Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1859693, 19 pages

https://doi.org/10.1155/2017/1859693

## Magnetohydrodynamic Three-Dimensional Couette Flow of a Maxwell Fluid with Periodic Injection/Suction

^{1}Department of Mathematics and Statistics, Riphah International University, Sector I-14, Islamabad, Pakistan^{2}COMSATS Institute of Information Technology, Kamra Road, Attock 43600, Pakistan

Correspondence should be addressed to Y. Ali; moc.liamg@5risayr

Received 26 December 2016; Accepted 14 March 2017; Published 13 April 2017

Academic Editor: Eusebio Valero

Copyright © 2017 Y. Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A mathematical model for magnetohydrodynamic (MHD) three-dimensional Couette flow of an incompressible Maxwell fluid is developed and analyzed theoretically. The application of transverse sinusoidal injection at the lower stationary plate and its equivalent removal by suction through the uniformly moving upper plate lead to three-dimensional flow. Approximate solutions for velocity field, pressure, and skin friction are obtained. The effects of flow parameters such as Hartmann number, Reynolds number, suction/injection parameter, and the Deborah number on velocity components, skin friction factors along main flow direction and transverse direction, and pressure through parallel porous plates are discussed graphically. It is noted that Hartmann number provides a mechanism to control the skin friction component along the main flow direction.

#### 1. Introduction

In recent years, the problem of LFC (laminar flow control) has gained considerable importance due to its importance in the reduction of drag and hence in improving the vehicle power by a considerable amount. To control the boundary layer artificially, several methods have been proposed. One of the effective techniques for the reduction of the drag coefficient which causes large energy losses is the boundary layer suction method. It has been established theoretically as well as experimentally that the laminarization of boundary layer over a profile reduces the drag and hence the vehicle power requirements by a very significant amount. According to boundary layer, suction method slowed that fluid particles in the boundary layer are removed through the holes and slits in the wall into the interior of the body and, therefore, the transition from laminar to turbulent flow causing increase of drag coefficient may be deferred or prevented [1]. Many workers have considered the numerous aspects of fluid flow problems with suction but most of these studies cope with two-dimensional flows only. Gersten and Gross [2] considered the viscous fluid and studied the effect of transverse sinusoidal suction velocity on flow with heat transfer over a porous wall. Singh [3] studied the effect of transpiration cooling in the presence of the transverse sinusoidal suction/injection velocity. Chaudhary et al. [4] analyzed three-dimensional Couette flow in the presence of transpiration cooling between the plates and reported the effects of suction/injection velocity on the flow field, skin friction, and heat transfer. Guria and Jana [5] investigated unsteady three-dimensional fluctuating Couette flow through porous plates with heat transfer and found that the main flow velocity decreases with increase in frequency parameter; however, the magnitude of the cross flow velocity increases with increase in frequency parameter. Sharma et al. [6] considered radiation effect in three-dimensional Couette flow with suction/injection on temperature distribution. Chauhan and Kumar [7] investigated heat transient effects in a three-dimensional Couette flow between partly filled channels by a porous material. Various workers [8–11] also investigated three-dimensional flow viscous fluid past a porous plate under different physical conditions. Many technological problems and natural phenomena are vulnerable to magnetohydrodynamic (MHD) analysis. In the design of heat exchangers and pumps and flow meters, thermal protection, control, and reentry, in space propulsion and so forth, MHD principle is used by engineers. It has been proven theoretically and experimentally that the transition from laminar to turbulent flow which causes the drag coefficient to increase may be prevented/delayed by suction of the fluid by the application of transverse magnetic field and by heat and mass transfer from the boundary layer to the wall. Das [12] studied three-dimensional MHD Couette flow of a viscous incompressible fluid with heat transfer through a porous plate and reported effects of constant suction and sinusoidal injection on the flow. Sharma and Chaudhary [13] presented MHD effect on viscous incompressible flow between two horizontal parallel porous plates and heat transfer with periodic injection/suction. It was observed that forward flow is developed in the region near the stationary plate, while backward flow is developed in the region near the moving plate. Goyal and Naraniya [14] analyzed theoretically three-dimensional free convection Couette flow of a viscous incompressible fluid with transpiration cooling in the presence of transverse magnetic field. The static plate and the plate in uniform motion are subjected to transverse sinusoidal injection and uniform suction of the fluid. Recently, many workers [15–17] studied three-dimensional Couette flow of an incompressible fluid.

All the above studies have been performed in viscous fluid. Although the Navier-Stokes equations can cope with the flows of viscous fluids, these equations are inadequate to describe the characteristics of non-Newtonian fluids. Shoaib et al. [18–22] analyzed theoretically three-dimensional non-Newtonian fluids flow along an infinite plane with periodic suction.

However, to the best of the authors’ knowledge, the application of transverse sinusoidal injection/suction velocity for the flow of a second-grade fluid between parallel plates has not appeared in the literature. Therefore, in the present work, magnetohydrodynamic three-dimensional Couette flow of a Maxwell fluid with periodic injection/suction is analyzed. A constant suction velocity at the wall leads to two-dimensional flow [2]; however, due to variation of suction velocity in transverse direction on wall, the problem becomes three-dimensional. The solution of the problem is presented using regular perturbation technique. The results obtained are evaluated for various dimensionless parameters such as suction/injection parameter , the Deborah number , Hartmann number , and Reynolds number Re. The article is organized as follows: Section 2 presents description of the problem, Section 3 gives formulation of the problem, Section 4 approximates solutions, and Section 5 incorporates results and discussion, while Section 6 includes conclusion.

#### 2. Description of the Problem

Consider steady three-dimensional fully developed laminar Couette flow of an incompressible electrically conducting Maxwell fluid between two parallel porous plates having separation “” between them. The -plane is taken along the lower plate and the -axis perpendicular to the plates as shown in Figure 1. The magnetic field of uniform strength normal to the plates is applied. The injection/suction velocity distribution [2] of the formis assumed, where is suction/injection velocity and is its amplitude. The lower plate is kept stationary, while the upper plate is moving with uniform velocity along the positive -axis. The transverse sinusoidal injection of the fluid at the lower plate with its corresponding removal by periodic suction through the upper plate is considered. The velocity components along the -, -, and - directions are , , and , respectively. Since the flow is assumed to be fully developed and laminar, all the physical quantities are independent of ; of course, the flow remains three-dimensional due to injection/suction velocity (1).