Simultaneous effects of MHD and partial slip on peristaltic flow of Jeffery fluid in a rectangular duct

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Highlights

  • This paper analyses the peristaltic flow of Jeffery fluid.

  • Simultaneous effects of MHD and partial slip conditions are taken into account.

  • Closed form solutions are obtained by the method of separation of variable.

  • Effects of sundry parameters are presented through graphs and trapping bolus discipline

  • Comparison with the existing studies is also made.

Abstract

The purpose of this paper is to study the closed-form solutions of peristaltic flow of Jeffery fluid under the simultaneous effects of magnetohydrodynamics (MHD) and partial slip conditions in a rectangular duct. The influence of wave train propagation is also taken into account. The analysis of mathematical model consists of continuity and the momentum equations are carried out under long wavelength (0<<) and low Reynolds number (Re0) assumptions. The governing equations are first reduced to the dimensionless system of partial differential equation using the appropriate variables and afterwards exact solutions are obtained by applying the method of separation of variables. The role of pertinent parameters such as Hartmann number M, slip parameter β1, volumetric flow rate Q, Jeffery parameter λ1 and the aspect ratio β against the velocity profile, pressure gradient and pressure rise is illustrated graphically. The streamlines have also been presented to discuss the trapping bolus discipline. Comparison with the existing studies is made as a limiting case of the considered problem.at the end.

Introduction

Over the last thirty years, or so, a very hot topic, namely non-Newtonian fluid flow appeared in many application in industrial processes, biology, medicine, catalytic chemistry and environmental applications [1], [2], [3], [4], [5], [6]. In bio-fluids the physiological system has been investigated by several researchers in order to find the treatment of diagnostic problems that arise during circulation in a human body. There are several models which have been proposed to describe such physiological fluids, however their full potential has not been exploited yet and a lot of questions remain unresolved. Among several models, non-Newtonian Jeffrey model is significant because Newtonian fluid model can be deduced from it as a special case by taking λ1=0. It is also speculated that the physiological fluids such as blood exhibit Newtonian and non-Newtonian behaviors simultaneously.

The peristaltic models with reactions are also encountered in the physiological study that is why among all continuous system taking place in a human's body and becoming the main reason for one's life, peristalsis is one of them. Basically peristaltic word comes from Greek word peristaltikos which means clasping and compressing. The importance of peristalsis can be assumed by the fact that, it is an automatic series of muscular contraction and relaxation, which takes place in human's body, such as digestive tract or digestive system. Peristalsis does cause the movement of food through digestive system, chyme in the gastrointestinal tract, urine from kidney to bladder and bile from gallbladder to duodenum. Also, transport of lymph in the lymphatic vessels and vasomotion of small blood vessels like arterioles, venules and capillaries involve the peristaltic motion are the common examples of peristaltic motion. In addition peristaltic motion has been equally, playing an important and beneficial part in physiological sciences by devising mechanical and biomechanical instruments such as roller pumps and heart lung machine, which function mainly by using the principle of peristalsis. This has opened a new dimension for researchers to maneuver their equipment for obtaining better results in their respective field of interest. Different mathematical models have been employed by many authors to use peristaltic flow in Newtonian and non-Newtonian fluid [7], [8], [9], [10], [11].

Moreover, magnetohydrodynamics (MHD) effects on peristalsis are significant in magneto therapy, hyperthermia, arterial flow, compressor, etc. The controlled application of low intensity and frequency pulsating fields modify the cell and tissue. Magnetic susceptibility of chyme is also satisfied the ions contained in the chyme or with heat generated by the magnetic field. The magnets could heat inflammations, ulceration, several diseases of uterus and bowel (intestine). Furthermore, MHD is used in the study of electrically conducting fluids; examples of such fluids include electrolytes, saltwater, liquid metals and plasmas. A number of researchers have applied magnetic field on peristaltic mechanisms, one way or the other. An extensive literature on the said topics is now available but we can only mention a few recent interesting investigations here [12], [13], [14], [15], [16], [17], [18], [19].

Furthermore, the fluids that exhibit slip effect have many applications, for instance, the polishing of artificial heart valves and internal cavities. In all these studies the peristaltic flow problems have been extensively investigated with no slip condition. A very less emphasis has been given to such flows in the presence of a slip/partial slip conditions. Although, the application of slip condition in the peristaltic flows has special relevance in polymers and physiology. The case of slip effects may appear for two types of fluids namely fluids having much more elastic character and rare field gases. In these fluids, slippage appears subject to large tangential traction. It is found through experimental observations that the existence of slippage is possible in the non-Newtonian fluids, polymer solution and molten polymer. Additionally, a clear layer is occasionally detected next to the wall when flow of dilute suspension of particles examined. In experimental physiology such a layer is observed when blood flow through capillary vessels is considered. The peristaltic flows subject to different flow aspects and configurations have been examined. Some pertinent studies in numerous situations can be seen from the list of Refs. [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and several therein.

With all abovementioned studies, one can clearly observe that no analysis for magnetohydrodynamic peristaltic flow of Jeffery fluid in the presence of partial slip conditions is reported in the available literature yet. The present work puts forward to fill this gap. The flow analysis is performed under the constraints of long wavelength approximation and low Reynolds number. An exact solution for the expression for both velocity of the fluid and pressure gradient are obtained by using the method of separation of variables. The pumping characteristics such as pressure rise and trapping phenomena are obtained numerically by using software Mathematica. The physical features of pertinent parameters have been comprehensively elaborated through graphs. The organization of paper is in the following fashion: After introduction, Section 2 contains the formulation of the problem. Solution of the problem is given in Section 3. Section 4 is devoted for results and discussion and finally the main outcomes are concluded in Section 5.

Section snippets

Mathematical formulation

Let us consider the peristaltic flow of an incompressible, Jeffrey fluid in a duct of rectangular cross section having the channel width 2d and height 2a. Cartesian coordinates system is considered in such a way that X-axis is taken along the axial direction, Yaxis is taken along the lateral direction and Zaxis is along the vertical direction of a rectangular duct.

The peristaltic waves on the wall are represented asZ=H(X,t)=±a±bcos[2πλ(Xct)],where a and b are the amplitudes of the waves, λ

Solution of the problem

In this section the analytical solutions will be determined by using the following transformation [32]:u(y,z)=v(y,z)+w1(y)

In view of above transformation, Eq. (19) along with the associated conditions yields the following system of differential equations:β21+λ1d2w1dy2M2w1=M2+dpdxandβ21+λ12vy2+11+λ12vz2M2v=0.w1(y)=1,aty=±1,v(y,z)=0,aty=±1,v(y,z)=β11+λ1vz1w1(y),atz=h,v(y,z)=β11+λ1vz1w1(y),atz=h.

One can clearly observe that Eq. (24) is a second order linear but non-homogenous ODE.

Results and discussion

In this section, the relative changes in the behavior of the fluid flow, caused by the variation in the different parameters, are displayed and discussed graphically. The pertinent parameters for the considered problem are, aspect ratio β, Jeffrey's parameter λ1, volumetric flow rate Q, slip parameter β1 and the Hartmann number M.

Fig. 1, Fig. 2, show the velocity profile u of the peristaltic wave, for different values of slip parameter β1, Hartmann’s number M. From these figures it is observed

Conclusion

The peristaltic flow of non-Newtonian flow of Jeffery fluid with MHD and partial slip condition has been studied under the assumptions of long wavelength and low-Reynolds number. The governing partial differential equations corresponding to their boundary condition has been solved by using the method of separation of variables. It is noted that an increase in the slip parameter decreases the pressure gradient whereas the size of trapped bolus increases by increasing the slip parameter. It is

Acknowledgements

Farooq Hussain is grateful to the Higher Education Commission (HEC) Pakistan to award him indigenous scholarship for his Ph.D. studies.

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