Unsteady Peristaltic Pumping of a Jeffrey Fluid in a Finite Length Tube With Permeable Wall

Unsteady Peristaltic Pumping of a Jeffrey Fluid in a Finite Length Tube With Permeable Wall

P. Vinod Kumar

Department of Mathematics, JNTU College of Engineering, Karimnagar, Hyderabad, INDIA

Y. V .K. Ravi Kumar

Practice School Division, Birla Institute of Technology (BITS) Pilani, Hyderabad, INDIA

Shahnaz Bathul

Department of Mathematics, JNTU College of Engineering, Kukatpally, Hyderabad, INDIA.

Abstract – Investigation concerning Peristaltic pumping of a Jeffrey fluid in a finite length tube with permeable wall is made. The walls of the finite length tube are subjected to the contraction waves that do not cross the stationary boundaries. Mathematical analysis is expressed through long wavelength and low Reynolds number approximation in non-dimensional form . Saffman slip boundary condition used in this analysis. Exact analytical expressions of axial velocity, radial velocity, non-dimensional flow rate are obtained. The effect of various parameters on the flow pattern is discussed through graphs. When = the results are in agreement with Ravi Kumar.Y.V.K [2010].

Keywords: Peristaltic Pumping, finite length tube, Jeffrey fluid, Permeable wall.

1. INTRODUCTION

   satisfactory efferentus of the male reproductive tract and in the transport of spermatozoa in the most of the physiological fluids behave like non-Newtonian fluids. Li and Brasseur [6] considered the non-steady peristaltic transport in a finite length tube by considering Newtonian fluid. Misra and Pandey [8-9] extended the model for peristaltic flow of power-law fluids in a finite length circular cylindrical peristaltic pumping in a finite length tube with permeable wall.

   Now, we proposed to study the unsteady peristaltic pumping in a finite length tube with permeable wall with Jeffrey fluid model, Saffman boundary conditions are used at the permeable wall of the tube.

2. JEFFREY FLUID MODEL

   The equation for an incompressible Jeffrey fluid are

   Peristalsis is a mechanism of pumping of viscous

   fluids in ducts against an adverse pressure gradient by means of a series of moving contractile rings on the wall.

   T pI S

   (1)

   Most of the transportations in physiology are due to the

   2

   peristaltic pumping mechanism. The flow of blood through arteries and veins, the flow of urine through ureter, the passage of bile from the gall bladder to the duodenum, the

   S

   1 1

   ( t

   2

   t 2 )

   (2)

   movement of chime in the gastrointestinal tract and the

   where T and S are Cauchy stress tensor and extra stress

   transportation of the food bolus through the alimentary

   canal are some examples of peristaltic pumping.

   The study of the fluid mechanism of peristaltic

   tensor, P is the pressure, I is the identity tensor,

   1 is the

   transport has been confirmed experimentally by Latham[1] and Weinberg et al.[2]. Several research proposed different models by considering various geometries with relevant

   ratio of the relaxation to retardation times, retardation time and is the shear rate.

   2 is the

   boundary conditions [10-20].Shapiro et.al [3] made a detailed investigation of peristaltic pumping of a Newtonian fluid in a flexible channel and a circular tube.

   Most of the theoretical investigations have been carried out by assuming blood and the other physiological fluids behave like a Newtonian fluid. Although this approach may provide a satisfactory understanding of the peristaltic mechanism in the ureter ,it fails to provide a

3. MATHEMATICAL FORMULATION

   We consider the peristaltic pumping of a non- Newtonian fluid namely Jeffrey fluid in a finite length tube with permeable walls. Sinusoidal waves of constant speed propagate along the channel boundaries.

   The wall deformation of the peristaltic wave is given by

   1 (rV ) U

   r h(z,t ) mz 0.5aA(1 cos[ 2 (Z ct )])

   0

   r r (1 1 ) z

   (10)

   (3)

   p 1

   r U p

   where is the minimum tube occlusion, a is the average radius of the bolus, A is the amplitude, is the wavelength

   z r r

   (

   (1 1 )

   r )

   (11) r 0

   of the peristaltic wave, m is a constant whose magnitude depends on the length of the tube, exist, and inlet dimensions, and c is the Velocity of the peristaltic wave.

   Introducing a wave frame ( , ) moving with velocity c away from the fixed frame ( , ) by the following transformations

   = , = , = , = , where ( ,

   ) and ( , ) are velocity components in wave and fixed frame ,respectively.

   After using these transformations, the equations of motion

   (12)

   The non dimensional boundary conditions are discussed bellow

   The wall itself undergoes radial vibrations with the velocity

   H

   i.e. V t at r = H (13)

   are

   V

   r

   [V

   U V 0

   z r

   V U V ] p

   [

   2V

   (4)

   1 V 2V V

   ]

   The Saffman condition is imposed on permeable wall of the tube

   i.e U U at r = H (14)

   r z

   r r 2

   r r

   z 2 r 2 r

   (5)

   U U p 2U 1 U 2V

   The centre line velocity and velocity gradient in the radial direction are taken as zero

   [V U ] [ ]

   r z

   z r 2

   r r

   (6)

   z 2

   i.e V=0 at r = 0 (15)

   where and are the velocity components in the and

   directions respectively.

   U 0

   r

   at r = 0 (16)

   Now we introduce the following non-dimensional variables

   r r , Z z , t ct , H h , a ,U U

   The finite tube length requires the pressures at the two ends of the tube

   a a

   V pa2 b ca

   c

   am 1

   where

   i.e

   P p0

   at z = 0 (17)

   V c , p c , a , Re (

   ) , ( )

   k

   U and V are the axial and radial velocity components in the laboratory frame and k is the permeability of the wall.

   After nondimensionalization, equations (4) (6) become

   P pL

   at z = L (18)

4. SOLUTION OF THE PROBLEM

   1 (rV )

   U 0

   (7)

   Solving equation (10) subject to the boundary conditions

   given by (14) and (16), we get axial velocity as

   r r (1 1 ) z

   2 V V p 2

   1 (rV ) 4 2V

   U (1 1 ) p (r 2 2H H 2 )

   (19)

   Re [V r

   U z ] r

   r [

   ]

   r r (1 1 )

   (8)

   z2

   4 z

   Re[V U

   r

   U U

   z

   ] p

   z

   1

   r r

   ( r U

   (1 1 ) r

   (9)

   )

   2 2U

   z2

   where H Mz 0.5A(1 cos 2(z ct)),

   a

   M m

   By considering long wavelength and low Reynolds number approximations and dropping terms of order and higher, it follows from equations (7) and (8) that the appropriate equations describing the flow in the frame are

   a

   Using equation (19) in equation (10) and solving together with the boundary condition given by (15), we get the radial velocity as

   2 r3

   Hr

   Hr2

   2 p

   2 p H r

   V (1 1 ) (

   16

   ) 4 8

   z2 (1 1 ) ( H ) z

   t 4

   1. Substituting equation (11) in equation (8) and integrating with respect

      p 8Q0 [ H 1

      0

      2 L

      H (H 4 3)

      ]dt]1

      to z ields

      p 1

      16 z (H 4 )2 H (s, t)

      CQ (1 1 )

      T 0 (4 H )

      1 ds

      1

      T

      2

      0

      z H (H 4 )3 [G0 (t) (1 ) H

      t ds]

      where 0 is the pressure difference required to maintain

      1 0

   2. where 0 at most depends on time.Integrating again yields a relationship between intraluminal pressure, wall geometry, and wall velocity as

      zero net flow rate at z.

5. NUMERICAL RESULTS AND DISCUSSION

Figs 1-4 are drawn for the time varying pressure distribution

z

p(z, t) p0 (t)

p(s, t)dz

for a integral number of peristaltic waves in the tube for

L =2. From fig.1 , at t = 0 ,it is observed that the

0 t

z 16 G (t)

0 H (1 )(H 4 )3 H (H 4 )3

s1 1

H

H

t 2 1

pressure rises very sharply at the initial end reaches the cusp

, then decreases to a lower rate at the midpoint of the bolus and it finally rises sharply to meet the leading end of the

p(z, t) p (t)

0 1

0

(H 4 )2

1. ds ]ds

   bolus to transport it under huge control. Similar

   0 is determined by evaluating equation (22) together with the equation (18) as

   phenomenon is observed for different values of . It is

   noticed that there is a large decrease in the peak pressures

   p(t)16

   (

   L 16 s1 1

   H (1 )(H 4 )3 H

   H (s , t)

   (H 4 )2

   2

   ds )ds

   t 2 1

   (maximum and minimum pressures) with increasing

   permeability parameter.

   G (t) 0 1 0

   0 L 1

   0

   H (H 4 )3 ds

   where p(t) pL (t) p0 (t)

2. From figs1-4, it is observed that the increase in time shifts the bolus to the right of the axis for increasing t . The pressure distribution exhibits how the two boluses are carried along the length by changing the pressure distribution.

   4.1 Volume Flow Rate

   The non dimensional flow rate is given as

   Figs 5-7, are drawn by considering the propagation of a non

   integral number of waves in the train. It can be observed

   H

   Q(z, t) 2CQ Urdr 2CQ

   0

   p [ H

   4

   z 16

   H 3

   ]

   4

   that decreases with an increase . From both the cases, it is observed that the peaks of the pressures are identical in the integral case while different in the non-integral case. fig 8 is drawn to study the effect of Jeffrey parameter for non-

3. where = 1 for train wave and = for single wave transport.

   Pumping performance is characterized by the relationship between a time averaged the volume flow rate is =

   1 , and the pressure difference between the end

   integral case when 1 = 0 the results are in agreement with that of Ravi Kumar [2010].

   From figs. 18 and fig. 9 we can observe that the effect of

   and 1 on as decreases with an increase in and

   1.

   0

   of the tube , where is fixed in time. Then we get

   0

   p

   Q Q0[1 p ]

4. where 0 is the maximal flow rate at z,

   p p [1 Q ]

   Q

   0

   0

2CQ (1 1 ) T p H 2

Q Q0 ( T

) [

2 L

0 16(4 H )

1 ds

]dt

0

H (H 4 )3

Fig 1 Pressure distribution with z along the tube for = 0.0, = 0.1, = 0.07, with = 0.5 , A = 1.609, = 2 ,

when t = 0 and 1 = 1.

Fig 2 Pressure distribution with z along the tube for = 0.0, = 0.1, with = 0.5 , A = 1.609, = 2 , when t =

Fig 5 Pressure distribution with z along the tube for =

0.02 and

= 1.

0.0, = 0.1, with = 0.5 , A = 1.609, = 1.69, when

1

t = 0 and 1 = 1.

Fig 3 Pressure distribution with z along the tube for = 0.0, = 0.1, with = 0.5 , A = 1.609, = 2 , when t =

0.04 and 1 = 1.

Fig 6 Pressure distribution with z along the tube for =

0.0, = 0.1, with = 0.5 , A = 1.609, = 1.69,

when t = 0.04 and 1 = 1.

Fig 4 Pressure distribution with z along the tube for =

Fig 7 Pressure distribution with z along the tube for = 0.05, = 0.06, = 0.07, with = 0.5 , A =

0.0, = 0.1, with = 0.5 , A = 1.609, = 2 , when t =

0.06 and 1

= 1.

1.609, = 1.69 , when t = 0.3 and 1 = 1.

Fig 8 Pressure distribution with z along the tube for = 0.1, with = 0.5 , A = 1.609, = 1.89, when t = 0.0 and

1 = 0, 1 = 1.

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