Combined effects of thermal radiation and Hall current on MHD Casson nanofluid exerted by peristaltic transport

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WSN 155 (2021) 1-22 EISSN 2392-2192

Combined effects of thermal radiation and Hall

current on MHD Casson nanofluid exerted

by peristaltic transport

S. K. Asha^1,* and Chandrashekhar Hadapad²

Department of Mathematics, Karnatak University, Pavate Nagar, Dharwad 580003, India

1,2E-mail address: as.kotnur2008@gmail.com , chandruh.h4@gmail.com

ABSTRACT

The present work intends to analyse the MHD Casson nano fluid inside an asymmetric channel.

The equations governing the fluid flow admit both Thermophoresis and Brownian motion alongside thermal radiation. We acquired closed form analytical solutions for temperature, axial velocity, nanoparticle concentration and pressure gradient profiles. The ascendency of governing parameters like thermal radiation parameter, thermophoresis, Brownian, Hartmann number and Hall parameter on dimensionless temperature, concentration and pressure gradient profiles have been explained in detailed manner with the aid of plots and table. The prevailed results in this article are verified with prior available literature for specific cases and they are in good agreement.

Keywords: thermal radiation, nanofluid, MHD, Hall current, peristaltic transport

1. INTRODUCTION

The stereotyped heat transfer fluids (HTF) like ethylene glycol, oil and water, shows an intrinsic property of poor thermal behaviour. Because of their low thermal capabilities in heat transfer phenomena there is an advantage for metals and non-metals like copper, silver, SiC, Cuo shows improvised and ultra-high thermal conductivities in comparison with traditional heat

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transfer fluids (HTF). An offbeat idea of immersing solid materials in base fluids to overcome the limitation of HTFs in showing low capabilities of heat transfer was initiated by Maxwell [1]. The idea of using small 10^³or 10^⁶ meter sized solid particles, provided a route to issues like rapid settling in fluids, jamming in micro tubes, laceration of surface and also the high pressure descent which restricted them in practical applications. The present area nanotechnology furnished an opportunity to process and make particles with standard opaque sizes less than 50 nanometre which brings in base fluids an enhanced thermal behaviour in lowering substantial heat loss in heating and cooling processes.

Choi [2] had come up with fresh idea of nano-fluids by incorporating these materials which are ranges from 1-100 nm. The term nano-fluids were originated by Choi and Eastman [3]. These fluids refers to a kind of nanotechnology oriented heat transfer fluids made by immersing mili-micron sized materials inside traditional heat transfer fluids also with a substance which tends to reduce the surface tension of a fluid to enhance stability in thermal conduction usually the oxides of metals, nitrides, carbon ceramics, semiconductors, nitrides etc.

are used as nanoparticles.

Vassalo et al. [4] and You et al. [5] brought out some of the work related to fact that thermal capabilities in heat transfer processes are more improved for nano-fluids contrast to orthodox fluids. They have established that immersing less amount of nanoparticles to ordinary fluids payoff advancement in the conduction capabilities of the fluids. For further reading see refs [6-10].

In commercial applications where requirements regarding rate of cooling cannot be fulfilled by the normal heat conductive fluids due to their ineffective thermal conductivity, it can be enhanced by the application of nanoparticles wherein Brownian motion of the particle increases the capabilities which helps in thermal conductivity. Moreover the Magneto- hydrodynamic (MHD) nanofluids display a vital role in these applications. These fluids can be widely employed in modulating processes; filters which use fibres, switches, in cooling process etc. for further reading refer these [11-15].

The classification of non-Newtonian behaviour whose viscidity declines under shear strain is the casson fluid model. This is a shear-thinning or pseudoplastic fluid in which observable fluid viscosity diminishes with increased stress. The best suited example for shear thinning fluid is human blood as it permits the blood viscosity to decline when there is a rise in rate of shear strain.

Similar behaviour can be seen in materials like paint, sand in water, ketchup, soup, honey etc. this fluid model behaves like a kind of a flexible solid beyond a threshold shear stress and low shear strain. At a negligibly small rate of stress this fluid has an infinite viscidity, but the viscosity reduces to zero when raised to an infinite shear stress. For more refs [16-22].

Thermal conduction or convection flows strengthened the need of thermal radiation in day to day biomedical applications such as IR (Infrared Radiation) in treating various parts of the human body.

In compliance with the fact that radiative energy transfer does not require any intervening medium for radiant exchange between two body and also its dependency over the absolute temperature differences, radiation effect can be seen substantially in many applications like in power plants, Engines, measuring the thermal effects in Rocket nozzles, some technology incuding the utililization of solar energy radiation etc. for further reading see refs [23-28].

We have used Rosseland’s approximation for linearization of thermal radiation.

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2. PROBLEM FORMULATION

Consider a viscous, incompressible, steady two dimensional peristaltic motion of Casson nanofluid model in an asymmetric channel of fixed thicknessd₂d₁. The fluid motion inside the walls is fabricated by propagation of sinusoidal waves of less amplitudes a₁ and b₁with steady speed c of the channel walls. The mathematical description of the geometry of the conduit (Figure 1) can be written as

1 1

1

. . . . 2

, Y h a Cos X c t  d



  

      (1)

2 2

2

. . . . 2 .

, Y h a Cos X c t   d



  

        (2)

Figure 1. Schematic diagram of physical model

We have assumed Cartesian co-ordinate system ( , )X Y^{. .} where X^. and Y^. are perpendicular to one another,  is the wavelength, t^.is the time, ^. is the phase difference and with d₁ and d₂

satisfying the condition:

2 2 2 2

1 1 1 1 1 2

.

2 .

a b  a b Cosd d (3)

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Which helps in keeping the walls not intersecting each other. The effect of uniform applied magnetic field with density vector B(0, 0,B₀) can be neglected if it is assumed to be low Reynolds number. The equation for J (current density) comprising the Hall effect, ion slip, and thermoelectric is given by Ref [29].

. 1

( ) ,

e

J E V B J B

^ e n ^

      

  (4)

2 0 2

. . . .

( ), ( ), .

1

J B B U mV mU V O

m

  

     

   (5)

where J is the current density,  is the electric conductivity of the fluid, U^. and V^. are the coordinates of the velocity and ⁰

e

m e B

 e n is the Hall term. We have considered that E 0(no applied voltage), eis the electric charge and n_eis the electron density.

Velocity V^. in vector form can be written as,

. . . . . . . . . ( ( , , ), ( , , ))

V U X Y t V X Y t (6)

where U X Y t^.(^{. . .}, , ) and V X Y t^.(^., , )^{. .} are the velocity field components.

The Constitutive equation of a Casson fluid model is

( 0 )2 ,

ij 2 ij

c

y e

  

   When   _c, (7)

( 0 )2 ,

2

y

ij ij

c

e

  

   When   _c. (8)

where _y ⁰ ² ,

  the yield stress of the fluid is, ₀is the plastic dynamic viscosity of the non- Newtonian fluid,is the product of the component of the rate of deformation with itself. (i.e.

ij. ij

 e e ), e_ij -( , )i j ^thcomponent of the rate of deformation and _cis the critical value based on the non-Newtonian model.

The radiative heat flux in accordance with Rosseland’s approximation [29], _qr^. can be modelled as

3 1

. . . .

16 . 3

T T

qr y

k

 

   . (9)

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where k^. denotes the Rosseland mean absorption, and ^. is the Stefan –Boltzmann constant.

The governing equations portraying the peristaltic motion of a two dimensional steady Casson nanofluids are as follows.

. .

. . 0

U V

X Y

 

 

 

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2 2 2

0 0 0

2 2

. . . . . .

. . 1 . . . . .

1 ( ) ( ),

. . . . . .2 1 ^c

U U U P U U B

U V U gB T Tr gB C C

t X Y x X Y m

    



 

 

  

     

              

     

     

    

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2

2 2

0

2 2 2

. . . . . .

. . 1 .

1 ,

. . . . . . 1

B

V V V P V V

U V V

t X Y Y X Y m

  



 

 

 

      

          

     

    

    

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2 2

. . . . . . .

. .

( ) ( )

. . . . . . .

. . . . .

( ) . . . . .

p T m

p B r

T T T T T D T T

ct U V kT c D

t X Y X Y X Y

q

C T C T

c D

X X Y Y Y

 



 

           

           

        

       

         

  



   

  

       

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2 2 2 2

. . . . . . .

. .

. . . . . . . .

B T

m

C C C C C D T T

U V D

t X Y X Y D X Y

   

           

   

        

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where U^. and V^. are the components of velocity in X^. and Y^. axes being perpendicular to each other. PressureP^. , Hall termm, temperature distributionT^. , _p is the base fluid density, Electrical conductivity of the fluid , coefficient of volumetric expansion, strength of the applied magnetic fieldB₀, k_Tis the thermal conductivity of the fluid, reference temperatureT^.₁, concentrationC^. ,(c)_p is the nanoparticles effective heat capacity, D_Tis the coefficient of thermophoresis diffusion, D_Bcoefficient of Brownian diffusion, _Tmmean fluid temperature, _qr^. radiative heat flux,T^. is the temperature of the fluid, q is the gravity.

The corresponding dimensional boundary conditions are

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0 0

. . . . .

, , ,

U c TT CC at ₁ ₁

1

. . . . 2

Y h d a Cos X c t 



  

      ,

1 1

. . . . .

, , ,

U c TT CC at ₂ ₁

2

. . . . 2 .

Y h d b Cos X c t  



  

        . (15)

Relation between wave frame and lab frame are established by

. . . . . . . . .

, , , .

x X c t u U c yY vV (16)

The dimensionless parameters are

1 2 1

0 1

1 1 1 1

2 1 1

2 1

2 1 1

1 0 1 0

3 1 0

1 1

. . . .

, , , , , , , , ,

. . . . . . .

0, 0 , , , , , Pr , ,

. . . . ( )

. .

. . ( ) ( ) D ( )

16 . , , ( ) ,

3

f

p B

b t

f

d d a

X Y c t U

x y t d u M B d a

d d c d cd

T T C C h h V b C k

c h h v b

d d c d c

T T C C

c C C c

T cd

Rd Re N N

c v k

 

 

   

  

 

 

 

 

        

 

       

 

     ¹ ⁰

3 3

1 1 0 1 1 0

. .

( ) D

( ) ,

. . . .

( ) ( )

, , , .

p T

f m

r r

T T c T gld T T gld C C

G B u v

c c y x



  

 









 



    

       

(17)

In view of (17) equations (11)-(14), reduces to

2 2

1 1 0,

1

p u M

u Gr Br

x y m

 

 

 

       

    (18)

p 0 , y

 

 (19)

2 2 2

2 N P rt N P r_b Rd P r 2 0,

y y y

y y

    

    

       

    (20)

2 2

2 Nt 2 0.

y Nb y



  

 

  (21)

The dimensionless boundary conditions are

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1, 0, 0 1,

u  C  at yh (22)

1, 1, 1 2,

u  C  at yh (23)

3. CLOSED FORM OF SOLUTION Integrating once equation (21), we get

1,

Nt C

y Nb y



 

  

  (24)

Substituting equation (24) in equation (20) and after simplification we get,

2

2 A .

y y

 

 

 

 (25)

Implementing Adomian decomposition method for equation (25) gives

1

3 4 _yy ,

C C y L A

y

   ^ ^ ^^

Now,

0 C3 C y4 ,

  

1

1 ⁿ , 0.

n A Lyy n

y

 _  ^ ^^ ^  (26)

Following equation (26) we get,

2

1 4

2 3

2 4

3 4

4 5

4 4

2!

3!

4!

5!

AC y

A C y



 



 



 



 

and so on.

Decomposition of  as . 0 ⁿ n

 ^ 

 gives,

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3 5 2 4 4 3 4

( ) ( ) ( ) ( ) ( )

3! 5! 2! 4! 4! ,

C Ay Ay Ay Ay Ay

C Ay

  A ^^^       ^{ }^{ }    ^^^

    

 

i.e., 3 C⁴ .

C SinhAy CoshAy

   A  (27)

Integrating once again equation (24) we get

1 2.

Nt C y C Nb

     (28)

Substituting equation (27) in above equation (28) for,

 

3 4 1 2.

t b

N C

C SinhAy CoshAy C y C

N A

 

       

  (29)

Equation (18) can be rewritten as

0 0 0

2 ,

yy

L u N u A p B C

x 

  

      (30)

where L_yybeing a differential operator of second order, so L^_yy¹ is a inverse second order integration operator defined by

1(.) (.) .

yy 0 0 y y

L^    dydy (31)

Operating with L^_yy¹, equation (24) becomes

   

1 1 1 2

5 6 _yy 0 p _yy 0 _yy 0 .

u C C y L A L B L C N u

x 

     

         (32)

Equation (26) gives,

 

2 2

6 0 0 0 0

5 2 2 2

1 1

2 3

0 0 0 0

2 2 3

1 1 2 1

0 1

2

2 1

( ) ( )

( )

0 2! 2!

( )

( ) ( )

2! ( ) 3!

( )

t t t b

b b b

t b

b

C A p Ny B B F Ny

u C Ny SinhAy CoshAy

N N x A F N F

C N F Ny C N C N N Ny

SinhAy CoshAy

N N F N A F N N h h

C N N h N N h h

   

    

        

    

    

   

      

    

   

 



( )2

2! ,

 Ny 

 

 

 

 

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2 1

1 ( ), 0.

n yy n

u _ N L^ u n (33)

Following the equation (33), we get

 

2 3 4 2 4

6 0 0 0 0

1 5 2 4 2

1

4 2 5

0 0 0 0

2 4 3

1 1 2 1

0

( ) ( ) ( ) ( )

2! 3! 4! 4!

1

( )

( ) ( )

4! ( ) 5!

(

t t t b

b b b

t b

C A B N B F

Ny Ny p Ny Ny

u C SinhAy CoshAy

N N x A F N F

C N F Ny C N N C N N Ny

SinhAy CoshAy

C N N

   

    

        

    

    

   

      

    

   

  ¹ ⁴

2 2 1

) ( )

4! ,

( )

b

h Ny N N h h

 

 

 

  

 

4 5 6 4 6

6 0 0 0 0

2 5 2 6 2

1 1

4

6 7

0 0 0 0

2 6 3

1 1 2 1

0

( ) ( ) ( ) ( )

4! 5! 6! 6!

( )

( ) ( )

6! ( ) 7!

(

t t t b

b b b

t b

C A B N B F

Ny Ny p Ny Ny

N N x A F N F

SinhAy CoshAy

C N N

   

    

        

    

    

   

      

    

   

  ¹ ⁶

2 2 1

) ( )

6! ,

( )

b

h Ny N N h h

 

 

 

  

 

6

6 7 8 8

6 0 0 0 0

3 5 2 8 2

1 1

8 6 9

0 0 0 0

2 8 3

1 1 2 1

0

( ) ( ) ( ) ( )

6! 7! 8! 8!

( )

( ) ( )

8! ( ) 9!

(

t t t b

b b b

t b

C A B N B F

Ny Ny p Ny Ny

N N x A F N F

SinhAy CoshAy

C N N

   

    

        

    

    

   

      

    

   

  ¹ ⁸

2

2 1

) ( )

8! ,

( )

b

h Ny N N h h

 

 

 

  

 

2

2 2 1 2 2

6 0 0 0 0

5 2 2 2 2

1 1

0 3

2 1

( ) ( ) ( ) ( )

2 ! (2 1)! 2 ! 2 !

2 2 ( )

( ) (

0 0 0

2 1 2 ! 2 2 1 ( )

n

n n n n

n n

t b

b

C A B N B F

Ny Ny p Ny Ny

u C SinhAy CoshAy

n N n N x n A F N F n

n n

C N Ft Ny C N Nt C N N N

SinhAy CoshAy

n n N N h h

N N Fb N Ab F



   

    

          

  

 

      

2 1

0 1 2

2

2 1

) (2 1)!

( ) ( )

2 ! ,

( )

n

t b n

b

y n C N N h Ny

N N h h n

  

 

  

 

 

 

  

  

  

By the method of Decomposition by Adomian we get

. 0 ⁿ

u u

n

 

 (34)

(10)

Solution in closed form can be written as

     

 

0 0 0

6 0

5 2 2 2 2

1 1

0 0 0 0

2 2 2 3

1 1 2 1

0 1

2

2 1

1 1

( )

1 ( )

( ) ( )

( ) 1

.

( )

t t t b

b b b

t b

b

B SinhAy CoshAy B F CoshNy

C A p

u C CoshNy SinhNy CoshNy

N N x A N F N F

C N F CoshNy C N SinhAy CoshAy C N N SinhNy Ny

N N F N A N F N N h h

C N N h CoshNy N N h h

 

      

 

   

  

 

 

(35)

The volume flow rate is given as

2

1

h q u dy

h

  (36)

The instantaneous flux at any axial is given by

2

2 1

1

( , ) (1 ) .

h

Q x t u dy h h q h

      (37)

The mean volume rate of flow over one period T c

  of the peristaltic wave is

1 1

2 1

( ) 1 .

0 0

QQ dt  h h q dt  d q (38)

From equation (19), the expression for pressure gradient is,

2 2

1 1 ,

1 ^r ^r

p u M

u G B

x y m

 

 

 

      

    (39)

Table 1. Illustrates comparison of output of present study correlating with previous work for variations in Brownian motion termN_band thermophoresis termN_ton temperature and

concentration fields for a0.1,b0.2,d1,0.1,0.6,Rd0.3, Pr6.7. &

0.2, 0.5, 0.1, 0, 0.2, Pr 0.1.

b d a x  Rd  respectively.

t b

N N ( , )x y

Beg and Tripathi [30]

( , )x y



Present work

( , )x y



Beg and Tripathi [30]

( , )x y



Present work

1 0.9845 0.9764 0.8715 0.8579

2 0.9942 0.9828 0.8125 0.8020

3 1.0024 0.9971 0.7261 0.7037