Physics Letters A 372 (2008) 4271–4278 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Effect of the induced magnetic field on peristaltic flow of a couple stress fluid Kh.S. Mekheimer Mathematical Department, Faculty of Science (Men), Al-Azhar University, Nasr City 11884, Cairo, Egypt article info abstract Article history: Received 31 January 2008 Received in revised form 28 February 2008 Accepted 31 March 2008 Available online 9 April 2008 Communicated by F. Porcelli Keywords: Couple stress fluid Induced magnetic field MHD We have analyzed the MHD flow of a conducting couple stress fluid in a slit channel with rhythmi- cally contracting walls. In this analysis we are taking into account the induced magnetic field. Analytical expressions for the stream function, the magnetic force function, the axial pressure gradient, the axial induced magnetic field and the distribution of the current density across the channel are obtained using long wavelength approximation. The results for the pressure rise, the frictional force per wave length, the axial induced magnetic field and distribution of the current density across the channel have been computed numerically and the results were studied for various values of the physical parameters of in- terest, such as the couple stress parameter γ , the Hartmann number M, the magnetic Reynolds number R m and the time averaged mean flow rate θ . Contour plots for the stream and magnetic force functions are obtained and the trapping phenomena for the flow field is discussed.  2008 Elsevier B.V. All rights reserved. 1. Introduction The study of a couple stress fluid is very useful in understand- ing various physical problems, because it possesses the mechanism to describe rheologically complex fluids such as liquid crystals, col- loidal fluids, liquids containing long-chain molecules as polymeric suspensions, animal and human blood and lubrication. The micro-continuum theory of couple stress fluid proposed by Stokes [1], defines the rotational field in terms of the velocity field for setting up the constitutive relationship between the stress and strain rate. Stokes micro-continuum theory is the simplest gener- alization of the classical theory of fluids, which allows for polar effects such as the presence of couple stresses, body couples and a non-symmetric stress tensor. Some theoretical studies [2–6] of blood flow indicate that some of the non-Newtonian flow prop- erties of blood may be explained by assuming the blood to be a fluid with couple stress. The couple-stress fluid may be consid- ered as a special case of a non-Newtonian fluid which is intended to take into account the particle size effects. Moreover, the couple stress fluid model is one of the numerous models that proposed to describe response characteristics of non-Newtonian fluids. The constitutive equations in these fluid models can be very complex and involving a number of parameters, also the outcoming flow equations lead to boundary value problems in which the order of differential equations is higher than the Navier–Stokes equations. Some recent investigations regarding such fluids are mentioned in the studies [7–15]. E-mail address: kh_mekheimer@yahoo.com. Magnetohydrodynamics (MHD) is the science which deals with the motion of a highly conducting fluids in the presence of a mag- netic field. The motion of the conducting fluid across the magnetic field generates electric currents which change the magnetic field, and the action of the magnetic field on these currents gives rise to mechanical forces which modify the flow of the fluid [16]. The magnetohydrodynamic (MHD) flow of a fluid in a channel with elastic, rhythmically contracting walls (peristaltic flow) is of in- terest in connection with certain problems of the movement of conductive physiological fluids, e.g., the blood, blood pump ma- chines and with the need for theoretical research on the operation of a peristaltic MHD compressor. Effect of a moving magnetic field on blood flow was studied by Stud et al. [17], and they observed that the effect of suitable moving magnetic field accelerates the speed of blood. Srivastava and Agrawal [18] considered the blood as an electrically conducting fluid and constitutes a suspension of red cell in plasma. Also Agrawal and Anwaruddin [19] studied the effect of magnetic field on blood flow by taking a simple math- ematical model for blood through an equally branched channel with flexible walls executing peristaltic waves using long wave- length approximation method and observed, for the flow blood in arteries with arterial disease like arterial stenosis or arteriosclero- sis, that the influence of magnetic field may be utilized as a blood pump in carrying out cardiac operations. Moreover, the principle of magnetic field in the form of a device called Magnetic Resonance Imaging (MRI). Now MRI is used for diagnosis of brain, vascular diseases and all the human body. It has now been accepted that most of the physiological fluids behave like a non-Newtonian fluids. This approach provides a sat- isfactory understanding of the peristaltic mechanism involved in small blood vessels, lymphatic vessels, intestine, ductus efferentes 0375-9601/$ – see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.03.059