Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 3085249, 13 pages

https://doi.org/10.1155/2017/3085249

## Unsteady Micropolar Fluid over a Permeable Curved Stretching Shrinking Surface

^{1}Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia^{2}Institute For Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia^{3}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor, Malaysia^{4}Department of Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania

Correspondence should be addressed to Norihan Md Arifin

Received 26 September 2016; Accepted 15 December 2016; Published 20 February 2017

Academic Editor: Mohammad D. Aliyu

Copyright © 2017 Siti Hidayah Muhad Saleh 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

This work deals with the unsteady micropolar fluid over a permeable curved stretching and shrinking surface. Using similarity transformations, the governing boundary layer equations are transformed into the nonlinear ordinary (similarity) differential equations. The transformed equations are then solved numerically using the shooting method. The effects of the governing parameters on the skin friction and couple stress are illustrated graphically. The results reveal that dual solutions exist for stretching/shrinking surface as well as weak/strong concentration. A comparison with known results from the open literature has been done and it is shown to be in excellent agreement.

#### 1. Introduction

The fundamental nature of the theory of micropolar fluid flow lies in the extension of the constitutive equations for Newtonian fluids, so that more complex fluid such as particle suspensions, liquid crystals, animal blood, and turbulent shear flows can be expressed by this theory. The micropolar fluid theory which takes into account the inertial characteristics of the substructure particles which are permitted to endure rotation has been intended by Eringen [1] and was further discussed by Eringen [2]. This theory has caused much attention and many aspects of flows being reexamined to verify the effect of the fluid microstructure. A detailed study on the applications of microcontinuum fluid mechanics has been accessed by Ariman et al. [3]. Since then, many other studies related to the ideas of micropolar fluid have been done. Ahmadi [4] analyzed self-similar solution for incompressible micropolar flow over semi-infinite plate. A study on stagnation flows of micropolar fluids with strong and weak interactions was reported by Guram and Smith [5]. Then, it seems Gorla [6] has obtained numerical results for the micropolar boundary layer flow at a stagnation point on moving wall using fourth-order Runge Kutta method.

During the past several years considerable interest has been verified in the study of steady and unsteady flows of a viscous incompressible fluid determined by a linearly stretching surface through a quiescent fluid. Such flow situations are encountered in a number of manufacturing practices, such as the cooling of metallic plate in a cooling bath, polymer sheet extrusion, and heat-treated materials that are on a conveyor belt. Crane [7] found a closed-form exact solution for a two-dimensional laminar flow of an incompressible viscous fluid over a linearly stretching sheet, and Wang [8] discovered similarity solutions for the axisymmetric case. In another paper, Wang [9] has analyzed the viscous flow due to a stretching sheet with surface slip and suction. The combination of both stagnation point-flows past a stretching surface was solved numerically by Mahapatra and Gupta [10, 11] and analytically by Mahapatra et al. [12]. The effects of radiation with magnetic field on stretching sheet were obtained by Mat Yasin et al. [13]. Also, in a very interesting recently published paper, Turkyilmazoglu [14] has mathematically resolved the domain of existence for magnetohydrodynamic mixed convection flow of a micropolar fluid past a heated or cooled stretching permeable surface by taking into account the heat generation and absorption effects. Furthermore, few studies have been done on the reaction of permeable stretching sheet in nanofluid which was conducted by Ibrahim and Shankar [15], Bachok et al. [16], and Zaimi et al. [17]. Moreover, Turkyilmazoglu [18] analyzed the flow of a micropolar fluid due to a porous stretching sheet and heat transfer.

Furthermore, the work on boundary layer flow due to a shrinking sheet has also attracted much interest. It seems that Miklavčič and Wang [19] investigated the flow over a shrinking sheet and found an exact solution of the Navier–Stokes equations. It was discovered that mass suction is needed to sustain the flow over a shrinking sheet. From physical point of view, vorticity of the shrinking sheet is not restricted within a boundary layer, and the flow is improbable to subsist unless an adequate suction on the boundary is discussed [19]. Further, the velocity of the shrinking sheet is confined in the boundary layer with stagnation flow was discovered by Wang [20]. This new type of shrinking sheet flow is fundamentally a backward flow as discussed by Goldstein [21]. On the other hand, Turkyilmazoglu [22] has analytically studied the steady flow of micropolar fluid and heat transfer over a shrinking sheet under porous wall conditions. He has verified mathematically the bounds of multiple existing solutions of purely exponential kind. It is worth mentioning that the importance and new results on the flow induced by a shrinking sheet in a various fluid and geometrical approach were recently presented by Rosali et al. [23], Naramgari and Sulochana [24], Merkin et al. [25], and Soid et al. [26]. In addition, Roşca and Pop [27] investigated combined effects of micropolar fluid towards a permeable shrinking sheet in the presence of slip effect and Aurangzaib et al. [28] extend it with MHD mixed convection effects. Next, a study of the viscous flow over a shrinking sheet analytically solved using a second-order slip flow model was by Fang et al. [29] where they obtained that the result has two branches, or dual solutions, in a certain range of the parameters, while Bhattacharyya et al. [30] studied that the velocity and thermal boundary layer thicknesses for the second solutions are always larger than the first solutions.

Instead of plane boundary layer, there has been growing interest in the study of the effects of curvature by several investigators recently. Few papers have been discussed on this topic and found that its presence inside the boundary layer is no more negligible as in the case of a flat stretching sheet. Sajid et al. [33] considered stretching curved surface viscous fluid flow and discovered that the boundary layer thickness increases for a curved surface compared to flat surface. Moreover, the drag force to move the liquid on curved surface is less than on a flat surface. Then, they extend the problem to micropolar fluid (see Sajid et al. [34]). Abbas et al. [31] discussed the heat transfer analysis by considering the two heat processes, namely, prescribed surface temperature and described heat flux on MHD boundary layer flow on a curved stretching surface. Further, they continued with slip effect with generation and thermal radiation in Abbas et al. [35]. Homogeneous-heterogeneous reactions in MHD flow due to an unsteady curved stretching surface were being observed by Imtiaz et al. [36].

It is worth mentioning that there are also many examples of unsteady flow and in fact, there is no real flow situation, natural or artificial, that does not occupy some unsteadiness. The flows in all manufacturing applications were randomly assumed to be steady. In many engineering instruments, unsteadiness is an integral part of the problem. The helicopter rotor, the cascades of turbomachinery blade, the ship propeller, and so forth usually work in an unsteady environment. Most of the elementary ideas of unsteady viscous flows are described by Telionis [37]. Quite different flow behavior is observed for an unsteady shrinking sheet that is for an unsteady stretching sheet. Roşca and Pop [32] performed a study on the unsteady boundary layer flow over a permeable curved stretching/shrinking sheet. Boundary layer flow of a viscous fluid over an unsteady curved stretching sheet with magnetic field is discussed by Naveed et al. [38].

The aim of the present paper is to study the flow problem due to the unsteady plane, two-dimensional laminar flow caused by a permeable curved stretching/shrinking sheet in a micropolar fluid. The governing partial differential equations are first transformed into a system of ordinary differential equations, before being solved numerically. The expressions for the skin friction coefficients and couple stress coefficients, including velocity and microrotation velocity, are determined to understand the flow characteristics. It is hoped that the results obtained will not only provide useful information for applications, but also serve as a complement to the previous studies. To the best of our knowledge these results are new and original.

#### 2. Basic Equations

Consider the unsteady two-dimensional boundary layer flow of a micropolar fluid over a permeable curved stretching/shrinking surface coiled in a circle of radius about the curvilinear coordinates and , where is normal to tangent vector at any point on the surface and is the arc length coordinate along the flow direction, as depicted in Figure 1(a), so that large values of represent a slightly curved sheet. The geometry for the stretching and shrinking sheet is shown in Figures 1(b) and 1(c), respectively. A constant magnetic field, , is applied in the -direction. By assuming that the magnetic Reynolds number is low, thus, we have neglected the effects of induced magnetic field. Also the applied electric field is assumed to be zero. Moreover, it is assumed that the surface is stretched/shrinked with the velocity along the -direction. It should be mentioned that is the mass flux velocity, where corresponds to suction and corresponds to injection, respectively, and is the time.