Fluid flow over a nonlinearly stretching sheet

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Abstract

Numerical solutions are obtained for a class of nonlinear third order differential equations arising in fluid flows over a nonlinearly stretching sheet, using a similarity transformation which is different from that of the linearly stretching sheet problem. Furthermore, using the Schauder theory, existence of a solution of the third order differential equation over a large but finite interval is established. Moreover, the analytical solutions are obtained and are compared with the corresponding numerical solutions. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of nonlinear stretching on the flow characteristics.

Introduction

The flow of a viscous fluid over a stretching sheet has important industrial applications, for example, in the extrusion of a polymer sheet from a dye or in the drawing of plastic films. During the manufacture of these sheets, the melt issues from a silt and is subsequently stretched to achieve the desired thickness. The desired characteristics of the final product strictly depend on the rate of cooling and the process of stretching.

In view of these applications, Sakiadis [1] initiated the study of boundary layer flow over a continuous solid surface moving with constant speed. Due to entrainment of the ambient fluid, this boundary layer flow is quite different from boundary layer flow over a semi-infinite flat plate. Erickson et al. [2] extended this problem to the case in which the transverse velocity at the moving surface is non-zero, with heat and mass transfer in the boundary layer being taken into account.

These investigations have a definite bearing on the problem of a polymer sheet extruded continuously from a dye. It is often tacitly assumed that the sheet is inextensible, but situations may arise in the polymer industry in which it is necessary to deal with a stretching-plastic sheet, as pointed out by Crane and Angew [3]. Danberg and Fansler [4] investigated the non-similar solution for the flow in the boundary layer past a wall that is stretched with a velocity proportional to the distance along the wall. Gupta and Gupta [5] analyzed the heat and mass transfer corresponding to the similarity solution for the boundary layer over a stretching sheet subject to suction or blowing. Chen and Char [6] investigated the heat transfer characteristics over a continuous stretching surface with variable surface temperature.

The above investigations are restricted to flows of Newtonian fluid, however, of late non-Newtonian fluids have become more important industrially. The laminar boundary layer on an inextensible continuous flat surface moving with a constant velocity in its own plane in a non-Newtonian fluid characterized by a power-law model (Ostwald–de Waele fluid) is studied by Fox et al. [7], using both exact and approximate methods. The power-law model does not include any elastic properties (such as normal stress differences in shear flow), but in certain polymer processing applications one deals with flow of a viscoelastic fluid over a stretching sheet. Because of this, Rajagopal et al. [8] studied the flow behavior of a viscoelastic fluid (suggested by Coleman and Noll [9]) over a stretching sheet and gave an approximate solution to the flow field. Recently, Troy et al. [10] gave the exact solution for this problem.

Motivated by these analyses, Vajravelu and Rollins [11] studied heat transfer in a viscoelastic fluid over a continuous stretching sheet with power-law surface temperature or power-law surface heat flux. A series solution to the energy equation in terms of Kummer’s and parabolic cylinder functions was developed and some asymptotic cases were studied.

Very recently, Vajravelu and Roper [12] studied the flow and heat transfer in a viscoelastic fluid over a stretching sheet with power-law surface temperature, including the effects of viscous dissipation, internal heat generation or absorption, and work due to deformation in the energy equation. They augmented the missing boundary condition, used the proper sign for the material constant (α1  0), and analyzed the salient features of the flow and heat transfer characteristics.

The physical situation discussed in all the above studies is related to the process of linearly stretching sheet case. Another physical phenomenon is the case in which the sheet is stretched in a nonlinear fashion. Hence, Vajravelu [13] studied the flow and heat transfer phenomenon over a nonlinearly stretching sheet. The resulting nonlinear differential equations along with the pertinent boundary conditions were solved numerically.

Analytical solutions, which could be used to validate the numerical solutions of this nonlinear problem are clearly desirable. In this paper, we study the existence and behavior of exact solutions of third order nonlinear differential equations arising in viscous fluid flows. In Section 2, we shall consider the mathematical model; in Section 3, we shall prove the existence results; in Section 4, we shall present the numerical analysis; and in Section 5, we shall present the numerical solution (through graphs) and discuss the results.

Section snippets

Flow analysis

Consider the flow of a viscous fluid adjacent to a wall coinciding with the plane y = 0, the flow being confined to y > 0. Two equal and opposite forces are introduced along the x-axis so that the wall is stretched keeping the origin fixed. The basic boundary layer equations for the steady flow in usual notation, areux+vy=0,uux+vuy=ν2uy2,where u and v are the velocity components in x and y directions respectively, and ν the kinematic viscosity. The boundary conditions to the problem areu=

Numerical analysis and validation of the solution

We first transform the differential equation (2.6) to a first-order system:y1=y2,y2=y3,y3=(2n/(n+1))y22-y1y3,where y1 = f, y2 = f′, y3 = f″ and a prime denotes differentiation with respect to the independent variable η. The boundary conditions arey1=0,y2=1,atη=0,y20,asη.Then a numerical integration scheme is used to solve a two-point boundary value problem for the following system of m first-order ordinary differential equation in the domain (ηL, ηR):dyidη=Fi(η,y1,y2,,ym),i=1,2,,m.The

Discussion of the results

Fig. 2, Fig. 3 describe respectively the behaviors of the functions f(η) and f′(η) for some values of the parameter n. These figures show that both f and f′ decrease with an increase in n. This decrease in f and f′ is almost negligible for large n because the coefficient 2n/(n + 1) in the differential equation (2.6) approaches 2 as n approaches infinity. However, the velocity components u and v are strongly affected by the parameter n (see the definition of u, v, and η in (2.5)).

In Table 1 values

References (13)

  • C.K. Chen et al.

    J. Math. Anal. Appl.

    (1988)
  • K. Vajravelu et al.

    J. Math. Anal. Appl.

    (1991)
  • K. Vajravelu et al.

    Int. J. Nonlinear Mech.

    (1999)
  • K. Vajravelu

    Appl. Math. Comput.

    (2001)
  • B.C. Sakiadis

    Am. Inst. Chem. Eng. J.

    (1961)
  • L.E. Erickson et al.

    Ind. Eng. Chem. Fund.

    (1966)
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