Effect of Inclined Magnetic Field on the Peristaltic Flow of Non-Newtonian Fluid with Partial Slip and Couple Stress in a Symmetric Channel

This paper focus on the variable inclined magnetic field on the peristaltic flow of non-Newtonian fluid. The angle between the magnetic and velocity fields is variable and all the flow variables are functions of two parameters coordinates and time. The equations which describing the diffusion of inclined magnetic field are derived and solved by using long wavelength approximation. We addressed some of the important physical concepts that have a direct relationship such as the pressure gradient, Pressure rise, velocity field, the stream line and trapping phenomena. This study is done through drawing many graphs by using the MATHEMATICA package.


INTRODUCTION
Peristaltic problems have gained a considerable importance because of it applications in physiology, engineering, and industry. Such applications include urine transport, swallowing food through the esophagus, movement of chime in the gastrointestinal tract, movement of ovum in the female fallopian tubes, vasomotor of small blood vessels, transport of slurries, corrosive

Original Research Article
fluids, sanitary fluids, and noxious fluids in nuclear industry. In view of these applications, several researchers investigated such fluid flow problem considering different subject like Noreen Sher Akbar et al. [1] have studied the Peristaltic flow of a Williamson fluid in an inclined asymmetric channel partial slip and heat transfer, Rama et al. [2] discussed the Second Law Analysis for Combined with Convection in Non-Newtonian Fluids over a Vertical Wedge Embedded in a Porous Medium, the results show that the Bejan number increases with the viscosity index and the buoyancy parameter, R. Ali et al. [3] analyzed Mixed convection heat and mass transfer of non-Newtonian fluids from a permeable surface embedded in a porous medium . N. Sandeep et al. [4] have shown the Effect of Inclined Magnetic Field on Unsteady free Convective flow of Dissipative Fluid past a Vertical Plate, N. Sandeep et al. [5] analyzed Radiation and inclined magnetic field effects on unsteady MHD convective flow past an impulsively moving vertical plate in a porous medium. N. Sandeep et al. [6] studied the Aligned magnetic field, radiation and rotation effects on unsteady hydro magnetic free convection flow past an impulsively moving vertical plate in a porous medium, Masoud A. Frand et al. [7] Study the Effect of Magnetic Field on Free Convection in Inclined Cylindrical Annulus Containing Molten Potassium. A. Malvandi et al. [8] analyzed MHD mixed convection in a vertical annulus filled with Al2O3water nanofluid considering nano-particle migration, M. Modather et al. [9] studied MHD Mixed Convection Stagnation-Point Flow of a Viscoelastic Fluid towards a Stretching Sheet in a Porous Medium with Heat Generation and Radiation, C.S.K. Raju et al. [10] analyzed Radiation, Inclined Magnetic field and Cross-Diffusion effects on flow over a stretching surface. C.S.K. Raju et al. [11] discussed the Effects of aligned magnetic field and radiation on the flow of Ferro fluids over a flat plate with nonuniform heat source/sink, C. Sulochana et al. [12] discussed the Radiation and Chemical Reaction Effects on MHD Nanofluid Flow over a Continuously Moving Surface in Porous Medium with Non-Uniform Heat Source/Sink.
With the above introduction the objective of this investigation is to analysis the effect of inclined magnetic field on the peristaltic flow of non-Newtonian fluid with partial slip and couple stress in a symmetric channel. The relevant equations are modeled and simplified using long wavelength approximation and the expressions for pressure rise has been calculated using numerical integration by software MATHEMATICA package, some of the important physical concepts that have a direct relationship like the pressure gradient, Pressure rise, velocity field, the stream line and trapping phenomena have been addressed.

MATHEMATICAL FORMULATION AND ANALYSIS
Consider MHD flow of an electrically conducting viscous fluid in asymmetric channel through porous medium. The lower wall of the channel is maintained at temperature T1 while the upper wall has temperature T0 as shown in the Fig. 1.

Fig. 1. Schematic diagram of a tow-dimensional asymmetric channel
The geometry of the wall surface is define as: U, V are the velocities in X and Y directions in fixed frame, ρ is constant density, p is the pressure, υ is the kinematics viscosity, σ is the electrical conductivity, K is the permeability parameter, Introducing a wave frame (X,Y) moving with velocity c away from the fixed frame (x, y) by the transformation x=X-ct, y=Y, u=U-c, v=V, p(x) =P(X, t) Defining Using the above non-dimensional quantities and neglecting the terms of order δ and higher, the resulting equations in terms of stream function ψ ( ) can be written as: Since we are considering the partial slip on the wall, therefore, the corresponding boundary conditions for the present problem can be written as The finishing couple stress boundary condition is: Where q is the flux in the wave frame, a, b, φ and d satisfy the relation The solution of the momentum equation straight forward can be written as , amplitude ratioφ , partial slip L. We find that pressure gradient is maximum at X=0.5 for α =1. increases.
As shown in Fig. 4 and Fig. 7, the pressure gradient decreases when the other parameters increase. ,φ the p ∆ will be decreases. In Fig. 8 p ∆ increases for Q < -0.3and for Q > -0.3, p ∆ has an opposite behavior. In Fig. 9 p ∆ decreases for Q<-0.4 and for Q> -0.4, p ∆ has an opposite behavior.

CONCLUSIONS
We have discussed the results with inclined magnetic field on the peristaltic flow of a non-Newtonian fluid with couple stress and partial slip. The results are discussed through graphs. We have concluded the following observations: