Mixed Convection on Stagnation Point Flow of MHD Casson ...

Mixed Convection on Stagnation Point Flow of MHD Casson Fluid

Towards Stretched Sheet

M. Sreedhar Babub*, V. Venkata Ramanac,

b,cDepartment of Applied Mathematics, Yogi Vemana University, Vemana Puram,

Ganganapuram, Kadapa, Andhra Pradesh-516005, India

Email: [email protected], [email protected]

Abstract

In this article we investigated non-Newtonian fluid and its MHD stagnation point flow

with the thermal and mass buoyancy effects. A non-Newtonian fluid model obtained from the

Casson fluid model. By adopting similarity metamorphosis, the governing equations are

transferred as nonlinear ordinary differential equations. Those equations are solved

numerically by applying the R K Gill method forward with shooting technique. The nature of

Local Nusselt number, skin friction coefficient, Sherwood number and the profiles of

velocity, temperature and the concentration exhibited pictorially and deliberated by taking

assisting and opposing flow constraints into consideration. The numerical results turned up

are distinguished with formerly published work concealed by limiting cases and found to be

in remarkable agreement.

Keywords: MHD, stagnation point flow, heat and mass transfer, Casson fluid model,

assisting and opposing flows.

*Corresponding Author E-mail: [email protected]

Journal of Xi'an University of Architecture & Technology

Volume XII, Issue III, 2020

Issn No : 1006-7930

Page No: 2704

mailto:[email protected]



1. Introduction

Issues identified with convective boundary layer flow assume a key job in engineering

and industrial exercises. These flows are utilized to oversee thermal impacts in numerous

modern outlets, for example, motor cooling frameworks, gadgets, and PC power. Sakiadis

first concentrated the boundary layer flow on a ceaseless strong surface moving at a

consistent speed [1]. On a piece, the boundary layer flow is totally not quite the same as the

Blasius flow as far as diversion condition fluids. Tsou et al, They checked the hypothetical

forecasts of Newtonian liquids proposed by Sarkidis. [2] Viscous fluid stream in past

extended movies was an exemplary issue in fluid elements. Crane initially examined the

convective boundary layer stream on a stretchable sheet [3]. Guptha and others examined. [4]

On the extended sheet, suction or passing up heat and mass exchange. They likewise got

temperature and fixation profiles by considering isothermal moving plates. Chen and Char [5]

examined the stream and heat move of a layered limit layer, which is performed by suction or

blow trim of a ceaseless and stretchable direct sheet under a recommended divider

temperature and heat stream. Into, Gangadhar et al. [6] examined the progression of

nanofluids in an unsteady limit layer outside the extended surface. Wang [7] thought about

the state of the sanctioned divider temperature and concentrated the progression of stale

stream to the therapist film. Ishak et al. [8-9] contemplated the mixed convection of vertical

and persistent extended tablets, the progression of MHD stagnation focuses to extended

tablets, and the progression of slacking focuses to vertical and straight extended tablets.

While thinking about a recommended divider temperature or heat stream, numerous analysts

proceed with this kind of model utilizing viscoelastic fluids or micropolar and viscous

liquids. As of late, Khan et al. contemplated the progression of stagnation focuses on stretch

sheets. [10] Hayat et al. They considered the progression of a second-degree liquid on an

extended surface under Newtonian heating. [11]

In fluid mechanics, non-Newtonian fluid stream issues assume a significant job in

different fields, for example, therapeutic innovation and natural fluid mechanics. What's

more, the heat move of non-Newtonian fluids has a wide scope of uses in polymer

arrangements, compound designing, power building examined by Ibrahim and Gaddisa,

obstruction decrease, sweat cooling, and polymer smell compression. Powell limit layer

streams, for example, plastic preparing Non-Newtonian Eyring-based liquids containing

nanoparticles contain nanoparticles Furthermore, Hirschhorn et al. They analyzed MHD

stream under slip limit conditions. [13] On the tablet. As of late, Bear et al. They considered



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viscoelastic fluids on unbending bodies. [14] Li et al. [15] numerically inspected non-

Newtonian liquids in micro channels. Additionally, Reddy et al, They examined the impacts

of the entropy heat of non-Newtonian liquids [Sixteen]. Casson [17] presented another liquid

called Casson viscoplastic fluid, which is a non-Newtonian liquid. Casson presented the

investigation while concentrating the stream condition of printing ink shade suspensions.

Milk, gelatin, soup, ketchup, human blood, concentrated natural product juice, nectar are

entering the Casson fluid classification, and this fluid is more appropriate for rheological

information than common place viscoplastic models numerous materials. A Casson fluid

shear conveyance fluid was acquired. The fluid has unbounded consistency and zero shear

rate, and no stream happens beneath as far as possible and zero thickness endless shear rate

[18]. A few specialists [19, 20] have contemplated the issue of Casson fluids by utilizing

fitting changes. Abbas et al, They dissected comparative issues with the progression of heat

move fluid over the surface/stretch sheet for each situation. [21] and Mustafa et al. [22]

Mythili and Sivaraj assessed the physical impacts of Casson fluids in various geometries

under the requirements of transient stream [23], while Das et al. Created MHD consequences

for plates under a similar transient stream. [24] Nadeem et al. They let MHD's Casson fluid

move through a bit of paper. [25] Raju et al. Raju et al. They dissected the heat and mass

exchange of Casson fluids from porous extended hot surfaces. [26] Gangadhar et al.

contemplated the dissemination and heat impacts of digression hyperbolic fluids on extended

surfaces. [27] An ongoing report by Gangadhar et al. [28] The Cattaneo-Christov heat stream

model has been utilized to instigate thick dissemination impacts in non-Newtonian Maxwell

fluids.

Right now, impacts of heat and mass lightness on non-Newtonian fluids are consolidated

to concentrate on the investigation of heat and mass exchange. The Casson fluid model was

utilized to analyze the rheological properties of shear diminishing and shear thickening of

non-Newtonian fluids. This examination might be valuable for comparable investigations in

heat, mass and modern fields.

2. Mathematical Formulation

Consider a Cartesian facilitate framework and a limit layer of a Casson fluid. The fluid

can't go through a consistently moving extended surface situated on the y = 0 plane. The fluid

can't consolidate through laminar free convection close to the stagnation point and is



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incompressible Figure 1. The outer stream speed to the limit layer is xcue , and the direct

speed of the stretch sheet is xauw . At the point when two comparable and inverse powers

are presented, the sheet extends at a speed relative to the good ways from the starting point.

At the point when a consistent attractive field B0 is applied, the initiated attractive field is

disregarded on the grounds that its Reynolds attraction is little. It is expected that the x-pivot

is toward the extended surface and that the hub is symmetrical to the extended surface.

Rheological conditions speak to isomorphic progressions of Casson fluids (Eldabe and Salwa

[30])

cij

y

B

c

y

B

ij

ep

eijp

,2

2

,2

2

(1)

Here, ijijee,……, ije speaks to the

thji ),( part of the strain rate, py speaks to the fluid wet

blanket power, speaks to the result of the strain rate segment and itself, c speaks to the

basic worth, and the consistency of the item dependent on non-Newtonian model and B

speak to Dynamic plastic thickness of non-Newtonian liquids.

The conditions for the mass, force, energy and types of the limit layer are as per the

following.

0

y

v

x

u , (2)

),()()(

11

2

0

2

20

CCgTTguuB

y

u

dx

duu

y

uv

x

uu

CTe

e

e

(3)

,2

2

y

T

y

Tv

x

Tu

(4)

,2

2

y

CD

y

Cv

x

Cu

(5)

Where speed segments in x -and y -bearings spoke to by u and v separately. In eq. (3), the

assisting flow represented by ‘+’ symbol while the opposing flow represented by ‘_’ symbol

and also the parameter δ can be positive or negative which is in eq. (3), as assumed by Bejan

and Khair [30] and Khair and Bejan [31], the kinematic viscosity /0 , the viscosity of



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the fluid represented by0 , the non-Newtonian Casson parameter represented by

ycB p/2 , the electrical conductivity represented by σ, the coefficient of thermal

expansion represented by T , the coefficient of expansion with concentration represented by

C , fluid temperature represented byT , fluid concentration represented by C , fluid density,

heat capacity and thermal conductivity represented by , pc , k respectively and thermal

diffusivity represented by pck /

The boundary conditions in the present issue are

www CCTTvxaxuu ,,0,)( , at 0y (6)

,,, CCTTuu e As y (7)

In the equation given above, wT and wC are temperature and fixation at the divider individually.

The accompanying dimensionless amounts are utilized to change the administering

conditions.

CC

CC

TT

TTfxvay

v

a

ww

)(,)(,, (8)

In the above equations the stream function represented by

Equations (3) - (5) are altered as

01

13

32

2

2

2

22

f

a

cfM

a

cfff (9)

0Pr

12

2

ff (10)

01

2

2

ff

Sc (11)

Here the subsidiary as for notation was prime. The Prandtl number /Pr , where

denotes thermal diffusivity, the Hartmann number a

BM

2

02 , the local Reynolds number

)(Re

xux w

x , the thermal buoyancy parameter2Re x

Gr ,the Grashof number



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2

3)(

xTTgGr wT

, the solutal buoyancy parameter2Re x

Gc , the local Grashof number

2

3)(

xCCgGc wC

and the Schmidt number DSc / .

The boundary conditions which are represented by equations (6) and (7) are transformed as

,1)0(,1)0(,1)0(

,0)0(

ff (12)

,0)(,0)(,)(

a

cf (13)

The physical quantities skin friction coefficient, local Nusselt number and Sherwood number

are represented by Cf, Nu and Sh respectively and are defined as

,)(

,)(

,2

CCD

qxSh

TTk

qxNu

uC

w

m

w

w

w

w

f

(14)

Here w stands for surface shear stress, wq denotes surface heat flux and mq denotes surface

mass flux and are prearranged by

000

,,2

y

m

y

w

yc

y

Bwy

CDq

y

Tkq

y

up

(15)

In the above equations denotes dynamic viscosity, k and D denotes thermal and mass

conductivity’s.

By applying similarity transformations to equation (7), we get

,)0(

Re,

)0(

Re

,)0(1

1Re2

2

xx

fx

ShNu

fC

(16)

Where

2

Rexa

x denotes local Reynolds number.

3.1The Runge–Kutta–Gill method:

By utilizing R-K-Gill strategy along the shooting system and Newton-Raphson method, the

nonlinear coupled conditions (9)–(11) with the limit conditions (12)- (13) are fathomed



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numerically. To this technique, we characterize

765432

2

21 ,,,,,, YYYYYf

Yf

Yf

(17)

72

2

652

2

433

3

22

2

1 ,,,,,, FFFFFf

Ff

Ff

(18)

Substitute into conditions (9) - (11) are transforms into an arrangement of synchronous

conditions with the qualities which are characterize in conditions (17) and (18),

,21 YF (19)

,32 YF (20)

642

2

2

2

31

2

23

11 YY

a

c

a

cYMYYYF

(21)

,54 YF (22)

),Pr( 51425 YYYYF (23)

,76 YF (24)

),( 71627 YYYYScF (25)

The boundary conditions given in (12) and (13) are altered as

,1)0(,1)0(,1)0(,0)0( 6421 YYYY (26)

,0)(,0)(,/)( 642 YYacY (27)

Here, taken as 10 , based on physical parameters.

The initial conditions are noted as tYY )0(,)0( 53 and sY )0(7 . By the utilization of

Newton–Raphson method, we calculate the values of t, and s such that they satisfy

solutions of equations (19)-(25) and also fulfill the outer boundary conditions (26) – (27). We

start this with the initial estimated values ))0(),0(),0(( st by the shooting method.

Calculations of Newton–Raphson strategy are extended to consolidate the halfway

subsidiaries concerning measurement of every factor. Coming up next are the subsidiaries of

).,..,,( 521 FFFF with respect to t, and s are as follows:

),...,,(),...,,(),...,,( 2017161512111076 FFFFFFFFFFFF tt (28)

So, we need to find ,0,0,0 st FFF respectively. Following Cebeci and Keller [32],

yields a system of algebraic equations which satisfy the boundary conditions when ξ = 0.

0,0,0 ststfsftff ststst (29)



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The system of equations which are represented by equation (29) can be written in matrix

equation AX = B:

f

s

t

fff

st

st

st

(30)

The above system of equations was solved by using Cramer’s rule and the next

approximation of t, and s can be calculated by utilizing the below formulas:

)det(

),(det

,)det(

),(det,

)det(

),(det

)()(

)()()()(

A

JIAss

A

JIAtt

A

JIA

Boldnew

BoldnewBoldnew

(31)

By the values of t, and s which are evaluated by the above equations, we apply the fourth-

order R-K-Gill method to evaluate the solutions of first order ordinary differential equations

F1, F2, . . . , F20. Following Gill [33] and R-K formula is

,6

1

2

22

3

1

2

22

3

1

6

143211 hkhkhkhkYY ii

(32)

iYFk 1 (33)

12

2k

hYFk i (34)

213

2

2121

2

1kkYFk i (35)

323

2

21

2

2kkYFk i (36)

In the above conditions h speaks to step size. Right now, acquired numerical arrangements

are seen as good when step size h= 0.01. The relative contrast between two emphases is

greatest by the pre-allotted resilience 610 to assembly. The iterative plan is abrogated

when the distinction meets the assembly criteria.

3.2 Validation of the Numerical Procedure:

To state these ends, they are connected to ends previously recorded in the overview.

Connection is performed for explicit qualities of stream parameters. These related

outcomes are appeared in Table 1, and Shehzad et al. have distributed these outcomes.



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[34] Relevance is adequate and is built up in close participation with existing ends and

with subtle blunders.

Table 1 Examination of skin friction coefficient with the reachable endings in writing for

various estimations of β and M on account of non-Newtonian fluid

).0/Pr( caSc

β

M

−(1 +1

𝛽) 𝑓′′(0)

Shehzad et al. [34] Present study

0.5

0.8

1.4

2.0

0.8

0.5

0

0.5

1.0

1.5

1.93649

1.67705

1.46385

1.36931

1.50000

1.67705

2.12132

2.70416

1.936768

1.677122

1.463867

1.369314

1.500364

1.677122

2.121321

2.704163

4 Analysis and computations:

By utilizing the RK Gill numerical strategy technique, the change conditions existing

apart from everything else, energy, and material conditions spoke to by conditions (8), (9),

and (10) are spoken to by conditions (11) and (12). The count aftereffects of RK Gill strategy

were gotten by utilizing MATLAB. The heat and mass exchange stream qualities are

demonstrated graphically, and determined and showed as tables as different parameter

esteems, specifically a/c, β, M, λ, δ, Pr, and Sc. The information Pr = 0.7, β = 0.5, M = 0.5,

λ = 1, δ = 0.5, and Sc = 0.66 (except if in any case noted) are utilized in all figures spoke to

by Figure 2-19.

Fig. 2 shows the speed bends of various impact esteems of the speed proportion c/a,

where “a” speaks to the extending pace of the limit surface and “c” (c> 0) speaks to the

pressure speed c of the stagnation point stream. At the point when c/a> 0, the stream is made

out of a transformed limit layer structure with a similar thickness as c/a. At the point when c/a

<1, the speed of the stretch sheet c�̅� is littler than the outside speed of the surface a�̅�. On the

off chance that c/a <1 the thickness of the limit layer speeds up proportion c/an increments.

Figure 3 talks about the adjustments in the Casson parameter β in the assistant and convection

speed bends. Here, all qualities of β speak to Casson liquid, and endlessness for β speaks to

Newtonian liquid. It very well may be seen that for the contrary stream case, a diminishing in

the estimation of the Casson parameter brings about a decrease in the thickness and energy of



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the speed constraining layer, and it is seen that the contrary pattern is useful for the stream

case. It has likewise been seen that Casson liquids have lower transient speed esteems than

Newtonian liquids. For the helper stream and the contrary stream, the speed bends of various

qualities of M are appeared in FIG. 4. It very well may be seen that the speed circulation

diminishes on account of helper stream and increments on account of inverse stream. Here, M

= 0 implies that there is no attractive field. This is because of the way that the use of an

attractive field to an undermined conductive liquid makes an obstruction called the Lorentz

power; this power makes the liquid speed decline. The difference in the speed bend regarding

the parameters of heat lightness (λ) and mass (δ) is appeared in Figures 5 and 6, separately.

These charts show that the speed increment with expanding λ and δ, which helps the stream,

while the contrary pattern is watched for the contrary stream. Effect of the Prandtl Pr is

number appeared in Figure 7 on the assistant and inverse procedures. An expansion in the

Prandtl number diminishes the speed of the liquid and the thickness existing apart from

everything else limit layer, which helps the stream, and the contrary movement is seen in the

contrary stream. Right now, synthesis of hydrogen is considered as follows: Sc = 0.22, water

fume SC =0.66 and carbon dioxide Sc = 0.94. Figure 8 talks about the impact of these

synthetic concoctions on the speed profile. He additionally called attention to that within the

sight of heavier species (Schmidt Lager number), the speed of the liquid in the limit layer

diminishes.

Figure 9 delineates the connection between the temperature bend and β, assistant and

inverse streams. These temperature bends start at the point of confinement of θ and afterward

reduction to zero. To help with the stream, the temperature bend is diminished as the Casson

β parameter increments. What's more, the temperature profile of Casson liquid is higher than

that of Newtonian liquid. For Carson liquids, this change is more noteworthy than for

Newtonian liquids. It was likewise recognized that the temperature profile of Carson liquid

(β> 0) was fundamentally not the same as Newtonian liquid (β = ∞). On account of

countercurrent, the temperature appropriation and the thickness of the heat limit layer become

more slender as β increments. It tends to be seen from Fig. 10 that on account of helper

stream (λ> 0), the temperature and thickness of the heat limit layer decline with the expansion

of λ, in spite of the fact that the contrary pattern will be distinguished on account of inverse

stream (λ<0). In FIG. 11, it is recognized that the thickness of the heat limit layer diminishes

with the expansion of δ on account of the helper stream (δ> 0), and switches on account of

the countercurrent (δ <0). From Figure 12, the adjustment in Pr can be distinguished in the



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temperature bend to help and contradict stream. What's more, higher Pr gauges bring down

the heat conductivity and in this way the temperature of the two streaming fluids.

The impact of β on the fixation conveyance of helper and convection is appeared in Figure

13. It tends to be seen from this figure as the helper stream area and β in β increment, the

convergence of the liquid declines. A contrary pattern was seen in the contrary stream zone.

It very well may be seen from FIGS. 14 and 15 that the convergence of the limit layer

thickness diminishes with the expansion of the heat lightness and mass lightness parameters,

and the contrary example is seen on account of invert stream. FIG. 16 shows a fixation profile

of Schmidt Sc. With the advancement of Sc, the diffusivity of the substance diminishes, so

the focus circulation diminishes. Figures 17-19 show the obstruction conduct, heat move and

mass exchange rate at various qualities of the assistant stream and the Carson parameter β in

inverse ways. An expansion in the Carson parameter β brings about a decline in the

estimation of the skin contact coefficient, a diminishing in the pace of heat and mass

exchange to convection, and a convective conduct is recognized under the state of assistant

stream.

5. Conclusion:

The motivation behind this investigation was to read mixed convection MHD for heat and

mass exchange on extended surfaces. To this end, a non-Newtonian liquid was acquired from

the Casson liquid model. The finishes of this examination are as per the following.

1. The thickness and speed appropriation of the momentum boundary layer brings about

a decrease in Hartmann number, thermal buoyancy, mass buoyancy, Prandtl number,

or Schmidt number to help stream, and the other way around on account of inverse

stream. Furthermore, for helper and inverse stream limitations, the speed circulation

and the thickness of the beat restricting layer increment as the speed proportion

parameter increments.

2. As far as possible stream, where Casson parameters are utilized, decrease the

thickness of the limit layer and lessen the temperature of the liquid, and increment the

heat or mass lightness parameters. For the contrary stream confinement, the

temperature of the liquid will increment with the Casson parameter, while the

temperature of the liquid will increment with the heat or mass convection parameter.

For the assistant stream limit and the contrary stream limit, as the quantity of Prandtl

expands, the temperature of the liquid and the thickness of the heat limit layer decline.



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3. To help stream conditions, on account of Casson parameters, both the grouping of the

liquid and the thickness of the limit layer are expanded, while the heat lightness or

mass lightness parameters are expanded. For inverse stream conditions, the

temperature of the liquid increments as the Casson parameter, heat convection, or

mass convection parameter increments. For assistant stream conditions and inverse

stream conditions, liquid fixation and focus boundary layer thickness decline with

expanding Schmidt number.

4. At last, as far as possible stream, expanding Casson parameters builds surface shear

pressure and heat and mass exchange rates. For inverse stream limitations, these

outcomes are turned around.

Fig. 1: Physical model and facilitate framework



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Fig. 2: Velocity distribution )(f for dissimilar standards of a/c.

Fig. 3: Velocity distribution )(f for dissimilar standards of β.



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Fig. 4: Velocity distribution )(f for dissimilar standards of M.

Fig. 5: Velocity distribution )(f for dissimilar standards of λ.



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Fig. 6: Velocity distribution )(f for dissimilar standards of δ.

Fig. 7: Velocity distribution )(f for dissimilar standards of Pr.



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Fig. 8: Velocity distribution )(f for dissimilar standards of Sc.

Fig. 9: Temperature distribution )( for dissimilar standards of β.



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Fig. 10: Temperature distribution )( for dissimilar standards of λ.

Fig. 11: Temperature distribution )( for dissimilar standards of δ.



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Fig. 12: Temperature distribution )( for dissimilar standards of Pr.

Fig. 13: Attentiveness distribution )( for dissimilar standards of β.



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Fig. 14: Attentiveness distribution )( for dissimilar standards of λ.

Fig. 15: Attentiveness distribution )( for distinct values of δ.



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Fig. 16: Attentiveness distribution )( for dissimilar standards of Sc.

Fig. 17: Skin friction coefficient fx C2/1Re for dissimilar standards of β for both flows.



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Fig. 18: Nusselt number Nux

2/1Re for dissimilar standards of β for both flows.

Fig. 19: Sherwood number Shx

2/1Re for dissimilar standards of β for both flows.



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Table 1 Examination of skin friction coefficient with the reachable endings in writing for

various estimations of β and M on account of non-Newtonian fluid

).0/Pr( caSc

β

M

−(1 +1

𝛽) 𝑓′′(0)

Shehzad et al. [34] Present study

0.5

0.8

1.4

2.0

0.8

0.5

0

0.5

1.0

1.5

1.93649

1.67705

1.46385

1.36931

1.50000

1.67705

2.12132

2.70416

1.936768

1.677122

1.463867

1.369314

1.500364

1.677122

2.121321

2.704163

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