Peristaltic Transport of a Couple-stress Fluid with nanoparticles having permeable

walls

K. Maruthi Prasad1, N. Subadra*2 and S. Karunakar Reddy3

1Department of Mathematics, School of Technology, GITAM University, Hyderabad Campus, Hyderabad, Telangana, India-502329.

Email: kaipa_maruthi@yahoo.com2Department of Mathematics, Geethanjali College of Engg.& Tech., Cheeryal (V),

Keesara (M), R.R. Dist., Telangana, India-501301.Email: nemani.subhadra@gmail.com

3Department of Mathematics, JNTUH, Kukatpally, Hyderabad, Telangana, India-500085.Email: karunakarreddy.shadagonda@gmail.com

* Corresponding author: nemani.subhadra@gmail.com

Abstract: The paper deals with a theoretical investigation of the peristaltic transport of a

couple-stress fluid with heat and mass transfer effects. The velocity, pressure drop, time

averaged flux, frictional force, mechanical efficiency, temperature profile, nano particle

phenomena, heat transfer coefficient and mass transfer coefficient of the fluid are

investigated, when the Reynolds number is small and wave length is large by using

appropriate analytical methods. Effects of different physical parameters like couple-stress

fluid parameters, Brownian motion parameter, thermophoresis parameter, local temperature

Grashof number as well as local nano particle Grashof number on pressure drop

characteristics, frictional force, heat transfer coefficient, mass transfer coefficient and steam

line patterns of the fluid are studied. The expressions for pressure drop, temperature profile,

nano particle phenomenon, heat transfer coefficient and mass transfer coefficients are

sketched through graphs. The streamlines are drawn to discuss trapping phenomena for some

physical quantities.

Keywords: Peristalsis, Couple-stress fluid, Brownian motion parameter, Thermophoresis

parameter, Heat transfer coefficient, Mass transfer coefficient.

1. Introduction

Peristalsis is an important mechanism for fluid transport which can be generated by the

propagation of waves along the walls of a flexible tube containing liquid. Physiologically, it

is known as an automatic and vital process that drives the urine from kidney to bladder

through the ureters, food through the digestive tract, bile from the gall bladder into the

duodenum, vasomotion in small blood vessels and so on. The peristaltic transport is also

exploited in industrial pumping as it provides efficient means for sanitary fluid transport in

nuclear industries and in roller pumps. Several investigations have analyzed the peristaltic

motion of both Newtonian and non-Newtonian fluids in physiological as well as mechanical

systems (Fung & Yih, [1], Shapiro, Jaffrin, & Weinberg, [2], Shehawey & Sebaei, [3],

Radhakrishnamacharya & Srinivasulu, [4], Alemayehu & Radhakrishnamacharya, [5],

Prasad et al. [6]).

Couple-stress fluid is a special case of non-Newtonian fluid which was developed by

Stokes, [7]. The important feature of these fluids is that the stress tensor is not symmetric and

their accurate flow behaviour cannot be predicted by the classical Newtonian theory. The

main effect of couple stresses will to be introducing a size dependent effect that is not present

in the classical viscous theories. Shehawey & Mekheimer, [8] studied Couple-stresses in

peristaltic transport of fluids. Effect of peripheral layer on peristaltic transport of a couple-

stress fluid was investigated by Prasad & Radhakrishnamacharya, [9]. Alemayehu &

Radhakrishnamacharya, [5] discussed dispersion of a solute in peristaltic motion of a couple-

stress fluid through a porous medium with slip condition. Hydromagnetic effect on inclined

peristaltic flow of a couple stress fluid was investigated by Shit & Roy, [10].

Nanofluid is a fluid containing nano-sized particles called nanoparticles. These fluids are

engineered colloidal suspensions of nanoparticles in a base fluid. The nanoparticles used in

nanofluids are typically made of metals, oxides, or carbon nanotubes. Nanofluids have new

properties that make them potentially useful in many applications in heat transfer, including

microelectronics, fuel cells, pharmaceutical processes and hybrid powered engines (Das,

Putra, & Roetzel, [11]). They exhibit enhanced thermal conductivity and the convective heat

transfer coefficient compared to the base fluid. Many researchers done their research in

nanofluid technology (Noreen Sher Akbar, [12]), Akbar & Nadeem, [13], Prasad et al. [14].

Most of the researchers have done their study using no slip boundary condition at the

walls of the vessels. But the blood vessel walls may be movable, flexible and permeable in

nature. Chu & Fang, [15] investigated peristaltic transport in a slip flow. Slip effects on

peristaltic transport of power-law fluid through an inclined tube was studied by Naby &

Shamy, [16]. Hayat et al. [17] studied slip effects on peristaltic transport in an inclined

channel with mass transfer and chemical reaction.

Motivated by the above studies, peristaltic transport of a couple-stress fluid with

nanoparticles having permeable walls under the assumption of long wavelength and low

Reynolds number have been investigated. Homotopy perturbation method is used to solve

coupled equations of temperature profile and nanoparticle phenomena. The analytical

solutions of velocity, pressure drop and frictional force are obtained. The effects of various

parameters on these flow variables are investigated.

2. Mathematical Formulation

Consider the peristaltic transport of an incompressible couple stress fluid with

nanoparticles in a tube of uniform cross section of radius ‘a’ with sinusoidal wave travelling

along the boundary of the tube with speed c , amplitude band wave lengthλ. Choosing the

cylindrical polar coordinate system (R ,θ ,Z ) , the wall deformation due to the propagation of

an infinite train of peristaltic waves is given by

R=H ( z , t )=a+b sin 2πλ

(Z−ct) (1)

The governing equations in the fixed frame for an incompressible couple-stress fluid

flow with nano particles in the absence of body moment and body couple are given as (Maiti

et al. [18])

T ji . j=ρd w i

dt(2)

e ijkT jkA +M ji. j=0 (3)

lij=−pδij+2μ ijd ij (4)

μij=4ηω j .i+4 η'ωij (5)

(ρc )fdT '

dt=k∇2T '+( ρc )p[DB∇C

' .∇T '+DT

T0' ∇T

' .∇T ' ] (6)

dC '

dt=DB∇

2C '+[DT '

T0' ]∇2T ' (7)

where w i is the velocity vector, T ij and T ijA are the symmetric and antisymmetric parts of

the tensor T ij respectively, M ij is the couple-stress tensor, μij is the deviatoric part ofM ij, ωij is

the vorticity vector, d ij is the symmetric part of the velocity gradient, η and η' are constants

associated with the couple-stress, p is the pressure and other terms have their usual meaning

from tensor analysis.ρ f is the density of the fluid, ρpis the density of the particle, Cis the

volumetric volume expansion coefficient, f represents the body forces, ddt represents the

material time derivative, Cis the nano particle concentration, DBis the Brownian diffusion

coefficient and DT is the thermophoretic diffusion coefficient. The ambient values of T∧C as

r tend to h are denoted by T oand Co .

Using the transformation

r=R , z=Z−ct , u=U ,w=W−c ,θ=θ

From a stationary to a moving frame of reference, the equations (2) – (7) are converted to

μ∇2[1− 1α2 ∇

2]w'=d p'd z '+ρgβ (T '−T 0

' )+ρgβ (C'−C 0' ) (8)

[u' ∂T '∂r '+w' ∂T

'

∂ z ' ]=β [ ∂2T '

∂r '2+ 1r '∂T '

∂ r '+ ∂

2T '

∂z '2 ]+τ ¿(9)

[u' ∂C'∂ r '+w' ∂C

'

∂ z ' ]¿DB[ ∂2C '

∂ r '2+ 1r '∂C '

∂r '+ ∂

2C'

∂ z'2 ]+DT '

T0' [ ∂2T '

∂ r '2+ 1r '∂T '

∂ r '+ ∂

2T '

∂ z '2 ] (10)

with∇2=1r∂∂r (r ∂∂ r )

whereτ=(ρC)P(ρC )f

is the ratio between the effective heat capacity of the nanoparticle material

and heat capacity of the fluid.

Introducing the following non-dimensional quantities:

r=r'

a, z= z '

λ,w=w

'

c, p=a

2 p '

λcμ, t= c t

'

λ,u= λu '

ac,θ t=

T '−T 0'

T 0' ,

δ=aλ,R e=

2 ρcaμ

,σ=C'−C0

'

C0' , β= k

( ρc )f, Nb=

(ρc )pDBC 0'

(ρc )f,

N t=( ρc )pDT 'T 0

'

(ρc )f β,Gr=

gβa3T 0'

γ 2 , B r=gβa3C 0

'

γ2 ,

α=aα=√ μη a ,h'=haUsing the non-dimensional quantities in equations (8)-(10) and applying the long

wavelength and low Reynolds number approximations and neglecting the inertial terms, the

equations (8)-(10) are converted to

1r∂∂r (r ∂∂ r (1− 1

α 2 ∇2)w)=dpdz +Grθt+B rσ (11)

0=1r∂∂ r (r ∂θ t∂ r )+N b

∂σ∂r∂θt∂ r

+N t ( ∂θ t∂ r )2

(12)

0=1r∂∂ r (r ∂σ∂ r )+ N t

Nb(1r∂∂ r (r ∂θ t∂ r )) (13)

whereθt , σ ,Nb , N t ,Gr ,B r are the temperature profile, nanoparticle phenomena, Brownian

motion parameter, Thermophoresis parameter, local temperature Grashof number and local

nano particle Grashof number, and w is the axial velocity, r is the radial coordinate,α is the

couple-stress fluid parameter.

The non-dimensional boundary conditions are:

∂w∂r

=0 ,∂ θt∂r

=0 , ∂ σ∂ r

=0at r=0, (14)

w=−k ∂w∂r,θt=0 , σ=0at r=h ( z )=1+ε sin 2π z , (15)

∂2w∂r2 −η

r∂w∂r

=0at r=h (z )=1+ε sin 2πz, (16)

∂2w∂r2 −η

r∂w∂ris finite at r=0. (17)

whereε (¿ba) is the amplitude ratio and η'=

ηη is a couple-stress fluid parameter.

Boundary conditions (16) and (17) indicate that the couple-stresses vanish at the tube

wall and is finite at the tube axis respectively.

3. Solution of the Problem

Homotopy perturbation method is used to solve the coupled equations of temperature

profile and nanoparticle phenomena i. e. Eqs. (12) and (13).

The homotopy for the equations (12) and (13) are as follows Akbar & Nadeem, [13]:

H (ζ , θt )=(1−ζ ) [L (θ t )−L (θ t10 ) ]+ζ [L (θt )+N b∂σ∂r∂θ t∂r

+N t( ∂θt∂r )2] (18)

H (ζ , σ )= (1−ζ ) [L (σ )−L (σ10 ) ]+ζ [L (σ )+N t

Nb(1r∂∂ r (r ∂θ t∂r ))] (19)

where,L=1r∂∂ r (r ∂∂ r ) is taken as linear operator for convenience.

θ1 0 (r , z )=( r2−h2

4 ) ,σ 1 0 (r , z )=−(r 2−h2

4 ) (20)

are chosen as initial guess which satisfy the boundary conditions.

Define

θt (r , z )=θ t0+ζ θt1+ζ2θ t2+−−−−¿ (21)

σ (r , z )=σ 0+ζ σ1+ζ2σ2+−−−−¿ (22)

The series (21) and (22) are convergent for most of the cases. The convergent rate

depends on the nonlinear part of the equation.

Adopting the same procedure as done by Akbar & Nadeem, [13], the solution for

temperature profile and nano particle phenomena can be written for ζ=1 as

θ (r , z )=( r 4−h4

64 ) (Nb−N t ) (23)

σ (r , z )=−( r2−h2

4 ) N t

N b (24)

Substituting Eqs. (23) and (24) in Eq. (11), and solving Eq. (11) using the boundary

conditions (14)-(17), the expression for velocity is

w=B ( I 0 (α r )−I 0 (α h )−k α I 1 (α h ) )−C+ dpdz ( r

2

4−h

2

4− kh

2 )+Gr (N b−N t ) ( 1

α6 −h4

64α 2 +r2

4 α 4 −r2h2

256+ r4

64 α 4 +r6

2304 )+Br( N t

N b)(−1α 4 + h2

4 α2 −r 2

4 α2 +r2h2

16− r4

64 ) (25)

where

A=α [α I 0 (α h )−(1+ηh )I 1(α h)]

B=−Gr

A (Nb−N t )(( 12α 4 (1−η ))+( h

4

192(1+η ))+( h2

16 α2 (1−3η )))

−B rA ( N t

N b)(( 1

2α 2 (η−1 ))−( h2

16(1+η )))

C=Gr (Nb−N t )( 1α6 +

h2

4α 4 −h4

256+ h6

2304+ hk

2α 4 −k h3

128+ k h3

16α2 + k h5

384 )+Br( N t

N b)(−1α 4 + 3h4

64− kh

2α2 + k h3

16 )The dimension less flux in the moving frame is given as

q=∫0

h

2 rwdr (26)

Substituting Eq. (25) in Eq. (26) and solving, the flux is

q= dpdzS+2B(h I 1 (α h )− k h

2α2

I 1 (α h )− h2

2I 0 (α h ))−C h2

+Gr (N b−N t ) ( h2

α6 + h4

8α4 − h6

192α 2 +h8

9216 )+Br( N t

Nb)(−h2

α 4 + 5h6

192 ) (27)where, S=−h4

8− k h

3

2

From Eq. (27), the expression for dpdz is

dpdz

=1Sq−2B (h I1 (α h )− k h

2α2

I 1 (α h )− h2

2I0 (α h ))+C h2

−Gr (Nb−N t )( h2

α6 +h4

8 α4 −h6

192α2 + h8

9216 )−B r( N t

N b)(−h2

α 4 + 5h6

192 ) (28)

The pressure drop over the wavelength ∆ Pλ is defined as

∆ Pλ=−∫0

1 dPdzdz . (29)

Substituting the expression dpdz in Eq. (29), the pressure drop is

∆ Pλ=q L1+L2 (30)

where, L1=∫0

1 1Sdz

and

L2=∫0

1 2BS (h I 1 (α h )− k h

2α2

I1 (α h )−h2

2I 0 (α h ))dz−∫

0

1 C h2

Sdz

+Gr (Nb−N t )∫0

1

( h2

α6 + h4

8α4 − h6

192α 2 +h8

9216 ) 1Sdz

+Br( N t

N b)∫

0

1

(−h2

α 4 + 5h6

192 ) 1Sdz

Following the analysis of Shapiro et al. (1969), the time averaged flux over a period in the

laboratory frame Q is given as

Q=1+∈2

2+q (31)

Substituting Eq. (31) in equation Eq. (34), the time averaged flux is

Q=1+∈2

2+∆ P λL1

−L2

L1 (32)

The dimensionless frictional force F at the wall is

F=∫0

1

h2(−dPdz )dz (33)

Heat Transfer Coefficient

The heat transfer coefficient at the wall is given as

Zθ (r , z )=( ∂h∂z )(∂θ t∂ r ) (34)

Mass Transfer Coefficient

The mass transfer coefficient at the wall is as follows

Zσ (r , z )=( ∂h∂ z )( ∂σ∂r ) (35)

4. Results and Discussion

Analytical expressions for the pressure drop, time averaged flux, frictional force, heat

transfer coefficient and mass transfer coefficient have been calculated. Various graphs are

depicted by using Mathematica software.

4.1 Pressure rise characteristics

Figs. 1.1-1.7 illustrate the variation of pressure rise(−∆ pλ)with time averaged flux (Q) of

peristaltic waves for different values of couple-stress fluid parametersα ,η, Brownian motion

parameter(N b), thermophoresis parameter(N t), local temperature Grashof number (Gr ) , local

nano particle Grashof number(Br )and slip parameter (k ) .

From Figs. 1.1 – 1.3 it is clear that, pressure rise(−∆ pλ ) decreases with the couple-stress

fluid parameterα , ηand Brownian motion parameter(N b ) . From Figs. 1.4-1.7 it is noticed that

pressure rise(−∆ pλ ) increases with the increase of thermophoresis parameter(N t), local

temperature Grashof number(Gr ) ,local nano particle Grashof number(Br )and with slip

parameter (k ).

It is also observed that pressure rise (−∆ pλ ) increases with the increase of time averaged

flux (Q ) for fixed values of α ,η ,Nb , N t ,Gr ,Br ,k .

Fig.1.1: Effect of Q∧α on(∆ pλ )(ϵ=0.9 , η=0.2 ,Gr=0.5 ,Br=0.3 ,N b=1.3 , N t=1.8 , k=0.6 )

Fig.1.2: Effect of Q∧ηon(∆ pλ)(ϵ=0.4 , α=1 ,G r=0.5 , Br=0.3 ,N b=1.3 ,N t=1.8 , k=0.6 )

4.2 Frictional Force

The variation of frictional force (F)with various parameters is shown graphically from

Figs. 2.1-2.6. The frictional force (F) increases with couple-stress fluid parameterη,

thermophoresis parameter (N ¿¿ t) ,¿local temperature Grashof number (Gr ) , local Nano

Fig.1.3: Effect of Q∧N bon(∆ pλ)(ϵ=0.1 , α=3.2, η=1.2,Gr=0.5 ,B r=0.3 , k=0.6 , N t=1.8 )

Fig1.4: Effect of Q∧N t on(∆ pλ)(ϵ=0.1 , α=1.2 , η=0.8 ,G r=0.5 , Br=0.3 , k=0.6 , Nb=1.3 )

Fig.1.5: Effect of Q∧Gr on(∆ pλ)(ϵ=0.3 , α=2.2 ,η=1.8 ,N b=1.3 ,B r=0.3 , k=0.6 , N t=1.8 )

Fig.1.6: Effect of Q∧Br on(∆ pλ)(ϵ=0.9 , α=2.2 ,η=0.8 , k=0.6 ,G r=0.5 , Nb=1.3 ,N t=1.8 )

Fig.1.7: Effect of Q∧k on(∆ pλ)(ϵ=0.9 , α=2.2 ,η=0.8 ,Gr=0.5 ,Nb=1.3 , ,N t=1.8 ,B r=0.3 )

particle Grashof number (Br )and with slip parameter (k )but decreases with Brownian motion

parameter¿¿) (Figs. 2.1-2.6).

4.3Temperature Profile

Effects of temperature profile (θt) with respect to the Brownian motion parameter (Nb)

and thermophoresis parameter (N t) has been shown from Figs. 3.1-3.2. It can be seen that,

temperature profile(θt) increaseswith the increase of Brownian motion parameter (Nb) and

Fig.2.4: Effect of Q∧N t onF(ϵ=0.5 , α=3.2 ,η=1.2 ,Gr=0.5 ,B r=0.3 ,Nb=1.3 , k=0.6 )

Fig.2.1: Effect of Q∧ηonF(ϵ=0.9 , α=1.8 ,Gr=0.5 ,B r=0.3 ,N b=1.3 ,N t=1.8 , k=0.6 )

Fig.2.3: Effect of Q∧N bonF(ϵ=0.5 , η=0.2 ,Gr=0.5 ,Br=0.3 , α=1.2 ,N t=1.8 , k=0.6 )

Fig.2.5: Effect of Q∧Gr onF(ϵ=0.1 , α=1.8 ,η=1.5 ,N b=1.3 ,B r=0.3 , , N t=1.8 , k=0.6 )

Fig.2.6: Effect of Q∧Br onF(ϵ=0.1 , α=0.8 , η=0.5 ,Gr=0.5 ,N b=1.3 , ,N t=1.8 , k=0.6 )

Fig.2.2: Effect of Q∧k onF(ϵ=0.1 , α=0.8 , η=0.5 ,Gr=0.5 ,N b=1.3 ,N t=1.8 , Br=0.3)

decreases with the increase of thermophoresis parameter(N t ) .It is interesting to observe that

value of temperature profile(θ) is maximum in the range[−0.5 ,0.5 ].

4.4Nano particle phenomenon

Figs. 4.1-4.2 explain the nature of nano particle phenomena (σ ) for different values of

Brownian motion parameter (Nb )and thermophoresis parameter(N t ). It can be seen that, the

nano particle phenomena(σ ) increases with the increase of Brownian motion parameter (N b )

and decreases with thermophoresis parameter (N t ) .It is also observed that nano particle

phenomena (σ ) attains maximum value atr=0.

4.5 Heat Transfer Coefficient

Figs.5.1-5.3 indicate the variation of heat transfer coefficient (Zθ) for various values of

Brownian motion parameter(Nb), thermophoresis parameter (N t) and amplitude ratio(ϵ ).

From Figs. 5.1-5.3, it can be observed that, the value of the heat transfer coefficient (Zθ)

Fig. 3.1: Variation in Temperature profile with N b

Fig. 3.2: Variation in Temperature profile with N t

Fig. 4.1: Variation in Nano particle with N b

Fig. 4.2: Variation in Nano particle with N t

increases with Brownian motion parameter (Nb )and thermophoresis parameter(N t ) , amplitude

ratio(ϵ ) and then decreases after attaining a constant value.

4.6 Mass Transfer Coefficient

Figs. 6.1-6.3 illustrate the effect of various parameters on mass transfer coefficient(Zσ). It

is interesting to observe that, mass transfer coefficient(Zσ) decreases with the increase of

Brownian motion parameter(N b), thermophoresis parameter(N t )and then increases after

attaining a constant value. However, mass transfer coefficient(Zσ)shows an opposite

behaviour with respect to amplitude ratio(ϵ ).

Fig. 5.1: Variation in heat transfer coefficient with Nb

Fig. 5.2: Variation in heat transfer coefficient with N t

Fig. 5.3: Variation in heat transfer coefficient with ϵ(z¿3 , N t=2.8 ,Nb=0.2¿

4.7 Streamline patterns

Trapping is a phenomenon where the streamlines on the center line in the moving frame

are divided in order to encircle a bolus of fluid particles circulating along closed streamlines

under certain given conditions. Trapping is a characteristic of peristaltic motion. As the bolus

emanate to be trapped by the wave, the bolus moves at equal speed with that of the wave.

Figs. 7.1-7.4 illustrates the streamline patterns and trapping for different values of couple-

stress fluid parameters, Brownian motion parameter, thermophoresis parameter, local

temperature Grashof number, local nano particle Grashof number and amplitude ratio.

From Fig. 7.1 it is clear that, the volume of the trapped bolus increases with the increase

of couple-stress fluid parameterα . From figs. 7.2-7.4 it is observed that the volume of the

trapped bolus decreases with the decrease of couple-stress fluid parameter η, slip parameter

(k ) and with Brownian motion parameter N b .

Fig. 6.1: Variation in Mass transfer coefficient with N b

Fig. 6.2: Variation in Mass transfer coefficient with N t

Fig. 6.3: Variation in Mass transfer coefficient with ϵ

5 Conclusion

Fig.7.1: Stream line patterns for different values ofα(ϵ=0.9 ,Q=−1.5 , η=0.2 ,Gr=0.5 ,Br=0.3 , Nb=5 , N t=4 , k=0.4 )

Fig.7.2: Stream line patterns for different values of η(ϵ=0.9 ,Q=−1.5 , α=0.2 ,Gr=0.5 , Br=0.3 ,Nb=5 ,N t=4 , k=0.4 )

Fig.7.3: Stream line patterns for different values of Nb

(ϵ=0.9 ,Q=−1.5 , α=0.2 ,Gr=0.5 , Br=0.3 , η=0.2 ,N t=4 , k=0.4 )

Fig.7.4: Stream line patterns for different values of k

(ϵ=0.7 ,Q=−1.5 , α=0.2 ,Gr=0.5 ,Br=0.3 , η=0.2 , Nb=5 , N t=4 )

The major premise of the paper is peristaltic transport of a couple-stress fluid with

nanoparticles having permeable walls. The study particularly pertains to a situation when the

Reynolds number is low and wavelength is large. Emphasis was laid to investigate the

pressure drop characteristics, frictional force, temperature profile, nanoparticle phenomena,

heat transfer coefficient, mass transfer coefficient and stream line patterns of the couple-

stress fluid parameters, Brownian motion parameter, thermophoresis parameter. Homotopy

perturbation method is used to solve the coupled equations of temperature profile and

nanoparticle phenomena and analytical methods have been applied to the present study to

find the other variables.

The main points of the analysis are as follows:

a. Pressure rise decreases with the couple-stress fluid parameterα , ηand Brownian

motion parameter. Pressure rise increases with the increase of thermophoresis

parameter, local temperature Grashof number, local nano particle Grashof number and

with slip parameter.

b. Frictional force (F) increases with couple-stress fluid parameterη, thermophoresis

parameter, local temperature Grashof number, local Nano particle Grashof number

and with slip parameter but decreases with Brownian motion parameter.

c. Temperature profile increases with the increase of Brownian motion parameter and

decreases with thermophoresis parameter and the value of temperature profile is

maximum/minimum forr∈ [−0.5,0 .5 ] .

d. Nano particle phenomenon increases with the increase of Brownian motion parameter

and decreases with thermophoresis parameter and it reaches maximum atr=0.

e. Heat transfer coefficient increases with Brownian motion parameter and

thermophoresis parameter, amplitude ratio and after attaining a constant value, the

heat transfer coefficient starts decreasing.

f. Mass transfer coefficient decreases with the increase of Brownian motion parameter,

thermophoresis parameter and after attaining a constant value atr=0, the mass

transfer coefficient starts increasing. However, mass transfer coefficientshows an

opposite behaviour with respect to amplitude ratio.

g. The volume of the trapped bolus increases with the increase of couple-stress fluid

parameterα and decreases with the increase of couple-stress fluid parameter η ,

Brownian motion parameter and with slip parameter.

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