View of Free Convective Couette Flow of a Dusty Fluid through a Porous Medium with Periodic Permeability in the Absence of Electrically Magneto Hydro Dynamic Model

4553

Free\Convective Couette Flow1of1a1Dusty1Fluid1through1a Porous Medium1with1Periodic1Permeability1in1the1Absence1of Electrically Magneto

Hydro Dynamic Model

K. Rajesh¹, A. Govindarajan², M. Vidhya³

1,2

Department of; Mathematics, College of; Engineering; and; Technology, SRM; Institute; of science and technology, Kattankulathur-603 203,

Kancheepuram District, Tamil Nadu, India.

³

Department of Mathematics, Sathyabama Institute of science and technology Tamilnadu, India

Email: [email protected], [email protected], [email protected]

*Corresponding Author: A. Govindarajan

Abstract

“The purpose of this investigation stands to discuss the effects of periodic permeability on1the; free1convective flow of a dusty viscous; incompressible1fluid through a1highly1porous1channel. The porous1medium is confined by an infinite perpendicular porous plate supercilious the free stream velocity to be uniform. Analytical solutions are gained for the dusty flow field,the1temperature field, the1skin1friction and the rate1of heat1transfer. when there is an increase in mass concentration1of dust1particles, it is found that the1velocity profile of fluid and dust particles reduces.”

Keywords - Couette flow; free convective; Porous medium; Periodic Permeability; Dust Parameters; Heat transfer 1. Introduction

“ Free1convection1flow1in1enclosures1has1become1increasingly significant in1engineering1applications1in recent1years outstanding1to1fact1growth1of technology, effecting1cooling1of1electronic1equations1range1from individual1transistors to1mainframe1computers.The1study1of1flow1through1porous medium finds application1in geophysics, agricultural engineering and technology to study the underground water resources.The1flow”of viscous fluids through porous medium is very much prevalent in nature: therefore, such studies have been attracting the considerable attention of engineers and scientist all over the world. Convection in porous media finds applications in oil extraction, thermal energy storage and flow through filtering devices.”1Govindarajan etal [3] discussed”3D Couette,flow of dusty fluid with transpiration cooling.”Vidhya etal [4] studied Laminar convection through,porous medium between two,vertical parallel plates with heat source. Reddy etal [5] analyzed about”Heat transfer in hydro magnetic rotating flow of viscous fluid through non-homogeneous porous medium with, constant heat”source/sink.

Raju etal [6]”described Unsteady MHD free convection and chemically reactive flow past an infinite, vertical porous plate.”Umamaheswar etal [7] deliberated that”unsteady MHD free, convective visco-elastic fluid flow bounded,by an infinite inclined porous plate in the presence of heat source, viscous dissipation,and ohmic”heating.

Ravikumar et al [8] analyzed same method for Heat,and1mass transfer. Raju etal [9] investigated Analytical study of MHD free convective, dissipative boundary layer flow past a porous vertical surface in the, presence of thermal radiation, ,chemical reaction and constant suction. Seshaiah etal [10]”studied about,the effects of chemical reaction and,radiation on unsteady MHD free convective fluid flow embedded in a porous medium,with time-dependent suction,with temperature gradient heat source.”Gurivireddy etal [11] analyzed”Thermal diffusion1effect on MHD heat and mass1transfer flow”rections.Reddy etal [12] undertaken research on Unsteady MHD Free Convection, Flow Characteristics of a,Viscoelastic Fluid Past a Vertical,Porous Plate. Raju etal [13] investigated”Unsteady free convection flow past a periodically1accelerated vertical plate with Newtonian heating.”Umamaheswar etal [14]

spotted that Combined, Radiation in fluid flow. Mamatha etal [15] reported Thermal diffusion in MHD. Rao etal [16] conveyed about”MHD,transient free convection and chemically reactive,flow past a porous vertical plate with radiation and temperature gradient dependent heat source in slip flow”regime. Reddy etal [17] confirmed that Magneto,convective flow with non-Newtonian fluid ,porous with

variable suction. Raju etal [18] established the Radiation”absorption effect on MHD free convection chemically,reacting visco-elastic fluid past an oscillatory vertical porous plate,in”slip regime. Raju etal [19] studied about Unsteady MHD thermal diffusive, radiative and free,convective flow past a vertical porous plate,through non- homogeneous porous medium., Raju etal [20] investigated on Heat transferAeffects magnetic field. M.Vidhya etal [21] discussed that”Free,convective and oscillatory flow of a dusty fluid through a porous,medium.”Sharma etal [22]

reported Rotational impactAon”micro polar fluid past a1semi-infinite vertical1porous plate with1suction.”Mohan etal [24] states”that Thermal Radiation,and Chemical Reaction Effects on Unsteady MHD Free Convection Flow of a Viscous Dissipative Casson Fluid Past an Exponentially Infinite Vertical Plate Through Porous Medium with TGHS.”Kumar etal [25] undertaken research,on Unsteady MHD free convective flow of a radiating fluid past an inclined permeable plate in the presence of heat source. Swapna etal [26] analyzed chemical reaction and thermal vertical porous plates. Chitra etal [27] states that Heat and Mass transfer on unsteady MHD flow past an infinite vertical plate through a porous1medium with time varying pressure gradient. Lalitha etal [28] deliberated Effects of chemical reaction and heat generation on MHD free convective oscillatory1couette flow through a variable porous medium. Swarnalathamma etal [29] described about Combined Effects on Unsteady MHD Convective flow of Rotating Viscous Fluid through a Porous Medium,over a Moving Vertical Plate. Mohan etal [30] found that Thermal Diffusion1and TGHS Effect on MHD Viscous1Dissipative Kuvshinski´ S Fluid Past an Inclined1Plate Through Porous Medium with Thermal,Radiation and Chemical Reaction. Veeresh etal [31] presented about”Thermal Diffusion Effects on Unsteady Magnetohydrodynamic”Boundary”Layer Slip Flow past a Vertical Permeable Plate.”Varma etal [32] investigated”MHD‎ rotating heat and mass transfer, free convective flow”in mass‎ diffusion.

Raju etal [33] described about”Thermal diffusion and rotational effects”on magneto1hydrodynamic”mixed convection flow of heat absorbing/generating visco-elastic fluid through a porous channel.1Frontiers in Heat and Mass Transfer.”Sivaiah etal [34] investigated the”Numerical study of mhd boundary layer flow of a viscoelastic1and dissipative fluid past a porous plate,in the presence of thermal radiation.”

“The purpose1of this present paper is to study the effect of1transverse1periodic variation1of1the1permeability on the1heat1transfer1and1the1free1convective flow of1a dusty viscous1incompressible1fluid. The purpose of this is to study the effect of permeability parameter, Grashof number and dust parameters1on1the1velocity1field, temperature1field, skin1friction and nusselt number. Analytical solutions remain given through graph.”

2. MATHEMATICAL MODELLING:

“We consider1the1flow of a1viscous incompressible dusty fluid1through a highly porous1medium1bounded by an infinite1vertical1porous1plate, with constant suction. The1plate1lying1vertically1on x* - z*1plane1with x*1axis occupied along1the plate1in upward1direction.1The y*-axis is taken1normal to the plane1of the plate and directed into the dusty fluid flowing1laminarly with a uniform1free stream velocity1U.”The permeability of the1porous medium1is assumed to be of the form,”



 



 



 



 



l z z K

K ^o

cos * 1

*) * (

*  

(1)

The1problem becomes 3-Dimensional1due1to such1a1permeability1variation. All the fluid assets1are supposed constant1except that1the1influence of the density1variation with1temperature being1measured only in the body energy term.

Thus1denoting the1velocity1components by1u*, 1v*,1 w* 1in x*,1 y*,1 z* - directions for the fluid and up*, 1vp*, wp* 1to be the1components,in,x*, y*, z* direction for the dust particles, the flow 1through1 a highly porous 1medium is1 governed by the following1 equations:

4555 Equation of Continuity in Fluid Phase:

0

*

*

*

* 







 z w y

v (2)

Equation of Motion in x direction in Fluid phase

2 2

0

2 2

* * * *

* * ( * *) ( * ) ( * *)

* * * * * ^p

KN

u u u u

v w g T T u U u u

y z y z K

  

 

 

   

         

     (3)

Equation of Motion in y direction in Fluid phase

2 2

0

2 2

* * 1 * * * *

* * ( * *)

* * * * * * ^p

KN

v v p v v v

v w v v

y z y y z K

 

 

 

          

      (4)

Equation of Motion in z-direction in Fluid phase

2 2

0

2 2

* * 1 * * * *

* * ( * *)

* * * * * * ^p

KN

w w p w w w

v w w w

y z z y z K

 

 

 

           (5)

Energy Equation is

2 2

0

2 2

* * * *

* * ( * *)

* * * *

s p

p p T

N mC

T T k T T

v w T T

y z C y z  C

 

        (6)

Equations of Continuity in Particles Phase:

* 0

*

*

* 



 





z w y

v_{p p}

(7)

Equation of Motion in x direction in Particle1phase

* *

* * ( * *)

* *

p p

p p p

u u K

v w u u

y z m

 

  

  (8)

Equation of Motion in y direction in Particle phase

*)

*

* (

* *

*

*

^p

*

_p ^p _p

p v v

m K z

w v y

v v  



 



 (9)

Equation of Motion in z direction in Particle phase

*)

*

* (

* *

*

*

^p

*

_p ^p _p

p w w

m K z

w w y

v w  



 



 (10)

Energy Equation in Particle phase



 



 







 







 





T p p p

p p p

T T z

w T y v T



*

*

*

* *

*

* * (11)

The boundary1conditions for,fluid phase:

y* = 0 : 1u* = 0, 1v* = -V, 1w* = 0, 1T* = TW*,1 y*→∞ :1 u* = U, 1w* = 0, 1p* = p∞*,1T* = T∞*.

(12) The boundary conditions for particle phase:

y* = 0: up* = 0, vp* = +V, wp* = 0, Tp* = TW*, y*‎→‎∞‎:‎up* = U, wp* = 0, Tp* = T∞*. (13) We introduce the following Non dimensional Variables for fluid phase:

* ,

* ,

*,

*,

*,

*,

V2

p p

V w w V v v U u u l z z l y y













*

*

*

*







 

T T

T T

W

 ,

p p s

C C l

m V

f N  

^{0 , ,}

For Particle phase:

*,

*,

*,

V w w V v v U

u_p  u^p _p  ^p _p  ^p

*

*

*

*







 

T T

T T

W p

p

Equations (2) to (11) take the following,forms

After introducing the Non-Dimensional quantities, the equation of motion for Fluid Phase is obtaining as

0







 z w y

v (14)

2 2

2 2

0

1 ( 1)[1 cos ]

Re ( )

Re Re ^p

u u u u u z f

v w G u u

y z y z K

 

  

           

     

(15)

) Re (

] cos 1 [ Re

1

0 2

2 2 2

v f v K

v z z

v y

v y p z w v y

v v _p



 



 











 





 

 



   (16)

) Re (

] cos 1 [ Re

1

0 2

2 2 2

w f w K

w z z

w y

w z

p z w w y

v w _p



 



 











 





 

 



   (17)

) Pr(

3 2 Pr

Re 1

2 2 2

2   



 _

 



 











 



 





p

f z

z y y w

v (18)

After introducing the Non-Dimensional quantities, the equation of motion for Particle Phase is obtaining as

0









z w y v_{p p}

(19)

) 1(

u z u

w u y

v_p u^p _p ^p _p 



 

 



 (20)

4557 )

1(

v z v

w v y

v_p v^p _p ^p _p 



 

 





(21)

) 1(

w z w

w w y

v_p w^p _p ^p _p 



 

 





(22)

) (  



  



 





p p

p p

p

a

w z

v y

(23)

where

2

*)

* (

UV T g T

G  ^W  ^ ,



 Vl

Re ,

k C_p

 

Pr ,

2 0 0

* l k  K

The1corresponding1boundary1conditions become

y = 0: 1 u = 0, 1v = -1, 1w = 0,  = 1, 1up = 0, 1vp = 1, 1wp = 0, 1p =1, (24)

y‎→∞:‎1u = 1, 1 w = 0, 1p = p, 1 = 0, 1up = 1, 1wp = 0, 1p = 0 (25) In order to obtain the solution of problem we expand the velocity components, pressure and temperature fields in

powers of the amplitude.

h(y,z) = h0(y) +h1(y,z) + ²h2(y,z) (26) Where h positions for u, up, 1v, 1vp, 1w, 1wp, 1,1p, 1p

when  = 0, the”problem1reduces1to the1two-dimensional1free1convective1dusty1flow1through1a1porous medium with1constant1permeability1which1is1governed,by1the1equations and the matching1boundary1conditions.The solution1of this 2- dimensional1problem is”

v0 = -1, vp0 = 1,p0 = p∞, w0 = 0, wp0 = 0, (27) e^my

0 

 (28)

m a

me ae ^{my ay}

P 

 ^  ^



0

(29)

Ry

my G e

e G

u₀ 1 ₀ ^ ( ₀1) ^ (30)

1 )

1

(

_{1 2 2 1}

0   ^  ^Ry  ^my

y

p a a e a e ae

u (31)

when   0, substituting (25) and the non-dimensional eqn.

z z k

K( ) 1  cos⁰ 

 

(32) For”periodic1permeability mad about the1equations (14) to (23) and comparation1the coefficients of1identical powers1of1, neglecting1those of ², ³ etc., we get the first orde1requations and the corresponding boundary conditions with help of (26). These equations are the partial1differential1equations which describe free1convective 3-Dimensional dusty flow. For solution we shall first consider

equations are the temperature field and independent of the main dusty flow. We assume v1, 1 v p1, 1w1, 1 wp1, 1p11of1the1form:

z y

v z y

v₁( , ) ₁₁( ) cos



(33)

z y v z y w   ^{( ) sin} ) 1 , ( ₁₁ 1   (34)

z y p z y p₁( , ) ₁₁( ) cos



(35)

z y v z y v_P1( , ) _P11( ) cos



(36)

z y v z y w_{P P}

  ^{( ) sin} ) 1 , (

₁₁ 1   (37)

“wherever the1prime in v₁₁ (y) 1enotes the differentiation1with1respect1to‎„y‟‎Expressions1for1v1(y,z), w1(y,z) 1and vp1(y,z), wp1(y,z) have1been1chosen1so that1the equations1of,continuity1are1satisfied. Substituting the expressions (32) to (36) into continuity equations and explaining under the corresponding1altered boundary conditions” as we get solutions1of v1, vp1, w1, wp1, p1 as: k z e z e y v y y         ^{ }

) cos )( 1 ( ) ) ( , (

0 1 2       ^{ } (38) 1 2 0 1 y y w ( y,z ) ( e e ) sin z ( k ) ( )            (39)

M k z yz e p y ) 1 )( ( ) cos ( 0 1  2    ^      ^ (40)

z a e a e a e a a v

^{y y} y p

1 (

^{ }

) cos 

6 5 4 3 7 1



^



^



^



(41)

    a e

^

a e

^

^z e a a w

^{y y} y p

) sin 1 (

5 4 8 7 1    

   

(42)

for the main flow1and1temperature1field1solution we1assume u1, up1, 1, P1 as per u1 (y,z) 1= 1u11 (y) cos z (43)

1 (y,z) 1= 1₁₁ (y) cos z (44)

up1 (y,z) = 1up11 (y) cos z (45)

P1 (y,z) = 1P11 (y) cos z (46) Substitution of (42), (43), (44) & (45) into the partial1differential equations and reduce1them to the1ordinary

4559 differential1equations.

2 1

11 11 11 11 11 0

2 2

3 3 ^P

f f

Re Pr  Re Re Re PrV

 

               (47)

' 11 11

) 11

(Da_p a v_p _po

(48) with corresponding1boundary1conditions

y = 0 : u11 = 0, 11 = 0, up11 = 0, p11 = 0,

y‎→∞‎:‎up11 = 0, u11 = 0, 11 = 0, p11 = 0 (49)

1 1

( ) ( )

( ) ( ) ( )

1 [c e1 ^y A e2 ^{ m y} A e3 ^my A e4 ^{ a y} A e5 ^{a y} A e6 ^{ a y} A e7 ^ay A e8 ^{m y}]cos z

  ^  ^{ }  ^  ^{ }  ^{ }  ^{ }  ^  ^  (50)

1 2 1 2 3 4 5 6 7

1 1 1 1 2

8 9 10 11 12 13 14

1

15 16 17

y my Ry ( m ) y ( m ) y ( R ) y y ( a ) y

( a ) y ( a ) y ay ( m ) y ( ) y ( ) y y

y ( R ) y ( R ) y

u [ c e T e T e T e T e T e T e T e

T e T e T e T e T e T e T e

T e T e T e ] cos z

           

           

    

        

     

   

(51)

θp11=



^A₁ ^yA₂



^{e3Λ2fy A e}₃ ^{(λ r)y A e}₄ ^{(π r)y A e}₅ ^{λ 3Λ2f y}

(π 3Λ2f)y ry my y m)y

A e6 A e7 A e8 A e9 A10e

cos z (

 

 

 

 

 

  

    



          

 

 

 

 

 

 

(52)

up11=

19 20 21 1 22 23

24 25 26 27 28

2

29 30 31 32 33 3

2 2

3 3

34 35

( m )y ( m )y m y ( Re Pr)y ( Re Pr)y

A e A e A e A e A e

Re Pr y y ( r )y ( m )y ( m r )y

A e A e A e A e A e

( f )y

( m )y ry my y

A e A e A e A e A e

f f

( )y y

A e A e

           

          

      

     

     

 











cos z











(53)

Where A₁ to A₃₅ are known constants and not presented here for the sake of brevity 3. RESULTS AND DISCUSSION

(a) VELOCITY PROFILES FOR THE FLUID PHASE:

We make the following conclusions from fig (1and 2)

“The1velocity1profiles of1both1the1fluid and the1dust1decrease with either1an1increase1in1the mass concentration of1the1dust1particles (or) increase in the Grashof number (or)”increase in the permeability of the porous medium (or) increase in the Reynolds number.”All the profiles obtain their maximum value very near the lower plate as it1maintain an increasing1trend near the lower plate and thereafter they become constant and reach the value 1 at the other plate. This graph is drawn for (Gr>0) which corresponds to1cooling of1the1plate. While (Gr

< 0) corresponds to heating1of1the1plate for which1the profiles maintain a reversing trend in its behaviour. That is

the reason for not drawing a separate graph for (G<0).”

Both the dust particles and1the fluid perform in the1equal manner. But the1profiles of the dust are at a lower height as compared with the fluid. The entire curve coincides very near the upper plate for both fluid and dust.

(b) SKIN FRICTION:

“From1table11 it1is clear that1the coefficient of skin friction Tx increases1with an increase1in mass concentration of the dust particles (or) an1increase in Prandtl number while it decreases with an increase1in Grashof number (or) increase1in the1permeability1of the1porous1medium (or) increase in Reynolds number.

For clean fluid (f = 0) the skin friction1coefficient decreases with decreasing permeability k0 of the porous medium it also decreases with increase of Grashof” number. The values of Tx are less in air,( Pr == 0.7) 1and more in water 1 (Pr = 7.0). An1increase in the Reynolds1number1leads1to decrease1in Tx.”

(c) NUSSELT1NUMBER:

From table 2 it clear that the coefficient of Nusselt number Nu in the case of water (Pr = 7.0) increases1with an1increase1in1mass1concentration1of1the1dust1particles (or) an1increase in Reynolds1number. While it1decreases1with an1increase in the permeability of the porous medium

From table 3 it is clear that the coefficient1of Nusselt number Nu in the case of air (Pr = 0.7) increase1in the1permeability1of1porous1medium (or) increase in1the Reynolds number (or)decreases 1with1an1increase1in mass1concentration1of1the1dust1particles. Nu is positive1in1the1case1of1water1and1negative1in1the1case1of air.

Nu for air is less when compared with Nu for water.

For clean fluid (f = 0) an increasing in the1permeability of1the porous1medium1leads1to1a decrease Nu. The values of Nu are much less in1the1case1of1water (Pr=7.0) 1than1in1the1case1of1air (Pr=0.7). Nu decreases with increasing Reynolds number.

Pr = 0.7,  =‎0.2,‎ε‎=‎0.2

4561 Pr = 0.7,  =‎0.2,‎ε‎=‎0.2

Table – 11Values1of1Skin1friction1coefficient1at1the plate1when ε = 0.2, 1 z = 0, 1 = 0.2

Tx f G k0 Pr

Re

0.5 1 1.5

TX1 0.2 1 0.5 0.7 3.7680 3.0076 2.2522 TX2 0.2 3 0.5 0.7 0.6118 -1.0163 -2.8819 TX3 0.4 1 0.5 0.7 3.4020 3.2015 3.1029 TX4 0.4 1 0.6 0.7 3.2479 3.1642 3.1228 TX5 0.4 1 0.6 7.0 3.3395 3.2610 3.1894

Table – 21Values1of1Nusselt1Number1at1the1plate1when ε = 0.2, 1z = 0,  = 0.2, 1Pr = 7.0

Nu f k0

Re

0.5 1 1.5

Nu1 0.2 0.5 2.0843 4.0392 6.4689

Nu2 0.4 0.5 2.1811 4.1182 6.5534

Nu3 0.4 0.6 1.2234 2.7440 4.7101

Table – 3 Values1of1Nusselt1Number1at1the1plate1when ε = 0.2, 1 z = 0, 1  = 0.2, 1Pr = 0.7

Nu f k0

Re

0.5 1 1.5

Nu1 0.2 0.5 -0.2431 -0.9030 -1.3434

Nu2 0.4 0.5 -2.2646 -2.5484 -2.7220

Nu3 0.4 0.6 -2.6507 -2.8996 -3.0537

6. Conclusion

“” Velocity profile for both fluid and dust decreases after there1is1an1increase1in1mass1concentration1of1the dust1particles Grashof1number., Permeability and Reynolds number. As a result the profile of the1dust are at lower height as compared with the fluid Tx decreases with decreasing k0 and increasing Gr for clean fluid (f=0). For increasing k0 leads to decrease in Nu for clean (f=0). The values of T are less in air (Pr=0.7) and more in water (Pr=7.0) whereas the values in Nu is more in air (Pr=0.7) and less in water (Pr=7.0).”

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