Journal of Engineering

Volume 2016 (2016), Article ID 9718786, 10 pages

http://dx.doi.org/10.1155/2016/9718786

## Similarity Solution for High Weissenberg Number Flow of Upper-Convected Maxwell Fluid on a Linearly Stretching Sheet

Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran

Received 19 November 2015; Revised 9 March 2016; Accepted 24 April 2016

Academic Editor: Oronzio Manca

Copyright © 2016 Meysam Mohamadali and Nariman Ashrafi. 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

High Weissenberg boundary layer flow of viscoelastic fluids on a stretching surface has been studied. The flow is considered to be steady, low inertial, and two-dimensional. Upon proper scaling and by means of an exact similarity transformation, the nonlinear momentum and constitutive equations of each layer transform into the respective system of highly nonlinear and coupled ordinary differential equations. Numerical solutions to the resulting boundary value problem are obtained using an efficient shooting technique in conjunction with a variable stepping method for different values of pressure gradients. It is observed that, unlike the Newtonian flows, in order to maintain a potential flow, normal stresses must inevitably develop. The velocity field and stresses distributions over plate are presented for difference values of pressure gradient and Weissenberg numbers.

#### 1. Introduction

The flow of a liquid within a thin film over stretching plate is often encountered in most manufacturing processes. Examples include extrusion of plastic sheets, fabrication of adhesive tapes, and application of coating layers onto rigid substrates. Coating processes demand a smooth glossy surface to meet the requirements for best appearance and optimum service properties such as low friction, transparency, and strength. Due to the moving surface, the main flow is closed to the extruded material while the far field stays almost stagnant.

Most structured liquids, like polymers, show strong viscoelastic effects at small deformations, and their measurement is very useful as a physical probe of the microstructure. The subject of the viscoelastic flow was applied to developing pipeline designs such that the fluidity of high wax content crude oils is maintained [1]. More applications specific to the viscoelastic model include the modeling of plastics such as PET resins, which are used in, for example, film casting [2]. It has also been used to simulate the flow of polymer which is used for the process of wire-coating [3].

The analytical study of boundary layer flow due to a stretching sheet was initiated by Crane [4]. He assumed the velocity of the sheet to vary linearly as the distance from the slit and obtained an analytical solution. The work of Crane was subsequently extended mostly on both Newtonian and non-Newtonian (inelastic) boundary layer flows (see, e.g., [5–8]) and only a few works on viscoelastic and elastic boundary layer flows [9]. There is no report of such work on the stretching sheet. In this connection, however, Hassanien [10] studied the second-grade fluid boundary layer over a linearly stretching sheet. The study was on boundary layer approximations of Newtonian flows [11] in order to simplify the governing equations. Here, boundary layer equations were solved by a similarity method for elastic flows of Deborah numbers of up to 0.2.

The upper-convected Maxwell fluid is a class of viscoelastic fluid that can explain characteristics of fluid relaxation time. It excludes complicated effects of shear-dependent viscosity and thus allows one to emphasize the influence of fluid’s elasticity on characteristics of its boundary layer. So far, the exact solution corresponding to the unsteady flow of a Maxwell fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate is developed employing the Fourier sine transforms [12]. It is reported, in the mentioned research, that the velocity decreases by increasing the relaxation time while the magnitude of shear stress increases. Furthermore, Shateyi [13] studied the MHD flow of UCM past a vertical stretching sheet in a Darcian porous medium under the influence of thermophoresis, thermal radiation, and a uniform chemical reaction for Deborah as high as unity.

Moreover, the unsteady flow of Maxwell fluid induced over oscillating accelerated sheet was investigated in [14]. In another work, Ashraf et al. [15] used Homotopy method to simulate the flow and heat transfer of UCM fluid over a moving surface in a parallel free stream with the convective surface boundary condition. Recently time-dependent three-dimensional boundary layer flow of a Maxwell fluid over a stretched sheet has been investigated by Homotopy method [16].

In the above viscoelastic flows the governing equations are scaled by Reynolds number only (similar to Newtonian fluids). Effect of viscoelasticity is, therefore, not realistically presented. It is already reported that even at low Reynolds and high Weissenberg numbers a boundary layer develops in the flow of viscoelastic fluid [17]. This fact is also backed by experiments [18]. High Weissenberg flows mean long relaxation time in which the velocity of fluid vanishes at the wall and particles away from the wall travel long distances within one relaxation time so that particles close to the wall travel only a short distance. This leads to boundary layer in the shear stress [19]. The viscoelastic boundary layer is formed in a thin region closer to the wall in which the relaxation terms are recovered.

Up to now the boundary layer equations for the UCM fluids in two-dimensional creeping flow along a flat boundary for high Weissenberg numbers are derived [19, 20]. It was shown that scaling parameters in view of the high Weissenberg condition and taking the leading terms of the upper-convected Maxwell fluid governing equations result in the viscoelastic boundary layer development of order* Wi*^{−1}. Similar studies on the Phan-Thien-Tanner (PTT) and the Giesekus fluids result in the boundary layer development of order* Wi*^{−1/3} and order* Wi*^{−1/2}, respectively [21]. This phenomenon can also be physically interpreted that “elastic” boundary layers for the Phan-Thien-Tanner and Giesekus fluid are similar to those for the upper-convected Maxwell model and arise when the dimensionless parameter measuring the size of the quadratic term is small. In fact, if the quadratic term is not small, the PTT model will have “viscometric” boundary layers in which it behaves like a generalized Newtonian fluid [22]. For the Giesekus model, the viscometric behavior is different in that the shear stress remains bounded at infinite shear rate. Furthermore, Lie group theory is used in [9] by Atalık to obtain point symmetries of the boundary layer equations derived for the high Weissenberg number flow of UCM and Phan-Thien-Tanner (PTT) fluids.

Using an implicit function, the existence of solutions for viscoelastic boundary layer which arises from spatially periodic perturbations of uniform shear flow was addressed [23]. Also, the well-posedness of boundary layer equations for time-dependent flow of UCM fluid in the limit of high Weissenberg and Reynolds numbers was analyzed [24]. Furthermore, a systematic perturbation procedure to solve the initial value problem for creeping flow of the UCM fluid at high Weissenberg number is formulated [25].

For instance, citing an analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field, Ogilvie and Proctor [26] showed that the dynamics of the Oldroyd-B fluid in the limit of large Deborah number correspond to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. In some aspects, the problem of high Weissenberg number asymptotic for viscoelastic flows is similar to high Reynolds number asymptotic for Newtonian flows. The boundary layer also arises in high Weissenberg number flows since the convected derivative terms become essential at a short distance from the wall, leading to the formation of the aforementioned sharp boundary layer in the stresses [19].

The aim of this work is study of the boundary layer formation in high Weissenberg creeping flow of UCM fluids past a stretching plate using similarity transformation. The stretching rate is assumed to be proportional to the ratio of horizontal distance on the direction. Using similarity transformation the partial differential governing equations are transformed into a set of ordinary differential equations. The ordinary differential equations are then integrated numerically using a Runge-Kutta subroutine and shooting technique. Typical results for the velocity and stress profiles are presented.

#### 2. Governing Equations

The steady flow of a viscoelastic fluid over a (linearly) stretching sheet () is brought to attention here. Consider two-dimensional steady flow of an upper-convected Maxwell fluid occupying the half-plane . The fluid is flown by the movement of a thin elastic sheet emerging from a narrow slit at the origin of a Cartesian coordinate system under investigation shown schematically in Figure 1. The continuity and momentum equations for creeping flow are written aswhere is the extrastress tensor, is the velocity field, and and are the pressure and density, respectively.