An Analytical Study of Weakly Nonlinear Dynamics of a Walters ’ Liquid B around a Flexible Sheet Undergoing Super Linear Stretching

The paper discusses the boundary layer flow of Walters’ liquid B over a stretching sheet. The stretching is assumed to be a quadratic function of the coordinate along the direction of stretching. The study encompasses within its realm bothWalters’ liquid B and second order liquid. The velocity distribution is obtained by solving the nonlinear governing differential equation. Analytical expressions are obtained for stream function and velocity components as functions of the viscoelastic and stretching related parameters. It is shown that the viscoelasticity goes hand in hand with quadratic stretching in enhancing the lifting of the liquid as we go along the sheet.


Introduction
Polymer extrusion, drawing of copper wires, continuous stretching of plastic films, and artificial fibers, hot rolling, wire drawing, glass-fiber, metal extrusion, and metal spinning are some of the examples where the problem of a stretching sheet arises.Ever since the pioneering works of Sakiadis 1,2 , Liao 15 , and Dandapat and Gupta 16 .The core assumption in most of the reported problems is that the stretching is linearly proportional to the axial distance.This is valid provided the stretching process is delicate and slow, leading to the assumption of constant rate of stretching.It is not difficult to see that the above assumption is quite idealistic and impractical.In the strictest sense the stretching has to be nonlinearly proportional to the axial distance.In the present paper, as a first step in the general modelling exercise, we make use of a simple quadratic stretching model.

Mathematical Formulation
We consider a steady state two-dimensional boundary layer flow of an incompressible isothermal viscoelastic liquid, of the type Walters' liquid B, over a quadratic stretching sheet see Figure 1 .The Walters' liquid B represents an approximation for short or rapidly fading memory liquids and is thus an approximation to first order in elasticity.The liquid is at rest and the motion is created by pulling the sheet on both ends with equal forces parallel to the sheet and with a speed u, which varies quadratically with the distance from the slit as u αx βx 2 .The resulting motion of the otherwise quiescent liquid is thus caused solely by the moving sheet.On assuming β, and thereby δ, quite small we can make use of the boundary layer theory see Rajagopal et al. 17 .
The steady two-dimensional conservation of mass and the momentum boundary layer equation for the quadratic stretching sheet problem involving Walters' liquid B are see Beard and Walters 18 : subject to the boundary conditions: u αx βx 2 at y 0, v δx at y 0, u 0 as y −→ ∞.

2.2
Here, u and v are the components of the liquid velocity in the x and y directions, respectively, μ is the limiting viscosity at small rates, and k 0 is the first moment of the distribution function of relaxation times.Further we assume β is quite small, that facilitates the assumption of a weakly two-dimensional flow as considered in the paper.As pointed out by Vleggaar 9 , in a polymer processing application involving spinning of filaments without blowing, laminar boundary layer occurs over a relatively small length of the zone 0.0-0.5 m from the die which may be taken as the origin of Figure 1.This is in fact the zone over which the major part of the stretching takes place.In such a process the initial velocity is low about 0.3 m/s but not very low, enough always to assume linear stretching.Thus a good approximation of the velocity of the sheet is u α x βx 2 at any rate for the first 10-60 cm of the spinning zone , where α and β are the constants velocity gradients.We have adopted the quadratic stretching model in our problem.Using the dimensionless variables where k 1 αk 0 /μ is the viscoelastic parameter.The parameter k 1 represents a measure of the relative importance of elastic and viscous effects and can thus be identified with the Weissenberg number.
Introducing the stream function ψ X, Y , we get Using 2.5 in 2.6 , we get The boundary conditions to be satisfied by ψ can be obtained from 2.2 , 2.3 , and 2.5 as follows:

2.7
The similarity solution to 2.6 , subject to 2.7 , may be taken as where prime denotes differentiation with respect to Y .Substituting 2.8 into 2.6 and equating the coefficients of X, X 2 , and X 3 , we get the following three ordinary differential equations: Equation 2.10 turns out redundant as it can be obtained by differentiating 2.9 once with respect to Y .In the subsequent analysis we show that 2.9 can, in fact, be obtained from 2.11 , by a suitable transformation, which in turn implies consistency.The boundary conditions, for solving 2.9 for f, given by 2.7 can be obtained in the form where s β * /δ * .One can easily see that 2.11 is a differential equation for f Y and we can also verify that f Y e −sY is a solution of 2.11 , and this satisfies the derivative boundary conditions in 2.12a -2.12c .Thus an appropriate solution of 2.9 is which satisfies the boundary condition 2.12a -2.12c provided We also note that 2.13 can be a solution of the nonlinear differential equation 2.9 if and only if

2.15
We may now write f Y from 2.13 -2.15 as

2.16
Reverting to the symbol "s" we may easily see that Using this in 2.11 we can arrive at 2.9 .This proves the "consistency" of the 3 equations 2.9 -2.11 forf Y .
The expression for the streamline pattern of the flow in the region around the stretching sheet can be obtained from 2.8 as follows: where C is a constant.The streamline ψ C can be written in the functional form as Substituting 2.8 into 2.5 , we get

2.20
Having obtained the analytical expression for the stream function ψ and the velocity components U and V , we now move on to discuss the results obtained in the study.

Results and Discussion
The problem of a flexible sheet undergoing quadratic stretching is investigated for the flow it generates in its immediate neighbourhood.The stretching sheet is the sole reason for the liquid flow, and liquid viscoelasticity significantly influences the flow.The flow is studied with the help of streamline patterns and also the axial and transverse velocity distributions.
The results are analyzed against the background of the classical linear stretching problem δ * 0 involving Newtonian liquids k 1 0 .Before we discuss the results of the study, we make some general observations.From 2.15 it is clear that the k 1 range of applicability of the solution is −∞, 1 .This can further be substantiated as follows.Differentiating equation 2.9 with respect to Y , and subject to condition 2.12a -2.12c , one gets From the above equation, we see that f 0 0 for k 1 1.In conjunction with the condition f 0 − s in 2.12a , this would mean Obviously for k 1 1, the quadratic stretching problem ceases to exist.For k 1 > 1 we note that s 1/ 1 − k 1 is complex.Hence it stands reiterated that the range of applicability of k 1 must be −∞, 1 .We note that negative values of k 1 give us the results of a second order liquid and positive values of k 1 those of a Walters' liquid B model.We now discuss the results of the study on Walters' liquid B followed by those on the second order liquid.
Figure 2 is a plot of the streamline ψ X, Y 1 for different values of δ * and k 1 0.2.Increasing value of δ * indicates the increasing rate of quadratic stretching.We find from the figure that increasing rate of stretching restricts the dynamics in the axial direction to regions close to the slit.
Figure 3 is a plot of various stream lines ψ X, Y C when δ * 0.1 and k 1 0.2.As is depicted in the figure, at large axial distances the streamlines converge together and are lifted up due to quadratic stretching.Figure 4 is a plot of the streamline ψ X, Y 1 for different values of k 1 and δ * 0.1.It is evident from the aforementioned 3 figures that the viscoelastic parameter k 1 and the quadratic stretching parameter δ * work against each other in the lifting of the liquid as we go downstream.We now discuss the axial and transverse velocity distributions with an observation that that is, the flow outside the boundary layer becomes uniform and is directed perpendicular to the sheet.The boundary layer thickness δ 1 , defined as the distance from the sheet at which the streamwise velocity U has been reduced to one percent of the velocity X β * X 2 of the quadratic stretching surface, can be expressed as Thus, we see that the boundary layer thickness δ 1 is coordinate-independent for all permissible values of k 1 .Figure 5 shows the variation of δ 1 with k 1 .We see from the figure that the effect of increasing k 1 is to decrease δ 1 .Figures 6-15 that are three-dimensional plots of the velocity components U X, Y and V X, Y reveal more than the conventional two-dimensional projections on the U-Y and V -Y planes.Figure 6 is a plot of the Crane 19 profile of the linear stretching problem.One can easily see from the figure that the horizontal and vertical extent of the dynamics on the stretching sheet increases as we go along axial direction.
Figure 7 brings out the effect of the quadratic stretching of the sheet as well as the viscoelasticity of the liquid.Clearly both the above effects give rise to an extended dynamic region compared to the linear stretching problem of a Newtonian liquid.Comparing Figures 7 and 8 it is obvious that the quadratic stretching increases the vertical extent of the dynamic region.Comparing Figures 8 and 9 of quadratic stretching we find that the effect of viscoelasticity is to initiate lifting of the liquid more closer to the slit compared to that of a Newtonian liquid.The transverse velocity profile brings out the fact that quadratic stretching greatly influences the vertical velocity compared to that in the case of linear stretching.Figure 10 depicts the X-independence of V while Figure 11 spells out that quadratic stretching induces the X-dependence of the transverse velocity component V .Figures 12 and 13 explain the nature of the influence of k 1 on V for the problem of quadratic stretching.The influence of k 1 on V X, Y is similar to its influence on U X, Y and the same is demonstrated by   The influence of quadratic stretching on U X, Y and V X, Y of a secondorder liquid shows that the vertical variation is comparatively less than axial variation.Figures 14 and 15 are the axial and transverse velocity profiles for the quadratic stretching sheet problem in a second-order liquid.The corresponding graphs for a Walters' liquid B model are Figures 7 and 11.It is clear that the lifting is initiated closer to the slit in the case of a Walters' liquid B model compared to the second-order liquid.In the case of the latter the flow is more strongly two dimensional than in the former case.This is clearly seen on comparing Figures 11 and 15.

Figure 1 :
Figure 1: Schematic diagram of the stretching sheet problem.

Figure 7 :Figure 8 :
Figure 7: Axial velocity U X, Y for the quadratic stretching sheet problem.

Figure 9 :Figure 10 :
Figure 9: Axial velocity U X, Y for the quadratic stretching sheet problem.

Using 2 .Figure 11 :Figure 12 :
Figure 11: Transverse velocity V X, Y for the quadratic stretching sheet problem.

Figure 13 :Figure 14 :
Figure 13: Transverse velocity V X, Y for the quadratic stretching sheet problem.

Figure 15 :
Figure 15: Transverse velocity V X, Y for the quadratic stretching sheet problem.

Figures 12 and 13
Figures12 and 13.The influence of quadratic stretching on U X, Y and V X, Y of a secondorder liquid shows that the vertical variation is comparatively less than axial variation.Figures14 and 15are the axial and transverse velocity profiles for the quadratic stretching sheet problem in a second-order liquid.The corresponding graphs for a Walters' liquid B model are Figures7 and 11.It is clear that the lifting is initiated closer to the slit in the case of a Walters' liquid B model compared to the second-order liquid.In the case of the latter the flow is more strongly two dimensional than in the former case.This is clearly seen on comparingFigures 11 and 15.
Figures12 and 13.The influence of quadratic stretching on U X, Y and V X, Y of a secondorder liquid shows that the vertical variation is comparatively less than axial variation.Figures14 and 15are the axial and transverse velocity profiles for the quadratic stretching sheet problem in a second-order liquid.The corresponding graphs for a Walters' liquid B model are Figures7 and 11.It is clear that the lifting is initiated closer to the slit in the case of a Walters' liquid B model compared to the second-order liquid.In the case of the latter the flow is more strongly two dimensional than in the former case.This is clearly seen on comparingFigures 11 and 15.