Effect of magnetic field on peristaltic flow of Walters –B fluid through a porous medium in a tapered asymmetric channel

6889 | P a g e  J a n u a r y 2 0 1 7 w w w . c i r w o r l d . c o m  Effect of magnetic field on peristaltic flow of Walters –B fluid through a porous medium in a tapered asymmetric channel.  Ahmed M.Abdulhadi, Tamara S. Ahmed  College of science 1 , university of Baghdad  ahm6161@yahoo.com,  Collage of education for pure science (ibn al Haitham),university of Baghdad  Tamaraalshareef@yahoo.com  ABSTRACT


INTRODUCTION
Peristaltic transport is a form of material transport included by a progressive wave of contraction or expansion along the length of distensible tube mixing and transporting the fluid in the direction of the wave propagation. This kind of phenomenon is termed as peristaltic. It plays an indispensable role in transporting many physiological fluids in the body under various situations as urine transport from kidney to bladder, the movement of chyme in the gastrointestinal tracts, transport of spermatozoa in the ductus efferent's of the male reproductive tract, movement of ovum in the fallopian tubes, swallowing of food through esophagus and the vasomation of small blood vessels many modern mechanical devices have been designed on the principle of peristaltic pumping to transport the fluids without internal moving parts, for example the blood pump in the heart-lung machine and peristaltic transport of naxious fluid in nuclear industry. The mechanism of peristaltic transport has attracted the attention of many investigators since its investigation by Latham [13], Burns and pareks [2], shapero et all. [22], Fung and yih [3], Takabatake and Ayukawa [25], Akram and Nedeem [1], mekheimer and Elkot [16], Mekheimor and al -arabi [15], mekheimer [14], Nadeamand akbar [19], Kothandapaniet al. [10], of peristaltic flow for different fluids have been reported under various conditions with reference to physiological and mechanical situations. Most of these investigations are confined to the peristaltic flow only in a symmetric channel or tube . consideration of wall properties in peristalsis is of special value in study of blood flow in arteries and veins, urine flow in the urethras and air flow in the lungs. Peristaltic motion in a complaint wall channel has also been investigated by some researchers. Radhakrishnamacharya and Srinivasulu [20] analyzed the influence of wall properties on peristaltic motion of Newtonian fluid with heat transfer. Peristaltic motion of micro polar fluid in circular cylindrical tubes with wall properties is discussed by muth et al [17], Hayat et al. [5,6] examined the MHD peristaltic flow of Jeffery and Johnson -Segalman fluids with compliant walls.srinivas and kothandapani [23] analyzed the heat and mass transfer effects on MHD peristaltic flow of Newtonian fluid in a porous channel with compliant walls. Riaz et al. [21] investigated the peristaltic motion of prandtl fluid in rectangular duct with wall properties. Recently, peristaltic flow of burgers fluid in complaint walls channel was investigated by javed et al. [9]. Peristaltic flow with complaint walls and hall current was studied by gad [4]. Very recently, the combined influence of heat and mass transfer on the peristaltic motion of pseudeplastic fluid with wall properties was analytically explored by hina et al. [7]. Slip effects on the peristaltic flow of eyring-powell fluid with wall properties were examined by hina [8]. Amongst the many suggested models, walters [26] has developed a physically accurate

MATHEMATICAL MODELS
Let us consider the MHD flow of an incompressible and electrically conducting walters -B fluid through a porous medium of two -dimensional tapered a symmetric channel. We assume that infinite wave train traveling with velocity c along the non -uniform walls. We choose a rectangular coordinate system for the channel with X along the direction of wave propagation and parallel to the centre line and Y transverse to it. The wall of the tapered a symmetric channel are given in fig.( aaare the amplitudes of the waves,  is the wave length, 2d is the width of the channel at the inlet, . In which S is the Cauchy stress tensor, -PI is the spherical part of the stress due to constrain of in compressibility,  is the extra stress tensor, ( ) the stress components are given by in which  is the fluid density ,P is the pressure, ( , ) UV are velocity components in the direction of the laboratory frame

( , )
XY ,  is the electrically conductivity and B0 is the constant magnetic field.
We employ the following dimensionless variables in the governing equations of motion: , , (7) is satisfied automatically and Eqs.(8),(9),(10), (11) and (12) Now, in the laboratory frame ( , ) XY the flow is unsteady, However if treated it as steady flow in the wave frame (x, y), so the equations (13), (14), (15), (16) and (17) can be written as:    Under the long wave length and low Reynolds number approximations, equations (18) and (19) can be written as : From equations (24) indicates that P is the independent of y, from equation (23) We note that h1(x, t) and h2(x ,t) represent the dimensionless form of the surfaces of peristaltic walls:     , , , a a a a are constants can be determinates by using the boundary conditions in Eq.(33).

Velocity distribution
Influences of geometric parameters on the velocity distribution have been illustrated in Fig.(8-14) these figures are scratched at the fixed values of x=0.3, t=0.5 . the change in values of m on the axial velocity u is shown in fig.(8), it can be found that the axial velocity u decrease with an increase in m at the centre of channel and after y=-0.5 at the lower wall of the channel but after y=+0.5 at the upper wall of channel the velocity u increase with an increase in m . Fig.(9) shows the influence of  on the axial velocity u, it observed that an increase in  causes an increase in u at the centre of channel and decreasing in u after y=-0.5 at the lower part of channel. Fig.(10) displays the effect of a on the axial velocity u, it examined that an increase in a causes an increase in u after y=-0.5 at the lower part of channel and decreasing in u at the centre of the channel . Fig.(11) shows the effect of b on the axial velocity u, it observed that an increase in b causes decreasing in u at the centre of channel and upper wall of channel. The influence of M on the axial velocity u is shown in Fig.(12), it noticed that an increase in M causes decrease in u at the centre of channel which is the same behavior of effect of K on axial velocity u and it is platted in Fig.(13) and conversely behavior of effect of  on axial velocity u which is shown in Fig.(14).

Trapping phenomenon
The phenomenon of trapping is another interesting topic in peristaltic transport. The formation of an internally circulating bolus of fluid through closed stream lines is called trapping and this trapped bolus is pushed a head along with

CONCLUDING REMARKS
In this paper, we investigated the peristaltic transport of Walters -B fluid through aporous medium in the tapered a symmetric channel under the influence of magnetic field, the channel a symmetry is produced by choosing the peristaltic waves train on the non-uniform walls to have different amplitudes and phases along-wave length and low Reynolds number approximations are adopted. A regular perturbation method is employed to obtain the expression for stream function, axial velocity and pressure gradient-numerical study has been conducted for average rise in pressure over a wave length. The effects of Hartmann number (M), the inverse of Darey number (k), wave amplitudes (a &b), channel width (m) and phase angle  on the pressure rise, axial velocity and stream lines are also investigated in detail. It found that :