Chemical Reaction Impact on MHD free convection Flow of an

Chemical Reaction Impact on MHD free

convection Flow of an Incompressible

Viscous Fluid Past a Vertical Plate under

Oscillatory Suction Velocity

M.Anil kumar

Department of Mathematics

Anurag Group of Institutions, Venkatapur(V), Ghtkesar (M), Hyderabad, Telangana State, India

Y.Dharmendar Reddy



V.Srinivasa Rao



Abstract - The aim of this research paper is to study the effects of unsteady hydromagnetic heat and mass transfer MHD

flow of an electrically conductive incompressible viscous fluid, past an infinite vertical porous plate together with a porous

medium of time dependent permeability under a normal oscillatory plate suction velocity. The influence of the uniform

magnetic field is believed to act normally on the flow and the permeability of the porous medium fluctuates over time.

The problem is solved, numerically with the finite element method for velocity, temperature, concentration fields and also

the expressions for skin friction, Nusselt number and Sherwood number are shown in tabular form.

Keywords – Heat and Mass transfer, MHD flow, Vertical plate, Suction Velocity, Viscous fluid, Chemical

Reaction.

I. INTRODUCTION

The investigation of heat and mass exchange has been widely done by numerous scientists during most

recent couple of decades because of utilization in science and innovation. Such wonders are seen in numerous

physical conditions like lightness actuated movements in the environment, semi strong bodies like earth and some

more.

Examination of hydromagnetic characteristic convection stream with heat and mass exchange in permeable

and non-permeable media has drawn extensive considerations of a few specialists attributable to its applications in

geophysics, astrophysics, aeronautics, meteorology, electronics, chemical, and metallurgy and oil ventures.

Magnetohydrodynamic (MHD) natural convection flow of an electrically conducting fluid with porous medium has

also been successfully exploited in crystal formation. In addition to it, the thermal physics of hydromagnetic

problems with mass transfer is of much significance in MHD flow-meters, MHD energy generators, MHD pumps,

controlled thermo-nuclear reactors, MHD accelerators, etc. keeping in view the importance of heat and mass

transfer, Several researchers have analyzed the free convective and mass transfer flow of a viscous fluid through

porous medium. The permeability of the porous medium is assumed to be constant while the porosity of the medium

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may not be necessarily being constant. Kim [1] studied the unsteady MHD convective heat past a semi-infinite

vertical porous moving plate with variable suction. The problem of three-dimensional free convective flow and heat

transfer through porous medium with periodic permeability has been discussed by Singh and Sharma [2]. Singh et

al. [3] have analyzed the heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under

oscillatory suction velocity. Hossain and Mandal [4] investigated the mass transfer effects on unsteady

hydromagnetic free convection flow past an accelerated vertical porous plate. Jha [5] studied the hydromagnetic free

convection and mass transfer flow past a uniformly accelerated vertical plate through a porous medium when

magnetic field is fixed with the moving plate. Elbashbeshy [6] discussed heat and mass transfer along a vertical plate

in the presence of magnetic field.

Chemical reactions in a combined heat and mass transfer problems has great importance in many chemical

engineering processes and consequently drawn a lot of attention in the past few decades. Formation and dispersion

of fog, damage of crops due to freezing, food processing, cooling towers, distribution of temperature and moisture

over agriculture fields and groves of fruits trees are few examples of such types of processes. 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. In processes such as drying, evaporation at the surface of a water

body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur

simultaneously. Possible applications of this type of flow can be found in many industries. For example, in the

power industry, among the methods of generating electric power is one in which electrical energy is extracted

directly from a moving conducting fluid. Chambre and Young [7] have presented a first order chemical reaction in

the neighbourhood of a horizontal plate. Kesavaiah et al. [8] investigated the effect of the first order homogeneous

chemical reaction on the process of an unsteady flow past a vertical plate with a constant heat and mass transfer.

Muthucumaraswamy and Meenakshi Sundaram [9] investigated theoretical study of chemical reaction effects on

vertical oscillating plate with variable temperature and mass diffusion. Kandasamy et al. [10] studied the impact of

nonlinear MHD flow with heat and mass transfer characteristics of an incompressible, viscous, electrically

conducting and Boussinesq’s fluid on a vertical stretching surface with chemical reaction and thermal stratification

effects. Dulal pal et al. [11] made an explanatory examination for the problem of unsteady mixed convection with

thermal radiation and first-order chemical reaction on magnetohydrodynamics boundary layer flow of viscous,

electrically conducting fluid past a vertical permeable plate has been presented.

Unsteady oscillatory free convective flow plays a crucial role in the field of chemical engineering,

aerospace technology etc. such type of flow arises due to unsteady motion of a boundary or boundary temperature.

S.Das et.al[12] studied the effect of mass transfer on free convective flow and heat transfer of a viscous

incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time

dependant permeability and oscillatory suction in presence of a transverse magnetic field and heat source. S.R

Mishra et. al [13] analysed the effect of heat and mass transfer on free convective flow of a visco-elastic

incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time

dependent oscillatory permeability and suction in presence of a uniform transverse magnetic field and heat source.

Ambethkar [14] examined the impact of heat and mass transfer in an unsteady MHD free convective flow past an



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infinite vertical plate with constant suction. Modather et al. [15] invistigated the problem of heat and mass transfer

of an oscillatory 2-dimensional viscous, electrically conducting micropolar fluid over an infinite moving permeable

plate in a saturated porous medium in the presence of a transverse magnetic field. A.Bakr [16] studied the impact of

steady and unsteady MHD micropolar flow and mass transfers flow with constant heat source in a rotating frame of

reference in the presence chemical reaction of the first-order, taking an oscillatory plate velocity and a constant

suction velocity at the plate.

The current study is mainly focused on the effects of unsteady hydromagnetic heat and mass transfer MHD flow of an electrically conductive incompressible viscous fluid, past an infinite vertical porous plate together with a porous medium of time dependent permeability under a normal oscillatory plate suction velocity.

II. MATHEMATICAL FORMULATION

An unsteady hydromagnetic flow of viscous, incompressible, electrically conducting fluid past an infinite

vertical porous plate in a porous medium of time dependent permeability and suction velocity is considered as

shown in Fig. a.

Figure-a. Physical sketch and geometry of the problem

Figure (a): Physical Interpretation of the problem

In Cartesian co-ordinate system x - axis is assumed to be along the plate in the direction of the flow and

y -axis normal to it. A uniform magnetic field is introduced normal to the direction of the flow. In the analysis it is

assumed that the magnetic Reynolds number is much less than unity so that the induced magnetic field is neglected

in comparison to the applied magnetic field. Further all the fluid properties are assumed to be constant except that of

the influence of density variation with temperature. Therefore, the basic flow in the medium is entirely due to

buoyancy force caused by temperature difference between the wall and the medium. Initially at t ≤0, the plate as

well as fluid is assumed to be at the same temperature and the concentration of species is very low so that the Soret

and Dufour effects are neglected.

When t >0 the temperature of the plate is instantaneously raised (or lowered) to wT * and the concentration of

species is raised (or lowered) to wC *.under the started assumptions and taking the usual Boussinesq’s

approximation into account, the governing equations for momentum, energy and concentration are



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Continuity Equation:

0v

t

(1)

Momentum Equation:

22 2* 0

2

0

( ) ( )B uu u u vu

v g T T g C C vt y y K

(2)

Energy Equation

2

2

p

T T k Tv

t y C y

(3)

Concentration Equation

2

2( )

C C Cv D Kr C C

t y y

(4)

Where u and v are the velocity components along the x -axis and y . g is the acceleration due to gravity, β and

* -the thermal and concentration expansion coefficient respectively. is the kinematic viscosity, ρ is the fluid

density, B o is magnetic induction, oK is the permeability of porous medium, σ is the electrical conductivity of the

fluid, T is the fluid temperature ,T is temperature of the fluid at infinity, C is the species concentration, C

is

the species concentration at infinity, k is the thermal conductivity of the fluid , pC is the specific heat at constant

pressure, rK is the chemical reaction parameter, t is the time, D is the chemical molecular diffusivity.

The corresponding Boundary conditions are:

,0: 0,t u T T C C for all y

,

0, ( ) ,

0 : ( ) 0

0,

in t

w w

in t

w w

u T T T T e

t C C C C e at y

u T T C C as y

(5)

From the continuity equation, it can be seen that v is either a constant or a function of time. So assuming suction

velocity to be oscillatory about a non-zero constant mean, one can write (1 )in t

ov v e where ov is the mean

suction velocity, n is frequency of oscillation and 0, 1ov is a positive constant. The negative sign indicates



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that the suction velocity is directed towards the plate. The permeability of the porous medium is considered to be

( ) (1 )in t

o oK t K e .

The non-dimensional quantities introduced in these equations are defined as:

u=

o

u

v

, t=

2

4

ot v

, y=

0y v

, T=

w

T T

T T

, C=

w

C C

C C

, Gm=

*

3

0

( )wg C C

v

,

Gr =3

0

( )wg T T

v

, K 0 =

2

0 0

2

v K

, Pr =

pc

k

, M =

0

oB

v

, Sc =

D

Kr=2

0

rK

v

, n=

2

0

4vn

(6)

Where Gr is the Grashof Number, Gm is the Modified Grashof number, Sc is the Schmidt Number, Pr is the Prandtl

number, M is the Magnetic Parameter, n the frequency of oscillation

The Governing Equations for momentum energy and concentration in dimensionless form are

uMeK

u

y

uCGTG

y

ue

t

umr

2

int

0

2

2int

)1()1(

4

1

(7)

2

2int 1

)1(4

1

y

T

Py

Te

t

T

r

(8)

CKy

C

Sy

Ce

t

Cr

c

2

2int 1

)1(4

1 (9)

The relevant boundary conditions in dimensionless form are:

u=0, intint 1,1 eCeT at y=0

u 0, T ,0 C 0 as y (10)

III. METHOD OF SOLUTION BY FINITE ELEMENT METHOD

Finite element Method (FEM): The (FEM) is a numerical and computerized technique to solve a diversity of

practical engineering problems. The primary component of FEM is its competence to explain the geometry or the

support of the problem to be studied with great flexibility. This is due to the discretization of the area of the problem

that is carried out using highly flexible uniform or non-uniform elements that can easily illustrate complex shapes.

The method consists mainly of assuming the continuous function by parts of the solution and procuring the

parameters of the functions to diminish the error in the result. The steps given in the FEM are detailed below.

i: Discretization: The ultimate thought of (FEM) is to split the area of the problem into subdomains named finite

elements. The assembly of elements is called finite element network. These finite elements are connected non-

superimposed to completely cover the entire problem space.



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ii: Formulation of the Element Equations:

a) An illustrative element is isolated from the network and the variational construction of the given problem is made

on the typical element.

b) On an element, an assessed result of the variational problem is created and, when replacing it in the system, the

equations of the element are generated.

c) The element matrix, which is otherwise called the stiffness matrix, is created using the interpolation functions of

the element.

iii: Assembly of the Element Equations: The algebraic equations thus obtained are assembled imposing the

conditions of continuity between elements. This generates an enormous number of mathematical equations known as

the comprehensive finite element model, which administers the entire domain.

iv: Imposition of the Boundary Conditions: On the Assembled algebraic equations, the Dirichlet's and Neumann

boundary conditions are imposed

v: Solution of assemble equations: The assembled equations thus attained can be resolved by any of the numerical

methods, specifically, the Gaussian elimination procedure, the LU decomposition procedure and the final matrix

equation can be resolved with an iterative procedure.

IV. Skin- friction, Rate of Heat and Mass transfer:

Skin-friction coefficient ( ) at the plate is

0)(

y

y

u

Heat transfer coefficient (N u ) at the plate is

0)(

yu

y

TN

Mass Transfer coefficient (S h ) at the plate is

S 0)(

yh

y

C

V. Results and Discussions In figures 1 to 6, the velocity profile u has been plotted against y for magnetic parameter (M), Schmidt number (Sc),

Prandtl number (Pr), Grashof number (Gr) , Modified Grashof number (Gm) and Chemical reaction parameter

K r for both cooling of the plate (Gr>0) and heating of the plate(Gr<0). In figures 7 ,8 and 9 the temperature

distribution T and the concentration distribution C have been plotted against y for different values of Prandtl number

and different values of Schmidt number, respectively. In all these figures the values of Prandtl number are chosen

for mercury(Pr=0.025) air(Pr=0.71) water(Pr=7) and water at 4o

C (Pr=11.4).The values of Schmidt number are

taken for hydrogen (Sc=0.22) helium(Sc=0.30) water vapour (Sc=0.60) oxygen(Sc=0.66),ammonia(

Sc=0.78),methanol(Sc=1.0) and



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Propyl benzene at 20 c0(Sc=2.62). Here the real parts of complex quantities are dealt with as these have physical

significance in the problem and t=1, stable values for velocity, temperature and concentration fields are got.

Figure 1(a): Effect of magnetic parameter M on velocity Field u for cooling of the plate Gr = 10.0, Gm =

10.0, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .

Figure 1(b): Effect of magnetic parameter M on velocity Field u for heating of the plate

Gr = -10.0, Gm = 10.0, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .



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Figure 2(a): Effect of Schmidt number Sc on velocity Field u for cooling of the plate

Gr = 10.0, Gm = 10.0, M=0.5, Pr = 0.71, 100 K , = 0.005, 2

nt .

Figure 2(b): Effect of Schmidt number Sc on velocity Field u for heating of the plate

Gr = -10.0, Gm = 10.0, M=0.5, Pr = 0.71, 100 K , = 0.005, 2

nt .



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Figure 3(a): Effect of prandtl number Pr on velocity Field u for cooling of the plate

Gr = 10.0, Gm = 10.0, M=0.5, Sc = 0.22, 100 K , = 0.005, 2

nt .

Figure 3(b): Effect of prandtl number Pr on velocity Field u for heating of the plate

Gr = -10.0, Gm = 10.0, M=0.5, Sc = 0.22, 100 K , = 0.005, 2

nt .



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Figure 4(a): Effect of Grashof number Gr on velocity Field u for cooling of the plate

Gm = 10.0, M=0.5, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .

Figure 4(b): Effect of Grashof number Gr on velocity Field u for heating of the plate

Gm = 10.0, M=0.5, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .



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Figure 5(a): Effect of modified Grashof number Gm on velocity Field u for cooling of the plate Gr = 10.0,

M=0.5, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .

Figure 5(b): Effect of modified Grashof number on velocity Field u for heating of the plate Gr = 10.0,

M=0.5, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .



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Figure 6(a): Effect of chemical Reaction parameter Kr on velocity Field u for cooling of the plate Gr =

10.0, Gm = 10.0, M=0.5, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .

Figure 6(b): Effect of chemical Reaction parameter Kr on velocity Field u for heating of the plate Gr = -

10.0, Gm = 10.0, M=0.5, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .

From Figures 1(a) to 6(a), it is observed that an increase in magnetic parameter, Schmidt number Prandtl number,

chemical reaction parameter decreases the velocity field and increase in Grashof number and modified Grashof

number increases the velocity field for cooling of the plate (Gr>0).

A comparison of velocity field curves due to cooling of the plate show that in the neighborhood of the plate the

velocity raises very rapidly and there after decreases steadily indicating that the curves falls gradually after attain

minimum value near the plate



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Figures 1(b) to 6(b) show that an increase in magnetic parameter, Schmidt number and chemical reaction parameter

decreases the velocity field for heating of the plate (Gr<0) and an increase in prandtl number, increases the velocity

field for heating of the plate (Gr<0)

From figures 3(a) and (3b), it is seen that in both cooling of the plate and heating of the plate for water (Pr =7.0) and

water at 4o

C (Pr=11.4) the velocity is almost same.

Figure (7): Effect of prandtl number Pr on temperature Field T

Gr = 10.0, Gm = 10.0, M=0.5, Sc = 0.22, 100 K , = 0.005, 2

nt .

Figure (8): Effect of Schmidt number Sc on concentration Field C

Gr = 10.0, Gm = 10.0, M=0.5, Pr = 0.71, 100 K , = 0.005, 2

nt .



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Figure 9: Effect of chemical reaction parameter Kr on concentration Field C

Gr = 10.0, Gm = 10.0, M=0.5, Sc = 0.22, Pr = 0.71, 100 K , = 0.005, 2

nt .

An increase in Prandtl number decreases the temperature field (Figure 7). Also, temperature field falls more

rapidly for water in comparison to air and the temperature field curve is exactly linear for mercury which is more

sensible towards change in temperature. From this observation it is concluded that mercury is most effective for

maintaining temperature difference and can be used efficiently in the laboratory. Air can replace mercury, the

effectiveness of maintaining temperature changes are much less than mercury. If temperatures are maintained air can

be better and cheap replacement for industrial purposes.

From Figure 8 shows that an increase in Schmidt number decreases the concentration field. Also,

concentration field falls slowly and steadily for hydrogen and helium but falls very rapidly for oxygen and ammonia

in comparison to water-vapour. Thus water-vapour can be used for maintaining normal concentration field and

hydrogen can be used for maintaining effective concentration field. In Figure 9 shows that increase of chemical

reaction parameter decreases the concentration field.

An interesting velocity field flow phenomenon was observed for cooling of the plate (Gr>0) when negative

sign was applied to Gr and Gm the same numerical values for velocity field but opposite in sign were got.

Table 1 represents the numerical values of skin-friction coefficient ( ) for variation in Gr, Gm, M, Sc, Pr and

K 0 respectively, corresponding to cooling of the plate. An increase in (Gr) or (Gm) or K 0 leads to an increase in

the value skin-friction coefficient while an increase in M or Sc or Pr leads to a decrease in the value of skin-friction

coefficient.



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Table-1: skin-friction coefficient ( ) for cooling of the plate

Gr Gm M Sc Pr K 0

10 4 0.5 0.22 0.71 10 10.494717

20 4 0.5 0.22 0.71 10 16.568954

10 8 0.5 0.22 0.71 10 14.915193

10 4 0.5 0.66 0.71 10 8.632302

10 4. 1.0 0.22 0.71 10 6.409066

10 4 0.5 0.22 0.71 20 11.049682

10 4 0.5 0.22 7.0 10 5.227210

Table 2 represents the numerical values of skin friction coefficient ( ) for variations in Gr, Gm, M, Sc, Pr and K 0

respectively, corresponding to heating of the plate. An increase in Gr or M or Pr leads to an increase in the value

skin friction coefficient while an increase in Gm or Sc or K 0 leads to a decrease in the value of skin friction

coefficient.

Table-2: skin-friction coefficient ( ) for heating of the plate

Gr Gm M Sc Pr K 0

-10 4 0.5 0.22 0.71 10 -2.900642

-20 4 0.5 0.22 0.71 10 -9.924895

-10 8 0.5 0.22 0.71 10 1.222970

-10 4 0.5 0.66 0.71 10 -4.105411

-10 4. 1.0 0.22 0.71 10 -2.267819

-10 4 0.5 0.22 0.71 20 -2.964878

-10 4 0.5 0.22 7.0 10 3.093894

Table 3 represents the numerical values of heat transfer coefficient (N u ) for different values of Prandtl number Pr,

an increase in Pr leads to an increase in heat transfer coefficient. Also, the value of (N u ) is least for mercury and

highest for water at 4 c0.

Table-3: Heat transfer coefficient in terms of Nusselt number

Pr Nu

0.25 0.212059

0.71 0.626849

7.00 2.545455

11.4 2.961039

Table 4 represents the numerical values of mass transfer coefficient S h for different values of Schmidt number Sc.

An increase in Sc leads to an increase in mass transfer coefficient. Also, the value of S h is least for hydrogen and

highest for propyl benzene.



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Table-4: Mass transfer coefficient for different values for Sherwood number

Sc S h

0.22 0.317268

0.30 0.364993

0.60 0.555692

0.66 0.599557

0.78 0.671769

1.00 0.809331

2.62 1.583148

VI.CONCLUSIONS

In the present study the “chemical reaction impact of MHD free convective flow of a viscous fluid past a vertical plate under oscillatory suction velocity” is studied. The effects of velocity, temperature and concentration for different parameters like Gr, Gm, M. Sc, Pr and Kr are studied. The study concludes the following results:

1. The velocity decreases with the increasing of Hartmann number M.

2. The velocity decreases with the increase of Prandtl number Pr and Schmidt number Sc for cooling of the

plate (Gr > 0) and the velocity increases with the increase of Prandtl number Pr and Schmidt number Sc for

heating of the plate (Gr < 0).

3. The velocity increases with the increasing of Grashof number Gr and Modified Grashof number Gm.

4. The velocity and concentration profiles will decrease with the increase in chemical reaction parameter.

5. The temperature and Concentration decays with increasing values of Prandtl number Pr and Schmidt

number Sc respectively.

REFERENCES [1] Kim, Y.J., Unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. International

journal of engineering science, 2000. 38(8): p. 833-845.

[2] Singh, K. and R. Sharma, Transfer through a porous medium with periodic permeability. Indian J. pure appl. Math, 2002. 33(6): p. 941-949.

[3] Singh, A.K., A.K. Singh, and N. Singh, Heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Indian Journal of Pure and Applied Mathematics, 2003. 34(3): p. 429-442.

[4] Hossain, M. and A. Mandal, Mass transfer effects on the unsteady hydromagnetic free convection flow past an accelerated vertical porous plate. Journal of physics D: Applied physics, 1985. 18(7): p. L63.

[5] Jha, B.K., MHD free-convection and mass-transform flow through a porous medium. Astrophysics and Space science, 1991. 175(2): p. 283-289.

[6] Elbashbeshy, E., Heat and mass transfer along a vertical plate with variable surface tension and concentration in the presence of the magnetic field. International Journal of Engineering Science, 1997. 35(5): p. 515-522.

[7] Chambré, P.L. and J.D. Young, On the diffusion of a chemically reactive species in a laminar boundary layer flow. The Physics of fluids, 1958. 1(1): p. 48-54.

[8] Kesavaiah, D.C., P. Satyanarayana, and S. Venkataramana, Effects of the chemical reaction and radiation absorption on an unsteady MHD convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate embedded in a porous medium with heat source and suction. Int. J. of Appl. Math and Mech, 2011. 7(1): p. 52-69.

[9] Muthucumaraswamy, R. and S. Meenakshisundaram, Theoretical study of chemical reaction effects on vertical oscillating plate with variable temperature. Theoret. Appl. Mech, 2006. 33(3): p. 245-257.

[10] Kandasamy, R., K. Periasamy, and K.S. Prabhu, Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects. International Journal of Heat and Mass Transfer, 2005. 48(21-22): p. 4557-4561.

[11] Pal, D. and B. Talukdar, Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. Communications in Nonlinear Science and Numerical Simulation, 2010. 15(7): p. 1813-1830.

[12] Das, S., et al., Mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source. International journal of heat and mass transfer, 2009. 52(25-26): p. 5962-5969.

[13] Mishra, S., G. Dash, and M. Acharya, Mass and heat transfer effect on MHD flow of a visco-elastic fluid through porous medium with oscillatory suction and heat source. International Journal of Heat and Mass Transfer, 2013. 57(2): p. 433-438.

[14] Ambethkar, V., Numerical solutions of heat and mass transfer effects of an unsteady MHD free convective flow past an infinite vertical plate with constant suction. Journal of Naval Architecture and Marine Engineering, 2008. 5(1): p. 27-36.



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[15] Modather, M. and A. Chamkha, An analytical study of MHD heat and mass transfer oscillatory flow of a micropolar fluid over a vertical permeable plate in a porous medium. Turkish Journal of Engineering and Environmental Sciences, 2010. 33(4): p. 245-258.

[16] Bakr, A., Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference. Communications in Nonlinear Science and Numerical Simulation, 2011. 16(2): p. 698-710



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Chemical Reaction Impact on MHD free convection Flow of an