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

Abstract

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 both Walters’ 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.

1. 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], several works have appeared to consider various aspects of the problem (see Siddheshwar and Mahabaleswar [3, 4], Andersson [5–7], Rollins and Vajravelu [8], Vleggar [9], Ming-I and Cha’o-Kuang [10], Kelly et al. [11], Vajravelu and Hadjinicolaou [12], Liao and Pop [13], Magyari et al. [14], 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.

2. 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+𝛽𝑥². The resulting motion of the otherwise quiescent liquid is thus caused solely by the moving sheet. On assuming𝛽, and thereby 𝛿, quite small we can make use of the boundary layer theory (see Rajagopal et al. [17]).

Figure 1 

Schematic diagram of the stretching sheet problem.

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]):𝜕𝑢𝑢𝜕𝑢𝜕𝑥+𝜕𝜕𝑣𝑦=0,𝜕𝑥+𝑣𝜕𝜕𝑦𝑢=𝜈𝜕²𝑢𝜕𝑦²−𝑘₀𝑢𝜕³𝑢𝜕𝑥𝜕𝑦²+𝑣𝜕³𝑢𝜕𝑦³+𝜕𝑢𝜕𝑥𝜕²𝑢𝜕𝑦²−𝜕𝑢𝜕𝑦𝜕²𝑢𝜕𝑥𝜕𝑦,(2.1) subject to the boundary conditions:𝑢=𝑢𝑣=𝛼=𝑥0+𝛿𝑥a𝛽s𝑥a²𝑦t⟶𝑦at=𝑦∞0=,.0,(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 𝑘₀ 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=𝛼𝑥+𝛽𝑥² (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(𝑋,𝑌)=𝛼𝜈(𝑥,𝑦),(𝑈,𝑉)=(√𝑢,𝛼𝑣𝜈),𝛽^∗=𝛽𝛼𝜈𝛼,𝛿^∗=𝛿2𝛼.(2.3) Equation (2.1) take the form𝜕𝑈𝑈𝜕𝑈𝜕𝑋+𝜕𝜕𝑉𝑌=0,𝜕𝑋+𝑉𝜕𝜕𝑈𝑌=𝜕²𝑈𝜕𝑌²−𝑘₁𝑈𝜕³𝑈𝜕𝑋𝜕𝑌²+𝑉𝜕³𝑈𝜕𝑌³+𝜕𝑈𝜕𝑋𝜕²𝑈𝜕𝑌²−𝜕𝑈𝜕𝑌𝜕²𝑈𝜕𝑋𝜕𝑌,(2.4) where 𝑘₁=𝛼𝑘₀/𝜇 is the viscoelastic parameter. The parameter 𝑘₁ 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 get𝑈=𝜕𝜓𝜕𝑌,𝑉=−𝜕𝜕𝑋𝜓.(2.5) Using (2.5) in (2.6), we get𝜕³𝜓𝜕𝑌³+𝜕(𝜓,𝜕𝜓/𝜕𝑌)𝜕(𝑋,𝑌)+𝑘¹𝜕𝜓,𝜕³𝜓/𝜕𝑌³𝜕(𝑋,𝑌)−𝜕𝜕𝜓/𝜕𝑌,𝜕²𝜓/𝜕𝑌²𝜕(𝑋,𝑌)=0.(2.6)

The boundary conditions to be satisfied by 𝜓 can be obtained from (2.2), (2.3), and (2.5) as follows:𝜕𝜓𝜕𝑌=𝑋+𝛽^∗−𝜕𝜓𝑋²at𝑌=0,𝜕𝑋=2𝛿^∗𝜕𝜓𝑋at𝑌=0,𝜕𝑌=0as𝑌⟶∞.(2.7) The similarity solution to (2.6), subject to (2.7), may be taken as𝜓=𝑋𝑓(𝑌)−𝛿^∗𝑋²𝑓^(𝑌),(2.8) where prime denotes differentiation with respect to 𝑌. Substituting (2.8) into (2.6) and equating the coefficients of 𝑋,𝑋², and 𝑋³, we get the following three ordinary differential equations:𝑓^𝑓²−𝑓𝑓^=𝑓^−𝑘₁2𝑓^𝑓^−𝑓𝑓^−𝑓^²,(2.9)_𝑓𝑓^−𝑓𝑓^=𝑓^−𝑘₁𝑓^𝑓^−𝑓𝑓^{},(2.10)_²−𝑓^𝑓^=𝑘₁𝑓^𝑓^{}−2𝑓^𝑓^+𝑓^².(2.11)

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 form𝑓𝑓^(0)=1𝑓,(0)=𝑓0^(,0)=−𝑠,((22..1122ba))_(∞)=0,𝑓^(∞)=0,(2.12c)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) is𝑓(𝑌)=𝐴+𝐵𝑒^−𝑠𝑌,(2.13) which satisfies the boundary condition (2.12a)–(2.12c) provided𝐴=𝛿^∗𝛽^∗,𝐵=−𝛿^∗𝛽^∗,𝑠=1𝐴.(2.14) We also note that (2.13) can be a solution of the nonlinear differential equation (2.9) if and only if𝑠=𝛽^∗𝛿^∗=1√1−𝑘₁.(2.15) We may now write 𝑓(𝑌) from (2.13)–(2.15) as𝑓(𝑌)=√1−𝑘₁1−𝑒^{√−𝑌/1−𝑘1}.(2.16)

Reverting to the symbol “s” we may easily see that𝑓^(𝑌)=1−𝑠𝑓(𝑌).(2.17) 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:𝜓=𝑋𝑓(𝑌)−𝛿^∗𝑋²𝑓^(𝑌)=𝐶,(2.18) where 𝐶 is a constant. The streamline 𝜓=𝐶 can be written in the functional form as𝑌=1𝑠Ln𝑋/𝑠+𝛿^∗𝑋²𝑋/𝑠−𝐶.(2.19) Substituting (2.8) into (2.5), we get𝑈𝑉==𝑋−𝑓𝑓^((𝑌𝑌))+−2𝛿𝛿^∗_∗𝑋²𝑓^(𝑌),𝑋𝑓^(𝑌).(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.

3. 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 (𝑘₁=0). Before we discuss the results of the study, we make some general observations. From (2.15) it is clear that the 𝑘₁ 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 gets𝑓^(0)=1−𝑘₁𝑓^(0).(3.1) From the above equation, we see that 𝑓^(0)=0 for 𝑘₁=1. In conjunction with the condition 𝑓^(0)=−𝑠 in (2.12a), this would mean𝑓^(0)=−𝑠=0.(3.2) Obviously for 𝑘₁=1, the quadratic stretching problem ceases to exist. For 𝑘₁>1 we note that 𝑠=1/√(1−𝑘₁) is complex. Hence it stands reiterated that the range of applicability of 𝑘₁ must be (−∞,1).

We note that negative values of 𝑘₁ give us the results of a second order liquid and positive values of 𝑘₁ 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 𝑘₁=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 2 

Streamline 𝜓(𝑋,𝑌)=1 for different values of quadratic stretching parameter 𝛿∗.

Figure 3 is a plot of various stream lines 𝜓(𝑋,𝑌)=𝐶 when 𝛿^∗=0.1and𝑘₁=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 3 

Streamlines 𝜓(𝑋,𝑌)=𝐶 for different values of 𝐶.

Figure 4 is a plot of the streamline 𝜓(𝑋,𝑌)=1 for different values of 𝑘₁ and 𝛿^∗=0.1. It is evident from the aforementioned 3 figures that the viscoelastic parameter 𝑘₁ and the quadratic stretching parameter 𝛿^∗ work against each other in the lifting of the liquid as we go downstream.

Figure 4 

Streamline 𝜓∗(𝑋,𝑌)=1 for different values of 𝑘1.

We now discuss the axial and transverse velocity distributions with an observation that𝑉=𝑈−=1𝑋−𝑒𝑒^−𝑠𝑌_−𝑠𝑌+𝛿^∗𝑠𝑋²𝑒^−𝑠𝑌⟶0as𝑌⟶∞,𝑠+2𝛿^∗𝑋𝑒^−𝑠𝑌⟶−1𝑠as𝑌⟶∞,(3.3) that is, the flow outside the boundary layer becomes uniform and is directed perpendicular to the sheet. The boundary layer thickness 𝛿₁, defined as the distance from the sheet at which the streamwise velocity 𝑈 has been reduced to one percent of the velocity 𝑋+𝛽^∗𝑋² of the quadratic stretching surface, can be expressed as𝛿₁=Ln100𝑠=Ln100𝛽𝛿^∗∗.(3.4)

Using (2.15) in the above equation, we get𝛿₁=Ln100√1−𝑘₁.(3.5) Thus, we see that the boundary layer thickness 𝛿₁ is coordinate-independent for all permissible values of 𝑘₁. Figure 5 shows the variation of 𝛿₁ with 𝑘₁. We see from the figure that the effect of increasing 𝑘₁ is to decrease 𝛿₁.

Figure 5 

Variation of boundary layer thickness 𝛿1 with 𝑘1.

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 6 

Axial velocity 𝑈(𝑋,𝑌) for the linear stretching sheet problem of Crane [19].

Figure 7 

Axial velocity 𝑈(𝑋,𝑌) for the quadratic stretching sheet problem.

Figure 8 

Axial velocity 𝑈(𝑋,𝑌) for the quadratic stretching sheet problem.

Figure 9 

Axial velocity 𝑈(𝑋,𝑌) for the quadratic stretching sheet problem.

Figure 10 

Transverse velocity 𝑉(𝑋,𝑌) for the linear stretching sheet problem of Crane [19].

Figure 11 

Transverse velocity 𝑉(𝑋,𝑌) for the quadratic stretching sheet problem.

Figure 12 

Transverse velocity 𝑉(𝑋,𝑌) for the quadratic stretching sheet problem.

Figure 13 

Transverse velocity 𝑉(𝑋,𝑌) for the quadratic stretching sheet problem.

Figure 14 

Axial velocity 𝑈(𝑋,𝑌) for the quadratic stretching sheet problem.

Figure 15 

Transverse velocity 𝑉(𝑋,𝑌) for the quadratic stretching sheet problem.

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 𝑘₁ on 𝑉 for the problem of quadratic stretching. The influence of 𝑘₁ on 𝑉(𝑋,𝑌) is similar to its influence on 𝑈(𝑋,𝑌) and the same is demonstrated by Figures 12 and 13. The influence of quadratic stretching on 𝑈(𝑋,𝑌) and 𝑉(𝑋,𝑌) of a second-order 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.