approximate explicit analytical expressions of friction factor for flow of bingham fluids in smooth...

Commun Nonlinear Sci Numer Simulat 16 (2011) 239–251

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Commun Nonlinear Sci Numer Simulat

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Approximate explicit analytical expressions of friction factor for flowof Bingham fluids in smooth pipes using Adomian decomposition method

Mohammad Danish, Shashi Kumar, Surendra Kumar *

Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee 247 667, Uttarakhand, India

a r t i c l e i n f o

Article history:Received 15 January 2010Received in revised form 16 March 2010Accepted 17 March 2010Available online 21 March 2010

Keywords:Adomian decompositionFriction factorBingham fluids

1007-5704/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.cnsns.2010.03.013

* Corresponding author. Tel.: +91 1332 285714; fE-mail address: [email protected] (S. Kumar).

a b s t r a c t

This study concerns with the development of explicit expressions of friction factor for theflow of Bingham fluids in smooth pipes in both laminar and turbulent regimes by using apowerful analytical technique, i.e. Adomian decomposition method (ADM) and one of itsvariant namely restarted ADM (RADM). The results from these expressions match very wellwith those obtained by well-known correlations and are found to be valid for all practicalranges. It is hoped that the derived expressions will be useful for the researchers workingin the area.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Pressure drop requirements in an established piping network or the design of new networks need evaluation of frictionfactor (f). Once f is known, it becomes quite easier to handle different pipe-flow problems, viz. pressure drop evaluation forestimating the pump size or to find the flow-rate in a piping network for a given pressure drop.

In chemical engineering, the following definition of Fanning friction factor is popular [1]:

f ¼ Fk=ðAKÞ ð1Þ

where Fk is the force on the object due to relative motion between object and fluid, A is the characteristic area (usually wet-ted surface area), K is the kinetic energy per unit volume and f is the friction factor. For flow in circular pipes this equationleads to the following relation between pressure drop and friction factor [1]:

f ¼ 12

DLð�DpÞqv̂2 ð2Þ

Beside f, there is another popular friction factor called Darcy or Moody friction factor denoted by fD. These two frictionfactors are related as follows:

fD ¼ 4f

In a huge piping network, involving different pipe elements, large number of algebraic equations are generated for eval-uating friction factor [2]. This task becomes tedious as the network size increases and numerical approaches require a goodinitial guess of friction factor for each pipe element so that lesser computational efforts are needed. Therefore, approximateexplicit expressions are used to avoid these complex numerical iterative processes.

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Nomenclature

D pipe diameter (m)

K1 ¼ 16Re þ 16He

6Re2

� �constant in Eqs. (18) and (19)

K2 ¼ � 16He4

3Re8

� �constant in Eqs. (18) and (19)

L length of the pipe (m)m iteration number in RADMnT number of terms in Adomian series expansion�Dp pressure drop (N/m2)v̂ average velocity (m/s)

Greek letterse pipe absolute roughness (m)c shear rate (1/s)k hypothetical parameterlB viscosity of Bingham fluids (Pa s)q fluid density (kg/m3)s0 yield stress (Pa)s shear stress (Pa)

Mathematical functionslog10 logarithm on base ‘10’ln logarithm on base ‘e’

240 M. Danish et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 239–251

Many industrially important non-Newtonian fluids, e.g. various suspensions, slurries, pastes, gels, plastics, etc., are rep-resented by Bingham fluids [1]. Bingham fluids are characterized by two parameters namely, the yield stress ðs0Þ and theplastic viscosity of Bingham fluid ðlBÞ. At steady state and constant temperature and pressure conditions, the following rela-tionship exists between shear stress (s) and shear rate (c) for Bingham fluids:

lBc ¼ s� s0 if s > s0

c ¼ 0 if s 6 s0

The prediction of frictional losses for the flow of Bingham fluids becomes a vital issue for the piping design problem. Thefriction factor (Fanning friction) for these fluids in laminar regime in smooth pipes, depends upon Bingham Reynolds number

Re ¼ qv̂DlB

� �and Hedstrom number He ¼ s0qD2

l2B

� �and is given by the following implicit (quartic) equation [3,4]:

f16¼ 1

Reþ He

6Re2 �He4

3Re8f 3ð3Þ

Since, Eq. (3) is a quartic equation, a closed form analytical solution is possible. However, due to complexity it is rarelyemployed and as per the authors’ knowledge, no approximate explicit relation for Eq. (3) is available and numerical tech-niques are used for obtaining the solution.

For turbulent flow of Bingham fluids in smooth pipe, the famous implicit NPK (Nikuradse–Prandtl–Karman) equation isapplicable [4] and can be deduced from Colebrook–White equation [5] under smooth pipe condition [6].

1ffiffiffif

p ¼ 4:0 log Reffiffiffif

p� �� 0:40 ð4Þ

Further, it may be mentioned that Eq. (4) is the same equation as used for Newtonian fluids in turbulent flow in smoothpipes and therefore corresponding explicit correlations can also be used for these fluids. Out of the many existing correla-tions, we have selected only those relations whose predictions are reasonably close to Eq. (4) and so that a meaningful com-parison of the results can be made. Although, there are some other popular relations, e.g. [7–9] but due to relatively moreerror in their predictions, these have not been discussed. A brief description of the selected correlations is given below.

(i) Serghides [10] proposed the following friction factor correlation valid for Re > 2100 and for any value of e=D.

1ffiffiffiffifD

p ¼ A� ðB� AÞ2

ðC � 2Bþ AÞ2

! !ð5Þ

where A ¼ �2:0 log � e=D3:70þ 12

Re

� �, B ¼ �2:0 log e=D

3:70þ 2:51ARe

� �and C ¼ �2:0 log e=D

3:70þ 2:51BRe

� �.

M. Danish et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 239–251 241

(ii) Manadilli [11] proposed the following correlation for fD by using his so-called sigmoidal equation. This relation isapplicable for 5000 < Re < 108 and for any value of e=D.

1ffiffiffiffifD

p ¼ �2:0 loge=D3:70

þ 95Re0:983 �

96:82Re

� �ð6Þ

(iii) Romeo et al. [6], after developing different expressions on the lines of various available models, have given an explicitrelation for friction factor using multivariable nonlinear regression for turbulent flow of Newtonian fluids. The rele-vant range for Re and e/D lies in 3000—1:5� 108 and 0–0.05, respectively.

1ffiffiffiffifD

p ¼ �2:0 loge=D

3:7065� 5:0272

Relog

e=D3:827

� 4:567Re

� logðAÞ� ��

ð7Þ

where A ¼ e=D3:7065

� �0:9924þ 5:3326

208:815þRe

� �0:9345.

In another approach, Sablani et al. [4] have developed an explicit procedure for finding the friction factor for Binghamfluids in smooth pipes using an empirical soft computing tool, i.e. artificial neural network (ANN), but no explicit relationwas provided. Goudar and Sonnad [12] and More [13] have obtained explicit expressions of friction factor for turbulent flowof Newtonian fluids by treating NPK and Colebrook–White equations using Lambert W function, respectively. A similar use ofLambert W function for obtaining the expression for friction factor has been given by Keady [14]. A brief discussion on theLambert W function has been presented in appendix.

Recently, Adomian decomposition method (ADM) has proved to be a powerful analytical technique for solving differenttypes of nonlinear equations, e.g. algebraic equations, ordinary and partial differential equations and integral equations, etc.[15,16]. Because of its versatility, ADM has received a lot of attention and a vast amount of the literature is available [17–31].This method possesses sound mathematical background and does not involve empiricism unlike ANN. Moreover, a variety ofnonlinear equations can be tackled using this technique unlike Lambert W function which is suitable only for a particularform of nonlinear equation.

In the present work, different explicit expressions of friction factor for Bingham fluids under laminar as well as turbulentflow conditions have been obtained using a well-known decomposition technique namely Adomian decomposition method[15,16] and one of its effective variants, i.e. restarted Adomian decomposition method [24–26]. The predictions of the de-rived analytical expressions have been compared with those obtained by the above formulae for friction factor and are foundto be in excellent agreement.

2. Adomian decomposition method

The basic philosophy of the powerful ADM is to break the nonlinearity into components called Adomian polynomials, Ai

and the nonlinear equation is decomposed into infinite simple linear problems [15,16]. The Adomian polynomials can be eas-ily generated for any type of nonlinearity NðyÞ with the help of robust softwares like Mathematica, Maple, etc. and the infi-nite set of simpler linear equations can be solved sequentially using these softwares. The obtained analytical solution is aninfinite series called Adomian series and shows better convergence as compared to the Taylor series [18,29]. Hence, smallernumbers of terms are sufficient to represent the solutions. This parameter based method is attractive since it can deal withall types of nonlinearities whether weak or strong and does not require the presence of any small or large parameter unlikeother methods, e.g. perturbation method, etc.

2.1. Adomian polynomials

In this section, the generation of Adomian polynomials for any type of nonlinearity has been described with the specialemphasis on algebraic equations. Although on similar lines, one can easily obtain the Adomian polynomials for the nonlin-earities appearing in ordinary or partial differential equations. Adomian [15] has outlined a detailed procedure for generatingthe Adomian polynomials for different types of nonlinearities as well as described the steps for finding the solutions of dif-ferent types of equations. Expressions of Adomian polynomials for most types of nonlinearities, viz. yn; ey2

; sin y2, etc. havealso been given in Adomian [15].

For generating the Adomian polynomials, we consider a nonlinear function NðyÞ of unknown variable y. y is assumed to bedependent on a hypothetical parameter k 2 ½0;1�, i.e. y ¼ yðkÞ. Using Taylor series y is expanded around k ¼ 0, i.e.

yðkÞ ¼X1i¼0

kiyi ¼ y0 þ ky1 þ k2y2 þ � � � ð8Þ

where yi ¼ 1i!

oiyoki

��k¼0

.

Similarly, NðyðkÞÞ is expanded around k ¼ 0, i.e.

NðyÞ ¼X1i¼0

kiAi ¼ A0 þ kA1 þ k2A2 þ � � � ð9Þ


where Ai ¼ 1i!

oiNðyÞoki

��k¼0

are the Adomian polynomials and can be found for any nonlinear function NðyÞ by substituting Eq. (8)

in Eq. (9) and collecting the terms having same powers of k. For example, Adomian polynomials for NðyÞ ¼ y2 are obtained asfollows:

(i) From Eq. (8), one assumes

yðkÞ ¼ y0 þ ky1 þ k2y2 þ � � �

(ii) Substituting the above expression in the concerned nonlinearity, i.e. NðyÞ ¼ y2

NðyÞ ¼ y2 ¼ y0 þ ky1 þ k2y2 þ � � �� 2

Or

NðyÞ ¼ y20 þ 2ky0y1 þ k2 y0y2 þ y2

1

� þ � � �

(iii) On comparing the above expanded form of nonlinearity NðyÞ ¼ y2 with Eq. (9), one finds

A0 ¼ y20

A1 ¼ 2y0y1

A2 ¼ y0y2 þ y21

� � � �

Once the expressions for Adomian polynomials are known they are properly substituted into the concerned equation(algebraic/differential/integral equations). Thereafter, each type of equation is solved in their respective manner. The follow-ing subsections discuss the treatment of algebraic equation using ADM, which is applicable to the present work.

2.2. ADM for nonlinear equation

Consider a nonlinear equation NðyÞ ¼ 0. Expressing it in the following canonical form [15,24,25], one obtains

y ¼ C0 þ F0ðyÞ ð10Þ

Now for Eq. (10), the following homotopy is constructed [30]:

y ¼ C0 þ kF0ðyÞ ð11Þ

where C0 is a constant, F0ðyÞ is some function of y and k is the same embedding parameter considered in the previous section.Using Eqs. (8) and (9), y and F0ðyÞ can be replaced in Eq. (11) by their respective decomposed forms as shown below:

y0 þ ky1 þ k2y2 þ � � � ¼ C0 þ k A0 þ kA1 þ k2A2 þ � � ��

ð12Þ

where yis are the decomposed solutions and Ais are Adomian polynomials. On comparing one finds y0 ¼ C0; y1 ¼A0ðy0Þ; y2 ¼ A1ðy0; y1Þ; . . . ; yi ¼ Ai�1ðy0; y1; . . . ; yi�1Þ. It is clear that as k! 0; y ¼ y0 ¼ C0 and whereas, if k! 1; y ¼y0 þ y1 þ y2 þ � � � ¼ C0 þ A0 þ A1 þ A2 þ � � � ¼ FðyÞ. This means as k tends from 0 to 1 the initial guess y ¼ y0 ¼ C0 approachesthe exact solution of Eq. (10), i.e. y ¼

P1i¼0yi ¼ C0 þ

P1i¼0Ai. Obviously, there are many ways in which the canonical form, i.e.

Eq. (10) can be formed and therefore the convergence of the resultant solution series strongly depends on the form of theequation and particularly on the initial guess, i.e. C0. Besides, the final series solution form will be different for each ofthe equation forms. However, if the series is convergent it will converge to the smallest magnitude root as proved by Ado-mian [15]; other roots can be found by the deflation method.

2.3. RADM for nonlinear equation

Sometimes the series obtained by ADM converges slowly or gets diverged (due to the poor initial guess or due to the pres-ence of complex roots). To overcome this problem as well as to get the results more accurately and quickly, Babolian andBiazar [24,25] and Babolian and Javadi [26] modified the ADM by devising an iterative procedure for updating Eq. (10). Thismodified method has been named as restarted ADM (RADM). The steps involved are summarized below and the details ofthese modifications can be found in their original work.

Following updating procedure is adopted for Eq. (10):

Step 1 : Cm ¼ C0 for m ¼ 1 and Cm ¼ yðm�1Þ for m > 1 ð13aÞ

Step 2 : FmðyÞ ¼F0ðyÞ � yF 00ðCmÞ þ C0

1� F 00ðCmÞ� Cm for m P 1 ð13bÞ

Step 3 : yðmÞ ¼ Cm þ FmðCmÞ for m P 1 ð13cÞ

where yðmÞ denotes the RADM solution after mth iteration. Initially, C1 is taken equal to C0, i.e. for m = 1. The C0 and FðyÞ inEq. (10) are updated using the above steps; thereafter, the solution is obtained for each iteration by applying ADM to the


updated equation, i.e. Eq. (13c). In the next iteration, the value of solution so found is used to update Cm through Eq. (13a).This procedure is repeated until one gets the result of desired accuracy. Using the above step wise procedure of RADM wehave solved all the examples mentioned in Babolian and Javadi [26] and the results matched up to the last digit. It is ob-served that RADM successfully found the solutions in a few iterations except where canonical form cannot be formed. Ina few cases, this scheme may also diverge and for such a situation Basto et al. [29] have given a remedy. However, in thepresent problem this situation does not arise.

3. Friction factor for flow of Bingham fluids in a pipe

In this section, the nonlinear friction factor relations, i.e. Eqs. (3) and (4) have been solved using ADM and RADM and dif-ferent explicit expressions for friction factor are obtained. Accuracy of these expressions can be increased by taking moreterms in Adomian series.

3.1. Turbulent regime

Eq. (4) is transformed into the following canonical form, i.e. Eq. (10) by letting y ¼ 1 ffiffifp and the following equation is ob-

tained, i.e.

y ¼ 4:0log10ðReÞ � 0:40|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}C0

þð�4:0log10ðyÞÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}F0

ð14Þ

After getting the above transformed equation (14) and applying the ADM, one obtains the following decomposed solutions:

y0 ¼ C0

y1 ¼ A0 ¼ �4 lnðC0Þlnð10Þ

y2 ¼ A1 ¼16 lnðC0ÞC0ln2ð10Þ

y3 ¼ A2 ¼32 lnðC0Þð�2þ lnðC0ÞÞ

C20ln3ð10Þ

y4 ¼ A3 ¼128 lnðC0Þ 6� 9 lnðC0Þ þ 2ln2ðC0Þ

� �3C3

0ln4ð10Þ

From the above decomposed solutions, the two terms ðnT ¼ 2Þ Adomian expansion can be found as follows:

1ffiffiffif

p ¼ y ¼ y0 þ y1

Or

1ffiffiffif

p ¼ C0 � 1:73718 lnðC0Þ ð15aÞ

Similarly, five terms ðnT ¼ 5Þ Adomian solution is given by:

1ffiffiffif

p ¼ y ¼ y0 þ y1 þ y2 þ y3 þ y4

Or

1ffiffiffif

p ¼C4

0 � 1:73718ðC0 � 1:73718Þ 3:01779þ C20

� �ln C0 þ ð2:62122C0 � 13:6606Þln2C0 þ 3:03568ln3C0

� �C3

0

ð15bÞ

where C0 ¼ 4:0log10ðReÞ � 0:4.Subjecting the same Eq. (14) to RADM, the following updated canonical form similar to Eq. (13c) is obtained for first iter-

ation ðm ¼ 1Þ:

C0 ¼ C1 ¼ 4:0log10ðReÞ � 0:4

F1 ¼ �4ðC0 � yþ C0 lnðyÞÞ

4þ C0 lnð10Þ

Applying ADM to the above equations yields the following decomposed solutions for this iteration, i.e. m ¼ 1:

Table 1Compar

Re

4000

5000

10,0

50,0

100,

1,00


y0 ¼ C0

y1 ¼ A0 ¼ �4C0 lnðC0Þ

4þ C0 lnð10Þy2 ¼ A1 ¼ 0

y3 ¼ A2 ¼32C0ln2ðC0Þð4þ C0 lnð10ÞÞ3

y4 ¼ A3 ¼256C0ln3ðC0Þ

3ð4þ C0 lnð10ÞÞ4

Hence, the two terms ðnT ¼ 2Þ expansion after first iteration ðm ¼ 1Þ is found to be:

1ffiffiffif

p ¼ y ¼ y0 þ y1

Or

1ffiffiffif

p ¼ C0 �1:73718C0 lnðC0Þ

1:73718þ C0ð16aÞ

Similarly, the five terms ðnT ¼ 5Þ expansion after first iteration ðm ¼ 1Þ is found to be:

1ffiffiffif

p ¼ y ¼ y0 þ y1 þ y2 þ y3 þ y4

Or

1ffiffiffif

p ¼ C0 �1:73718C0 lnðC0Þ

1:73718þ C0þ 2:62122C0 lnðC0Þ½ �2

ð1:73718þ C0Þ3þ 3:03568C0½lnðC0Þ�3

ð1:73718þ C0Þ4ð16bÞ

Results from Eqs. (15a)—(16b) have been given in Tables 1 and 2 along with those obtained from known correlations. InTable 1, the results from ADM and RADM have been compared with the exact numerical solutions of NPK equation and it isfound that the RADM solution matched well even with two terms and two iterations. On comparing the number of iterationsrequired in Newton–Raphson method to achieve the same values, one finds that RADM with 2 terms outperformed the for-mer one; the starting guess for both the techniques were taken to be same and equal to 1=C2

0. This can be attributed to thefaster convergence properties of Adomian series as compared to the Taylor series [18,29]. Table 1 also depicts that for RADM,the effect of number of iterations (m) has a more pronounced effect on the result quality as compared to the number of termsðnTÞ.

ison of the friction factor values, obtained by ADM and RADM for turbulent flow, with exact numerical values.

fNPK Exactnumerical value

Iterations requiredin Newton–Raphson

Method Iterations requiredin RADM (m)

nT ¼ 2 nT ¼ 5

0.00998597 4 ADM – 0.01126290 0.00998333RADM 1 0.01014430 0.00998623

2 0.00998606 0.009985973 0.00998597 � � �

0.00935658 4 ADM – 0.01049160 0.00935432RADM 1 0.00949628 0.00935688

2 0.00935666 0.009356583 0.00935658 � � �

00 0.00772713 4 ADM – 0.00852989 0.00772571RADM 1 0.00782357 0.00772745

2 0.00772717 0.007727133 0.00772713 � � �

00 0.00522650 4 ADM – 0.00562191 0.00522598RADM 1 0.00527122 0.00522673

2 0.00522651 0.005226503 0.00522650 � � �

000 0.00450038 4 ADM – 0.00480185 0.00450003RADM 1 0.00453358 0.00450055

2 0.00450038 0.00450038

0,000 0.00291282 4 ADM – 0.00304959 0.00291271RADM 1 0.00292662 0.00291290

2 0.00291282 0.00291282

Table 2Comparison of friction factor values for turbulent flow obtained by using different correlations.

Re f

NPK (exact numerical value) Serghides [10] Manadilli [11] Romeo et al. [6] RADM (nT = 2, m = 1) RADM (nT = 5, m = 1)

4000 0.00998597 0.00997674 0.00997692 0.00999134 0.01014430 0.009986235000 0.00935658 0.00934814 0.00934837 0.00936171 0.00949628 0.0093568810,000 0.00772713 0.00772063 0.00772067 0.00773112 0.00782357 0.0077274550,000 0.00522650 0.00522271 0.00522276 0.00522863 0.00527122 0.00522673100,000 0.00450038 0.00449730 0.00449766 0.00450207 0.00453358 0.004500551000000 0.00291282 0.00291118 0.00291291 0.00291373 0.00292662 0.00291290


Table 2 compares the turbulent friction factor obtained by using different correlations used as alternatives of NPK equa-tion. Fig. 1 shows the percentage errors with respect to the NPK equation and it is clear that the expression resulted fromRADM with nT ¼ 5; m ¼ 1, i.e. Eq. (16b) outperformed all others and throughout exhibits an error of less than 0.005%. Thepredicted friction factor values using optimal ANN configuration of Sablani et al. [4] had mean relative error of 1.32% whereasthe best model of Romeo et al. [6] displayed errors between 0.02% and 0.05%. Therefore, it is evident that RADM solution fornT ¼ 5 and m ¼ 1 (Eq. (16b)), having very small error, is better as far as the explicit relations are concerned.

Beside measuring the accuracy of these correlations, it is also useful to measure the computational efforts for evaluatingthe turbulent friction factor by these correlations. For achieving this objective, the code developed in the programming lan-guage of Mathematica software has been run on a machine with the following configuration: Intel core 2duo processor withclock rate equal to 2.2 GHz and physical memory (RAM) equal to 4 GB. Two approaches were adopted for this purpose: (i)measuring the absolute CPU time and (ii) measuring the relative CPU time. For the first approach, most of the mathematicalsoftwares have several inbuilt commands, e.g. ‘‘Timing[. . .]” command is implemented in Mathematica. Using this command,the approximate absolute timings (CPU time), in evaluating the friction factor using various correlations, have been recordedand are shown in Table 3. However, no concrete conclusions can be drawn from these results as almost equal CPU time wasspent in evaluating friction factor for all the mentioned correlations. This may be due to the insignificant computer intensivenature of the problem as compared to the available powerful machine configuration and the robust software (Mathematica).Moreover, many a times the program resulted in zero CPU time. This is also true for other smaller operations because afterthe first session of the program, the variables and the data get stored in the system cache, and recalling and reprocessing thedata requires negligibly small time.

Because of the above disadvantages, the first approach was dropped. In the second approach, the program was instructedto run for many iterations for each correlation and for the same value of Re. The overall time consumed by CPU, now a muchhigher value, is then reported. Dividing it by the number of iterations yields the average CPU time spent in a single iterationfor each correlation. Though, in this case the system would be using data stored in system cache and thus would not give atrue picture of the time consumed by CPU. However, it definitely provides a relative estimate of the duration spent in theprocessing of each correlation. The results obtained by following this approach have been depicted in Fig. 2. One notes thatthe CPU time per iteration becomes constant as the number of iteration increases. It is clearly visible that the Colebrook–White relation [5] and the Serghides relation [10] consumed much greater CPU time per iteration as compared to the otherproposed correlations. The CPU time spent in the processing of Eq. (7) proposed by Romeo et al. [6] and Eq. (16b) proposed in

Fig. 1. Percentage error in friction factor values obtained by using various correlations for turbulent flow versus Reynolds number.

Table 3Comparison of absolute time consumed by CPU for evaluating friction factor for turbulent flow using different relations.

Re Sessions Approximate absolute time consumed by CPU (s)

NPK Manadilli RADM ðnT ¼ 2; m ¼ 1Þ RADM ðnT ¼ 5; m ¼ 1Þ

4� 103 1 0.016 0.015 0.015 0.0162 0.016 0.015 0.016 0.0153 0.015 0.015 0.015 0.0154 0.016 0.016 0.016 0.0165 0.016 0.015 0.015 0.016

106 1 0.016 0.015 0.015 0.0162 0.015 0.015 0.016 0.0153 0.016 0.016 0.015 0.0164 0.015 0.015 0.015 0.0165 0.016 0.015 0.016 0.015

Fig. 2. Relative comparison of time consumed by CPU per iteration for evaluating friction factor for turbulent flow using different correlations.


the present work, i.e. RADM ðnT ¼ 5; m ¼ 1Þ is approximately same. Similarly, Eq. (6) proposed by Manadilli [11] and Eq.(16a) proposed in the present work, i.e. RADM ðnT ¼ 2; m ¼ 1Þ took approximately the same amount of time in their process-ing. On comparing, it can be noticed that the time spent in the evaluation of Eq. (16b), i.e. RADM ðnT ¼ 5; m ¼ 1Þ is slightlygreater than the time spent in the evaluation of Eq. (6) proposed by Manadilli [11]. However, as evident in Fig. 1, the accuracyobtained by the former one is much higher than the latter one. It is worthwhile to note that the similar results for these twoapproaches were also obtained by using another effective mathematical tool, i.e. Matlab.

3.2. Laminar regime

To solve Eq. (3) by ADM, it is transformed into the following canonical form by taking y ¼ f :

y ¼ 16Reþ 16He

6Re2|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}C0

þ � 16He4

3Re8y3

!|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

F0

ð17Þ

Solving Eq. (17) with ADM one gets the following expanded terms:

y0 ¼ C0 ¼ K1

y1 ¼ A0 ¼K2

K31

y2 ¼ A1 ¼ �3K2

2

K71


y3 ¼ A2 ¼15K3

2

K111

y4 ¼ A3 ¼ �91K4

2

K151

where K1 ¼ 16Re þ 16He

6Re2 and K2 ¼ � 16He4

3Re8 .Hence, the two terms ðnT ¼ 2Þ expansion for laminar friction factor attains the following form:

f ¼ y ¼ y0 þ y1

f ¼ K1 þK2

K31

ð18aÞ

Likewise, the five terms ðnT ¼ 5Þ expansion for laminar friction factor gets the following form:

f ¼ y ¼ y0 þ y1 þ y2 þ y3 þ y4

f ¼ K1 þK2

K31

� 3K22

K71

þ 15K32

K111

� 91K42

K151

ð18bÞ

Solving Eq. (17) using RADM, the following canonical form is obtained for m ¼ 1:

C0 ¼ C1 ¼ K1

F1 ¼K2 K4

1 � 3K1y3 þ 3y4� �

K41 þ 3K2

� �y3

Application of ADM to the above equations, offers the following decomposed solutions:

y0 ¼ C0 ¼ K1

y1 ¼ A0 ¼K1K2

K41 þ 3K2

Therefore, the two terms expansion with one iteration is given below in explicit form:

f ¼ y ¼ y0 þ y1

f ¼ K1 þK1K2

K41 þ 3K2

ð19aÞ

In the next iteration of RADM, i.e. m ¼ 2, the following canonical form is obtained:

C0 ¼ K1

C2 ¼ K1 þK1K2

K41 þ 3K2

� �

F2 ¼ �C2 þK1 þ K2 y�3 þ 3yC�4

2

� �1þ 3K2C�4

2

Applying ADM to the above revised canonical form, one obtains the following two decomposed terms:

y0 ¼ C2 ¼ K1 þK1K2

K41 þ 3K2

y1 ¼ A0 ¼ �C2 þK1

1þ 3K2

C42

þ 4K2

C32 1þ 3K2

C42

� �
Therefore, the friction factor with two terms is given as:
f ¼ y ¼ y0 þ y1

Or

f ¼ K1 þ4K2

K1 þ K1K2K4

1þ3K2

� �3

0B@

1CA,

1þ 3K2

K1 þ K1K2K4

1þ3K2

� �4

0B@

1CA ð19bÞ

In the same way, the following canonical form is obtained in the first iteration of RADM ðm ¼ 1Þ for nT ¼ 5:


C0 ¼ C1 ¼ K1

F1 ¼K2 K4

1 � 3K1y3 þ 3y4� �

K41 þ 3K2

� �y3

The following decomposed terms are obtained by applying the ADM to the above equations:

y0 ¼ C0 ¼ K1

y1 ¼ A0 ¼K1K2

K41 þ 3K2

y2 ¼ A1 ¼ 0

y3 ¼ A2 ¼6K1K3

2

K41 þ 3K2

� �3

y4 ¼ A3 ¼ �10K1K4

2

K41 þ 3K2

� �4

The five terms solution after the first iteration of RADM is given by:

f ¼ y ¼ y0 þ y1 þ y2 þ y3 þ y4

Putting the values of yis in above expression and simplifying yields:

f ¼ K1 1� 10K42

K41 þ 3K2

� �4 þ6K3

2

K41 þ 3K2

� �3 þK2

K41 þ 3K2

� �0B@

1CA ð19cÞ

Similarly, for the next iteration of RADM, i.e. m ¼ 2, the following canonical form is obtained:

C0 ¼ K1; C2 ¼ T

F2 ¼ �T þ K1 þ K2ðy�3 þ 3yT�4Þ1þ 3K2T�4

where T ¼ K1 1� 10K42

K41þ3K2ð Þ4

þ 6K32

K41þ3K2ð Þ3

þ K2

K41þ3K2ð Þ

� �¼ fnT¼5;m¼1.

Applying the ADM one gets

y0 ¼ C2 ¼ T

y1 ¼TðK2 þ ðK1 � TÞT3Þ

3K2 þ T4

y2 ¼ 0

y3 ¼6K2TðK2 þ ðK1 � TÞT3Þ2

ð3K2 þ T4Þ3

y4 ¼ �10K2TðK2 þ ðK1 � TÞT3Þ3

ð3K2 þ T4Þ4

Hence, the five terms ðnT ¼ 5Þ RADM solution after two iterations ðm ¼ 2Þ is given by

f ¼ y ¼ y0 þ y1 þ y2 þ y3 þ y4

f ¼ 1

3K2 þ T4� �4 T 116K4

2 þ K1T15 þ 3K32T3ð11K1 þ 36TÞ þ 3K2

2T6 �4K21 þ 21K1T þ 4T2

� ��

þ K2T9 �10K31 þ 36K2

1T � 33K1T2 þ 20T3� ��

ð19dÞ

For the flow of Bingham fluids to be in laminar regime, the Reynolds number for a given Hedstrom number should be lessthan the corresponding critical Reynolds number. In Fig. 3, the plot of critical Reynolds number versus Hedstrom number hasbeen redrawn using the method described by Hanks and Pratt [32]. With the help of this figure, several possible combina-tions of Re and He for laminar flow, have been selected. Subsequently, the friction factor values for these sets of Re andHehave been obtained from the above expressions and are shown in Tables 4 and 5. The predicted values of friction factor

Fig. 3. Critical Reynolds number versus Hedstrom number.

Table 4Comparison of the friction factor values, obtained by ADM and RADM for laminar flow, with exact numerical values.

He Method nT Re ¼ 102 Re ¼ 103

102 Numerical – 0.186658 0.0162667ADM 2 0.186658 0.0162667

5 0.186658 0.0162667RADM 2 0.186658 (m = 1) 0.0162667 (1)

5 0.186658 (1) 0.0162667 (1)

104 Numerical – 2.47480 0.0419439ADM 2 2.59052 0.0419800

5 2.49409 0.04194392.51155 (1) 0.0419452 (1)

RADM 2 2.47554 (2) 0.0419439 (2)2.4748 (3)

5 2.48368 (1) 0.0419439 (1)2.4748 (2)


are found to be matching very well with their exact numerical counterparts. Though, not shown but here too, RADM withtwo terms was found to be superior to Newton–Raphson method. Hence, one may also use the numerical approaches ofADM/RADM without generating any such expressions.

In Fig. 4, the effect of number of terms in ADM/RADM solutions on the accuracy of the friction factor values has beenshown. Following observations can be made:

(i) Accuracy increases with the increase in number of terms irrespective of the type of flow, i.e. laminar or turbulent.Though, this effect is more pronounced in the latter case.

(ii) For the same number of terms, the RADM results are better than the corresponding ADM results.(iii) For the same number of terms in RADM, the accuracy in friction factor values increases with increase in number of

iterations.

4. Conclusions

In the present work, explicit analytical expressions of friction factor for the flow of Bingham fluids in smooth pipes havebeen derived by using ADM and RADM. Commendable results were obtained using these explicit relations. As shown in Fig. 1for turbulent flow, the predicted values from RADM ðnT ¼ 5; m ¼ 1Þ, i.e. Eq. (16b) were having a constant error of 0.005%, afar smaller value as compared to other cited correlations. The relative comparison of CPU time spent in the processing of Eq.(16b) is slightly higher than for Manadilli’s equation [11] or Romeo et al.’s equation [6], as shown in Fig. 2. Nevertheless, theaccuracy achieved by Eq. (16b) offsets the price paid in terms of marginally higher CPU time.

Table 5Comparison of the friction factor values, obtained by ADM and RADM for laminar flow, with exact numerical values.

He Technique nT Re ¼ 104 Re ¼ 5� 104 Re ¼ 105

106 Numerical – 0.02474800 – –ADM 2 0.02590520 – –

5 0.02494090 – –RADM 2 0.0251155 (m = 1) – –

0.0247554 (2)0.024748 (3)

5 0.0248368 (1) – –0.024748 (2)

108 Numerical – – 0.0837146 –ADM 2 – 0.0958373 –

5 – 0.089387 –RADM 2 – 0.0907662 (1) –

0.0856263 (2)� � �0.0837146 (5)

5 – 0.0879638 (1) –0.0841171 (2)� � �0.0837146 (4)

1010 Numerical – – – 2.012720ADM 2 – – 2.385630

5 2.219780RADM 2 2.25552 (1)

– – 2.11543 (2)� � �2.01272 (8)

5 – 2.18172 (1)2.06262 (2)� � �2.01272 (5)

–, combination not evaluated.

Fig. 4. Variation of percentage error in friction factor values with number of terms in ADM and RADM (laminar flow: Re = 105, He = 1010; turbulent flow:Re = 5 � 103).


For laminar flow, few explicit expressions have been derived by using ADM and RADM and are given by Eqs. (18a), (18b)and (19a)–(19d). The error in predictions of Eq. (19b), obtained by using RADM for nT ¼ 2 and m ¼ 2 was found to be within5.2%. Nonetheless, the accuracy of these expressions can be increased further by taking more terms into account or byincreasing the number of iteration in RADM as apparent in Fig. 4. Although, the number of iterations in RADM (m) has a morepronounced effect on the quality of results as compared to the number of terms ðnTÞ. It was also observed that the iterativescheme of RADM performed equally well or better than the classical Newton–Raphson method for both turbulent and lam-


inar cases. It is worthwhile to note that the presented analytical cum numerical approaches of ADM/RADM are equally appli-cable to other nonlinear algebraic equations.

Appendix A

A Lambert W function is defined as the inverse function of x ¼ yey, i.e. y = Lambert (x) and is symbolized by y ¼WðxÞ. Thisfunction is also sometimes called ProductLog function (as implemented in Mathematica). In general, the domain of the func-tion is the set of complex values and the outputs are also complex, however, for x 2 ½0;1Þ Lambert W function yields singlereal values. For x 2 ð�1;�1=eÞ, Lambert W function does not evaluate to any real value whereas, for x 2 ½�1=e;0Þ it com-putes two real values.

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approximate explicit analytical expressions of friction factor for flow of bingham fluids in smooth...

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