Effects of Slip and Hall on the Peristaltic Transport of a Hyperbolic Tangent Fluid in a Planar Channel

Effects of Slip and Hall on the Peristaltic Transport of a Hyperbolic Tangent Fluid in a Planar Channel

M. Anusha Bai1

1Research Scholar, Department of Mathematics,

Sri Krishnadevaraya University, Ananthapuramu-515003, A.P., India

Prof. R. Sivaprasad2

2Professor, Department of Mathematics,

Sri Krishnadevaraya University, Ananthapuramu-515003, A.P., India

Abstract:- The effects of slip and Hall on the peristaltic transport of a hyperbolic tangent fluid in a planar channel under the assumption of long wavelength. The expressions for the velocity and axial pressure gradient are obtained by employing perturbation technique. The effects of Weissenberg number, power-law index, Hall parameter, Hartmann number and amplitude ratio on the axial pressure gradient and time-averaged volume flow rate are studied with the aid of graphs.

Keywords: Peristaltic Transport, Hyperbolic Tangent fluid, Planar Channel and Slip and Hall parameter

1. INTRODUCTION

   Some fluids which are encountered in chemical applications do not adhere to the classical Newtonian viscosity prescription and are accordingly known as non-Newtonian fluids. One especial class of fluids which are of considerable practical importance is that in which the viscosity depends on the shear stress or on the flow rate. The viscosity of most non- Newtonian fluids, such as polymers, is usually a nonlinear decreasing function of the generalized shear rate. This is known as shear-thinning behavior. Such fluid is a hyperbolic tangent fluid (Ai and Vafai, 2005). Nadeem and Akram (2009) have first investigated the peristaltic flow of a hyperbolic tangent fluid in an asymmetric channel. Nadeem and Akbar (2011) have analyzed the peristaltic transport of a Tangent hyperbolic fluid in an endoscope numerically.

   Based on Experimental controls, it was shown that the controlled application of low intensity and frequency pulsing magnetic fields could modify cell and tissue behavior. Biochemistry has taught us that cells are formed of positive or negative charged molecules. This is why these magnetic fields applied to living organisms may induce deep modifications in molecule orientation and in their interaction. An impulse magnetic field in the combined therapy of patients with stone fragments in the upper urinary tract was experimentally studied by Li et al. (1994). It was found that impulse magnetic

   field (IMF) activates impulse activity of ureteral smooth muscles in 100% of cases. Elshahed and Haroun (2005) have investigated the peristaltic flow of a Johnson-Segalman fluid in a planar channel under the effect of a magnetic field. Hayat and Ali (2006) have investigated the peristaltic motion of a MHD third grade fluid in a tube. Hayat et al. (2007) have first investigated the Hall effects on the peristaltic flow of a Maxwell fluid trough a porous medium in channel. Magnetohydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer was studied by Nadeem and Akram (2011). Eldabe (2015) have studied the Hall Effect on peristaltic flow of third order fluid in a porous medium with heat and mass transfer. Hall effects on the peristaltic pumping of a hyperbolic tangent fluid in a planar channel were studied by Subba Narasimhudu and Subba Reddy (2017).

   The peristaltic flow of a Newtonian fluid through a two-dimensional micro channel with the slip effect was first investigated by Kwang (2000). El Sehaway et al. (2006) have studied the effect of slip on the peristaltic motion of a Maxwell fluid in a two-dimensional channel. The effects of slip and non- Newtonian parameters on the peristaltic transport of a third grade fluid in a circular cylindrical tube were investigated by Ali et al. (2009). Chaube et al. (2010) have analyzed the slip effects on the peristaltic flow of a micropolar fluid in a channel. Effects of slip and induced magnetic field on the peristaltic flow of pseudoplastic fluid were analyzed by Noreen et al. (2011). Subba Reddy et al. (2012) have investigated the slip effects on the peristaltic motion of a Jeffrey fluid through a porous medium in an asymmetric channel under the effect of magnetic field. Akbar et al. (2012) have discussed the peristaltic flow of a hyperbolic tangent fluid in an inclined asymmetric channel with slip and heat transfer. Slip effects on peristaltic transport of a Prandtl fluid in a channel under the effect of magnetic field was studied by Jyothi et al. (2015).

2. MATHEMATICAL FORMULATION

   Y

   H X ,t

   Y

   H X ,t

   b

   b

   c

   c

   a

   a

   We consider the peristaltic motion of a hyperbolic tangent fluid in a two-dimensional symmetric channel of width 2a under the effect of magnetic field. The flow is generated by sinusoidal wave trains propagating with constant speed c

   O

   O

   X

   X

   along the channel walls. A uniform magnetic field B0 is

   applied in the transverse direction to the flow. The magnetic Reynolds number is considered small and so induces magnetic field neglected. Fig. 1 represents the physical model of the channel.

   The wall deformation is given by

   Y H ( X ,t) a b cos 2 ( X ct) , (1)

   where b is the amplitude of the wave, – the wave length and

   along the axis of the channel and Y perpendicular to X . Let

   along the axis of the channel and Y perpendicular to X . Let

   X and Y – the rectangular co-ordinates with X measured

   n

   B0

   Fig. 1 The Physical Model

   B0

   Fig. 1 The Physical Model

   1 1n

   1 n 1

   0 0 0

   0 0 0

   (U ,V )

   be the velocity components in fixed frame of

   reference ( X ,Y ) .

   The flow is unsteady in the laboratory frame ( X ,Y )

   . However, in a co-ordinate system moving with the

   (5)

   The above model reduces to Newtonian for and n 0 .

   0

   propagation velocity c (wave frame (x, y)), the boundary shape

   is stationary. The transformation from fixed frame to wave frame is given by

   The equations governing the flow in the wave frame of reference are

   x X ct, y Y ,u U c, v V

   (2)

   u v 0

   (6)

   where (u, v)

   and (U ,V )

   are velocity components in the

   x y

   wave and laboratory frames respectively.

   The constitutive equation for a Hyperbolic Tangent fluid is

   u u

   u p yx B2

   tanh n

   v xx 0 mv u c

   v xx 0 mv u c

   (3)

   x

   y

   x x

   y 1 m2

   0

   (7)

   where is the extra stress tensor, is the infinite shear rate

   viscosity, o

   is the zero shear rate viscosity, is the time

   v

   u

   p

   B2

   constant, n is the power-law index and is defined as

   u

   v xy yy 0 mu c v

   x

   1 1

   y

   y x

   y 1 m2

   (8)

   2

   2

   2

   2

   ij ji

   i j

   (4)

   where is the density is the electrical conductivity, B0

   where is the second invariant stress tensor. We consider in the constitutive equation (3) the case for which 0 and

   is the magnetic field strength and m is the Hall parameter.

   The corresponding dimensional boundary conditions are

   1, so the Eq. (3) can be written as

   u xy c at y H (slip condition)

   (9)

   u 0

   at y 0

   2

   y p

   1 n We

   u 1

   u

   M u 1

   (symmetry condition) (10) here is the slip parameter.

   Introducing the non-dimensional variables defined by

   0

   0

   x y u v a pa2 b

   x y

   p 0

   y y

   y y

   1 m2

   x ,

   y ,

   a

   u ,

   c

   v ,

   c

   ,

   p c , a

   y

   From Eq. (15) and (16), we get

   h H , t ct , xx ,

   a

   , yy ,

   a c

   xx xy

   c xy

   c yy

   2 2 2

   0 0 0

   dp 1 n u nWe

   u

   M u 1

   ac

   c a q

   dx y2

   y y

   1 m2

   Re

   0

   , We

   ,

   a

   , q

   c ac

   (17)

   (11)

   into the Equations (6) – (8), reduce to (after dropping the bars)

   The corresponding non-dimensional boundary conditions in the wave frame are given by

   u v 0

   (12)

   x y

   u 1 n We u 1 u 1 at

   u u p

   M 2

   y y

   Re u

   v 2

   xx xy

   m v u 1

   y h 1 cos 2 x

   (18)

   x

   3 v

   y

   v

   x x

   p 2 xy

   y

   yy

   1 m2

   M 2

   (13)

   u 0

   y

   at y 0

   (19)

   Re

   u x v y y

   y

   mu 1 v

   y 1 m2

   (14)

   The volume flow rate q in a wave frame of reference is given by

   h

   q udy . (20)

   u

   u

   where xx 2 1 nWe 1 x ,

   0

   The instantaneous flow Q( X , t) in the laboratory frame is

   h h

   xy

   1 nWe 1 u 2 v ,

   Q( X ,t) UdY (u 1)dy q h

   (21)

   y

   2 1 n We 1 v ,

   x

   0 0

   The time averaged volume flow rate Q over one period

   yy y

   T

   1

   c

   c

   of the peristaltic wave is given by

   u 2

   u

   v 2

   v 2 2 1 T

   2 2

   2

   2 2

   Q Qdt q 1

   (22)

   x

   and

   y

   x y T 0

   M aB0

   is the Hartmann number.

   0

   Under lubrication approach, neglecting the terms of order and Re, the Eqs. (13) and (14) become

3. SOLUTION

   Since Eq. (17) is a non-linear differential equation, it

   u 1

   dp0 cosh Ny

   1

   is not possible to obtain closed form solution. Therefore, we employ regular perturbation to find the solution.

   0 N 2 1 n

   1

   dx a1

   For perturbation solution, we expand u, dp

   dx

   as follows

   and q

   where N M /

   1 n(1 m2 )

   and

   (32)

   u u Weu OWe2

   (23)

   0 1

   dp dp dp 2

   a1 cosh Nh N (1 n)sinh Nh .

   0 We 1 O We dx dx dx

   (24)

   The volume flow rate q is given by

   q q

   Weq

   • OWe2

   0

   (25)

   0 1 1 dp sinh Nh Nha

   q0 0 1 h

   Substituting these equations into the Eqs. (17) – (19),

   N 3 1 n dx a

   we obtain

   A. SYSTEM OF ORDER We0

   From Eq. (A) , we have

   1

   (33)

   dp 2u M 2

   dp q h N 3 1 na

   0 1 n 0 u

   1

   (26)

   0 0 1

   (34)

   dx y2 1 m2 0

   dx sinh Nh Nha1

   and the respective boundary conditions are

   D. SOLUTION FOR SYSTEM OF ORDER We1

   u 1 n u0 0 y

   1

   at y h

   (27)

   Substituting Eq. (32) in the Eq. (29) and solving the Eq. (29), using the boundary conditions (30) and (31), we obtain

   u0 0 at

   y 0

   (28)

   dp 2

   y

   0

   u 1

   dp1

   cosh Ny 1 n

   dx

   f y

   1 N 2 1 n dx a

   3 N 1 na 3

   B. SYSTEM OF ORDER We1

   1

   1

   (35)

   dp 2u

   u 2 M 2

   Where

   1 1 n 1

   2

   2

   o

   2 u1

   (29)

   dx y

   y y

   1 m

   sinh 2Nh 2sinh Nhcosh Ny 2sinh Ny sinh 2Ny cosh Nh

   f y

   2cosh 2Nh cosh Nhcosh Ny sinp Nh cosh Ny

   N 1 n

   and the respective boundary conditions are

   sinh Nh 2sinh Ny sinh 2Ny

   .

   u u 2

   u1 1 n 1 n 0

   0 at y h

   The volume flow rate q is given by

   y

   (30)

   y

   1

   1 dp sinh Nh Nha dp 2

   q1 1 1 a2 0

   N 3 1 n

   dx a

   dx

   u1 0

   y

   at y 0

   (31)

   1

   (36)

   C. SOLUTION FOR SYSTEM OF ORDER We0

   Solving Eq. (26) using the boundary conditions (27) and (28), we obtain

   where Fig. 2 shows the variation of the axial pressure

   1

   1

   4 3cosh Nh 2sinh 2Nh sinh Nh cosh 2Nh cosh Nh a

   a n 3

   dp

   gradient

   with We for n 0.5, m 0.2 , 0.1,

   2

   and

   6N 4 1 n3 a3

   dx

   M 1, 0.5 and Q 1. It is observed that, the axial

   dp

   3

   3

   a3 N 1 n3sinh Nhcosh 2Nh 1 2sinh Nh.

   From Eq. (36) and (34), we have

   pressure gradient number We .

   increases with increasing Wiessenberg

   dx

   dp

   dp q N 3 1 na

   dp 2

   The variation of the axial pressure gradient

   dx

   with

   1 1 1 a 0

   n for

   We 0.01,

   m 0.2,

   M 1,

   0.1,

   dx sinh Nh Nha

   4 dx

   1

   (37)

   0.5 and Q 1 is depicted in Fig. 3. It is found that,

   dp

   where

   4 3cosh Nh 2sinh 2Nh sinh Nh cosh 2Nh cosh Nh a

   the axial pressure gradient

   dx

   power-law index n .

   decreases with an increase in

   a4 n

   .

   6N 1 n2 a2 sinh Nh Nha

   1 1

   1 1

   Substituting Equations (16) and (37) into the Eq. (24)

   3

   gradient

   Fig. 4 illustrates the variation of the axial pressure

   dp

   dp

   with for n 0.5, We 0.01, M 1,

   dx

   0 1

   0 1

   dp dp dp

   and using the relation We

   dx dx dx

   and neglecting

   m 0.2, 0.5 and Q 1. It is noted that, the axial

   dp

   terms greater than O We , we get

   pressure gradient

   dx

   decreases with increasing slip

   dp q h N 3 1 na q h N 3 1 na 2

   1 Wea 1

   4

   4

   parameter .

   dx sinh Nh Nha1

   sinh Nh Nha1

   (38)

   dp

   The variation of the axial pressure gradient

   dx

   with

   m for

   n 0.5,

   We 0.01,

   M 1,

   0.1,

   The dimensionless pressure rise per one wavelength in the wave frame is defined as

   0.5

   and Q 1 shown in Fig. 5. It is noted that, the

   dp

   p 1 dpdx

   0 dx

   (39)

   axial pressure gradient parameter m

   decreases with increasing Hall

   dx

   Note that, as

   0

   our results coincide with the

   Fig. 6 depicts the variation of the axial pressure

   results of Subba Narasimhudu and Subba reddy (2017); as dp

   0 ,

   m 0 , M 0 , We 0

   and

   n 0 our

   gradient

   with M for

   n 0.5,

   m 0.2,

   results coincide with the results of Shapiro et al. (1969).

   We 0.01,

   0.1,

   0.5

   and

   Q 1. It is

4. RESULTS AND DISCUSSION

   In this section, we have carried out numerical calculations and plotted graphs to study effects of the Weissenberg number We, the power-law index n ,the slip

   observed that, on increasing Hartmann number M increases

   dp

   the axial pressure gradient .

   dx

   dp

   parameter , the Hall parameter m, the Hartmann number M

   The variation of the axial pressure gradient

   with

   dx

   and the/8- amplitude ratio on the axial pressure gradient and pumping characteristics.

   for

   n 0.5, m 0.2 , 0.1,

   M 1,

   We 0.01 and Q 1 is depicted in Fig. 7. It is found

   dp

   that, the axial pressure gradient

   increases with increasing

   The variation of the pressure rise

   p with Q for different

   dx values of with

   n 0.5,

   m 0.2 ,

   0.1,

   M 1

   amplitude ratio .

   Fig. 8 illustrates the variation of the pressure rise p

   and We 0.01 is depicted in Fig. 13. It is observed that, the

   time-averaged flow rate Q increases with increasing in both the pumping and free pumping regions, while it decreases

   with Q for different values of We with n 0.5,

   with increasing n in the co-pumping region for chosen

   m 0.2, 0.1, M 1 and 0.5 . It is noted that,

   the time-averaged volume flow rate Q increases with increasing Wiessenberg number We in pumping p 0

   p 0 .

   , free-pumping p 0

   regions.

   and co-pumping p 0

   The variation of the pressure rise

   p with Q for

   different values of n with We 0.01,

   m 0.2,

   0.1, M 1 and 0.5

   is shown in Fig. 9. It is

   found that, the time-averaged flow rate Q decreases with increasing n in both the pumping and free pumping regions, while it increases with increasing n in the co-pumping region.

   Fig. 10 shows the variation of the pressure rise p

   with Q for different values of with n 0.5,

   We 0.01,

   0.1,

   M 1

   and

   0.5 . It is

   observed that, the time-averaged flow rate Q decreases with increasing in both the pumping region and the free pumping region, while it increases with increasing in the co-pumping region.

   The variation of the pressure rise

   p with Q for

   different values of m with n 0.5, We 0.01,

   0.1, M 1 and 0.5 is illustrated in Fig. 11. It is

   observed that, the time-averaged flow rate Q decreases with increasing m in the pumping region, while it increases with increasing m in both the free pumping and co-pumping regions.

   Fig. 12 depicts the variation of the pressure rise p

   with Q for different values of M with n 0.5, 0.1

   , m 0.2, We 0.01 and 0.5 . It is noticed that,

   the time-averaged flow rate Q increases with increasing M in the pumping region, while it decreases with increasing M in both the free-pumping and co-pumping regions.

5. CONCLUSIONS

   In this paper, we investigated the effects of slip and Hall on the peristaltic transport of a hyperbolic tangent fluid in a planar channel under the assumption of long wavelength. The expressions for the velocity and axial pressure gradient are obtained by employing perturbation technique. It is found that, the axial pressure gradient and time-averaged flow rate in the pumping region increases with increasing the Weissenberg number We, the Hartmann number M and the amplitude ratio

   , while they decreases with increasing power-law index n ,

   slip parameter and Hall parameter m .

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