝑐𝑐𝑐𝑐𝑐𝑐𝜔𝜔′𝑡𝑡′, 𝐶𝐶′ = 𝐶𝐶0+ 𝜀𝜀(𝐶𝐶0− 𝐶𝐶𝑑𝑑)𝑐𝑐𝑐𝑐𝑐𝑐𝜔𝜔′𝑡𝑡′ 𝑎𝑎𝑡𝑡 𝑦𝑦 = 0 𝑢𝑢′ _{= 𝑈𝑈}′_(𝑡𝑡′_{) = 𝑈𝑈}

0(1 + 𝑐𝑐𝑐𝑐𝑐𝑐𝜔𝜔′𝑡𝑡′), 𝑇𝑇′= 𝑇𝑇𝑑𝑑 , 𝐶𝐶′= 𝐶𝐶𝑑𝑑 𝑎𝑎𝑡𝑡 𝑦𝑦 = 𝑑𝑑� (5)

Eliminating the modified pressure gradient under the usual boundary layer approximation equation (2) reduces to

𝜕𝜕𝑢𝑢′

𝜕𝜕𝑡𝑡′ + 𝑉𝑉0

𝜕𝜕𝑢𝑢′

𝜕𝜕𝑦𝑦′ =

𝜕𝜕𝑈𝑈′

𝜕𝜕𝑡𝑡′ + 𝜈𝜈

𝜕𝜕2_𝑢𝑢′

𝜕𝜕𝑦𝑦′ 2−

𝜎𝜎𝐵𝐵02(𝑢𝑢′−𝑈𝑈′)

𝜌𝜌 −

𝜈𝜈(𝑢𝑢′_−𝑈𝑈′₎

𝐾𝐾′ + 𝑔𝑔𝑔𝑔(𝑇𝑇′− 𝑇𝑇𝑑𝑑) + 𝑔𝑔𝑔𝑔𝑐𝑐(𝐶𝐶′− 𝐶𝐶𝑑𝑑) (6)

On introducing the following non dimensional parameters

𝑦𝑦 =𝑦𝑦_{𝑑𝑑 , 𝑡𝑡 =}′ 𝑡𝑡′_{𝑑𝑑 , 𝑢𝑢 =}𝑉𝑉0 _𝑈𝑈𝑢𝑢′ 0, 𝜃𝜃 =

𝑇𝑇′_{− 𝑇𝑇}

𝑑𝑑 𝑇𝑇0− 𝑇𝑇𝑑𝑑, 𝐶𝐶 =

𝐶𝐶′_{− 𝐶𝐶}

𝑑𝑑 𝐶𝐶0− 𝐶𝐶𝑑𝑑 , 𝑃𝑃𝑃𝑃 =

𝜌𝜌𝐶𝐶𝑝𝑝𝑉𝑉0𝑑𝑑 𝑘𝑘

𝑆𝑆𝑐𝑐 =_{𝐷𝐷 , 𝑅𝑅 =}𝜈𝜈 𝑅𝑅_𝑉𝑉′𝑑𝑑 0 , 𝐾𝐾 =

𝐾𝐾′_𝑉𝑉

0

𝜈𝜈𝑑𝑑 , 𝑀𝑀 = 𝐵𝐵0𝑑𝑑� 𝜎𝜎 𝜇𝜇 , 𝑅𝑅𝑃𝑃 =

𝑉𝑉0𝑑𝑑 𝜈𝜈 , 𝑆𝑆 =

𝑄𝑄′_𝑑𝑑

𝜌𝜌𝐶𝐶𝑝𝑝𝑉𝑉0, 𝑈𝑈 = 𝑈𝑈′

𝜔𝜔 =𝜔𝜔_𝑉𝑉′𝑑𝑑

0 , 𝐺𝐺𝐺𝐺 =

𝜈𝜈𝑔𝑔𝑔𝑔 (𝑇𝑇0−𝑇𝑇𝑑𝑑)

𝑈𝑈0𝑉𝑉02 , 𝐺𝐺𝐺𝐺 =

𝜈𝜈𝑔𝑔𝑔𝑔 (𝐶𝐶0−𝐶𝐶𝑑𝑑)

𝑈𝑈0𝑉𝑉02 , 𝑆𝑆𝑐𝑐 =

𝐷𝐷1(𝑇𝑇0−𝑇𝑇𝑑𝑑)

𝑑𝑑𝑉𝑉0(𝐶𝐶0−𝐶𝐶𝑑𝑑) (7)

in equations (1), (3), (4) and (6), we get the following non dimensional governing equations:

𝜕𝜕𝑢𝑢 𝜕𝜕𝑡𝑡 + 𝜕𝜕𝑢𝑢 𝜕𝜕𝑦𝑦= 𝜕𝜕𝑈𝑈 𝜕𝜕𝑡𝑡 + 1 𝑅𝑅𝑃𝑃

𝜕𝜕2𝑢𝑢 𝜕𝜕𝑦𝑦2− �𝑀𝑀

2

𝑅𝑅𝑃𝑃 + 1

𝐾𝐾� (𝑢𝑢 − 𝑈𝑈) + 𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃𝜃𝜃 + 𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃𝐶𝐶 (8)

𝜕𝜕𝜃𝜃 𝜕𝜕𝑡𝑡 + 𝜕𝜕𝜃𝜃 𝜕𝜕𝑦𝑦= 1 𝑃𝑃𝑃𝑃

𝜕𝜕2𝜃𝜃

𝜕𝜕𝑦𝑦2+ 𝑆𝑆𝜃𝜃 (9)

𝜕𝜕𝐶𝐶 𝜕𝜕𝑡𝑡 + 𝜕𝜕𝐶𝐶 𝜕𝜕𝑦𝑦= 1 𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃

𝜕𝜕2𝐶𝐶 𝜕𝜕𝑦𝑦2+ 𝑆𝑆𝑐𝑐𝜕𝜕

2_𝜃𝜃

𝜕𝜕𝑦𝑦2− 𝑅𝑅𝐶𝐶 (10)

With the following boundary conditions

𝑢𝑢 = 0, 𝜃𝜃 = 1 +𝜀𝜀₂(𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 _{+ 𝑃𝑃}−𝑖𝑖𝜔𝜔𝑡𝑡_{), 𝐶𝐶 = 1 +}𝜀𝜀

2(𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 + 𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡) 𝑎𝑎𝑡𝑡 𝑦𝑦 = 0

𝑢𝑢 = 1 +𝜀𝜀₂(𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 _{+ 𝑃𝑃}−𝑖𝑖𝜔𝜔𝑡𝑡_{), 𝜃𝜃 = 0, 𝐶𝐶 = 0 𝑎𝑎𝑡𝑡 𝑦𝑦 = 1}� (11)

To solve equation (8), (9) and (10) for purely oscillatory flow, we assume the solution of the form

𝜃𝜃 = 𝜃𝜃0(𝑦𝑦) +₂𝜀𝜀𝜃𝜃1(𝑦𝑦)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 +𝜀𝜀₂𝜃𝜃2(𝑦𝑦)𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡

𝐶𝐶 = 𝐶𝐶0(𝑦𝑦) +₂𝜀𝜀𝐶𝐶1(𝑦𝑦)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 +𝜀𝜀₂𝐶𝐶2(𝑦𝑦)𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡

𝑢𝑢 = 𝑢𝑢0(𝑦𝑦) +𝜀𝜀₂𝑢𝑢1(𝑦𝑦)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 +𝜀𝜀₂𝑢𝑢2(𝑦𝑦)𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡 (12)

Using equation (12) into (8) to (10) we get the following set of equations

𝜕𝜕2𝜃𝜃0

𝜕𝜕𝑦𝑦2 − 𝑃𝑃𝑃𝑃

𝜕𝜕𝜃𝜃0

𝜕𝜕𝑦𝑦 + 𝑆𝑆𝑃𝑃𝑃𝑃𝜃𝜃0= 0 (13)

𝜕𝜕2_𝜃𝜃1

𝜕𝜕𝑦𝑦2 − 𝑃𝑃𝑃𝑃

𝜕𝜕𝜃𝜃1

𝜕𝜕𝑦𝑦 + (𝑆𝑆 − 𝑖𝑖𝜔𝜔)𝑃𝑃𝑃𝑃𝜃𝜃1= 0 (14)

𝜕𝜕2_𝜃𝜃2

𝜕𝜕𝑦𝑦2 − 𝑃𝑃𝑃𝑃

𝜕𝜕𝜃𝜃2

𝜕𝜕𝑦𝑦 + (𝑆𝑆 + 𝑖𝑖𝜔𝜔)𝑃𝑃𝑃𝑃𝜃𝜃2= 0 (15)

𝜕𝜕2_𝐶𝐶0

𝜕𝜕𝑦𝑦2 − 𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃

𝜕𝜕𝐶𝐶0

𝜕𝜕𝑦𝑦 − 𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃𝑅𝑅𝐶𝐶0= −𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝜕𝜕2_𝜃𝜃0

𝜕𝜕𝑦𝑦2 (16)

𝜕𝜕2𝐶𝐶1

𝜕𝜕𝑦𝑦2 − 𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃𝜕𝜕𝐶𝐶_{𝜕𝜕𝑦𝑦}1− 𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃(𝑅𝑅 + 𝑖𝑖𝜔𝜔)𝐶𝐶1= −𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃𝜕𝜕 2_𝜃𝜃1

𝜕𝜕𝑦𝑦2 (17)

𝜕𝜕2_𝐶𝐶2

𝜕𝜕𝑦𝑦2 − 𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃

𝜕𝜕𝐶𝐶2

𝜕𝜕𝑦𝑦 − 𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃(𝑅𝑅 − 𝑖𝑖𝜔𝜔)𝐶𝐶2= −𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝜕𝜕2_𝜃𝜃2

𝜕𝜕𝑦𝑦2 (18)

𝜕𝜕2𝑢𝑢0

𝜕𝜕𝑦𝑦2 − 𝑅𝑅𝑃𝑃𝜕𝜕𝑢𝑢_{𝜕𝜕𝑦𝑦}0− �𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾� 𝑢𝑢0= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2𝜃𝜃0− 𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2𝐶𝐶0− �𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾� 𝑈𝑈0 (19)

𝜕𝜕2𝑢𝑢1

𝜕𝜕𝑦𝑦2 − 𝑅𝑅𝑃𝑃𝜕𝜕𝑢𝑢_{𝜕𝜕𝑦𝑦}1− �𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾 + 𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃� 𝑢𝑢1= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2𝜃𝜃1− 𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2𝐶𝐶1− �𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾 + 𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃� 𝑈𝑈0 (20)

𝜕𝜕2𝑢𝑢2

𝜕𝜕𝑦𝑦2 − 𝑅𝑅𝑃𝑃𝜕𝜕𝑢𝑢_{𝜕𝜕𝑦𝑦}2− �𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾 − 𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃� 𝑢𝑢2= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2𝜃𝜃2− 𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2𝐶𝐶2− �𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾 − 𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃� 𝑈𝑈0 (21)

Solving the equations (13) to (21), we get

𝜃𝜃 = 𝐴𝐴1𝑃𝑃𝐺𝐺1𝑦𝑦+ 𝐴𝐴2𝑃𝑃𝐺𝐺2𝑦𝑦+₂𝜀𝜀(𝐴𝐴3𝑃𝑃𝐺𝐺3𝑦𝑦+ 𝐴𝐴4𝑃𝑃𝐺𝐺4𝑦𝑦)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 +₂𝜀𝜀(𝐴𝐴5𝑃𝑃𝐺𝐺5𝑦𝑦+ 𝐴𝐴6𝑃𝑃𝐺𝐺6𝑦𝑦)𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡 (22)

𝐶𝐶 = 𝐴𝐴9𝑃𝑃𝐺𝐺7𝑦𝑦+ 𝐴𝐴10𝑃𝑃𝐺𝐺8𝑦𝑦+ 𝐴𝐴7𝑃𝑃𝐺𝐺1𝑦𝑦+ 𝐴𝐴8𝑃𝑃𝐺𝐺2𝑦𝑦

+𝜀𝜀₂(𝐴𝐴13𝑃𝑃𝐺𝐺9𝑦𝑦+ 𝐴𝐴14𝑃𝑃𝐺𝐺10𝑦𝑦+𝐴𝐴11𝑃𝑃𝐺𝐺3𝑦𝑦+ 𝐴𝐴12𝑃𝑃𝐺𝐺4𝑦𝑦)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡

𝑢𝑢 = 𝐴𝐴27𝑃𝑃𝐺𝐺13𝑦𝑦+ 𝐴𝐴28𝑃𝑃𝐺𝐺14𝑦𝑦+ 𝐴𝐴19𝑃𝑃𝐺𝐺1𝑦𝑦+ 𝐴𝐴20𝑃𝑃𝐺𝐺2𝑦𝑦+ 𝐴𝐴21𝑃𝑃𝐺𝐺7𝑦𝑦+ 𝐴𝐴22𝑃𝑃𝐺𝐺8𝑦𝑦+ 𝐴𝐴23𝑃𝑃𝐺𝐺1𝑦𝑦+ 𝐴𝐴24𝑃𝑃𝐺𝐺2𝑦𝑦+ 𝑈𝑈0+ 𝜀𝜀

2_𝜀𝜀(𝐴𝐴37𝑃𝑃𝐺𝐺15𝑦𝑦+ 𝐴𝐴38𝑃𝑃𝐺𝐺16𝑦𝑦+ 𝐴𝐴29𝑃𝑃𝐺𝐺3𝑦𝑦+ 𝐴𝐴30𝑃𝑃𝐺𝐺4𝑦𝑦+ 𝐴𝐴31𝑃𝑃𝐺𝐺9𝑦𝑦+ 𝐴𝐴32𝑃𝑃𝐺𝐺10𝑦𝑦+ 𝐴𝐴33𝑃𝑃𝐺𝐺3𝑦𝑦+ 𝐴𝐴34𝑃𝑃𝐺𝐺4𝑦𝑦+ 𝑈𝑈0)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 + 2(𝐴𝐴47𝑃𝑃𝐺𝐺17𝑦𝑦+ 𝐴𝐴48𝑃𝑃𝐺𝐺18𝑦𝑦+ 𝐴𝐴39𝑃𝑃𝐺𝐺5𝑦𝑦+ 𝐴𝐴40𝑃𝑃𝐺𝐺6𝑦𝑦+ 𝐴𝐴41𝑃𝑃𝐺𝐺11𝑦𝑦+ 𝐴𝐴42𝑃𝑃𝐺𝐺12𝑦𝑦+ 𝐴𝐴43𝑃𝑃𝐺𝐺5𝑦𝑦+ 𝐴𝐴44𝑃𝑃𝐺𝐺6𝑦𝑦+ 𝑈𝑈0)𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡

(24)

The non dimensional skin friction at the moving plate of the channel is given by

𝜏𝜏 = −𝜇𝜇 �𝜕𝜕𝑢𝑢_{𝜕𝜕𝑦𝑦}� 𝑦𝑦=1

= −(𝐴𝐴₂₇𝐺𝐺₁₃𝑃𝑃𝐺𝐺13_{+ 𝐴𝐴28𝐺𝐺14𝑃𝑃}𝐺𝐺14_{+ 𝐴𝐴19𝐺𝐺1𝑃𝑃}𝐺𝐺1_{+ 𝐴𝐴20𝐺𝐺2𝑃𝑃}𝐺𝐺2_{+ 𝐴𝐴21𝐺𝐺7𝑃𝑃}𝐺𝐺7_{+ 𝐴𝐴22𝐺𝐺8𝑃𝑃}𝐺𝐺8_{+ 𝐴𝐴23𝐺𝐺1𝑃𝑃}𝐺𝐺1₊

𝐴𝐴24𝐺𝐺2𝑃𝑃𝐺𝐺2) −𝜀𝜀₂�𝐴𝐴37𝐺𝐺15𝑃𝑃

𝐺𝐺15_{+ 𝐴𝐴38𝐺𝐺16𝑃𝑃}𝐺𝐺16 _{+ 𝐴𝐴29𝐺𝐺3𝑃𝑃}𝐺𝐺3_{+ 𝐴𝐴30𝐺𝐺4𝑃𝑃}𝐺𝐺4_𝑧𝑧

+𝐴𝐴31𝐺𝐺9𝑃𝑃𝐺𝐺9+ 𝐴𝐴32𝐺𝐺10𝑃𝑃𝐺𝐺10+ 𝐴𝐴33𝐺𝐺3𝑃𝑃𝐺𝐺3+ 𝐴𝐴34𝐺𝐺4𝑃𝑃𝐺𝐺4 � 𝑃𝑃 𝑖𝑖𝜔𝜔𝑡𝑡_’

− 𝜀𝜀

2� 𝐴𝐴47

𝐺𝐺17𝑃𝑃𝐺𝐺17+ 𝐴𝐴48𝐺𝐺18𝑃𝑃𝐺𝐺18 + 𝐴𝐴39𝐺𝐺5𝑃𝑃𝐺𝐺5+ 𝐴𝐴40𝐺𝐺6𝑃𝑃𝐺𝐺6 +𝐴𝐴41𝐺𝐺11𝑃𝑃𝐺𝐺11+ 𝐴𝐴42𝐺𝐺12𝑃𝑃𝐺𝐺12+ 𝐴𝐴43𝐺𝐺5𝑃𝑃𝐺𝐺5+ 𝐴𝐴44𝐺𝐺6𝑃𝑃𝐺𝐺6 � 𝑃𝑃

−𝑖𝑖𝜔𝜔𝑡𝑡_{’ (25)}

The rate of heat transfer at the moving plate of the channel in terms of non dimensional Nusselt number is given by

𝑁𝑁𝑢𝑢 = − �𝜕𝜕𝜃𝜃_{𝜕𝜕𝑦𝑦}�

𝑦𝑦=1= −(𝐴𝐴1𝐺𝐺1𝑃𝑃

𝐺𝐺1_{+ 𝐴𝐴2𝐺𝐺2𝑃𝑃}𝐺𝐺2_{) −}𝜀𝜀

2(𝐴𝐴3𝐺𝐺3𝑃𝑃𝐺𝐺3+ 𝐴𝐴4𝐺𝐺4𝑃𝑃𝐺𝐺4)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡 − 𝜀𝜀

2(𝐴𝐴5𝐺𝐺5𝑃𝑃𝐺𝐺5+ 𝐴𝐴6𝐺𝐺6𝑃𝑃𝐺𝐺6)𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡

(26)

The rate of mass transfer coefficient at the moving plate of the channel in terms of non dimensional Sherwood number is given by

𝑆𝑆ℎ = − �𝜕𝜕𝐶𝐶_{𝜕𝜕𝑦𝑦}�

𝑦𝑦=1 = −(𝐴𝐴9𝐺𝐺7𝑃𝑃 𝐺𝐺7_{+ 𝐴𝐴}

10𝐺𝐺8𝑃𝑃𝐺𝐺8+ 𝐴𝐴7𝐺𝐺1𝑃𝑃𝐺𝐺1+ 𝐴𝐴8𝐺𝐺2𝑃𝑃𝐺𝐺2) −₂𝜀𝜀(𝐴𝐴13𝐺𝐺9𝑃𝑃𝐺𝐺9+ 𝐴𝐴14𝐺𝐺10𝑃𝑃𝐺𝐺10+𝐴𝐴11𝐺𝐺3𝑃𝑃𝐺𝐺3+ 𝐴𝐴12𝐺𝐺4𝑃𝑃𝐺𝐺4)𝑃𝑃𝑖𝑖𝜔𝜔𝑡𝑡

−𝜀𝜀₂(𝐴𝐴17𝐺𝐺11𝑃𝑃𝐺𝐺11+ 𝐴𝐴18𝐺𝐺12𝑃𝑃𝐺𝐺12+𝐴𝐴15𝐺𝐺5𝑃𝑃𝐺𝐺5+ 𝐴𝐴16𝐺𝐺6𝑃𝑃𝐺𝐺6)𝑃𝑃−𝑖𝑖𝜔𝜔𝑡𝑡

Here 𝐺𝐺₁=𝑃𝑃𝑃𝑃+�𝑃𝑃𝑃𝑃2−4𝑆𝑆𝑃𝑃𝑃𝑃

2 , 𝐺𝐺2=

𝑃𝑃𝑃𝑃−�𝑃𝑃𝑃𝑃2_{−4𝑆𝑆𝑃𝑃𝑃𝑃}

2 , 𝐺𝐺3=

𝑃𝑃𝑃𝑃+�𝑃𝑃𝑃𝑃2_{−4(𝑆𝑆−𝑖𝑖𝜔𝜔 )𝑃𝑃𝑃𝑃}

2 , 𝐺𝐺4=

𝑃𝑃𝑃𝑃−�𝑃𝑃𝑃𝑃2_{−4(𝑆𝑆−𝑖𝑖𝜔𝜔 )𝑃𝑃𝑃𝑃}

2

𝐺𝐺5=𝑃𝑃𝑃𝑃+�𝑃𝑃𝑃𝑃

2_{−4(𝑆𝑆+𝑖𝑖𝜔𝜔 )𝑃𝑃𝑃𝑃}

2 , 𝐺𝐺6=

𝑃𝑃𝑃𝑃−�𝑃𝑃𝑃𝑃2_{−4(𝑆𝑆−𝑖𝑖𝜔𝜔 )𝑃𝑃𝑃𝑃}

2 , 𝐺𝐺7=

𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 +�𝑆𝑆𝑐𝑐2_{𝑅𝑅𝑃𝑃}2_{+4𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃𝑅𝑅}

2 , 𝐺𝐺8=

𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 −�𝑆𝑆𝑐𝑐2_{𝑅𝑅𝑃𝑃}2_{+4𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃𝑅𝑅}

2 ,

𝐺𝐺9=𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 +�𝑆𝑆𝑐𝑐

2_{𝑅𝑅𝑃𝑃}2_{+4𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅+𝑖𝑖𝜔𝜔 )}

2 , 𝐺𝐺10 =

𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 −�𝑆𝑆𝑐𝑐2_{𝑅𝑅𝑃𝑃}2_{+4𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅+𝑖𝑖𝜔𝜔 )}

2 ,

𝐺𝐺11=𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 +�𝑆𝑆𝑐𝑐

2_{𝑅𝑅𝑃𝑃}2_{+4𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅−𝑖𝑖𝜔𝜔 )}

2 , 𝐺𝐺12=

𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 −�𝑆𝑆𝑐𝑐2_{𝑅𝑅𝑃𝑃}2_{+4𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅−𝑖𝑖𝜔𝜔 )}

2 , 𝐺𝐺13=

𝑅𝑅𝑃𝑃+�𝑅𝑅𝑃𝑃2_+4(𝑀𝑀2₊𝑅𝑅𝑃𝑃 𝐾𝐾)

2 ,

𝐺𝐺14=

𝑅𝑅𝑃𝑃−�𝑅𝑅𝑃𝑃2_{+4�𝑀𝑀}2₊𝑅𝑅𝑃𝑃 𝐾𝐾�

2 , 𝐺𝐺15=

𝑅𝑅𝑃𝑃+�𝑅𝑅𝑃𝑃2_{+4�𝑀𝑀}2₊𝑅𝑅𝑃𝑃 𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �

2 , 𝐺𝐺16 =

𝑅𝑅𝑃𝑃−�𝑅𝑅𝑃𝑃2_{+4�𝑀𝑀}2₊𝑅𝑅𝑃𝑃 𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �

2 ,

𝐺𝐺17=

𝑅𝑅𝑃𝑃+�𝑅𝑅𝑃𝑃2_+4(𝑀𝑀2₊𝑅𝑅𝑃𝑃 𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 )

2 , 𝐺𝐺18=

𝑅𝑅𝑃𝑃−�𝑅𝑅𝑃𝑃2_+4(𝑀𝑀2₊𝑅𝑅𝑃𝑃 𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 )

2

𝐴𝐴1= −𝑃𝑃

𝐺𝐺 2

𝑃𝑃𝐺𝐺 1−𝑃𝑃𝐺𝐺 2, 𝐴𝐴2= 𝑃𝑃

𝐺𝐺 1

𝑃𝑃𝐺𝐺 1−𝑃𝑃𝐺𝐺 2, 𝐴𝐴3= −𝑃𝑃

𝐺𝐺 4

𝑃𝑃𝐺𝐺 3−𝑃𝑃𝐺𝐺 4, 𝐴𝐴4= 𝑃𝑃

𝐺𝐺 3

𝑃𝑃𝐺𝐺 3−𝑃𝑃𝐺𝐺 4, 𝐴𝐴5= −𝑃𝑃

𝐺𝐺 6

𝑃𝑃𝐺𝐺 5−𝑃𝑃𝐺𝐺 6, 𝐴𝐴6= 𝑃𝑃

𝐺𝐺 5

𝑃𝑃𝐺𝐺 3−𝑃𝑃𝐺𝐺 4

𝐴𝐴7= −𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐴𝐴1𝐺𝐺1

2

𝐺𝐺12−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐺𝐺1−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃𝑅𝑅, 𝐴𝐴8=

−𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐴𝐴2𝐺𝐺22

𝐺𝐺22−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐺𝐺2−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃𝑅𝑅, 𝐴𝐴9=

−𝑃𝑃𝐺𝐺 8+𝐴𝐴7(𝑃𝑃𝐺𝐺 8−𝑃𝑃𝐺𝐺 1)+𝐴𝐴8(𝑃𝑃𝐺𝐺 8−𝑃𝑃𝐺𝐺 2)

𝑃𝑃𝐺𝐺 7−𝑃𝑃𝐺𝐺 8

𝐴𝐴10=𝑃𝑃𝐺𝐺 7−𝐴𝐴7(𝑃𝑃𝐺𝐺 7_𝑃𝑃−𝑃𝑃_{𝐺𝐺 7}𝐺𝐺 1_−𝑃𝑃)−𝐴𝐴_{𝐺𝐺 8}8(𝑃𝑃𝐺𝐺 7−𝑃𝑃𝐺𝐺 2), 𝐴𝐴11= −𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐴𝐴3𝐺𝐺3

2

𝐺𝐺32−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐺𝐺3−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅+𝑖𝑖𝜔𝜔 ), 𝐴𝐴12=

−𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐴𝐴4𝐺𝐺42

𝐺𝐺42−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐺𝐺4−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅+𝑖𝑖𝜔𝜔 )

𝐴𝐴13=−𝑃𝑃𝐺𝐺 10+𝐴𝐴11(𝑃𝑃𝐺𝐺 10_{𝑃𝑃𝐺𝐺 9}−𝑃𝑃_−𝑃𝑃𝐺𝐺 3_{𝐺𝐺 10})+𝐴𝐴12(𝑃𝑃𝐺𝐺 10−𝑃𝑃𝐺𝐺 4), 𝐴𝐴14=𝑃𝑃𝐺𝐺 10−𝐴𝐴11(𝑃𝑃𝐺𝐺 9_{𝑃𝑃𝐺𝐺 9}−𝑃𝑃_−𝑃𝑃𝐺𝐺 3_{𝐺𝐺 10})−𝐴𝐴12(𝑃𝑃𝐺𝐺 9−𝑃𝑃𝐺𝐺 4)

𝐴𝐴15= −𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐴𝐴5𝐺𝐺5

2

𝐺𝐺52−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐺𝐺5−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅−𝑖𝑖𝜔𝜔 ), 𝐴𝐴16=

−𝑆𝑆𝑐𝑐𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐴𝐴6𝐺𝐺62

𝐺𝐺62−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 𝐺𝐺6−𝑆𝑆𝑐𝑐𝑅𝑅𝑃𝑃 (𝑅𝑅−𝑖𝑖𝜔𝜔 ),𝐴𝐴17=

−𝑃𝑃𝐺𝐺 12+𝐴𝐴14(𝑃𝑃𝐺𝐺 12−𝑃𝑃𝐺𝐺 5)+𝐴𝐴15(𝑃𝑃𝐺𝐺 12−𝑃𝑃𝐺𝐺 6)

𝐴𝐴18=𝑃𝑃

𝐺𝐺 11_{−𝐴𝐴14(𝑃𝑃}𝐺𝐺 11_−𝑃𝑃𝐺𝐺 5_{)−𝐴𝐴15(𝑃𝑃}𝐺𝐺 11_−𝑃𝑃𝐺𝐺 6) 𝑃𝑃𝐺𝐺 11−𝑃𝑃𝐺𝐺 12

𝐴𝐴19= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃

2_𝐴𝐴1

𝐺𝐺12−𝑅𝑅𝑃𝑃𝐺𝐺1−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾�, 𝐴𝐴20=

−𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2𝐴𝐴2

𝐺𝐺22−𝑅𝑅𝑃𝑃𝐺𝐺2−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾� , 𝐴𝐴21=

−𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃2𝐴𝐴9

𝐺𝐺72−𝑅𝑅𝑃𝑃𝐺𝐺7−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾�,

𝐴𝐴22= −𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃

2_𝐴𝐴10

𝐺𝐺82−𝑅𝑅𝑃𝑃𝐺𝐺8−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾�, 𝐴𝐴23=

−𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃2_𝐴𝐴7

𝐺𝐺12−𝑅𝑅𝑃𝑃𝐺𝐺1−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾�, 𝐴𝐴24=

−𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2_𝐴𝐴8

𝐺𝐺22−𝑅𝑅𝑃𝑃𝐺𝐺2−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾� ,

𝐴𝐴25= 𝐴𝐴19+ 𝐴𝐴20+ 𝐴𝐴21+ 𝐴𝐴22+ 𝐴𝐴23+ 𝐴𝐴24+ 𝑈𝑈0 ,

𝐴𝐴26= 𝐴𝐴19𝑃𝑃𝐺𝐺1+ 𝐴𝐴20𝑃𝑃𝐺𝐺2+ 𝐴𝐴21𝑃𝑃𝐺𝐺7+ 𝐴𝐴22𝑃𝑃𝐺𝐺8+ 𝐴𝐴23𝑃𝑃𝐺𝐺1+ 𝐴𝐴24𝑃𝑃𝐺𝐺2+ 𝑈𝑈0 ,

𝐴𝐴27=−1+𝐴𝐴_{𝑃𝑃𝐺𝐺 14}26−𝐴𝐴_−𝑃𝑃25_{𝐺𝐺 13}𝑃𝑃𝐺𝐺 14, 𝐴𝐴28=1−𝐴𝐴_{𝑃𝑃𝐺𝐺 14}26+𝐴𝐴_−𝑃𝑃25_{𝐺𝐺 13}𝑃𝑃𝐺𝐺 13 , 𝐴𝐴29= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃

2_𝐴𝐴3

𝐺𝐺32−𝑅𝑅𝑃𝑃𝐺𝐺3−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �,

𝐴𝐴30= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃

2_𝐴𝐴 4

𝐺𝐺42−𝑅𝑅𝑃𝑃𝐺𝐺4−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �, 𝐴𝐴31=

−𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃2𝐴𝐴13

𝐺𝐺92−𝑅𝑅𝑃𝑃𝐺𝐺9−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �, 𝐴𝐴32=

−𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃2𝐴𝐴14

𝐺𝐺10 2−𝑅𝑅𝑃𝑃𝐺𝐺10−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �

𝐴𝐴33= −𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃

2_𝐴𝐴11

𝐺𝐺32−𝑅𝑅𝑃𝑃𝐺𝐺3−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃�, 𝐴𝐴34=

−𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃2_𝐴𝐴12

𝐺𝐺42−𝑅𝑅𝑃𝑃𝐺𝐺4−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾+𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �,

𝐴𝐴35= 𝐴𝐴29+ 𝐴𝐴30+ 𝐴𝐴31+ 𝐴𝐴32+ 𝐴𝐴33+ 𝐴𝐴34+ 𝑈𝑈0 ,

𝐴𝐴36= 𝐴𝐴29𝑃𝑃𝐺𝐺3+ 𝐴𝐴30𝑃𝑃𝐺𝐺4+ 𝐴𝐴31𝑃𝑃𝐺𝐺9+ 𝐴𝐴32𝑃𝑃𝐺𝐺10+ 𝐴𝐴33𝑃𝑃𝐺𝐺3+ 𝐴𝐴34𝑃𝑃𝐺𝐺4+ 𝑈𝑈0

𝐴𝐴37=−1+𝐴𝐴_{𝑃𝑃𝐺𝐺 16}36−𝐴𝐴_−𝑃𝑃35_{𝐺𝐺 15}𝑃𝑃𝐺𝐺 16, 𝐴𝐴38=1−𝐴𝐴_{𝑃𝑃𝐺𝐺 16}36+𝐴𝐴_−𝑃𝑃35_{𝐺𝐺 15}𝑃𝑃𝐺𝐺 15, 𝐴𝐴39= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃

2_𝐴𝐴5

𝐺𝐺52−𝑅𝑅𝑃𝑃𝐺𝐺5−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 � ,

𝐴𝐴40= −𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃

2_𝐴𝐴6

𝐺𝐺62−𝑅𝑅𝑃𝑃𝐺𝐺6−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �, 𝐴𝐴41=

−𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃2_𝐴𝐴17

𝐺𝐺11 2−𝑅𝑅𝑃𝑃𝐺𝐺11−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �, 𝐴𝐴42=

−𝐺𝐺𝐺𝐺𝑅𝑅𝑃𝑃2_𝐴𝐴18

𝐺𝐺122−𝑅𝑅𝑃𝑃𝐺𝐺12−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �,

𝐴𝐴43= −𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃

2_𝐴𝐴15

𝐺𝐺52−𝑅𝑅𝑃𝑃𝐺𝐺5−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �, 𝐴𝐴44=

−𝐺𝐺𝐺𝐺 𝑅𝑅𝑃𝑃2𝐴𝐴16

𝐺𝐺62−𝑅𝑅𝑃𝑃𝐺𝐺6−�𝑀𝑀2+𝑅𝑅𝑃𝑃_𝐾𝐾−𝑖𝑖𝜔𝜔𝑅𝑅𝑃𝑃 �,

𝐴𝐴45= 𝐴𝐴39+ 𝐴𝐴40+ 𝐴𝐴41+ 𝐴𝐴42+ 𝐴𝐴43+ 𝐴𝐴44+ 𝑈𝑈0 ,

𝐴𝐴46= 𝐴𝐴39𝑃𝑃𝐺𝐺5+ 𝐴𝐴40𝑃𝑃𝐺𝐺6+ 𝐴𝐴41𝑃𝑃𝐺𝐺11+ 𝐴𝐴42𝑃𝑃𝐺𝐺12 + 𝐴𝐴43𝑃𝑃𝐺𝐺5+ 𝐴𝐴44𝑃𝑃𝐺𝐺6+ 𝑈𝑈0 ,

𝐴𝐴47=−1+𝐴𝐴46−𝐴𝐴45𝑃𝑃

𝐺𝐺 18

𝑃𝑃𝐺𝐺 18−𝑃𝑃𝐺𝐺 17 , 𝐴𝐴48=1−𝐴𝐴46+𝐴𝐴45𝑃𝑃

𝐺𝐺 17

𝑃𝑃𝐺𝐺 18−𝑃𝑃𝐺𝐺 17

RESULTS AND DISCUSSION

Physical significances of various parameters have been shown through graphs.

Figure.1: It is observed that velocity increases with increase in Reynolds number Re, Lorentz force parameter i.e. Hartmann number M, heat source parameter S, Grashoff number Gr and Modified Grashoff number Gm. It is also clear from this figure that for behaviour species i.e. with increase in Schmidt number, the velocity decreases.

Figure.2: It is evident that the velocity increases with increase in frequency of oscillation 𝜔𝜔, parameter Uo and peclet number Pe. Moreover it is observed that velocity shows opposite effects that of the parameters permeability K, soret muber So and Chemical reaction R.

Figure.3: Here it is noticed that temperature rises for the increase of the parameters S and Pe.

Figure.4: This shows the variations in concentration profiles with y for various parameters involved. It is noticed that increase in frequency of oscillation 𝜔𝜔 , concentration increases. On the other hand Re, Pe, S, Sc, R and So adversely affect the concentration profiles.

Figure.6: This figure also shows the periodic variation of skin friction profiles. Moreover it is observed that with the increase in Uo, R, So and K, skin friction increases.

Figure.7: This figure shows the variation of Sherwood number profiles with various parameters. These profiles are periodic in nature. With decrease in R and So Sherwood number increases. Further increase in the value of Re, Pe, Sc and S lead to decreased value in Sherwood number.

Figure.8: Variations of Nusselt number Nu are illustrated in this figure. It is noticed that increase in Pe, S and 𝜔𝜔 result an increase in the Nusselt number.

CONCLUSION

Results are presented graphically to illustrate the variation of velocity, temperature, concentration, Shearing stress and Nusselt number with parameters.In this study, the following conclusions are set out

• Velocity increases for the increase of Re, M, S, Gr, Gm, Uo, Pe and decreases when R, K, So, Sc increases. • Temperature rises when S and Pe increase.

• The parameters R, Re, S, Sc, Pe and So adversely affect the concentration profiles.

• Skin friction increases when Sc, Re, Uo, R, So and K increase and decreases for the increase of Gr, Gm, M, S. • Sherwood number increases with decrease in R, Re, Pe, So, Sc and S.

• Increase in Pe and S result an increase in the Nusselt number.

REFERENCES

[1] Ram G, Mishra R S, Indian J. Pure Appl. Math., 8, 637-647, 1977.

[2] V.M. Soundalgekar, ‘Free convection effects on the flow past a vertical oscillating plate’, Astrophysics and Space Science, 64, 165-172, 1979.

[3] S.T. Revankar, ‘Natural convection effects on MHD flow past an impulsively started permeable plate’ Indian J.Pure Appl. Math., 14, 530-539, 1983.

[4] Moreau R, Magneto hydrodynamics, Kluwer, Academic Publisher, Dordrecht, 1990.

[5] P.Mohapatra & N.Senapati, ‘Magneto hydrodynamic free convection flow with mass transfer past a vertical plate’ AMSE, 37(1), 1-23, 1990.

[6] W.L.Cooper, V.W.Nee and K.T.Yang, ‘Fluid mechanics of oscillatory and modulated flows and associated applications in heat and mass transfer’, Journal of Energy, Heat & Mass transfer, Vol.15, pp.119, 1993.

[7] N. Ahmed, D. Sarma, ‘Three dimensional free convective flow and heat transfer through a porous medium’, Indian J. Pure Appl. Math’s. vol.28, No.10, pp.1345-1353, 1997

[8] M.Anghel, H.S.Takhar, ‘Dufour effect and Soret effect on free convection boundary layer over a vertical surface embedded in porous medium’, Studiab Universities Babes-Boyai, Mathematica, 60(5), 2000, 11- 21.

[9] C. Geindreau, J. L. Auriault, ‘Effect of magnetic field in flow through porous medium’, J. Fluid Mech, 466, 343-363, 2002.

[10] R. Muthucumaraswamy, ‘Effect of heat and mass transfer on flow past an oscillatory vertical plate with variable temperature’, Int. J. Of Appl. Math and Mech, 4(1); 59-65, 2008.

[11] Muthucumaraswamy, R. and Janakiraman, ‘Mass transfer effect on isothermal vertical oscillating plate in presence of chemical reaction’, B. Int. J. Of Appl. Math and Mech, 4(1); 66-74, 2008.

[12] N. Ahmed, ‘MHD convection with soret and Dufour effect in a three dimensional flow past an infinite vertical porous plate’ Journal of Energy, Heat and Mass Transfer, 32, 55-70, 2010.

[13] N. Senapati & R.K Dhal, ‘Magnetic effect on mass and heat transfer of hydrodynamic flow past a vertical oscillating plate in presence of chemical reaction’, AMSE, B-2, 79(2); 60-66, 2011.