Magnetic Field a Heat Generation Effects on Second Grade ......Magnetic Field a Heat Generation...

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International Journal of Scientific Research in ___________________________ Research Paper . Mathematical and Statistical Sciences

Volume-7, Issue-2, pp.01-08, April (2020) E-ISSN: 2348-4519

Magnetic Field a Heat Generation Effects on Second Grade Fluid Flow

past an Oscillating Vertical Plate in Porous Medium

Rahul Mehta1*

, H. R. Kataria2

1Department of Mathematics, Sardar Vallabhbhai Patel Institute of Technology, Vasad, India

2Dept. of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara, India

*Corresponding Author: [email protected], Tel.: +91-98980-66500

Available online at: www.isroset.org

Received: 06/Mar/2019, Accepted: 14/Apr/2020, Online: 30/Apr/2020

Abstract— The present paper is concerned with the study of heat generation/absorption effect on unsteady natural

convective MHD Second grade fluid flow past an oscillating vertical plate in presence of thermal radiation and chemical

reaction. It is assumed that the bounding plate has ramped temperature with ramped surface concentration and isothermal

temperature with ramped surface concentration through porous medium. Governing non-dimensional equations are solved

using Laplace transform technique and analytic expressions are obtained of velocity, temperature and concentration

profiles. For both thermal plates, analytic expressions of Nusselt Number and Sherwood Number are derived and presented

in tabular form. The effects of Magnetic parameter M, second grade fluid , Heat generation/absorption parameter H,

thermal radiation parameter Nr, chemical reaction parameter Kr in time variable t on velocity, temperature and

concentration profiles are discussed through several graphs.

Keywords— MHD; Second grade fluid; Porous medium; Nusselt Number; Sherwood Number

Nomenclature:

Fluid velocity in direction Dimensionless fluid velocity in x direction

Time Dimensionless time

Fluid temperature Permeability parameter

Uniform magnetic field Magnetic parameter

Permeability of porous medium Thermal Grashof number

Concentration Mass Grashof number

thermal conductivity of the fluid Dimensionless fluid temperature

Specific heat at constant pressure Dimensionless concentration

Radiative heat flux Thermal radiation parameter

Heat absorption/generation coefficient Prandtl number

Mass diffusion coefficient Heat generation/absorption Parameter

Chemical reaction coefficient Schmidt number

Chemical reaction parameter N Nusselt number

Kinematic viscosity coefficient Fluid density

One of the material modules of second grade fluids. Acceleration due to gravity

Volumetric coefficient of thermal expansion Electrical conductivity

Volumetric coefficient of concentration expansion Porosity of the porous medium

Second grade parameter

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Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 7, Issue.2, Apr 2020


I. INTRODUCTION

As many fluids in engineering problems involves Non–

Newtonian fluids, study of Non–Newtonian fluids is

essential in engineering. Second grade fluid is one of such

fluid. MHD flow of many fluids like industrial oils, dilute

polymer solutions, slurry flows have been studied. Many

flow problems with various geometries and different

mechanical and thermal boundary conditions have also

been studied. Tan and Masuoka [1] considered the Stokes’

first problem for a second grade fluid and Rashidi et al. [2]

dealt with squeezing flow of a second-grade fluid.

Sheikholesmi [3] studied forced convection in a porous

cavity using Lattice Boltzmann Method. Sheikholesmi and

Bhatti [4] analyzed nanofluid heat transfer enhancement by

means of EHD. Sheikholeslami and Shehzad [5] discussed

effect of thermal radiation on ferrofluid flow considering

Lorentz forces. Sheikholeslami et al. [6] found numerical

expressions for forced convection heat transfer using

Lattice Boltzmann Method. Sheikholesmi and Shamlooei

[7] discussed natural convection in presence of thermal

radiation. Sheikholesmi [8] studied nanofluid in cavity

using Darcy law. Sheikholeslami [9] discussed nanofluid

forced convection heat transfer improvement in existence

of magnetic field using Lattice Boltzmann Method.

Katariya and Mittal [10] modelled velocity and

temperature of optically thick nanofluid. Katariya and

Mittal [11] analyzed velocity, mass and temperature of

optically thick nanofluid. Hayat et al. [12] studied MHD

flow of second grade fluid in porous channel whereas

Hatat et al. [13] solved MHD transient rotating flow of

second grade fluid. Hayat et al. [14] derived influenced of

heat transfer in second grade fluid. Samiqullah et al. [15]

studied unsteady MHD flow with ramped wall

temperature. Abolbashari et al. [16] analyzed entropy for

unsteady MHD flow. Rashidi et al. [17] studied MHD flow

for viscoelastic fluid in porous medium with radiation.

Rashidi et al. [18] derived analytical method for convective

and slip flow. Freidoonimehr et al. [19] observed free

convective flow in porous surface. Rashidi et al. [20]

approximated boundary layer viscoelastic fluid through

homotopy analysis method. Abbasbandy et al. [21] solved

Falkner-Skan Flow of MHD Oldroyd-B fluid numerically

as well as analytically. Zhou [22] et al. designed time

efficient optimization microchannel heat sink with wavy

channel. Song et al. [23] predicted hydrodynamic

properties of TiO2/water suspension. Ghasemi et al. [24]

studied blood flow containing nano particles in presence of

magnetic field. Fakour et al. [25] analyzed micro polar

fluid in permeable walls. Hatami et al. [26] investigated

nano fluid flow in non-parallel walls analytically. Kataria

and Patel [27 – 31] various MHD flows of Non-Newtonian

fluids under different conditions. Rassoulinejad-Mousavi

et al. [32] used two-equation energy model for heat transfer

in porous medium. Rassoulinejad-Mousavi et al. [33]

analyzed forced convection in circular tube. Saif and

Rassoulinejad-Mousavi [34] analytically studied fluid flow

in porous media with different wall conditioned moving or

stationary. Mohammdian et al. [35] studied thermal

management of lithium-ion battery. Rassoulinejad-

Mousavi et al. [36] analyzed Maxwell fluid through porous

medium. Rassoulinejad-Mousavi et al. [37] observed

viscous dissipation of non-linear drag term. Oztop and

Abu-Nada [38] numerically derived natural convection of

partially heated nanofluid.

Structure of this paper is follows. Section 1 which is this

section is introduction of the problem. Section 2 contains

Mathematical formulation of the Problem.

II. MATHEMATICAL FORMULATION OF THE

PROBLEM:

Figure 1: Physical sketch of the problem.

Fig. 1 gives sketch of the physical problem. Coordinate

system is selected in such a way that is taken as

the wall which is in the vertical direction and is

horizontal direction. As described in that figure, there

exists a magnetic field with strength in transverse

direction to the flow. Initially, at time , both the

fluid and the plate are at rest having a constant temperature

and the surface concentration is assumed to be

respectively. At the time , the temperature of

the plate is either increased or decreased to

⁄ when . For , it is maintained

constant . Mass transfer level at the wall surface is

elevated or reduced to

⁄ when

. For it is maintained constant . Viscous

dissipation effect, induced magnetic and electrical field

effects are neglected. In MHD flow one of the body force

term is the Lorentz force. Its formula is ,

Where B is the total magnetic field, J is the current density,

is electrical conductivity of the fluid, is the velocity

vector field

Governing equations of Boussinesq’s approximation under

above assumptions are as follows.

(

)

(

)

(1)

(2)



(3)

With following initial and boundary conditions:

{

⁄

{

⁄

(4)

Using the Rosseland approximation [39], the radiative heat

flux term is given by.

(5)

Where and are Stefan Boltzmann constant and mean

absorption coefficient respectively.

Assuming that the temperature difference between the fluid

within the boundary layer and free stream is small, so

can be expressed as a linear function of the temperature,

we expand about about Taylor's series and

neglecting higher order terms, we get

(6)

Thus we have

(7)

Using equations (6) and (7) in equation (3), we get

(8)

Introducing the following dimensionless quantities:

( )

( )

Using equation (8) and dimensionless quantities, equations

(1-4) becomes

(9)

(10)

(11)

With initial and boundary conditions

{

{

(12)

Where,

(

)

(

)

Exact solution for fluid velocity; Temperature and

Concentration is obtained for equations (9) to (11) with

initial and boundary condition (12) using the Laplace

transform technique.

II.I Solution of the Problem for ramped wall

temperature and ramped surface concentration:

(13)

(14)

[ ] (15)

II.II Solution of the Problem for isothermal

temperature and ramped surface concentration

In order to understand effects of ramped temperature of the

plate on the fluid flow, we must compare our results with

isothermal temperature. In this case, the initial and

boundary conditions are the same excluding Eq. (12) that

becomes .

(16)

(17)

[ ] (18)

Where

( √

)

(19)

⁄ ∫ (

√ )

⁄ (

√ )

∫ ∫ (

√ )

(

√ )

(20)

[ √ (

√ √ )

√ (

√ √ )] (21)

(22)



(23)

(24)

[(

√ ) √ (

√ √ )

(

√ ) √ (

√ √ )] (25)

(26)

∫

(27)

∫

(28)

∫

(29)

II.III Nusselt Number, Sherwood Number and Skin

friction:

Expressions of Nusselt Number Nr, Sherwood Number Sh

and Skin friction are calculated from equations (13-18)

using the relation

(

)

(

)

(

)

|

(30)

II.III.I For ramped wall temperature and ramped

surface concentration:

(31)

(32)

|

[ ] (33)

II.III.II For isothermal temperature and ramped

surface concentration:

(34)

(35)

|

[

] (36)

III. RESULTS AND DISCUSSION

We have graphed the fluid velocity, temperature and

concentration for several values of Second grade fluid with

diffusivity , Magnetic field parameter M, thermal

radiation parameter Nr, chemical reaction parameter Kr

and Heat generation/absorption parameter H described in

Figs. 2-10.

Figure 2: Velocity profile u for different values of y and

at 𝒌 = 𝟎. 𝟖, 𝑴 = 𝟓, 𝑷𝒓 = 𝟕, 𝑺𝒄 = 𝟎. 𝟔𝟔, 𝑮𝒎 = 𝟒, 𝑮𝒓 = 𝟓,

𝑲𝒓 = 𝟓, 𝑯 = 𝟑, 𝑵𝒓 = 𝟓 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Fig.2 describes effect of thermal diffusivity on velocity

for constant and variable wall temperature. It is seen that

velocity falls with increment in values of . It is also

observed that, the boundary layer thickness appraises with

reduction in diffusivity.

Figure 3: Velocity profile u for different values of y and 𝑴

at 𝒌 = 𝟎. 𝟖, = 𝟎. 𝟓, 𝑷𝒓 = 𝟕, 𝑺𝒄 = 𝟎. 𝟔𝟔, 𝑮𝒎 = 𝟒, 𝑮𝒓 =

𝟓, 𝑲𝒓 = 𝟓, 𝑯 = 𝟑, 𝑵𝒓 = 𝟓 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Fig.3 shows that Magnetic field parameter has negative

impact on velocity for both thermal situations. This is due

to Lorentz force on the fluid at boundary.



Figure 4: Velocity profile u for different values of y and 𝒌

at 𝑴 = 𝟓, = 𝟎. 𝟓, 𝑷𝒓 = 𝟕, 𝑺𝒄 = 𝟎. 𝟔𝟔, 𝑮𝒎 = 𝟒, 𝑮𝒓 = 𝟓,

𝑲𝒓 = 𝟓, 𝑯 = 𝟑, 𝑵𝒓 = 𝟓 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Fig.4 reflects that in constant or variable temperature,

Permeability of porous medium improves with velocity at

entire boundary.


𝑵𝒓 at 𝑴 = 𝟓, = 𝟎. 𝟓, 𝑷𝒓 = 𝟕, 𝑺𝒄 = 𝟎. 𝟔𝟔, 𝑮𝒎 = 𝟒, 𝑮𝒓 =

𝟓, 𝑲𝒓 = 𝟓, 𝑯 = 𝟑, 𝒌 = 𝟎. 𝟖 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Figure 6: Temperature profile 𝜽 for different values of y

and 𝑵𝒓 at 𝑷𝒓 = 𝟕, 𝑯 = 𝟑 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Fig. 5 and Fig. 6 are about radiative impact on velocity and

temperature for both isothermal and ramped thermal

conditions. It is derived that velocity and temperature have

positive correlation with thermal radiation. Temperature’s

correlation is obvious whereas increase in velocity with

radiation is due to generation of heat, bond holding

components of particles are broken.

Figure 7: Velocity profile u for different values of y and 𝑯

at 𝑴 = 𝟓, = 𝟎. 𝟓, 𝑷𝒓 = 𝟕, 𝑺𝒄 = 𝟎. 𝟔𝟔, 𝑮𝒎 = 𝟒, 𝑮𝒓 = 𝟓,

𝑲𝒓 = 𝟓, 𝑵𝒓 = 𝟓, 𝒌 = 𝟎. 𝟖 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Figure 8: Temperature profile 𝜽 for different values of y

and 𝑯 at 𝑷𝒓 = 𝟕, 𝑵𝒓 = 𝟓 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Fig. 7 and Fig. 8 are graphs of heat generation/absorption

coefficient H on velocity and temperature. In both figures

positive sign reflects the heat generation and negative sign

means heat absorption. Heat generation obviously

increases temperature which eventually increases flow of

the fluid. So, if parameter of heat source is increased, there

will be sudden rise in temperature. Results are very much

supported physically as heat generation at the surface will

increase porosity which rises fluid flow.




𝑲𝒓 at 𝑴 = 𝟓, = 𝟎. 𝟓, 𝑷𝒓 = 𝟕, 𝑺𝒄 = 𝟎. 𝟔𝟔, 𝑮𝒎 = 𝟒, 𝑮𝒓 =

𝟓, 𝑯 = 𝟑, 𝑵𝒓 = 𝟓, 𝒌 = 𝟎. 𝟖 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒

Figure 10: Concentration profile C for different values of y

and 𝑲𝒓 at 𝑺𝒄 = 𝟎. 𝟔𝟔, 𝐚𝐧𝐝 𝒕 = 𝟎. 𝟒.

Chemical reaction has a reverse impact on velocity and

concentration for both thermal conditions as shown in Fig.

9 and Fig. 10. This means that when we increase values of

Kr, buoyancy effect is reduced which eventually reduces

concentration. Hence, flow is reduced.

Table 1 and table 2 validates Nusselt number and

Sherwood number derived for different values by the

values derived by seth et al. [40] and [41] respectively.

Table 1: Comparison of Nusselt number with Ref. [40] at

Pr = 0.71

Nr

t

Nusselt

Number

Nu for ramped

temp.

Ref [40]

Nusselt Number

Nu for

ramped temp.

Nusselt

Number Nu for

isothermal

temp. Ref [40]

Nusselt Number

Nu for

isothermal temp.

2 3 0.3 0.38368 0.3837 0.89492 0.8949

2 3 0.5 0.55828 0.5583 0.85907 0.8591

2 3 0.7 0.72887 0.7289 0.84872 0.8487

2 1 0.5 0.44983 0.4498 0.56755 0.5675

2 3 0.5 0.55828 0.5583 0.85907 0.8591

2 5 0.5 0.65207 0.6521 1.09210 1.0921

2 3 0.5 0.55828 0.5583 0.85907 0.8591

4 3 0.5 0.43244 0.4324 0.66543 0.6654

6 3 0.5 0.36548 0.3655 0.56239 0.5624

Table 2: Comparison of Sherwood Number with Ref. [41]

T Kr Sc

Sherwood

Number

Sh for ramped

temp. Ref

[41]

Sherwood

Number

Nu for ramped

temp.

Sherwood

Number

Sh for isothermal

temp.

Ref [41]

Sherwood

Number

Nu for isothermal

temp.

0.3 0.2 0.22 0.295649 0.2956 0.525702 0.5257

0.5 0.2 0.22 0.386593 0.3866 0.428415 0.4284

0.7 0.2 0.22 0.463189 0.4632 0.379505 0.3796

0.3 2.0 0.22 0.344659 0.3447 0.839945 0.8399

0.5 2.0 0.22 0.488076 0.4881 0.785973 0.7860

0.7 2.0 0.22 0.625355 0.6254 0.757863 0.7579

0.3 5.0 0.22 0.416933 0.4169 1.1897 1.1897

0.5 5.0 0.22 0.628694 0.6287 1.12945 1.1294

0.7 5.0 0.22 0.838894 0.8389 1.09522 1.0952

IV. CONCLUSION AND FUTURE SCOPE

The objective of this research is to obtain analytical

solution for MHD flow in oscillating vertical plate through

porous medium of second grade fluid and observe

radiation, heat generation or absorption and chemical

reaction effects. Results are derived for constant and

variable temperature of the surface. Graphical description

is done for important parameters behaviors on velocity,

temperature and concentration.

Key remarks for the conclusions can be summarized as

follows.

Velocity, temperature and concentration in

constant temperature and constant surface

temperature is more than those in variable

temperature and variable surface

concentration.

Magnetic field parameter M, second grade

parameter and chemical reaction parameter

Kr have retarding effects with velocity.

Thermal radiation parameter Nr, permeability

of porous medium K and heat generation

parameter H have positive impacts with

velocity.



Temperature of the fluid has increase

tendency with heat generation parameter H

and thermal radiation parameter Nr.

Concentration profile decreases if there is

increment in chemical reaction parameter Kr.

V. APPENDIX:

√

√

√

√

|

|

|

|

|

|

|

|

|

|

|

|

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AUTHORS PROFILE

Mr. Rahul Mehta pursued B. Sc., M. Sc. And M. Phil. In

Mathematics from Sardar Patel University, Vallabh Vidyanagar

in 1999, 2001, 2007. He is currently working as an Assistant

Professor in Sardar Vallabhbhai Patel Institute of Technology,

Vasad and pursuing his Ph. D. from The Maharaja Sayajirao

University of Baroda, Vadodara under the guidance of Prof. H. R.

Kataria.

Prof. H. R. Kataria pursued is his B. Sc. From St. Xavier’s

College, Ahmedabad, M. Sc. From The M. S. University of

Baroda, Vadodara and Ph. D. from SVNIT, Surat. Currently he is

Dean of Faculty of Science, The M. S. University of Baroda,

Vadodara. He is member of many professional bodies. He has

published more than 50 research papers in reputed international

journals including Thomson Reuters (SCI & Web of Science) and

conferences. His main research work foucuses in Fluid

Dynamics.

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