A BOUBAKER POLYNOMIALS EXPANSION SCHEME FOR SOLVING THE

FLOW OF NONLINEAR POWER-LAW FLUID IN SEMI-SOLID STATE

Wang Kai-kun 1,a), Fu Jin-long 1,b), Si Xin-hui 2,c)

1 School of Materials and Engineering, University of Science and Technology Beijing, China

2 School of Mathematics and Physics, University of Science and Technology Beijing, China

a) kkwang@mater.ustb.edu.cn ; b) jinlongsjz@126.com; c) xiaoniustu@163.com

Corresponding author: WANG Kai-kun; +86-010-62332598-6117; E-mail: kkwang@mater.ustb.edu.cn

Key Words: Boubaker Polynomials Expansion Scheme; Power-law fluid; Unsteady stretching

surface; Semi-solid state.

Abstract

The problem of an uncompressible power-law fluid has long been the challenge in semi-solid

forming area. In this paper, the flow of a power-law fluid film on an unsteady stretching surface is

analyzed by the means of Boubaker Polynomials Expansion Scheme (BPES). Analytic solutions

were given and compared with the numerical results for some real power-law index and the

unsteadiness parameter in wide ranges. The good agreement between them showed BPES could be

used effectively to solve the flow of nonlinear power-law fluid in semi-solid state.

1.Introduction

In recent years, much attention was paid to the nonlinear boundary value equations in non-linear

mechanics, fluid dynamics and the slurry modeling of semi-solid metal. It was demonstrated that

the Boubaker Polynomials Expansion Scheme (BPES) [1]

could be applied to compute the

power-law fluid with boundary conditions numerically.

The flow of an uncompressible power-law fluid on an unsteady stretching surface has been

widely applied in the process of material engineering, for instance, the extrusion of aluminum and

magnesium alloys in semi-solid state. In fact, during semi-solid extrusion, the liquid fraction on the

surface of the extruded product is always higher than that in the center. At the same time, the

friction between the extruded slurry and the forming die is intense, much more attention should be

paid to the surface of the extruded product from semi-solid forming. Since the mushy state flow of

the extruded slurry is still a pending problem in semi-solid area, it is a great challenge for

transforming the non-Newtonian boundary layer problem into an ordinary differential equation.

Several models have been proposed to explain this fluid. Among these, the power-law,

differential-type and rate-type models gained were most important [2]

. In the forming process, the

material in the center is formed in flow state while the ambient material is stagnant. The quality of

the final products depends on the stretching rate and the rate of cooling.

The free-surface flow of power-law liquids in thin films is a widely occurring phenomenon in

various industrial applications, for example, polymer and plastic fabrication, food processing and in

coating equipment. Myers[3]

considered a number of aspects of the boundary layer flow of a

power-law fluid. Andersson and Irgens[4]

have presented a comprehensive review about the film

flow of power-law fluids. Andersson et al.[5]

carried out a numerical analysis of the hydrodynamic

problem of power-law fluid flow within a liquid film over a stretching sheet.

Solid State Phenomena Vols. 192-193 (2013) pp 276-280Online available since 2012/Oct/24 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/SSP.192-193.276

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 129.186.1.55, Iowa State University, Ames, United States of America-27/09/13,13:42:27)

Based on the Boubaker Polynomials Expansion Scheme (BPES) , in this paper, we examine the

flow of an incompressible power-law liquid film over an unsteady stretching surface, which was

once studied by Andersson et al[5]

and the model was proposed by C.Wang and I.Pop[6]

.

For real power-law index and the unsteadiness parameter in wide ranges, numerical solutions

were given and compared with other results. The numerical solutions from our study could be

helpful in understanding the flow behavior during semi-solid extrusion process.

2.The model of the fluid motion

Fig.1. Schematic representation of a slurry on an unsteady stretching surface.

Fig.1 is the fluid motion within the film due to the stretching of the elastic surface. The flow of

the incompressible fluid is a power-law model, the boundary layer flow is governed by

two-dimensional basic equations in the hydromechanics as follows [5]

:

���� +

����=0. (1)

���� + � ��

�� + ���� = �

� ������ �. (2)

Where (x , y) are respectively the Cartesian coordinates along and normal to the surface , u and v are

the velocity components along the x and y directions ,respectively ,t is the time ,� is the density of

the slurry, and ��� is the shear stress. In the model, �� �� ≤ 0⁄ . The shear stress could be

expressed as:

��� = −� − �����

� . (3)

where K is called the consistency coefficient and n is the power-law index . Combining (2) and (3)

we have

���� + � ��

�� + ���� = − ���

��� −

�����

�. (4)

For the particular parameter value n=1，we can get the Newtonian fluid model with viscosity

coefficient K. If n deviates from 1, deviations from Newtonian behavior occur. For example, n < 1

and n > 1 respectively stand for pseudo-plastic or shear thinning fluids and dilatant fluids or shear

thickening fluids. Pseudo-plastic fluids is the fluids which have viscosity functions that decrease

with increasing shear rate .The fluid becomes less viscous and tends to become “thinner” with

increasing shear rates. On the contrary, dilatant fluid has viscosity functions that increase with the

shear rate. The fluid becomes more viscous and tends to become “thicker” with increasing shear

rates. The flow is caused by the stretching of the elastic surface at y=0 with a velocity:

Solid State Phenomena Vols. 192-193 277

� = ��!"� . (5)

Where c and α are positive constants both with dimensions time-1. The solution is valid under the

condition time t <1 $⁄ .

By introducing the similarity transformation (see Andersson et al.[5]), we can get the formulation

as followed:

%&−' ′′(�!�' ′′′ + ) *−' ′+ + ,�

�-�� ''′′ − . /' ′ + ,!��-�� 0'

′′12 = 0. (6)

where ) = 3�-� is unknown , and . = $ 4⁄ is a dimensionless parameter of unsteadiness and

denotes differentiation with respect to η.

The boundary conditions are: no-slip on the stretching surface (0 = 0) and vanishing of the shear

stress at the free-surface (y=h(x , t) or 0 = 1) along with the constraint that the motion must satisfy

the following condition:

= � �5�� +

�5�� . (7)

at the free-surface . In dimensionless form these boundary conditions becomes:

'&0( = 0, ' ′&0( = 1, '&1( = 6&,!�(,� , ' ′′&1( = 0. (8)

3. BPES solutions

We solve equation (6) with boundary condition (8) under the frame of Boubaker Polynomials

Expansion Scheme (BPES) .The method has been already applied to some non-linear boundary

layer problem. All of these demonstrated the effectiveness of BPES to the boundary layer

problem[7]

.

The resolution protocol is based on the Boubaker Polynomials Expansion Scheme (BPES) [7]

, the

main advantage of these formulations is the effectiveness to the four boundary conditions in the

well-known standard boundary value problem (BVP) which is given by Geng and Cui[8]

.

The first step of this scheme consists of application of the expression:

'&0( = �,89

∑ ;<89<=� × ?@<&0 × A<(. (9)

where ?@< are the 4k-order Boubaker polynomials , A< are ?@< minimal positive roots, BC is a

prefixed integer , ;<|<=�⋯8C are unknown pondering real coefficients .

In order to solve this problem, we can introduce a new function F(x), which can be expressed as:

F&G( = '&G( − G + 1 − 6&,!�(,� . (10)

the value of F(x) changes with x under the given parameters S and n.

Clearly, we can get the value of F(x) at the point of 0 and 1:

F&0( = 1 − 6&,!�(,� , (11)

FH&0( = 0, (12)

F&1( = 0, (13)

FHH&1( = 0 . (14)

Then we can get the expression of the unknown function f(x):

'&G( = F&G( + G + /6&,!�(,� − 11. (15)

In order to solve the formulation (8), substitute '&G( into the equation.

Thus, we can get a formulation about unknown pondering real coefficients ;.

278 Semi-Solid Processing of Alloys and Composites XII

We call the formulation we have got:

I&;�, ;,⋯;8C( = 0. (16)

So we assume:

F&;�, ;,⋯;8C( = I,. (17)

We can solve the minimum value of F and the corresponding values of ;′ under the

properties(11)-(14), thus we get the expression of f(x).

The author calculate the values of;′ and the Fig.2, Fig.3 and Fig.4 show the value of 'H&0( versus the variable 0, where 'H&0( is the value of velocity at the point of 0. The following results

are obtained using MATLAB.

Fig.2. Velocity profiles for Newtonian fluid (n=1).

Fig.3.Solutions of pseudo-plastic fluid (n=0.8). Fig.4.Solutions of dilatant fluid (n=1.2).

We find that for a given parameter S, the pseudo-plastic fluid (n=0.8) has the greatest velocity,

and the dilatants fluid (n=1.2) the least. The pseudo-plastic fluid which is known as shear thinning

fluid has the viscosity function that decreases with the increasing shear rates, so it is common to see

the pseudo-plastic flow on the stretching surface in the semi-solid extrusion. The opposite

phenomenon can be observed in the dilatants fluid.

The coefficient S which is of great importance on the film thickness 3 was explored numerically

for different values of the power-law index n[6]

. There exists a critical value of the unsteady

parameter .C above which no solutions could be obtained. For positive values of S, S→0, means

the case of an infinitely thick layer, for example,3 → ∞, on the other hand , S→2 stands for the

case of a liquid film of infinitesimal thickness, for example, 3 →0.

Solid State Phenomena Vols. 192-193 279

5.Conclusions

With an attempt to solve the problem of an uncompressible power-law fluid in semi-solid state,

Boubaker Polynomials Expansion Scheme (BPES) was used. The results are in good agreement

with the numerical results for some real power-law index. We have demonstrated that through a

specific similarity transformation, the unsteady stretching surface problem encountered in the

semi-solid processes can be solved.

Acknowledgements

The author would like to thanks all the referees for their valuable comments and suggestions,

which improved this paper.

Foundation item: Project supported by The National Natural Science Foundation of China

(NO.51174028); Project supported by The Beijing Natural Science Foundation (NO.2102029)

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280 Semi-Solid Processing of Alloys and Composites XII

Semi-Solid Processing of Alloys and Composites XII 10.4028/www.scientific.net/SSP.192-193 A Boubaker Polynomials Expansion Scheme for Solving the Flow of Nonlinear Power-Law Fluid in

Semi-Solid State 10.4028/www.scientific.net/SSP.192-193.276

DOI References

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