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

Download Full-Text PDF Cite this Publication

Text Only Version

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 0at 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 .

  6. REFERENCES

  1. Abo-Eldahab, E.M., Barakat, E.I. and Nowar, K.I. Effects of Hall and ion-slip currents on peristaltic transport of a couple stress fluid, International Journal of Applied Mathematics and Physics, 2 (2) (2010), 145-157.

  2. Ai, L. and Vafai, K. An investigation of Stokes second problem for non-Newtonian fluids, Numerical Heat Transfer, Part A, 47(2005), 955-980.

  3. Akbar, N.S., Hayat, T., Nadeem, S. and Obaidat, S. Peristaltic flow of a Tangent hyperbolic fluid in an inclined asymmetric channel with slip and heat transfer, Progress in Computational Fluid Dynamics, an International Journal, 12(5) (2012), 363-374.

  4. Ali, N., Wang, Y., Hayat, T. and Oberlack, M. Slip effects on the peristaltic flow of a third grade fluid in a circular cylindrical tube, J. Appl. Mech., 76(2009), 1-10.

  5. Bhatti, M. M., Ali Abbas, M. and Rashidi, M. M. Effect of hall and ion slip on peristaltic blood flow of Eyring Powell fluid in a non- uniform porous channel , World Journal of Modelling and Simulation, 12(4) (2016), 268-279.

  6. Chaube, M.K., Pandey, S.K. and Tripathi, D. Slip Effect on Peristaltic Transport of Micropolar Fluid, Applied Mathematical Sciences, 4(2010), 2105 2117.

  7. Eldabe, N.T.M., Ahmed Y. Ghaly, A.Y., Sallam, S.N., Elagamy, K. and Younis, Y.M. Hall effect on peristaltic flow of third order fluid in a porous medium with heat and mass transfer, Journal of Applied Mathematics and Physics, 3(2015), 1138-1150.

  8. Elshahed, M. and Haroun, M. H. Peristaltic transport of Johnson- Segalman fluid under effect of a magnetic field, Math. Probl. Engng., 6 (2005), 663677.

  9. El-Shehawey, E.F., El-Dabe, N.T. and El-Desoky, I. M. Slip effects on the Peristaltic flow of a Non-Newtonian Maxwellian Fluid, Acta Mech., 186(2006), 141-159.

  10. Hayat, T and Ali, N. Peristaltically induced motion of a MHD third grade fluid in a deformable tube, Physica A: Statistical Mechanics and its Applications, 370(2006), 225-239.

  11. Hayat, T., Ali, N, and Asghar, S. Hall effects on peristaltic flow of a Maxwell fluid in a porous medium, Phys. Letters A, 363(2007), 397 – 403.

  12. Jyothi, B., Satyanarayana, B. and Subba Reddy, M.V. Slip effects on peristaltic transport of a Prandtl fluid in a channel under the effect of magnetic field, South Asian Journal of Mathematics, 5(1)(2015), 1 – 12.

  13. Kwang Hua Chu, W. and Fang, J. Peristaltic transport in a slip flow, Eur. Phys., J. B., 16(2000), 543-547.

  14. Li, A.A., Nesterov, N.I, Malikova, S.N. and Kilatkin, V.A. The use of an impulse magnetic field in the combined of patients with store fragments in the upper urinary tract. Vopr kurortol Fizide. Lech Fiz Kult, 3(1994), 22-24.

  15. Nadeem, S. and Akram, S. Peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel, Z. Naturforsch., 64a (2009), 559 567.

  16. Nadeem, S. and Akram, S. Magnetohydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer, Acta Mech. Sin., 27(2) (2011), 237250.

  17. Nadeem, S. and Akbar, S. Numerical analysis of peristaltic transport of a Tangent hyperbolic fluid in an endoscope, Journal of Aerospace Engineering, 24(3) (2011), 309-317.

  18. Noreen, S., Hayat, T. and Alsaedi, A. Study of slip and induced magnetic field on the peristaltic flow of pseudoplastic fluid, International Journal of Physical Sciences, 6(36)(2011), 8018-8026.

  19. Prasanth Reddy, D. and Subba Reddy, M.V. Peristaltic pumping of third grade fluid in an asymmetric channel under the effect of magnetic fluid, Advances in Applied Science Research, 3(6)(2012), 3868 3877.

  20. Shapiro, A.H., Jaffrin, M.Y and Weinberg, S.L. Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech. 37(1969), 799-825.

  21. Subba Narasimhudu, K. and Subba Reddy, M. V. Hall effects on the peristaltic pumping of a hyperbolic tangent fluid in a planar channel, Int. J. Mathematical Archive, 8(3) (2017), 70 – 85.

  22. Subba Reddy, M.V., Jayarami Reddy, B., Nagendra, N. and Swaroopa, B. Slip effects on the peristaltic moton of a Jeffrey fluid through a porous medium in an asymmetric channel under the effect magnetic field, Journal of Applied Mathematics and Fluid Mechanics, 4(2012), 59-72.

Leave a Reply

Your email address will not be published. Required fields are marked *