research article estimation of the shear stress parameter...

Research ArticleEstimation of the Shear Stress Parameter of a Power-Law Fluid

Samer S Al-Ashhab and Rubayyi T Alqahtani

Department of Mathematics and Statistics Al ImamMohammad Ibn Saud Islamic UniversityPO Box 90950 Riyadh 11623 Saudi Arabia

Correspondence should be addressed to Samer S Al-Ashhab ssashhabimamuedusa

Received 28 May 2016 Accepted 27 June 2016

Academic Editor Mohamed Abd El Aziz

Copyright copy 2016 S S Al-Ashhab and R T Alqahtani This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We apply the Adomian decomposition method to a power-law problem for solutions that do not change the sign of curvature Inparticular we consider solutions with positive curvature The power series obtained via the Adomian decomposition method isused to estimate the shear stress parameter as well as the instant of time where the solution reaches its terminal point of a steadystate We compare our results with estimates obtained via numerical integrators More importantly we illustrate that the error ispredictable and can be reduced without further effort or using higher order terms in the approximating series

1 Introduction

Due to its interestingmathematical properties andhuge num-ber of applications the problem of non-Newtonian fluid flowhas attracted attention from a large number of researchersThe most commonly used model for non-Newtonian fluidmechanics is the Ostwald-de Waele model with a power-lawrheologyThe value of the power-law index 119899 = 1 correspondsto aNewtonian fluidwhile 119899 gt 1describes a dilatant or shear-thickening fluid and 0 lt 119899 lt 1 describes a pseudo-plasticor shear-thinning fluid The corresponding nonlinear third-order problem is the following

(

100381610038161003816100381610038161198911015840101584010038161003816100381610038161003816

119899minus1

11989110158401015840)

1015840

+

1

119899 + 1

11989111989110158401015840= 0 (1)

subject to

119891 (0) = 0

1198911015840

(0) = 120598

1198911015840(120578) 997888rarr 1 as 120578 997888rarr infin

(2)

where the primes are derivatives with respect to 120578 To simplifynotation let 119910 = 119891 119909 = 120578 where these are referred to as

the similarity variables of the problem Equation (1) can bewritten for positive curvature solutions 11991010158401015840 gt 0 as

119910101584010158401015840= 120573119910 (119910

10158401015840)

2minus119899

120573 =

minus1

119899 (119899 + 1)

(3)

where solution domains with 11991010158401015840 = 0 are ruled out for 119899 gt 2(Those domains are naturally ruled out since the problem issolved until the solution reaches its terminal state 11991010158401015840 = 0 atsay 119909 = 119905

119891)

Earlier studies of this problem date back to the workby Blasius [1] where he considered the case where 119899 = 1Existence and uniqueness of similar power-law problemshave also been studied by Guedda and Hammouch [2 3]Nachman and Taliaferro [4] and Zheng et al [5] to mentiona few In particular in [5] the authors studied the problemwith identical boundary conditions to the ones consideredhere but with coefficient equal to 1 on the second terminvolving 11989111989110158401015840 in (1) aboveTheir approach however enabledthem to obtain bounds on an important parameter theskin friction coefficient in addition to proving existenceand uniqueness They compared their results to numericalestimates to illustrate their accuracy and it was observed thattheir bounds (upper and lower for skin friction) were veryclose for large 119899 so that an accurate estimate can readilybe obtained by taking for example the average of thosetwo bounds We will not be able to compare our results to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4729063 4 pageshttpdxdoiorg10115520164729063

2 Mathematical Problems in Engineering

theirs directly however due to the coefficient on the secondterm in (1) above which somewhat changes the ldquodynamicsrdquoof the governing equation and its solutions (even thoughthe deviation may be small for some smaller values of 119899)Howell et al [6] studied the problem in the context ofmomentum and heat transfer In [7] the authors Chen et alconsidered boundary layer flow of an electrically conductingnon-Newtonian fluid in the presence of a magnetic fieldTheir setting required two additional terms in the equationinvolving 1198911015840 and (1198911015840)2 explicitly but with slightly differentboundary conditions They proceeded to utilize Croccovariables and theAdomian decompositionmethod (ADM) toobtain approximate solutions to the problem (in the Croccovariables domain) as well as estimate the value of the skinfriction coefficient and they illustrated the accuracy of theirfindings Further numerical and approximate solutions canbe found in [8ndash11] for example

Existence and uniqueness were established for the exactcurrent problem at hand in [12] where all possible values ofthe power-law index 119899 were considered and it was assumedthat 11989110158401015840 ge 0 (or 11991010158401015840 ge 0) In [13 14] one finds extensivediscussion of the general theory of non-Newtonian boundarylayer flow fluids including the physical derivation of thepower-law problem Lastly we note that in [15] a unifiedequation governing positive and negative curvatures fornon-Newtonian power-law fluids was derived and analyticalsolutions were obtained

In Section 2 we derive the first few terms in a power seriesexpansion (using the ADM) for the solution to the problemand use those terms to approximate the solution where ourapproach will help in obtaining estimates not only for theshear stress parameter but also for the instant of time 119905

119891

where the solution virtually reaches its terminal point where1199101015840(119905119891) = 1 11991010158401015840(119905

119891) = 0 within 10minus4 In Section 3 we compare

our results to numerical estimates obtained via numericalcomputer integrators

2 The Series Solution

Tofind the power series solution to our problemwe utilize theAdomian decompositionmethod observe that the solution 119910satisfies

119910 = 120598119909 + 1205721199092+ 119871minus1

(119865) (4)

where the condition 11991010158401015840(0) = 2120572 replaces the condition atinfinity and where 119865 = 120573119910(119910

10158401015840)2minus119899 from (3) above 119871minus1 =

int

119909

0int

119909

0int

119909

0(sdot)119889119909 119889119909 119889119909 is an operator where 119871 = 11988931198891199093 is the

operator applied to 119910 on the left hand side of (3) as 119871[119910] Nowexpand 119910 in an infinite series of the form 119910 = sum

infin

119894=0119910119894 The

decompositionmethod is based on substituting this series for119910 into (4) above and extracting the terms out one by one bya linearization process of the operator In particular 119910

119894rsquos are

obtained via

119910119894= 119871minus1(119860119894minus1) 119894 = 1 2 3 (5)

where the 119860119894rsquos are referred to as the Adomian polynomials

and are given by

1198600= 119865 (119910

0)

1198601=

120597119865

120597119891

(1199100) sdot 1199101+

120597119865

12059711989110158401015840(1199100) sdot 11991010158401015840

1

1198602=

120597119865

120597119891

(1199100) sdot 1199102+

120597119865

12059711989110158401015840(1199100) sdot 11991010158401015840

2+

1205972119865

12059711989112059711989110158401015840(1199100)

sdot 119910111991010158401015840

1+

1205972119865

120597211989110158401015840(1199100) sdot

(11991010158401015840

1)

2

2

(6)

and where we have skipped writing (12059721198651205972119891)(1199100) sdot ((119910

1)2

2) = 0 from 1198602 This in turn yields the explicit expressions

(with 1199100= 120598119909 + 120572119909

2 and using 120574 = 2120572)

1198600= 120573 (120598119909 + 120572119909

2) (120574)2minus119899

1198601= 120573 (2120572)

2minus1198991199101+ 120573 (2 minus 119899) (120598119909 + 120572119909

2) (120574)1minus119899

11991010158401015840

1

1198602= 120573 (2120572)

2minus1198991199102+ 120573 (2 minus 119899) (120598119909 + 120572119909

2) (120574)1minus119899

11991010158401015840

2

+

120573 (2 minus 119899) (1 minus 119899) (120598119909 + 1205721199092) (120574)minus119899

(11991010158401015840

1)

2

2

+ 120573 (2 minus 119899) (120574)1minus119899

119910111991010158401015840

1

(7)

Proceeding into finding 1199101 1199102 1199103via (5) we obtain

1199101= 120573 (120574)

2minus119899

(

1205981199094

4

+

21205721199095

5

)

1199102= 1205732(120574)4minus2119899

(

1205981199097

7

+

21205721199098

8

) + 1205732(120574)3minus2119899

(2 minus 119899)

sdot (

312059821199096

6

+

201205721205981199097

7

+

4012057221199098

8

)

1199103= 1205733(120574)6minus3119899

(

12059811990910

10

+

212057211990911

11

) + 1205733(120574)5minus3119899

(2 minus 119899)

sdot (

312059821199099

9

+

2012057212059811990910

10

+

40120572211990911

11

) + 1205733(120574)5minus3119899

(2

minus 119899) (

612059821199099

9

+

5612057212059811990910

10

+

112120572211990911

11

) + 1205733(120574)4minus3119899

sdot (2 minus 119899)2(

1512059831199098

8

+

21012057212059821199099

9

+

1120120598120572211990910

10

+

2240120572311990911

11

) + 1205733(120574)4minus3119899

(2 minus 119899) (1 minus 119899) (

1512059831199098

8

+

21012057212059821199099

9

+

1120120598120572211990910

10

+

2240120572311990911

11

)

+ 1205733(120574)5minus3119899

(2 minus 119899) (

1512059821199099

9

+

11212057212059811990910

10

+

224120572211990911

11

)

(8)

Mathematical Problems in Engineering 3

Table 1 Comparison between estimated values for 120574 and 119905119891for 119899 = 15

120598 minus025 0 05 075120574 (numerical) 03241 03647 02748 01720120574 (series) 03243 03664 02754 01722Error 00002 00017 00006 00002Improved 120574 (|error| lt 00008) 03234 03655 02745 01713119905119891(series) 512 4095 3065 257

Relative error in 119905119891

27 36 50 32Improved 119905

119891(error lt 13) 5235 421 318 2685


120598 minus025 0 05 075120574 (numerical) 03995 04326 03539 02534120574 (series) 03945 04278 03497 02504Error minus0005 minus00048 minus00042 minus0003Improved 120574 (|error| lt 0001) 03985 04318 03537 02544119905119891(series) 374 3026 1985 1415


41 48 61 67Improved 119905

119891(error lt 18) 3615 2901 186 129

The solution must go on to 1199104 1199105 and so forth We

take 119910 asymp 1199100+ 1199101+ 1199102+ 1199103as an approximate solution

(which turns out to yield high accuracy relatively speakingin approximating shear stress) and use it to estimate thevalues of the shear stress parameter 120574 as well as the instant oftime where the solution virtually reaches a constant velocitynamely 119905

119891 where 1198911015840(119905

119891) = 1 11989110158401015840(119905

119891) = 0 (More precisely

and using the current notation 119905119891is the minimum value of

119909 gt 0 such that 1199101015840(119909) = 1 and 11991010158401015840(119909) = 0 and where the lasttwo conditions hold for all 119909 ge 119905

119891)

3 Numerical and Series Estimates

Using the series in the previous section as an approximationto the solution of the problem one obtains the followingestimates for the initial curvature at 119905 = 0 (also referred to asthe shear stress or vorticity parameter) namely 120574 = 2120572 aswellas 119905119891 We compare our results (referred to here as ldquoseriesrdquo)

with those obtained from numerical MATLAB integratorswhich are anticipated to have high accuracy (referred to hereas ldquonumericalrdquo) We expect future works will obtain resultsusing other numerical schemes and methods Table 1 showsresults for 119899 = 15 whereas Tables 2 and 3 show results for119899 = 25 and 119899 = 3 respectively We observe that for values of120598 lt 1 the estimated values of the shear stress parameter 120574 arequite accurate (especially for values of 120598 closer to 1) as long as119899 stays close to 119899 = 2 where the equation reduces to a linearequation

The ranges we considered here for 119899 and 120598 are verypractical which signifies the current results On the otherhand while the estimation of the values of 119905

119891may not be

as significant as that for the shear stress parameter 120574 welist two observations The needed accuracy for 119905

119891is not as

acutecrucial as that for 120574 in applications so that our estimatesare considered relatively accurate Secondly the error in 119905

119891is

one sided it is an underestimate for 119899 = 15 while it is anoverestimate for 119899 = 25 and 119899 = 3 Therefore a correctioncan be simply done The same observation holds true for thevalues of shear stress parameter 120574 for 119899 = 25 and 119899 = 3 Infact for 119899 = 25 adding 0004 to each estimated value of 120574willensure an error of less than 0001 On the other hand for 119899 =3 adding 00083 to each estimated value of 120574 will ensure anerror of less than 00018This shows that our approach proveshighly accurate as the predictable error provides means tosignificantly improve the estimates without furtheraddedwork and effort For 119899 = 15 the estimated values for 120574 arerelatively speaking very accurate

It is also noted that the error increases with decreasing120598 so that the error can further be reduced by looking for aquadratic or higher order interaction between the error and120598 to further reduce the error but we leave the details of thisanalysis

Lastly observe that for a wide range of 120598 le 0 the error canbe singled out as staying virtually constant an observationthat can help in reducing the error with virtually no effortto practically almost 0 once it is determined for one or twomeasurements for each value of interest of 119899

4 Conclusions

The Adomian decomposition method was used to obtainapproximate series solution for a power-law problem that hasrecently been studied in literature The approximate solutionwas used to obtain estimates for the shear stress parameter 120574as well as the time to reach terminal pointsolution 119905

119891 It was

noticed that only four terms of the infinite series were sufficientto obtain high accuracy for a certain very practical range of





70 80 99 107Improved 119905

119891(error lt 41) 34 2755 1735 117

the power-law index 119899 A crucial observation in this contextis that the error in the measured parameters is predictablewhich makes it possible to significantly reduce the error withvirtually no effort This is a peculiar phenomenon which weconjecture to apply to many other power-law problems ofnon-Newtonian fluids

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908

[2] M Guedda and Z Hammouch ldquoSimilarity flow solutions ofa non-Newtonian power-law fluidrdquo International Journal ofNonlinear Science vol 6 no 3 pp 255ndash264 2008

[3] M Guedda ldquoBoundary-layer equations for a power-law shear-driven flow over a plane surface of non-Newtonian fluidsrdquo ActaMechanica vol 202 no 1ndash4 pp 205ndash211 2009

[4] A Nachman and S Taliaferro ldquoMass transfer into boundarylayers for power law fluidsrdquo Proceedings of the Royal Society ofLondon Series A Mathematical and Physical Sciences vol 365no 1722 pp 313ndash326 1979

[5] L Zheng X Zhang and J He ldquoExistence and estimate of pos-itive solutions to a nonlinear singular boundary value problemin the theory of dilatant non-Newtonian fluidsrdquo Mathematicaland Computer Modelling vol 45 no 3-4 pp 387ndash393 2007

[6] T G Howell D R Jeng and K J De Witt ldquoMomentum andheat transfer on a continuous moving surface in a power lawfluidrdquo International Journal of Heat and Mass Transfer vol 40no 8 pp 1853ndash1861 1997

[7] X-H Chen L-C Zheng and X-X Zhang ldquoMHD boundarylayer flow of a non-newtonian fluid on a moving surface witha power-law velocityrdquo Chinese Physics Letters vol 24 no 7 pp1989ndash1991 2007

[8] X Su L Zheng and J Feng ldquoApproximate analytical solutionsand approximate value of skin friction coefficient for boundarylayer of power law fluidsrdquo Applied Mathematics and Mechanicsvol 29 no 9 pp 1215ndash1220 2008

[9] G Bognar ldquoSimilarity solution of a boundary layer flowsfor non-Newtonian fluidsrdquo International Journal of NonlinearSciences and Numerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[10] M C Ece and E Buyuk ldquoSimilarity solutions for free convec-tion to power-law fluids from a heated vertical platerdquo AppliedMathematics Letters vol 15 no 1 pp 1ndash5 2002

[11] S-J Liao ldquoA challenging nonlinear problem for numerical tech-niquesrdquo Journal of Computational andAppliedMathematics vol181 no 2 pp 467ndash472 2005

[12] D M Wei and S Al-Ashhab ldquoSimilarity solutions for non-Newtonian power-law fluid flowrdquo Applied Mathematics andMechanics vol 35 no 9 pp 1155ndash1166 2014

[13] H Schlichting Boundary Layer Theory McGraw-Hill PressNew York NY USA 1979

[14] G Bohme Non-Newtonian Fluid Mechanics North-HollandSeries inAppliedMathematics andMechanics Elsevier ScienceAmsterdam The Netherlands 1987

[15] S Al-Ashhab ldquoA curvature-unified equation for a non-Newtonian power-law fluid flowrdquo International Journal ofAdvances in Applied Mathematics and Mechanics vol 2 no 3pp 72ndash77 2015

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Mathematical Problems in Engineering

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Algebra

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Stochastic AnalysisInternational Journal of


theirs directly however due to the coefficient on the secondterm in (1) above which somewhat changes the ldquodynamicsrdquoof the governing equation and its solutions (even thoughthe deviation may be small for some smaller values of 119899)Howell et al [6] studied the problem in the context ofmomentum and heat transfer In [7] the authors Chen et alconsidered boundary layer flow of an electrically conductingnon-Newtonian fluid in the presence of a magnetic fieldTheir setting required two additional terms in the equationinvolving 1198911015840 and (1198911015840)2 explicitly but with slightly differentboundary conditions They proceeded to utilize Croccovariables and theAdomian decompositionmethod (ADM) toobtain approximate solutions to the problem (in the Croccovariables domain) as well as estimate the value of the skinfriction coefficient and they illustrated the accuracy of theirfindings Further numerical and approximate solutions canbe found in [8ndash11] for example

Existence and uniqueness were established for the exactcurrent problem at hand in [12] where all possible values ofthe power-law index 119899 were considered and it was assumedthat 11989110158401015840 ge 0 (or 11991010158401015840 ge 0) In [13 14] one finds extensivediscussion of the general theory of non-Newtonian boundarylayer flow fluids including the physical derivation of thepower-law problem Lastly we note that in [15] a unifiedequation governing positive and negative curvatures fornon-Newtonian power-law fluids was derived and analyticalsolutions were obtained

In Section 2 we derive the first few terms in a power seriesexpansion (using the ADM) for the solution to the problemand use those terms to approximate the solution where ourapproach will help in obtaining estimates not only for theshear stress parameter but also for the instant of time 119905

119891

where the solution virtually reaches its terminal point where1199101015840(119905119891) = 1 11991010158401015840(119905

119891) = 0 within 10minus4 In Section 3 we compare

our results to numerical estimates obtained via numericalcomputer integrators

2 The Series Solution

Tofind the power series solution to our problemwe utilize theAdomian decompositionmethod observe that the solution 119910satisfies

119910 = 120598119909 + 1205721199092+ 119871minus1

(119865) (4)

where the condition 11991010158401015840(0) = 2120572 replaces the condition atinfinity and where 119865 = 120573119910(119910

10158401015840)2minus119899 from (3) above 119871minus1 =

int

119909

0int

119909

0int

119909

0(sdot)119889119909 119889119909 119889119909 is an operator where 119871 = 11988931198891199093 is the

operator applied to 119910 on the left hand side of (3) as 119871[119910] Nowexpand 119910 in an infinite series of the form 119910 = sum

infin

119894=0119910119894 The

decompositionmethod is based on substituting this series for119910 into (4) above and extracting the terms out one by one bya linearization process of the operator In particular 119910

119894rsquos are

obtained via

119910119894= 119871minus1(119860119894minus1) 119894 = 1 2 3 (5)

where the 119860119894rsquos are referred to as the Adomian polynomials

and are given by

1198600= 119865 (119910

0)

1198601=

120597119865

120597119891

(1199100) sdot 1199101+

120597119865

12059711989110158401015840(1199100) sdot 11991010158401015840

1

1198602=

120597119865

120597119891

(1199100) sdot 1199102+

120597119865

12059711989110158401015840(1199100) sdot 11991010158401015840

2+

1205972119865

12059711989112059711989110158401015840(1199100)

sdot 119910111991010158401015840

1+

1205972119865

120597211989110158401015840(1199100) sdot

(11991010158401015840

1)

2

2

(6)

and where we have skipped writing (12059721198651205972119891)(1199100) sdot ((119910

1)2

2) = 0 from 1198602 This in turn yields the explicit expressions

(with 1199100= 120598119909 + 120572119909

2 and using 120574 = 2120572)

1198600= 120573 (120598119909 + 120572119909

2) (120574)2minus119899

1198601= 120573 (2120572)

2minus1198991199101+ 120573 (2 minus 119899) (120598119909 + 120572119909

2) (120574)1minus119899

11991010158401015840

1

1198602= 120573 (2120572)

2minus1198991199102+ 120573 (2 minus 119899) (120598119909 + 120572119909

2) (120574)1minus119899

11991010158401015840

2

+

120573 (2 minus 119899) (1 minus 119899) (120598119909 + 1205721199092) (120574)minus119899

(11991010158401015840

1)

2

2

+ 120573 (2 minus 119899) (120574)1minus119899

119910111991010158401015840

1

(7)

Proceeding into finding 1199101 1199102 1199103via (5) we obtain

1199101= 120573 (120574)

2minus119899

(

1205981199094

4

+

21205721199095

5

)

1199102= 1205732(120574)4minus2119899

(

1205981199097

7

+

21205721199098

8

) + 1205732(120574)3minus2119899

(2 minus 119899)

sdot (

312059821199096

6

+

201205721205981199097

7

+

4012057221199098

8

)

1199103= 1205733(120574)6minus3119899

(

12059811990910

10

+

212057211990911

11

) + 1205733(120574)5minus3119899

(2 minus 119899)

sdot (

312059821199099

9

+

2012057212059811990910

10

+

40120572211990911

11

) + 1205733(120574)5minus3119899

(2

minus 119899) (

612059821199099

9

+

5612057212059811990910

10

+

112120572211990911

11

) + 1205733(120574)4minus3119899

sdot (2 minus 119899)2(

1512059831199098

8

+

21012057212059821199099

9

+

1120120598120572211990910

10

+

2240120572311990911

11

) + 1205733(120574)4minus3119899

(2 minus 119899) (1 minus 119899) (

1512059831199098

8

+

21012057212059821199099

9

+

1120120598120572211990910

10

+

2240120572311990911

11

)

+ 1205733(120574)5minus3119899

(2 minus 119899) (

1512059821199099

9

+

11212057212059811990910

10

+

224120572211990911

11

)

(8)





27 36 50 32Improved 119905

119891(error lt 13) 5235 421 318 2685




41 48 61 67Improved 119905

119891(error lt 18) 3615 2901 186 129




119891 where 1198911015840(119905

119891) = 1 11989110158401015840(119905




119891)





119891may not be


119891is not as


119891is




4 Conclusions


119891 It was






70 80 99 107Improved 119905

119891(error lt 41) 34 2755 1735 117


Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













27 36 50 32Improved 119905

119891(error lt 13) 5235 421 318 2685




41 48 61 67Improved 119905

119891(error lt 18) 3615 2901 186 129




119891 where 1198911015840(119905

119891) = 1 11989110158401015840(119905




119891)





119891may not be


119891is not as


119891is




4 Conclusions


119891 It was






70 80 99 107Improved 119905

119891(error lt 41) 34 2755 1735 117


Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













70 80 99 107Improved 119905

119891(error lt 41) 34 2755 1735 117


Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article estimation of the shear stress parameter...

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