Computational Solutions for Mixed Convective Radiating ... · The effect of the relevant parameters on the velocity, ... electromagnetic casting, MHD power generation and targeting

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2212-2227 © Research India Publications. http://www.ripublication.com

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Computational Solutions for Mixed Convective Radiating Casson Fluid

Flow Past An Inclined Flat Plate In Presence of Hall Current

Ch. Krishna Sagar* and G. Srinivas1

*Department of Mathematics, Visvesvarya College of Engineering and Technology, Hyderabad, Telangana State, India. 1Department of Mathematics, Gurunanak Institute of Technology, Hyderabad, Telangana State, India.

*Corresponding Author

Abstract:

This study aims to study, when chemical reactions occur, heat transfer and mass transfer, the Hall current and heat radiation have a combined effect on mixing Casson convective, incompressible and conductive fluids on a vertical inclined plate. The Rosseland diffusion approximation describes the flow of radioactive heat in the energy equation. Using the implicit finite difference method, the numerical method is used to solve nonlinear equations that vary over time and energy based on the relevant initial and boundary conditions. The effect of the relevant parameters on the velocity, temperature and concentration fields is determined by the graph. Using these, the skin-friction coefficient, the term heat transfer rate and the mass transfer rate in the form of Nusselt numbers, is given in the form of a Sherwood number. Finally, it checks the effects of various physical flow parameters through graphs and tables. The comparison of the current numerical results (based on the finite difference method) is performed using the results previously published.

Keywords: Hall Current; MHD; Chemical reaction; Mixed convection; Thermal radiation; Finite difference method;

1. INTRODUCTION:

Magnetohydrodynamics is the knowledge of the magnetic properties of conductive fluids. Examples of such magnetic fluids are plasma, brine, electrolytes and fluid metals. Hannes Alfven [1] was the first to introduce magnetohydrodynamics (MHD). MHD has many practical applications in the field of art and technology. As crystal growth, reactor fluid metal cooling, plasma, magnetohydrodynamic sensors, electromagnetic casting, MHD power generation and targeting for magnetic drugs. The MHD depends on magnetic induction. When the magnetic force becomes stronger, the Hall effect caused by the Hall current cannot be ignored. Edwin Hall [2] was the first to propose the concept of Hall flow. The research on hydrodynamics is of great significance and attraction. The hydrodynamic problem has improved due to the influence of the Hall current. Pop and Soundalgekar [3] have studied the effect of the Hall effect on fluid magnetic viscous fluids independent of time. Ahmed and Zueco [4]

making a Hall effect study of the effect of heat and mass with penetration flow passage of rotation, and obtain an accurate solution of the model problem. Abdel Aziz [5] studied the effect of Hal on the viscous flow of nano fluidic fluids and the transfer of heat through an elongated film. Awais et al. [6] studied the effect of viscous dissipation with the Hall current and the ionic effects on the Jeffery convective fluid flow was studied. Sulochana [7] studied the transient flow of a permeable medium in a rotating parallel plate taking into account the Hall effect. Yih [8] studied the effect of free convection on heat and the mass transfer of MHD on a continuously moving permeable vertical surface. Hossain [9], Hossain and Mohammad [10] studied the free convection flow of MHD along the vertical plate and the effect of the Hall current on the porous plate. Salem and Abd El-Aziz [11] also studied the effects of Hall currents and chemical reactions on the flow of water on the expanding vertical surface. Recently, Ghosh et al. [12] studied the Hall effect in the parallel channel of the plate, while Abd El-Aziz [13] studied the effect of the Hall current on the flow and heat transfer of the conductive fluid on the transient expansion surface in presence of strong magnetic field was studied.

The effects of radiation are fundamental in space technology and high temperature processes. The latest developments in supersonic flight, missile re-entry, rocket combustion chambers, planetary power plants and gas-cooled nuclear reactors have focused on thermal radiation as an energy source, emphasizing the need to better understand radiation transfer in these processes. Cess [14] studied the interaction of convective heat transfer between radiation and vertical laminar flow, using singular perturbation techniques to absorb optically dense luminescent fluids. Arpaci [15] considered similar problems in both the thin and optically thick optical regions and used approximate integration techniques and first order contortions to solve the energy equation. Gnaneswara Reddy and Bhaskar Reddy [16] used finite element method to study the effects of radiation and mass transfer on transient MHD convection on vertical porous plates. Recently, Gnaneswara Reddy and Bhaskar Reddy [17] studied the mass transfer and heat generation of free convective flow on the appropriate vertical surface of MHD in porous media. Basiri Parsa [18] and others are the MHD boundary layer that flows


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through an expandable surface and has internal heat generation or absorption. Gnaneswara Reddy [19] analyzed thermophoresis, viscous dissipation and Joule heating to obtain constant MHD heat and mass transfer through oblique radiation with variable thermal conductivity - isothermally transmissive surface. Hossain et al. [20] used Rosseland approximation to determine the effect of radiation on the free convection of an optically dense, viscous, incompressible flow on a vertical perforated plate heated with a uniform surface temperature and a uniform aspiration rate. Hossain and Takhar [21] studied the effects of radiation on mixed convection along a vertical plate having a uniform surface temperature. Pal and Malashetty [22] proposed a solution similar to the boundary layer equation to analyze the effect of thermal radiation on the flow of the stagnation point on an elongated film that is generated or absorbed internally. Mukhopadhyay and Layek [23] studied free convective flux and radiant heat transfer of an incompressible viscous fluid on a porous vertical traction plate. Recently, Pal [24] studied the effect of thermal radiation on the heat and mass transfer of a two-dimensional support point of an incompressible viscous fluid on an elongated film in the presence of buoyancy. Shateyi and Motsa [25] studied the influence of thermal radiation of transient heat and mass transfer of the traction surface. Khan and others[26] analyzed the analysis of heat transfer of the MHD flow of non-Newtonian fluids in the presence of thermal radiation. Madani et al. [27] discusses the application of homotopy perturbations and numerical methods in the flow of magnetic field in the presence of radiation.

Under the auspices of their work, the aim of this work was to investigate the effect of the Hall current and thermal radiation on Casson fluid flow with unsteady magnetohydrodynamic convection through a vertical porous plate exposed in the presence of a chemical reaction dipped in a porous medium. Therefore, the aim of this study was to extend Sharma and Chaudhary [28] to study the instability involving first-order internal chemical reactions, Casson fluid, thermal radiation and inclination angles. The basic equations are solved numerically using the finite difference method. The temperature and concentration curves are displayed graphically for the various values of the radiation and inclination angles. The basic equations are solved numerically using the finite difference method. The temperature and concentration curves are displayed graphically for the various values of the parameters involved in the problem, Magnetic field parameter, Grashof number for heat transfer, Grashof number for mass transfer, Schmidt number, Prandtl number, Hall parameter, Permeability parameter, Casson fluid parameter, thermal radiation parameter, Angle of inclination parameter and Chemical reaction parameter, use the tables to discuss the influence of these related parameters on velocity, temperature and concentration field, and physically interpret the results.

Formation of Flow Governing Equations:

The equation for controlling the motion of an incompressible viscous conductive fluid in the presence of a magnetic field is:

Equation of continuity:

0 v (1)

Momentum Equation:

vk

vgBJpvvtv

2

(2)

Energy Equation:

yq

CTkTv

tTCp r

p

12

(3)

Species Diffusion Equation:

CCKCDCvtC

r2

(4)

Kirchhoff’s first law:

0 J (5)

General Ohm’s law:

e

e

ee Pe

BvEBJB

J

1

0 (6)

Gauss’s law of magnetism:

0 B (7)

Introducing a coordinate system zyx ,, with x -axis

vertically upwards, y -axis normal to the plate directed into

the fluid region and z -axis along the width of the plate. Let


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kwjviuv ˆˆˆ

be the velocity, kJjJiJJ zyxˆˆˆ

be the current density at the point tzyxp ,,, and

JBB ˆ0 be the applied magnetic field, kji ˆ,ˆ,ˆ being unit

vectors along x -axis, y -axis and z -axis respectively.

Since the plate is of infinite length in x and z directions, therefore all the quantities except possibly the pressure are independent of x and z .Now, the equation (2.1) gives

0

yv

(8)

which is trivially satisfied by

0Vv (9)

where 0V is a constant and 00 V . Therefore the velocity

vector v is given by

kwjViuv ˆˆˆ0 (10)

Again the equation (7) is satisfied by

jBB ˆ0 (11)

Also the equation (5) reduces to

0

yJ y

(12)

which shows that yJ constant. Since the plate is non-

conducting, 0yJ at the plate and hence 0yJ at all

points in the fluid. Thus the current density is given by

kJiJJ zxˆˆ (13)

Under the assumption ep , Electron pressure is constant and

0E , the electric field is zero then the equation (6) takes the form

BvBJBmJ

0 (14)

Where eem is the Hall parameter. The equations (10), (11), (13), and (14) yield,

mwumBJwmu

mBJ zx 2

02

0

1&

1

(15)

Consider when Hall currents, thermal radiation, heat transfer, mass transfer and chemical reactions occur, the incompressible, viscous and conductive Casson fluid flows through the transient magnetohydrodynamic limit layer of the vertically inclined plate. In this survey, we made the following hypotheses.

i. The x'-axis is taken along the vertically inclined plate in the direction of the motion with the slot as the origin, and the y'-axis is perpendicular to the plate in the outward direction towards the fluid of ambient temperature

T and concentration C . We choose a

stationary frame of reference (x',y',z') such that x'-axis is along the direction of motion of the vertical plate, y'-axis is normal to this surface and z'-axis is transverse to the x'y'-plane.

ii. A uniform magnetic field Bo is imposed along y'-axis and the effect of Hall currents is taken into account. The effects of Hall current give rise to a force in the z'-direction, which induces a cross flow in that direction, so the flow becomes three-dimensional.

iii. To simplify the problem, it is assumed that there is no variation of flow and heat transfer quantities in the z'-direction. The temperature and the species concentration are maintained at a prescribed constant values wT and wC at the plate.

iv. The fluid leads under a constant magnetic field. However, the Hall effect is taken into account and the Joule heating and viscosity loss are ignored. Except for the buoyancy effect, it is assumed that all the properties of the fluid are constant

v. We also assume the induced magnetic field is negligible with respect to the applied magnetic field. Since the magnetic Reynolds number is very small.

vi. The Cauchy stress tensor, S of a Casson’s non-Newtonian fluid [29] takes the form as follows:

11

S

(16)

Where is the dynamic viscosity, 1 is the ratio of relaxation to retardation times, dot above a quantity denotes the material time derivative and is the shear rate. The Casson model provides an elegant formulation that simulates the effects of delay and relaxation that occur in non-Newtonian polymer flows. The cutting rate and the cutting velocity gradient are defined in more detail according to the velocity vector,

V , as follows:

where TVV

(17)

and

.V

dtd

(18)


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x

CT

ww CT , rq

a b c

oV

z g

O Porous medium

oB

Casson fluid flow

y

u

w

v

O

a --- Momentum boundary layer, b --- Thermal boundary layer, c --- Concentration boundary layer

Fig. 1. Geometry of the problem

Equation of Continuity:

0

yv

(19)

Momentum Equations:

220

2 2

B u mwu u 1 uv 1t y y 1 m

ug T T cos g C C cosK

(20)


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K

wm

wumByw

ywv

tw

2

20

2

2

111 (21)

Energy Equation:

yq

CyT

CyTv

tT r

pp

12

2

(22)

Species Diffusion Equation:

CCKyCD

yCv

tC

r2

2

(23)

Subject to the boundary conditions

yastyCtyTtywtyuyatbetyCaetyTtywtyu

t

yallfortyCtyTtywtyuttiti

0,,0,,0,,0,0,,,,0,,0,

:0

0,,0,,0,,0,:0

(24)

For optically dense fluids, in addition to emission, there is self-absorption and the absorption coefficient usually depends on the wavelength and is large. Therefore, we can use the Rosseland approximation to obtain the radiation vector. It is believed that in the direction y' - direction in the presence of thermal radiation i.e., rq (transverse to the vertical surface). By using the Rosseland approximation [30] the radiative heat flux rq is given by:

yT

kq

e

sr

4

34

(25)

It should be noted that by using the Rosseland approximation, the present analysis is limited to optically thick fluids. If temperature differences within the flow are sufficiently small, then equation (25) can be linearized by expanding 4T in Taylor series about

T which after neglecting higher order terms takes the form:

43344 344 TTTTTTTT (26)

Using Eqs. (25) and (26) in the last term of Eq. (22), we obtain

2

23

316

yT

kT

yq

e

sr

(27)

Introducing (27) in the Eq. (22), the energy equation becomes

2

23

2

2

316

yT

CkT

yT

CyTv

tT

pe

s

p

(28)

For the sake of normalization of the flow model and facilitate numerical solutions, the author has to make the governing

equations from (20), (21), (23) and (28) under the boundary conditions (24) dimensionless by introducing the following dimensionless quantities:

2o o

o o w2 2o o

2 2 2w o 0

p w3o

w e r3 3 2o s o

y V t V T Tu wu , y , t , w , ,V 4 V T T

K V 4 BC C 4,K ,M , ,C C 4 V V

C 4 g T TPr ,Gr ,

V

4 g C C k 4KGc , N ,Sc ,Kr

DV 4 T V

(29)

All the physical variables are defined in the nomenclature. Equations (20), (21), (23) and (28) transform to the following non-dimensional forms

2

2 2

u u 1 u M4 4 1 u mwt y y 1 m

uGr cos Gc cosK

(30)

Kwmuw

mM

yw

yw

tw

22

2

11144

(31)

2

2

343

Pr44

yNN

yt

(32)

Kr

yScyt

2

244 (33)

The corresponding boundary conditions (24) in non-dimensional forms are

yaswuyateewu

t

yallforwuttiti

0,0,0,00,,0,0

:0

0,0,0,0:0

(34)

The skin-friction, Nusselt number and Sherwood number are important physical parameters for this type of boundary layer flow. The skin-friction at the plate, which is the non-dimensional form is given by

0

01

1111

yo

y

yu

Vyu

Cf

and

0

02

1111

yo

y

yw

Vyw

Cf

(35)

Where 1Cf and 2Cf are Skin-friction coefficients along wall

x -axis and z -axis respectively. The rate of heat transfer


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coefficient, which is the non-dimensional form in terms of the Nusselt number (Nu) is given by

0

yw yT

TTxNu

0

1Re

yx y

Nu

(36)

The rate of mass transfer coefficient, which is the non-dimensional form in terms of the Sherwood number (Sh), is given by

0

yw yC

CCxSh

0

1Re

yx y

Sh (37)

Where

xVo Re is the local Reynolds number.

2. Finite Difference Technique Solutions:

The non-linear momentum, energy and concentration equations given in equations (30), (31), (32) and (33) are solved under the appropriate initial and boundary conditions (34) by the implicit finite difference method. The transport equations (30), (31), (32) and (33) at the grid point (i, j) are expressed in difference form using Taylor’s expansion:

j 1 j j ji i i 1 i

j j jj ji 1 i i 1i i2 2

jj j ii i

u u u u4

t y

u 2u u1 M4 1 u mw1 my

uGr cos Gc cos

K

(38)

j 1 j j ji i i 1 i

j j ji 1 i i 1

2

jj j ii i2

w w w w4

t y

w 2w w14 1y

wM w muK1 m

(39)

j 1 j j ji i i 1 i

j j ji 1 i i 1

2

4t y

24 3N 4Pr 3N y

(40)

j 1 j j ji i i 1 i

j j jji 1 i i 1i2

4t y

24 KrSc y

(41)

Where the indices i and j refer to y and t respectively.

The initial and boundary conditions (34) yield

0,0,0,0&

0,,0,0

,0,0,0,0 0000

jM

jM

jM

jM

tiji

tiji

ji

ji

iiii

wuiateewuiallforwu

(42)

Thus the values of u, w,θ and ϕ at grid point t = 0 are known; hence the temperature field has been solved at time

ttt ii 1 using the known values of the previous time

itt for all 1........,,2,1 Ni . Then the velocity field is evaluated using the already known values of temperature and concentration fields obtained at ttt ii 1 . These processes are repeated till the required solution of u, w, θ and ϕ is gained at convergence criteria:

310,,,,,, numericalexact wuwuabs

(43)

3. RESULTS AND DISCUSSIONS:

Fig. 2. Effect of Gr on u

Fig. 3. Effect of Gc on u

0

0.075

0.15

0 3 6 9

u

y

Gr = 1.0, 2.0, 3.0,

4.0

0

0.075

0.15

0 3 6 9

u

y

Gc = 1.0, 2.0, 3.0, 4.0


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Fig. 4. Effect of M on u

Fig. 5. Effect of m on u

Fig. 6. Effect of K on u

Fig. 7. Effect of Pr on u

Fig. 8. Effect of Sc on u

Fig. 9. Effect of γ on u

0

0.075

0.15

0 3 6 9

u

y

M = 0.5, 1.0, 1.5, 2.0

0

0.075

0.15

0 3 6 9

u

y

m = 0.5, 1.0, 1.5, 2.0

0

0.075

0.15

0 3 6 9

u

y

K = 0.5, 1.0, 1.5, 2.0

0

0.075

0.15

0 3 6 9

u

y

Pr = 0.025, 0.71, 7.0, 11.62

0

0.075

0.15

0 3 6 9

u

y

Sc = 0.22, 0.30, 0.60, 0.78

0

0.075

0.15

0 3 6 9

u

y

γ = 0.5, 1.0, 1.5, 2.0


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Fig. 10. Effect of α on u

Fig. 11. Effect of N on u

Fig. 12. Effect of Kr on u

Fig. 13. Effect of Gr on w

Fig. 14. Effect of Gc on w

Fig. 15. Effect of M on w

0

0.075

0.15

0 3 6 9

u

y

α = π/6, π/4, π/3, π/2,

0

0.075

0.15

0 3 6 9

u

y

N = 0.5, 1.0, 1.5, 2.0

0

0.075

0.15

0 3 6 9

u

y

Kr = 0.5, 1.0, 1.5, 2.0

0

0.05

0.1

0 3 6 9

w

y

Gr = 1.0, 2.0, 3.0, 4.0

0

0.05

0.1

0 3 6 9

w

y

Gc = 1.0, 2.0, 3.0, 4.0

0

0.05

0.1

0 3 6 9

w

y

M = 0.5, 1.0, 1.5, 2.0


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Fig. 16. Effect of m on w

Fig. 17. Effect of K on w

Fig. 18. Effect of Pr on w

Fig. 19. Effect of Sc on w

Fig. 20. Effect of γ on w

Fig. 21. Effect of N on w

0

0.05

0.1

0 3 6 9

w

y

m = 0.5, 1.0, 1.5, 2.0

0

0.05

0.1

0 3 6 9

w

y

K = 0.5, 1.0, 1.5, 2.0

0

0.05

0.1

0 3 6 9

w

y

Pr = 0.025, 0.71, 7.0, 11.62

0

0.05

0.1

0 3 6 9

w

y

Sc = 0.22, 0.30, 0.60, 0.78

0

0.05

0.1

0 3 6 9

w

y

γ = 0.5, 1.0, 1.5, 2.0

0

0.05

0.1

0 3 6 9

w

y

N = 0.5, 1.0, 1.5, 2.0


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Fig. 22. Effect of Kr on w

Fig. 23. Effect of Pr on θ

Fig. 24. Effect of N on θ

Fig. 25. Effect of t on θ

Fig. 26. Effect of Sc on ϕ

Fig. 27. Effect of Kr on ϕ

0

0.05

0.1

0 3 6 9

w

y

Kr = 0.5, 1.0, 1.5, 2.0

0

0.35

0.7

0 3 6 9

θ

y

Pr = 0.025, 0.71, 7.0, 11.62

0

0.35

0.7

0 3 6 9

θ

y

N = 0.5, 1.0, 1.5, 2.0

0

0.5

1

0 3 6 9

θ

y

t = 0.5, 0.7, 0.9, 1.0

0

0.35

0.7

0 3 6 9

ϕ

y

Sc = 0.22, 0.30, 0.60, 0.78

0

0.35

0.7

0 3 6 9

ϕ

y

Kr = 0.5, 1.0, 1.5, 2.0


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Fig. 28. Effect of t on ϕ

The equations (30), (31), (32) and (33) are similar and solved numerically using the boundary conditions (34). A graphical representation of the numerical results is shown in Figs. (2) to (28) to show the effect of various numbers on the flow of the limit level. In this study, we examined the effects various material parameters like Grashof number for heat transfer, Grashof number for mass transfer, Magnetic field parameter, Hall parameter, Permeability parameter, Prandtl number, Schmidt number, Casson fluid parameter, Angle of inclination parameter and Thermal radiation parameter to determine their respective effects clearly observe the velocity, temperature and concentration distributions of the flow. The numerical results of the skin-friction coefficient, rate of heat and mass transfer coefficients expressed in terms of Nusselt numbers or Sherwood numbers, are presented in tabular forms. The primary velocity, secondary velocity, temperature and concentration of the numerical calculation, the selected value of the horizontal Prandtl (Pr = 0.025), the air is 25o C And atmospheric pressure ( Pr = 0.71), water ( Pr = 7.00) and water at 4oC( Pr = 11.40).In order to draw attention to the value of the results obtained in the study, the values of Scare hydrogen (Sc= 0.22), helium (Sc= 0.30), water vapour (Sc= 0.60) and ammonia (Sc= 0.78).For the physical meaning, the numerical discussion is in question and in t = 1.0,ωt = π/2 to obtain stable values for velocity, temperature and concentration fields. To find a solution to this problem, a finite vertical infinite plate was placed in the flow. This solves the entire limited problem. However, in the picture, y values range from 0 to 9. Velocity, temperature and concentration tend to zero at ymax = 9. They tend to be 9.This is true for every value y, this study considers a finite length. The temperature and concentration of the material are coupled at primary velocity by the Grashof number of heat transfer, and the Grashof number of mass transfer is shown in equation (30).The effects of Grashof numbers for heat and mass transfer on primary velocity profiles are shown in Figs. 2 and 3 respectively. Grashof number for heat transfer number refers to the relative influence of thermal buoyancy on the dynamics of the viscous fluid in the boundary layer. As expected, an increase in primary velocity was observed due to an increase in thermal force. If Gr increased, the peak velocity of the

primary velocity near the perforated plate increases rapidly and then descends smoothly at the velocity of free flow. The Grashof number of mass transfer determines the relationship between the buoyancy of the species and the dynamics of the viscous fluid. As expected, as the buoyancy of the species increases, the primary velocity of the fluid increases and the peak becomes more pronounced. The primary velocity profile reaches a significant maximum near the plate and is therefore appropriately reduced to approximate the value of the free flow. It should be noted that the primary velocity increases as the value of the Grashof mass transfer increases. Figs. 4 and 15 show the effect of the magnetic field parameter on primary and secondary velocities. As can be seen from these figures, with the increase of M, both the primary and the secondary velocities decrease. That is, the primary or secondary fluid is delayed due to the application of a transverse magnetic field. This phenomenon is clearly consistent with the fact that the Lorentz force due to the interaction of the magnetic field and the fluid velocity is subject to fluid movement. Fig. 12 shows the behaviour of the primary velocity distribution of various chemical reaction parameters Kr. Kr > 0 corresponding to a destructive chemical reaction. As can be seen from the profile, the primary velocity is reduced in the degraded chemical reaction in the boundary layer. This is because of the increase The rate of chemical reaction rate causes the pulses in the boundary layer to weaken in the degraded chemical reaction. The effects of Grashof number for heat and mass transfer numbers on the distribution of the secondary velocity are shown in Figs. 13 and 14. With increasing heat and mass transfer, this secondary velocity component also increases. In Fig. 27, as the concentration of the chemical reaction increases, the concentration decreases. The demonstration of a destructive chemical reaction indicates that the diffusion rate can be significantly modified by chemical reactions. This is because the chemical reaction has increased , it causes the concentration of the boundary layer to become thinner, thus reducing the concentration of the diffusive substance. This reduction in the concentration of the diffuser material reduces the mass diffusion. The effect of the Hall parameter m on the primary and secondary velocity profiles is shown in Figs. 5 and 16. We can see from these figures, the primary and secondary velocities increases Hall parameter increases. This is because the Hall current usually reduces the strength of the Lorentz force. This means that the Hall current tends to increase the fluid velocity component. Fig. 22 shows the effect of the chemical reaction parameters on the secondary velocity. It can be seen that as the value of this parameter increases, the secondary velocity profiles decreases. Fig. 23 shows the relationship between the temperature curve and y using different values of the number Prandtl (Pr).This number indicates that the temperature profile decreases as Prandtl number increases. This is because the fluid is highly conductive to the small values of the Prandtl number. Physically, as the Prandtl number increases, thermal diffusivity decreases and this phenomenon leads to a decrease in energy transfer capacity, which reduces the thermal interface.

Fig. 26 shows the concentration of various gases on the y curves, such as hydrogen (Sc = 0.22), helium (Sc = 0.30), water vapour (Sc = 0.60) and ammonia (Sc = 0.78).It has been

0

0.5

1

0 3 6 9

ϕ

y

t = 0.5, 0.7, 0.9, 1.0


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reported that the effect of increasing the Schmitt number (Sc) reduces the concentration of the concentration. This is consistent with the fact that the increase in Sc means a decrease in molecular diffusivity (D), which leads to a decrease in the concentration boundary layer. Therefore, for smaller Sc values, the concentration of the substance is greater and, for larger Sc values, the concentration of the substance is lower. Furthermore, it has been observed that the thickness of the concentration boundary layer has increased significantly with increasing frequency near the border, but the opposite tendency has been detected far from the plate. The influence of Schmidt number (Sc) on the primary and secondary velocities is shown in Figs. 8 and 19 respectively. From these figures, primary and secondary velocities decrease as Schmidt number increases. The Schmidt number represents the relationship between the moment and the mass diffusion coefficients. Schmidt's number then quantifies the relative validity of the momentum and mass transfer through the diffusion in the boundary layers of fluid dynamics (velocity) and concentration (species). The curves in Figs. 6 and 17 show the effect of the Permeability parameter (K) on primary and secondary velocities. As shown in Figs. 6 and 17, with this Permeability parameter increases, both the velocities in x' and z' directions increase. Figs. 11, 21 and 24 show an effect of thermal radiation (N) on the primary velocity, the secondary fluid temperature and the fluid velocity. From these figures, the thermal radiation of the door is reduced at each velocity and temperature. Physically, thermal radiation causes a decrease in the temperature of the fluid medium and, therefore, a decrease in the kinetic energy of the fluid particles. This results in a corresponding decrease in fluid velocity. Therefore, the Figs. 11, 21 and 24 are in good agreement with the laws of physics. So with the increase of N, θ, u and w will be reduced. Now, from these figures it can be inferred that the effect of radiation on temperature is greater than the effect on velocity. Therefore, thermal radiation does not have a significant effect on velocity, but it has a relatively more pronounced effect on the temperature of the mixture. Figs.25 and 28 show the effect of the time on temperature and concentration profiles. It is obvious from Figs. 25 and 28 where θ and ϕ increase as t increases. This means that the temperature and concentration of the fluid accelerates as time develops in the boundary layer region. The effect of Casson parameter (γ) on the profiles of fluid velocity in x'-direction (primary velocity) and z'-directions (secondary velocity) are shown graphically in Figs. 9 and 20 respectively. It is evident from these figures that on increasing the values of both Casson parameter, the fluid flow velocities (primary velocity and secondary velocity) decreases within the boundary layer region. The Casson parameter measures the yield stress and when it becomes large, the fluid behaves as a Newtonian fluid. The increase in the yield stress causes a stabilization effect. The effect of angle of inclination to the vertical direction on the velocity is shown in Fig. 10. From this figure we observe that the velocity is decreased by increasing the angle of inclination due to the fact that as the angle of inclination increases the effect of the buoyancy force due to thermal diffusion decreases by a factor of cosα. Consequently, the driving force to the fluid decreases as a result there is decrease in the velocity profile. From Figs. 7

and 18, we observe that as Pr increases, primary velocity profiles and secondary velocity profiles decrease respectively. This happens because when Pr increases the thermal boundary layer thickness rapidly decreases. This causes an increase in fluid viscosity. Consequently the primary velocity profiles and secondary velocity profiles decrease.

The influence of Gr,Gc,M,m,K, Pr,Sc,α, γ, N and Kr on skin-friction coefficient (Cf1) due to primary velocity profiles is discussed in tables 1 and 2. From these tables, we observed that the skin-friction coefficient is increasing with increasing values of Gr,Gc,m,K and the reverse effect is observed with increasing of M, Pr,Sc,α, γ, N, Kr. The influence of Gr,Gc,M,m,K, Pr,Sc,α, γ, N and Kr on skin-friction coefficient (Cf2) due to secondary velocity profiles is discussed in tables 1 and 3. From these tables, we observed that the secondary velocity skin-friction coefficient is rising with increasing values of Gr,Gc,M, m,K and the reverse effect is observed with increasing of Pr, Sc,α, γ, N, Kr. The influence of Pr , N and t on rate of heat transfer coefficient (Nu) due to temperature profiles is discussed in table 4. From this table, we observed that

the rate of heat transfer coefficient is increasing with increasing values of t and decreasing with increasing values of Pr and N. The influence of Sc, Kr and t on rate of mass transfer coefficient or Sherwood number (Sh) due to concentration profiles is discussed in table 5. From this table, we observed that the rate of mass transfer coefficient is increasing with increasing values of t decreasing with increasing values of Sc and Kr.

Table-1.: Numerical values of Skin-friction coefficients (Cf1) due to primary velocity profiles and (Cf2) due to secondary velocity profiles for different values of Gr, Gc, M, K and m

Gr Gc M K m Cf1 Cf2

2.0 2.0 0.5 0.5 0.5 1.2558412458 0.4512235698

4.0 2.0 0.5 0.5 0.5 1.4522018478 0.6652148921

2.0 4.0 0.5 0.5 0.5 1.5011248201 0.7884120155

2.0 2.0 1.0 0.5 0.5 1.1544833927 0.5582215486

2.0 2.0 0.5 1.0 0.5 1.3224619485 0.5214662489

2.0 2.0 0.5 0.5 1.0 1.3222160991 0.5211630188

https://www.sciencedirect.com/science/article/pii/S2211379718308076#f0010

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https://www.sciencedirect.com/topics/physics-and-astronomy/boundary-layer

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https://www.sciencedirect.com/topics/physics-and-astronomy/newtonian-fluids

https://www.sciencedirect.com/topics/engineering/velocity-profile

https://www.sciencedirect.com/topics/engineering/buoyancy-force

https://www.sciencedirect.com/topics/engineering/thermal-diffusion


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Table-2.: Numerical values of Skin-friction coefficient (Cf1) due to primary velocity profiles for different values of γ, α, Pr, N, Sc and Kr

γ α Pr N Sc Kr Cf1

0.5 45o 0.71 0.5 0.22 0.5 1.2558412458

1.0 45o 0.71 0.5 0.22 0.5 1.1885021349

0.5 90o 0.71 0.5 0.22 0.5 1.1799632045

0.5 45o 7.00 0.5 0.22 0.5 1.1055632478

0.5 45o 0.71 1.0 0.22 0.5 1.1662450124

0.5 45o 0.71 0.5 0.30 0.5 1.1553620188

0.5 45o 0.71 0.5 0.22 1.0 1.1662004875

Table-3.: Numerical values of Skin-friction coefficient (Cf2) due to secondary velocity profiles for different values of γ, Pr, N, Sc and Kr

γ Pr N Sc Kr Cf2

0.5 0.71 0.5 0.22 0.5 0.4512235698

1.0 0.71 0.5 0.22 0.5 0.3884122478

0.5 7.00 0.5 0.22 0.5 0.3011466947

0.5 0.71 1.0 0.22 0.5 0.3441501486

0.5 0.71 0.5 0.30 0.5 0.3221688412

0.5 0.71 0.5 0.22 1.0 0.3622854719

Table-4.: Numerical values of rate of heat transfer coefficient (Nu) due to temperature profiles for different values of Pr, N and t

Pr N t Nu 0.71 0.5 1.0 0.1544851256 7.00 0.5 1.0 0.0522148795 0.71 1.0 1.0 0.0877412516 0.71 0.5 2.0 0.2056221895

Table-5.: Numerical values of rate of mass transfer coefficient (Sh) due to concentration profiles for different values of Sc, Kr and t

Sc Kr t Cf2 0.22 0.5 1.0 0.1448522189 0.30 0.5 1.0 0.0755213369 0.22 1.0 1.0 0.0999521487 0.22 0.5 2.0 01999850125

1. Validation of Numerical Results:

To assess the correctness of the current finite difference method, the authors compared the results with published mass transfer rate data to understand the magnetohydrodynamic viscosity, the case where the incompressible fluid passes through the vertical porous plate and the porous plate exists in the porous medium. The hall current is calculated according to Sharma and Chaudhary of case [28] In the absence of Casson fluid, Angle of inclination, Thermal radiation and Chemical reactions, using different Schmidt values and phase angles, recorded other parameters. These results are all in table 6 shows these favourable comparisons show that the genius of numerical methods. Therefore, it is possible to use the code developed with great security to verify the problems involved in this document.

Table-6.: Sh is the Rate of mass transfer (Sherwood number) results obtained in the present study, and tC is the rate of mass transfer results obtained by Sharma and Chaudhary[28].

tC (Analytical results of Sharma and Chaudhary[28]) Sh (Present Numerical results)

ω ↓ Sc → 0.22 0.30 0.78 0.22 0.30 0.78 0.0 0.2200 0.3000 0.7800 0.2200000000 0.3000000000 0.7800000000 0.2 0.0800 0.1200 0.3800 0.0800124458 0.1200541846 0.3799995824 0.4 - 0.1700 - 0.2100 - 0.4100 - 0.1742122215 - 0.2055124551 - 0.4001251655 0.6 - 0.2700 - 0.3500 - 0.8100 - 0.2622154855 - 0.34112051894 - 0.8001452188 0.8 - 0.0800 - 0.1200 0.3900 - 0.0801145232 - 0.12022101577 0.3899952734 1.0 0.2100 0.2600 0.4400 0.2100154825 0.2511862015 0.4388521492

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2. Conclusions:

This work investigated the effect of a chemical reaction and thermal radiation on magneto hydrodynamics mixed convection through a vertically inclined perforated plate immersed in a porous medium with Hall current, heat and mass transfer. The linear coupled partial differential equation obtained is solved using the finite difference method. The authors studied a parametric study, illustrates the impact of different traffic parameters of velocity, temperature and concentration fields. The shear stress on the plate due to the primary and secondary velocity fields and the heat transfer and the mass transfer coefficients due to temperature and concentration are obtained in a dimensionless form. The results are displayed in graphical and tabular form. The authors concluded that the flow field and physical interest are significantly influenced by these parameters.

i. Because of the Lorentz magnetic force acting on the flow field, the action of the transverse magnetic field decreases the primary and secondary motions, which are accelerated by the Hall effect.

ii. The movement of the liquid is delayed due to chemical reactions. Therefore, the consumption of chemicals leads to a decrease in the concentration field, which in turn The buoyancy effect due to the concentration gradient is reduced. Therefore, the flow field is delayed.

iii. The concentration of the fluid is reduced due to the chemical reaction. This is due to the chemical consumption decreased range of concentrations of species.

iv. The Grashof number, which increases heat and mass transfer, has been found to improve the effects of thermal buoyancy and concentration, thus increasing primary and secondary velocities.

v. As the Schmidt number increases, the concentration distribution of the flow field decreases at all points. This means that the most common substances have a greater delay in the distribution of the concentration of the flow field.

vi. The Prandtl number reduces the flow field temperature at all points. The higher the Prandtl number, the sharper the flow field temperature decreases.

vii. The results obtained for the specific case of the problem were compared with the documents previously published and found in good agreement

3. Nomenclature:

List of variables:

B Magnetic Induction Vector

oB Intensity of the applied magnetic field ( 1mA )

C Dimensionless species concentration of the fluid

pC Specific heat at constant pressure

( KKgJ 1 )

C Concentration in the fluid far away from the plate

( 3mKg )

wC

Concentration of the fluid at the wall ( 3mKg )

D Chemical molecular diffusivity

( 12 sm )

TD Coefficient of chemical thermal diffusivity, 1111 KTLM

E Electric field

e Electron charge, Coloumb

Gr Grashof number for heat transfer

Gc Grashof number for mass transfer

Pr Prandtl number

Sc Schmidt number

rq Radiative heat flux

N Thermal radiation parameter

ep Electron pressure ( 2mN )

Kr Chemical reaction parameter

T Temperature of the fluid K

wT Temperature of the plate K

T Fluid temperature far away from the plate K

t Time s

u Velocity component in x direction ( 1sm )

V Velocity vector

oV Reference velocity ( 1sm )

w Velocity component in z direction

( 1sm )

g Acceleration due to gravity ( 2sm )

J Electric current density vector

K Permeability of the porous medium


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M Hartmann number or Magnetic field parameter

m Hall parameter

Nu Rate of heat transfer coefficient (or) Nusselt number

Sh Rate of mass transfer coefficient (or) Sherwood number

1Cf Skin-friction due to velocity (u) ( 2/ mN )

2Cf Skin-friction due to velocity (w) ( 2/ mN )

Greek symbols:

Coefficient of Volume expansion 1K

Density of the fluid 3/ mkg

* Volumetric Coefficient of expansion with

Concentration( 13 Kgm )

Kinematic Viscosity 12 sm

e Electron frequency (Hertz)

w Shear stress ( 2/ mN )

t Phase Angle (radians)

Angular frequency (Hertz)

Frequency parameter

Dimensionless Temperature K

Electrical conductivity, ( 11 m )

e Electron collision time (s)

i Ion collision time (s)

en Number of electron density

i Ion frequency (Hertz)

Thermal conductivity, mKW /

Angle of inclination parameter

(degrees)

Casson fluid parameter

Superscript:

/ Dimensionless properties

Subscripts:

w Conditions on the wall

Free stream conditions

p Plate

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