4553

**Free\Convective Couette Flow1of1a1Dusty1Fluid1through1a Porous ** **Medium1with1Periodic1Permeability1in1the1Absence1of Electrically Magneto **

**Hydro Dynamic Model**

**K. Rajesh**^{1}**, A. Govindarajan**^{2}**, M. Vidhya**^{3}

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 on* 1*the

*free1convective flow of a dusty viscous*

**;***incompressible*

**;***fluid through a*

**1***highly*

**1***porous*

**1***channel. The porous*

**1***medium 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, the1skin*

**1***friction and the rate1of heat1transfer. when there is an increase in mass concentration*

**1***of dust*

**1***particles, it is found that the*

**1****1**velocity profile of fluid and dust particles reduces.”

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

“ Free*1*convection*1*flow*1*in*1*enclosures*1*has*1*become*1*increasingly significant in*1*engineering*1*applications*1*in
recent*1*years outstanding*1*to*1*fact*1*growth*1*of technology, effecting*1*cooling*1*of*1*electronic*1*equations*1*range*1*from
individual*1*transistors to*1*mainframe*1*computers.The*1*study*1*of*1*flow*1*through*1*porous medium finds application*1*in
geophysics, agricultural engineering and technology to study the underground water resources.The*1*flow”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.”*1*Govindarajan 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,and*1*mass 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 porous*1*medium with time varying pressure gradient. Lalitha etal [28] deliberated Effects of
chemical reaction and heat generation on MHD free convective oscillatory*1*couette 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 Inclined*1*Plate 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 magneto*1*hydrodynamic”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 viscoelastic*1*and
dissipative fluid past a porous plate,in the presence of thermal radiation.”

“The purpose*1*of this present paper is to study the effect of*1*transverse*1*periodic variation*1*of*1*the*1*permeability
on the*1*heat*1*transfer*1*and*1*the*1*free*1*convective flow of*1*a dusty viscous*1*incompressible*1*fluid. The purpose of this is
to study the effect of permeability parameter, Grashof number and dust parameters*1*on*1*the*1*velocity*1*field,
temperature*1*field, skin*1*friction and nusselt number. Analytical solutions remain given through graph.”

**2. MATHEMATICAL MODELLING: **

“We consider*1*the*1*flow of a*1*viscous incompressible dusty fluid*1*through a highly porous*1*medium*1*bounded by
an infinite*1*vertical*1*porous*1*plate, with constant suction. The*1*plate*1*lying*1*vertically*1*on x* - z**1*plane*1*with x**1*axis
occupied along*1*the plate*1*in upward*1*direction.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)

The*1*problem becomes 3-Dimensional*1*due*1*to such*1*a*1*permeability*1*variation. All the fluid assets*1*are supposed
constant*1*except that*1*the*1*influence of the density*1*variation with*1*temperature being*1*measured only in the body
energy term.

Thus*1*denoting the*1*velocity*1*components by*1*u*, *1*v*,*1* w* *1*in x*,*1* y*,*1* z* - directions for the fluid and up*, *1*vp*,
wp* *1*to be the*1*components,in,x*, y*, z* direction for the dust particles, the flow *1*through*1* a highly porous
*1*medium is*1* governed by the following*1* 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 Particle*1*phase

* *

* * ( * *)

* *

*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 boundary***1***conditions for,fluid phase: **

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

* ,

* ,

*,

*,

*,

*,

*V*2

*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*

The*1*corresponding*1*boundary*1*conditions become

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

y→∞:*1*u = 1,* 1* w = 0, *1*p = p, *1* = 0, *1*up = 1, *1*wp = 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) + ^{2}h2(y,z) (26)
Where h positions for u, up,* 1*v,* 1*vp,* 1*w,* 1*wp,* 1*,*1*p,* 1*p

when = 0, the*”*problem*1*reduces*1*to the*1*two-dimensional*1*free*1*convective*1*dusty*1*flow*1*through*1*a*1*porous medium
with*1*constant*1*permeability*1*which*1*is*1*governed,by*1*the*1*equations and the matching*1*boundary*1*conditions.The
solution*1*of this 2- dimensional*1*problem 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*

*u*_{0} 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 cos^{0}

(32)
For”periodic*1*permeability mad about the*1*equations (14) to (23) and comparation*1*the coefficients of*1*identical
powers*1*of*1*, neglecting*1*those of ^{2}, ^{3} etc., we get the first orde*1*requations and the corresponding boundary
conditions with help of (26). These equations are the partial*1*differential*1*equations which describe free*1*convective
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, *1*w1,* 1* wp1,
*1*p1*1*of*1*the*1*form:

*z*
*y*

*v*
*z*
*y*

*v*_{1}( , ) _{11}( ) cos

###

_{ }(33)

*z*
*y*
*v*
*z*
*y*
*w*
^{(} ^{)} ^{sin}
) 1
,
( _{11}
1
(34)

*z*
*y*
*p*
*z*
*y*
*p*_{1}( , ) _{11}( ) cos

###

(35)*z*
*y*
*v*
*z*
*y*
*v*_{P}_{1}( , ) _{P}_{11}( ) cos

###

(36)*z*
*y*
*v*
*z*
*y*
*w*_{P}_{P}

### ^{(} ^{)} ^{sin} ) 1 , (

_{11}1 (37)

“wherever the*1*prime in *v*_{11} (*y*) *1*enotes the differentiation*1*with*1*respect*1*to„y‟Expressions*1*for1v1(y,z), w1(y,z)* 1*and
vp1(y,z), wp1(y,z) have*1*been*1*chosen*1*so that*1*the equations*1*of,continuity*1*are*1*satisfied. Substituting the expressions
(32) to (36) into continuity equations and explaining under the corresponding*1*altered boundary conditions” as we
get solutions*1*of 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*

^{z}

^{y}

^{y}*y*

*p*

### ) sin 1 (

5 4 8 7 1 ###

(42)for the main flow*1*and*1*temperature*1*field*1*solution we*1*assume u1, up1, 1, P1 as per
u1 (y,z) *1*= *1*u11 (y) cos z (43)

1 (y,z) *1*= *1*_{11} (y) cos z (44)

up1 (y,z) = *1*up11 (y) cos z (45)

P1 (y,z) = *1*P11 (y) cos z (46)
Substitution of (42), (43), (44) & (45) into the partial*1*differential equations and reduce*1*them to the*1*ordinary

4559
differential*1*equations.

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 corresponding*1*boundary*1*conditions

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

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

1 1

( ) ( )

( ) ( ) ( )

1 [*c e*1 ^{}^{y}*A e*2 ^{} ^{m y}*A e*3 ^{my}*A e*4 ^{} ^{a y}*A e*5 ^{a y}*A e*6 ^{} ^{a y}*A e*7 ^{ay}*A e*8 * ^{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=

##

^{A}

_{1}

^{yA}

_{2}

##

^{e}

^{3Λ}

^{2f}

^{y}

^{A e}

_{3}

^{(λ r)y}

^{A e}

_{4}

^{(π r)y}

^{A e}

_{5}

^{λ}

^{3Λ}

^{2f}

^{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 A_{1} to A_{35} 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)

“The*1*velocity*1*profiles of*1*both*1*the*1*fluid and the*1*dust*1*decrease with either*1*an*1*increase*1*in*1*the mass
concentration of*1*the*1*dust*1*particles (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 increasing*1*trend 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 to*1*cooling of*1*the*1*plate. While (Gr

< 0) corresponds to heating*1*of*1*the*1*plate for which*1*the 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 and*1*the fluid perform in the*1*equal manner. But the*1*profiles 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: **

“From*1*table*1*1 it*1*is clear that*1*the coefficient of skin friction Tx increases*1*with an increase*1*in mass
concentration of the dust particles (or) an1increase in Prandtl number while it decreases with an increase*1*in Grashof
number (or) increase*1*in the*1*permeability*1*of the*1*porous*1*medium (or) increase in Reynolds number.

For clean fluid (f = 0) the skin friction*1*coefficient 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) *1*and more
in water *1* (Pr = 7.0). An*1*increase in the Reynolds*1*number*1*leads*1*to decrease*1*in Tx.”

**(c) NUSSELT***1***NUMBER: **

From table 2 it clear that the coefficient of Nusselt number Nu in the case of water (Pr = 7.0) increases*1*with
an*1*increase*1*in*1*mass*1*concentration*1*of*1*the*1*dust*1*particles (or) an*1*increase in Reynolds*1*number. While
it*1*decreases*1*with an*1*increase in the permeability of the porous medium

From table 3 it is clear that the coefficient*1*of Nusselt number Nu in the case of air (Pr = 0.7) increase*1*in
the*1*permeability*1*of*1*porous*1*medium (or) increase in*1*the Reynolds number (or)decreases *1*with*1*an*1*increase*1*in
mass*1*concentration*1*of*1*the*1*dust*1*particles. Nu is positive*1*in*1*the*1*case*1*of*1*water*1*and*1*negative*1*in*1*the*1*case*1*of air.

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

For clean fluid (f = 0) an increasing in the*1*permeability of*1*the porous*1*medium*1*leads*1*to*1*a decrease Nu. The values
of Nu are much less in*1*the*1*case*1*of*1*water (Pr=7.0)* 1*than*1*in*1*the*1*case*1*of*1*air (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 – 1****1****Values****1****of****1****Skin****1****friction****1****coefficient****1****at****1****the plate****1****when ε = 0.2,**** 1**** z = 0, *** 1*

**= 0.2**

**T****x** **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 – 2****1****Values****1****of****1****Nusselt****1****Number****1****at****1****the****1****plate****1****when ε = 0.2,**** 1****z = 0, **** = 0.2,**** 1****Pr = 7.0 **

**Nu ** **f ** **k****0**

**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 Values****1****of****1****Nusselt****1****Number****1****at****1****the****1****plate****1****when ε = 0.2,**** 1**** z = 0,*** 1*

**= 0.2,**

**1****Pr = 0.7**

**Nu ** **f ** **k****0**

**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 there*1*is*1*an*1*increase*1*in*1*mass*1*concentration*1*of*1*the
dust*1*particles Grashof1number., Permeability and Reynolds number. As a result the profile of the1dust are at lower
height as compared with the fluid **T****x** decreases with decreasing **k****0** and increasing Gr for clean fluid (f=0). For
increasing **k****0** 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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