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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.

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 [

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. [

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 [

Consider steady three-dimensional fully developed laminar Couette flow of an incompressible electrically conducting Maxwell fluid between two parallel porous plates having separation “

Diagram of the problem.

The constitutive equation for a Maxwell fluid model is

Since

Substituting (

When

The components of shear stress

In this work, steady and fully developed laminar Couette flow of an incompressible Maxwell fluid through porous plates with periodic suction/injection is modelled and investigated analytically. The application of transverse sinusoidal injection at the lower plate remained stationary and its equivalent confiscation by suction through the uniformly moving upper plate leads to three-dimensional flow. The coupled highly nonlinear equations of motion are solved engaging perturbation method. The effects of various nondimensional parameters on velocity field, skin friction components, and pressure are presented graphically in Figures

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The main flow velocity profiles are depicted in Figures

The influences of injection/suction parameter

The transverse velocity component

The variations of skin friction components at the lower plate versus Reynolds number Re in the main flow direction and transverse directions are presented in Figures

Figures

The effects of injection/suction parameter

On the basis of above discussion, the following conclusions are made:

The main flow velocity decreases with increasing either injection/suction parameter or Reynolds number. It decreases with an increase in the Deborah number.

The velocity component

Reynolds number provides a mechanism to stabilize the skin friction components

The present analysis gives a better result as variable injection/suction velocity is considered at both plates because in natural practice injection/suction cannot be uniform in all cases.

Constants involved in this paper are

The authors declare that there are no conflicts of interest regarding the publication of this paper.