Impact of Couple Stress with Rotation on Walters , B Fluid in Porous Medium

: This study investigates the peristaltic flow of Walter's B fluid through an asymmetric channel using a couple stress model and a porous material. The constitutive equation is used to model mas balance, motion, trapped phenomena, velocity distribution

load on the peristaltic flow of an incompressible fluid was studied by the authors [16].The investigation of the effects of an angled magnetic field and the heat transfer from an asymmetric channel on hyperbolic tangent peristaltic flow with a porous medium in [17].Effects of the rotation on the mixed convection heat transfer analysis for the peristaltic transport of viscoelastic fluid in asymmetric channel.in[18] The goal of this research is to discuss the impact that an inclined magnetic field has on the peristaltic motion of Walter's B fluid.Because of its incompressibility, it fills an asymmetric channel.Parameters including rotation, density, wave amplitude, and channel taper are varied.Variations in the confined phenomenon, velocity distribution, and pressure gradient were also studied using varied values of the parameters.

PROBLEM DESCRIPTION
Assume the peristaltic transport of Walters B of an incompressible of tow dimensional in asymmetric channel considering width d1+d2 with electrically conduction fluid through a porous medium.The flow is produced by endless sinusoidal waves that repeat at a fixed speed c along the channel walls.As a result, the symmetric channel has variable wave amplitudes, phase angles, and channel widths.
Here is the modeling of the wall geometries )] where 1 a ,  , c, Ф are the wave's amplitude, length, and velocity, Ф is the phase difference various in the range (0 ≤ Ф ≤  ).Choose the tow dimension system, X along centerline channel, while Y is transverse to it.This demonstrates that a symmetric channel with out-of-phase waves is represented by a value of Ф=0 , whereas when Ф= .The waves are in phase.Further 1 a , 2 a , 1 d , 2 d , and Ф satisfies the condition. 2

3.THE CONSTITUTION EQUATION
Using a laboratory frame ( y x, ), the continuity and momentum partial differentials of the flow's constitutional equation are written as: The stress component are given   are the fluid density ,axial velocity , the transverse velocity, the transverse coordinate , the pressure, the viscosity , the material constant, the permeability parameter, and the constant magnetic field, The electrical conductivity is unsteady in natural peristaltic motion.Peristaltic motion is unsteady in nature, we can assume steady by using transformation from laboratory frame ( y x, ) to wave frame ( Y X , ) which defined as: The velocity elements represented by V U , and P of the pressure in the wave frame.
The following non-dimensional quantities are arranged in order to carry out the non-dimensionalanalysis: Where (Ha) is the Hartman number, (Re) is the Reynold number,( ) is the wave number, (K) is the viscoelastic parameter,(α) represent couple stress parameter.Thereafter, in light of Equation ( 11), Equations.(3) to ( 9) takes the form The components of extra stress tenser of walter's B are listed below The velocity elements are linked to the stream function (ѱ) by the relationships Substituted Equation (10) and Equation (18) in Equations (13)(14)(15)(16)(17), respectively Removal of p between Equations ( 19) and (20) produce The wave frame by the dimensionless boundary conditions: (33)

4.THE EMPACT OF COUPLE STRESS
This section provides the link between the viscoelastic and pair stress parameters.Using this connection, we can simplify the problem.The effect of all parameters, such as viscoelastic and pair stress, should be investigated in this study.At begging must find zero-th and first orders solutions, when us the relationship between viscoelastic and couple stress parameters we will find the zero-th order

6.RESULTS AND DISCUSSION
This section is divided in to sub sections.In the first place, we talk about traps.In the second, we look at the scatterplot of velocities.MATHEMATIC is used to graphically represent the pressure gradients stated in the third.6.1 TRAPPING SECTION Peristaltic motion's trapping is an intriguing phenomenon.In essence, it is the closed stream line creation of an internally circulating bolus of fluid.We testing the stream function profile with changing the values of parameters (k, α, β1, Ha, Ω (.The profile of stream lines behavior is increasing on the change for different parameters.Notes that the changing of the stream line and appearing the bolus for every parameter clearly in figures (2,4,5,6,7).Figure (3) We found that when the parameter () value increased, the trapped bolus shrank.The profile of stream line no effect of the size of trapped bolus with increasing the values of parameters (Q, β * ) clearly in figures.(8,9).

VELOCITY SECTION
The velocity distribution is represented by various properties.The change in velocity for different values of (Ω, α, β1, Ha, ɸ, K, Q, β * ) is explained in figures (10)(11)(12)(13)(14)(15)(16)(17).In figures (10,12) As we increase the various parameters (Ω, Ha), we see that the axial velocity in the channel's middle portion decreases.Whereas the axial velocity increases near the channel wall's edge.As the parameters (K), are increased, the axial velocity in the channel's central region increases, while.figure (11).shows that there is no influence from the channel's edge.As the parameters (α), are increased, the core part of the channel experiences a drop in axial velocity.while no effect of the boundary of channel that is shown in figure (14).The axial velocity decrease in the central region and near the right wall of the channel with increasing the parameter (ɸ), whereas increases near the left wall of the channel that is explained in figure (13).The axial velocity increases near the right wall of the channel with increasing the parameter (β1), whereas the velocity decreases in the middle and the left wall of channel that is in figure (15).From figures (16,17) shows that there is no effect on the axial velocity with increasing in the value of parameters (Q, β * ).

PRESSURE GRADIENT
The effect of different values of parameters (K, α, β1, Ha, Ω, ɸ, Q, β * ) on pressure gradient its clearly in figures (18)(19)(20)(21)(22)(23)(24)(25).Figures (18) that ascending values of the parameters (Ω) we see that the pressure gradient has no effect on the area close to the right wall of the channel, but it does increase the area close to the center and the left wall.In figures (19,22,23) the pressure gradient increases of the channel with increasing the parameters (K, β1, Ha).As the values of the parameters (α) are increased, the pressure gradient at the channel's wall boundary rises, while on effect of the central region that is explained in figure (21).Notes that the pressure gradient no effect of the channel with increasing of the parameters (Q, β * ) shown it in figures (24,25).The pressure gradient increases near the left wall of the channel with increasing the parameters (), while on effect of the central region and decreases near the right wall of the channel that is explained in figure (20).

CONCLUTION
The study focused on how rotating in an asymmetric channel and porous medium affects Walters' B fluid with couple stress and an angled magnetic field.Special attention was placed in this inquiry on studying trapping phenomena, velocity distribution and pressure gradient using a simple analytical solution.
-In case of stream function the trapped bolus size increases with increasing the value of parameters ( . -We saw that when the value of the parameter () increased, the size of the trapped bolus shrank.
-The amount of the trapped bolus has no impact on the stream line profile as parameters (Q, β*) values increase.
-The axial velocity decreases in the core part of the channel as each parameter (Ω, Ha) increases, whereas the axial velocity increases along the channel wall's parameter.
-As the fluid parameter (K) are raised, the axial velocity increases in the channel's center and no effect of the boundary of channel.
-As the fluid parameter (α) are raised, the axial velocity decreases in the channel's center and no effect of the boundary of channel.
mean flow rate (F) in the wave frame.In the laboratory frame the dimensionless time mean flow rate Q1 is related to F by through the expression:.= Q1 F+1+d(31) The h1(x) and h2(x) the dimensionless forms (32) where a ,b, Ф and d satisfy : Substituting Equations(8)-(23) in to Equation (24) and assumption the low Reynolds number and the long wavelength, we often the motion equation in frame of stream functionThe solution of the momentum equation is straight forward and can be written as ψ , C2, C3, C4, C5, C6 are large expression will mention in the end of this work.