an exact solution of mhd boundary layer flow of dusty

Research ArticleAn Exact Solution of MHD Boundary Layer Flow of Dusty Fluidover a Stretching Surface

Mudassar Jalil1 Saleem Asghar12 and Shagufta Yasmeen1

1Department of Mathematics COMSATS Institute of Information Technology Park Road Chak Shahzad Islamabad 44000 Pakistan2Nonlinear Analysis and Applied Mathematics (NAAM) Research Group Faculty of Science King Abdulaziz UniversityPO Box 80203 Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Mudassar Jalil mudassarjalilyahoocom

Received 12 October 2016 Accepted 8 February 2017 Published 6 March 2017

Academic Editor Nicolas Gourdain

Copyright copy 2017 Mudassar Jalil et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the boundary layer flow of electrically conducting dusty fluid over a stretching surface in the presence ofapplied magnetic field The governing partial differential equations of the problem are transformed to nonlinear nondimensionalcoupled ordinary differential equations using suitable similarity transformations The problem is now fully specified in terms ofcharacterizing parameters known as fluid particle interaction parameter magnetic field parameter and mass concentration ofdust particles An exact analytical solution of the resulting boundary value problem is presented that works for all values of thecharacterizing parameters The effects of these parameters on the velocity field and the skin friction coefficient are presentedgraphically and in the tabular form respectivelyWe emphasize that an approximate numerical solution of this problemwas availablein the literature but no analytical solution was presented before this study

1 Introduction

During the last few decades several exact analytical solutionsare developed for the one-dimensional flow of Newtonianand non-Newtonian fluids however only limited successhas been achieved for two- and three-dimensional flowsThis is because of the nonlinear nature of the Navier Stokesequations for the viscous fluid and more complex constitu-tive equations for non-Newtonian fluids Further difficultyarises when the governing equations are coupled nonlinearequations Although numerous numerical and approximatetechniques are available to compute solution of the fluiddynamics problems the advantage of the exact analyticalsolution lies in mathematical ingenuity and the strengthof the solution to explain physics of fluid flow The exactanalytical solution can also be considered as bench mark forfurther numerical and approximate investigations Some ofthe exact solutions for the laminar flow of Newtonian andnon-Newtonian fluids are given in [1ndash8]

Dusty fluid model flows have been a subject of specialinterest in recent studies due to their two-phase nature This

phenomenon occurs in fluid (liquid or gas) flows containinga distribution of solid particles For example motion of thedusty air in fluidization problems and the chemical processin which raindrops are formed by coalescence of small dustparticles Cosmic dust which is formed due to the mixingof dust particles and gas is primary precursor for planetarysystems The production of tails of comet 238 is due toemission of ionized gas and the dust particles from the cometbodyThe application of the dusty fluid can also be visualizedin the processes such as nuclear reactor cooling atmosphericfallout powder technology dust collection acoustics paintspray rain erosion sedimentation performance of solid fuelrock nozzles and guided missiles These facts have expeditedthe consideration of modeling solving and analyzing theflow of dusty fluids

Keeping interest in two-phase flows many researchersworked the dusty fluid model for various flow configura-tion and the boundary conditions However realizing thedifficulty of nonlinear coupled equations no attempt hasbeen made in working out any analytical solutionThereforethe solutions given by them are through numerical and

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2307469 5 pageshttpsdoiorg10115520172307469

2 Mathematical Problems in Engineering

approximate schemes A brief background history of dustyfluid is presented now The study on laminar flow of dustyfluid is initiated by Saffman [9] Chakrabarti [10] investigatedthe flow of dusty gas in the boundary layer region Theflow of dusty fluid over semi-infinite plate was discussed byDatta andMishra [11] Vajravelu andNayfeh [12] analyzed thehydromagnetic flow of dusty fluid over stretching surface inthe presence of suction velocity Gireesha et al [13] obtainedthe numerical solution for the two-dimensional boundarylayer flow and heat transfer of dusty fluid over stretchingsurface Numerical solution of MHD flow and heat transferof dusty fluid over a linearly stretching sheet was given byGireesha et al [14] Ramesh et al [15] considered MHDboundary layer flow of dusty fluid over inclined stretchingsheet Later on Gireesha et al [16] examined thermal radi-ation effects of MHD flow of dusty fluid over exponentiallystretching sheet We once again mention that all of thesestudies involve numerical or approximate solutions whereasno exact analytical solution is provided so far

The study of MHD flow is significant due to its manyengineering applications such as the cooling of reactorselectrostatic precipitation power generators MHD pumpsaccelerators petroleum industry and the design of heatexchangers Furthermore MHD flow plays important role inpetroleum industries agriculture geophysics astrophysicssolar physics metrology and the motion of earthrsquos coreKeeping in view the importance of MHD flow and theimportance of exact solution we present exact analyticalsolution for the MHD boundary layer flow of dusty fluidover a stretching surface Another distinctive aspect of thisstudy is the presentation of mathematical result for thenonlinear coupled partial differential equations which areusually handled numerically (as has been in the case ofdusty fluid) The mathematical methodology consists ofconverting the modeled equations into self-similar formcharacterized by dusty and magnetic parameters using theknown similarity transformationsThe coupled equations arethen solved analytically based on the solution given by Crane[1] The exact solution is calculated for the flow field andthe skin friction in terms of the characterizing parameterssuch asmass concentration of dust particles the fluid particleinteraction parameter and the magnetic parameter

2 Description of the Problem

Consider a steady of two-dimensional laminar boundarylayer flow of an electrically conducting viscous incompress-ible dusty fluid over a semi-infinite surface The surface isstretching with the velocity 119906119908(119909) = 119888119909 where the positiveconstant 119888 is the stretching rate Cartesian coordinate systemis located in such a way that 119909-axis and 119910-axis are takenalong (and normal to) the surface respectively while theorigin of the system is located at the leading edge The shapeof dust particles is assumed to be spherical with uniformsize and constant number density Considering these physicalassumptions along with the boundary layer approximations

the governing equations for the flow of dusty fluid togetherwith the boundary conditions are given as (see [14])

120597119906120597119909 + 120597V120597119910 = 0119906120597119906120597119909 + V

120597119906120597119910 = ]12059721199061205971199102 + 120574120591 (119906119901 minus 119906) minus 12059011986102120588 119906

120597119906119901120597119909 + 120597V119901120597119910 = 0119906119901 120597119906119901120597119909 + V119901

120597119906119901120597119910 = 1120591 (119906 minus 119906119901) 119910 = 0 119906 = 119906119908 (119909) V = 0119910 997888rarr infin 119906 = 0 119906119901 = 0 V119901 = V

(1)

where 119906 and 119906119901 (similarly V and V119901) are the 119909 and 119910 com-ponents of fluid and the dust particles velocity respectivelyParameter 120574 = 119898119873120588 denotes the mass concentration of dustparticles 120591 = 119898119870 is the relaxation time of particle phaseand 120588 ] 120590 and 1198610 represent the density kinematic viscosityelectrical conductivity of the fluid and induced magneticfield respectively The terms appearing in 120574 and 120591 are theStokes resistance (119870) the number density (119873) and the massof the dust particle (119898)

We define the following similarity transformations [14]

120578 = radic 119888]119910

119906 = 1198881199091198911015840 (120578) V = minusradic119888]119891 (120578) 119906119901 = 1198881199091198921015840 (120578) V119901 = minusradic119888]119892 (120578)

(2)

where prime (1015840) represents differentiation with respect to 120578Introducing transformations (2) in (1) we get the followingself-similar boundary value problem

119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 + 120573120574 (1198921015840 minus 1198911015840) minus1198721198911015840 = 011989211989210158401015840 minus 11989210158402 + 120573 (1198911015840 minus 1198921015840) = 0

119891 (0) = 01198911015840 (0) = 11198911015840 (infin) = 01198921015840 (120578) = 0119892 (120578) = 119891 (120578)

as 120578 997888rarr infin

(3)

Mathematical Problems in Engineering 3

Table 1 Comparison of skin friction coefficient for various values of 120573 andM with Gireesha et al [17]

119872 Gireesha et al [17]120573 = 0 Exact solution Gireesha et al [17]120573 = 05 Exact solution

02 1000 1000000 1034 103350502 1095 1095445 1126 112611405 1224 1224745 1252 125225110 1414 1414214 1438 143810112 1483 1483240 1506 150603215 1581 1581139 1602 160254020 1732 1732051 1751 1751609

where 120573 and 119872 are the fluid particle interaction parameterand magnetic parameter respectively and are defined by

120573 = 1119888120591 = 119870119888119898119872 = 12059011986102119888120588

(4)

The skin friction coefficient 119862119891 and the shear stress 120591119908 aredefined as

119862119891 = minus1205911199081205881199061199082 120591119908 = 120583(120597119906120597119910)

10038161003816100381610038161003816100381610038161003816119910=0 (5)

Using transformation (2) the dimensionless skin frictioncoefficient is defined by Re119909

minus12119862119891 = minus11989110158401015840(0) where Re119909 =1198881199092120592 is the local Reynolds numberIn the next section we will present the exact analytical

solution of these equations

3 Exact Analytical Solution

Based upon the solution given by Crane [1] we proposethat the nonlinear coupled equations (3) admit exponentiallydecaying solution of the form

119891 = 1205721 + 1205722119890minus1205723120578119892 = 1205724 + 1205725119890minus1205726120578 (6)

where 120572119894119904 (for 119894 = 1 6) are arbitrary constantsThe valuesof these constants can be determined by putting the solution(6) in (3) This yields

1205721 = minus1205722 = 1205724 = 1120577 1205723 = 1205726 = minus1205771205725 = minus 120573120577 (1 + 120573)

(7)

where the term 120577 is given as

120577 = radic1 +119872 + 1205731205741 + 120573 (8)

Substituting the values of 120572119894119904 (for 119894 = 1 6) from (7) into(6) we get closed form exact solution

119891 = 1120577 (1 minus 119890minus120577120578) 119892 = 1120577 (1 minus 1205731 + 120573119890minus120577120578)

(9)

It is worth noting that the solution given in (9) works forall values of 120573 120574 and 119872 The physical quantities like thevelocities of fluid and the dust particles and the skin frictioncoefficient are now expressed as

1198911015840 (120578) = 119890minus1205771205781198921015840 (120578) = 1205731 + 120573119890minus120577120578

minus11989110158401015840 (0) = 120577 = radic1 +119872 + 1205731205741 + 120573 (10)

The numerical solution of (3) exists in the literature butthe exact analytical closed from solution is presented herefor the first time Now using these results one can easilyinterpret the effects of dusty fluid parameters on the physicalquantities without using any numerical scheme It will not beout of place to make a comparison with the numerical resultspresented by Gireesha et al [17] using Runge Kutta Fehlbergfourth-fifth-order method (RKF45 Method) An excellentmatch is found between the two results as shown in Table 1

We also present the graphs for velocity profiles (Figures1(a) and 1(b)) for different values of parameters 119872 and 120573and compare these results with the numerical results givenin [14]This comparison will further validate the authenticityof the two solutions From Figure 1 we observe that both thefluid velocity and the velocity of dust particles decrease withincreasing 119872 This is because increasing magnetic field theopposing Lorentz force increases resulting in the decrease ofthe fluid velocity Effects of fluid particle interaction 120573 massconcentration of dust particles 120574 and themagnetic parameter


f㰀 (휂)g㰀 (휂)

M = 00 50 100

f㰀(휂)

g㰀(휂)

0

02

04

06

08

1

1 2 3 4 50휂

(a)f㰀 (휂)g㰀 (휂)

훽 = 01 05 10

1 2 3 4 50휂

0

02

04

06

08

1

f㰀(휂)

g㰀(휂)

(b)

Figure 1 Effects of119872 (a) and 120573 (b) on the velocity profiles

Table 2 Skin friction coefficient for various values of 120573 120574 and119872

120574 120573 119872 minus11989110158401015840(0)02 143759105 05 10 147196010 1527525

02 142009401 05 10 1425950

10 143178220 1741647

01 05 50 2456284100 3321646

119872 on the skin friction coefficient are given in Table 2 It isobserved that skin friction increases with increasing valuesof 120573 120574 and 119872 This happens because increase in theseparameters produces resistance to the flow and the skinfriction increases

4 Conclusion

An exact analytical solution for the MHD boundary layerflow of dusty fluid over a linearly stretching surface ispresented This result is uniformly valid for all values ofphysical characterizing parameters arising in the flow ofdusty fluid Expressions for the velocity and the skin frictionare presented for the flow configuration The results arecompared with the existing study giving good agreementThis work provides a base for further research in undertakinganalytical solutions of dusty fluid flow over stretching surfaceExact solution forMHDflow for viscoelastic non-Newtonian

fluid can be obtained using the analysis of this paper that willbe reported in a subsequent study

Nomenclature

1198610 Induced magnetic field (kgminus1 sminus2Aminus1)119888 Stretching rate (sminus1)119862119891 Skin friction coefficient119891 Dimensionless stream function for fluidvelocity119892 Dimensionless stream function forparticle velocity119870 Stokes resistance (kg sminus1)119898 Mass of the dust particle (kg)119872 Magnetic parameter119873 Number density of dust particle (mminus3)

Re119909 Local Reynolds number119906119908 Stretching velocity (m sminus1)119906 V Components of fluid velocity along 119909- and119910-axes (m sminus1)119906119901 V119901 Components of particle velocity along 119909-and 119910-axes (m sminus1)119909 119910 Cartesian coordinates (119909-axis is alignedalong the stretching surface and 119910-axis isnormal to it) (m)

Greek Symbols

120572119894 Arbitrary constants120573 Fluid particle interaction parameter120578 Similarity variable120574 Mass concentration of dust particles


] Kinematic viscosity of the fluid (m2 sminus1)120583 Dynamic viscosity of the fluid (kgmminus1 sminus1)120588 Fluid density (kgmminus3)120591 Relaxation time of particle phase (s)120590 Electrical conductivity of the fluid(kgminus1mminus3 s3A2)120591119908 Shear stress at the surface (kgmminus1 sminus2)120595 Stream function (m2 sminus1)

Superscripts

1015840 Differentiation with respect to 120578Subscripts

119901 Dust particle119908 Condition at surface

Competing Interests

Theauthors declare that they no conflict of interests regardingthe publication of this paper

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] B Siddappa and S Abel ldquoNon-Newtonian flow past a stretchingplaterdquo Zeitschrift fur Angewandte Mathematik und Physik vol36 no 6 pp 890ndash892 1985

[3] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1 pp 227ndash2301992

[4] H I Andersson ldquoAn exact solution of the Navier-Stokes equa-tions for magnetohydrodynamic flowrdquoActaMechanica vol 113no 1-4 pp 241ndash244 1995

[5] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[6] E Magyari and B Keller ldquoExact solutions for self-similarboundary-layer flows induced by permeable stretching wallsrdquoEuropean Journal of Mechanics B Fluids vol 19 no 1 pp 109ndash122 2000

[7] M Jalil S Asghar and M Mushtaq ldquoAnalytical solutions ofthe boundary layer flow of power-law fluid over a power-lawstretching surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 18 no 5 pp 1143ndash1150 2013

[8] M Jalil and S Asghar ldquoFlow of power-law fluid over a stretchingsurface a Lie group analysisrdquo International Journal of Non-Linear Mechanics vol 48 pp 65ndash71 2013

[9] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[10] K M Chakrabarti ldquoNote on boundary layer in a dusty gasrdquoAIAA Journal vol 12 no 8 pp 1136ndash1137 1974

[11] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[12] K Vajravelu and J Nayfeh ldquoHydromagnetic flow of a dusty fluidover a stretching sheetrdquo International Journal of Non-LinearMechanics vol 27 no 6 pp 937ndash945 1992

[13] B J Gireesha G K Ramesh H J Lokesh and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid overa stretching vertical surfacerdquo Applied Mathematics vol 2 pp475ndash481 2011

[14] B J Gireesha G S Roopa H J Lokesh and C S BagewadildquoMHD flow and heat transfer of a dusty fluid over a stretchingsheetrdquo International Journal of Physical and Mathematical Sci-ences vol 3 pp 171ndash182 2012

[15] G K Ramesh B J Gireesha and C S Bagewadi ldquoHeattransfer in MHD dusty boundary layer flow over an inclinedstretching sheet with non-uniform heat sourcesinkrdquo Advancesin Mathematical Physics vol 2012 Article ID 657805 13 pages2012

[16] B J Gireesha G M Pavithra and C S Bagewadi ldquoThermalradiation effect on MHD flow of a dusty fluid over an expo-nentially stretching sheetrdquo International Journal of EngineeringResearch amp Technology vol 2 pp 1ndash11 2013

[17] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Advances in Applied Science Research vol 3 no 4pp 2392ndash2401 2012

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Mathematical Problems in Engineering

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


approximate schemes A brief background history of dustyfluid is presented now The study on laminar flow of dustyfluid is initiated by Saffman [9] Chakrabarti [10] investigatedthe flow of dusty gas in the boundary layer region Theflow of dusty fluid over semi-infinite plate was discussed byDatta andMishra [11] Vajravelu andNayfeh [12] analyzed thehydromagnetic flow of dusty fluid over stretching surface inthe presence of suction velocity Gireesha et al [13] obtainedthe numerical solution for the two-dimensional boundarylayer flow and heat transfer of dusty fluid over stretchingsurface Numerical solution of MHD flow and heat transferof dusty fluid over a linearly stretching sheet was given byGireesha et al [14] Ramesh et al [15] considered MHDboundary layer flow of dusty fluid over inclined stretchingsheet Later on Gireesha et al [16] examined thermal radi-ation effects of MHD flow of dusty fluid over exponentiallystretching sheet We once again mention that all of thesestudies involve numerical or approximate solutions whereasno exact analytical solution is provided so far

The study of MHD flow is significant due to its manyengineering applications such as the cooling of reactorselectrostatic precipitation power generators MHD pumpsaccelerators petroleum industry and the design of heatexchangers Furthermore MHD flow plays important role inpetroleum industries agriculture geophysics astrophysicssolar physics metrology and the motion of earthrsquos coreKeeping in view the importance of MHD flow and theimportance of exact solution we present exact analyticalsolution for the MHD boundary layer flow of dusty fluidover a stretching surface Another distinctive aspect of thisstudy is the presentation of mathematical result for thenonlinear coupled partial differential equations which areusually handled numerically (as has been in the case ofdusty fluid) The mathematical methodology consists ofconverting the modeled equations into self-similar formcharacterized by dusty and magnetic parameters using theknown similarity transformationsThe coupled equations arethen solved analytically based on the solution given by Crane[1] The exact solution is calculated for the flow field andthe skin friction in terms of the characterizing parameterssuch asmass concentration of dust particles the fluid particleinteraction parameter and the magnetic parameter

2 Description of the Problem

Consider a steady of two-dimensional laminar boundarylayer flow of an electrically conducting viscous incompress-ible dusty fluid over a semi-infinite surface The surface isstretching with the velocity 119906119908(119909) = 119888119909 where the positiveconstant 119888 is the stretching rate Cartesian coordinate systemis located in such a way that 119909-axis and 119910-axis are takenalong (and normal to) the surface respectively while theorigin of the system is located at the leading edge The shapeof dust particles is assumed to be spherical with uniformsize and constant number density Considering these physicalassumptions along with the boundary layer approximations

the governing equations for the flow of dusty fluid togetherwith the boundary conditions are given as (see [14])

120597119906120597119909 + 120597V120597119910 = 0119906120597119906120597119909 + V

120597119906120597119910 = ]12059721199061205971199102 + 120574120591 (119906119901 minus 119906) minus 12059011986102120588 119906

120597119906119901120597119909 + 120597V119901120597119910 = 0119906119901 120597119906119901120597119909 + V119901

120597119906119901120597119910 = 1120591 (119906 minus 119906119901) 119910 = 0 119906 = 119906119908 (119909) V = 0119910 997888rarr infin 119906 = 0 119906119901 = 0 V119901 = V

(1)

where 119906 and 119906119901 (similarly V and V119901) are the 119909 and 119910 com-ponents of fluid and the dust particles velocity respectivelyParameter 120574 = 119898119873120588 denotes the mass concentration of dustparticles 120591 = 119898119870 is the relaxation time of particle phaseand 120588 ] 120590 and 1198610 represent the density kinematic viscosityelectrical conductivity of the fluid and induced magneticfield respectively The terms appearing in 120574 and 120591 are theStokes resistance (119870) the number density (119873) and the massof the dust particle (119898)

We define the following similarity transformations [14]

120578 = radic 119888]119910

119906 = 1198881199091198911015840 (120578) V = minusradic119888]119891 (120578) 119906119901 = 1198881199091198921015840 (120578) V119901 = minusradic119888]119892 (120578)

(2)

where prime (1015840) represents differentiation with respect to 120578Introducing transformations (2) in (1) we get the followingself-similar boundary value problem

119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 + 120573120574 (1198921015840 minus 1198911015840) minus1198721198911015840 = 011989211989210158401015840 minus 11989210158402 + 120573 (1198911015840 minus 1198921015840) = 0

119891 (0) = 01198911015840 (0) = 11198911015840 (infin) = 01198921015840 (120578) = 0119892 (120578) = 119891 (120578)

as 120578 997888rarr infin

(3)




02 1000 1000000 1034 103350502 1095 1095445 1126 112611405 1224 1224745 1252 125225110 1414 1414214 1438 143810112 1483 1483240 1506 150603215 1581 1581139 1602 160254020 1732 1732051 1751 1751609


120573 = 1119888120591 = 119870119888119898119872 = 12059011986102119888120588

(4)


119862119891 = minus1205911199081205881199061199082 120591119908 = 120583(120597119906120597119910)

10038161003816100381610038161003816100381610038161003816119910=0 (5)






119891 = 1205721 + 1205722119890minus1205723120578119892 = 1205724 + 1205725119890minus1205726120578 (6)



(7)


120577 = radic1 +119872 + 1205731205741 + 120573 (8)



(9)


1198911015840 (120578) = 119890minus1205771205781198921015840 (120578) = 1205731 + 120573119890minus120577120578

minus11989110158401015840 (0) = 120577 = radic1 +119872 + 1205731205741 + 120573 (10)





M = 00 50 100

f㰀(휂)

g㰀(휂)

0

02

04

06

08

1

1 2 3 4 50휂


훽 = 01 05 10

1 2 3 4 50휂

0

02

04

06

08

1

f㰀(휂)

g㰀(휂)

(b)



120574 120573 119872 minus11989110158401015840(0)02 143759105 05 10 147196010 1527525

02 142009401 05 10 1425950

10 143178220 1741647

01 05 50 2456284100 3321646


4 Conclusion



Nomenclature



Greek Symbols




Superscripts



Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












02 1000 1000000 1034 103350502 1095 1095445 1126 112611405 1224 1224745 1252 125225110 1414 1414214 1438 143810112 1483 1483240 1506 150603215 1581 1581139 1602 160254020 1732 1732051 1751 1751609


120573 = 1119888120591 = 119870119888119898119872 = 12059011986102119888120588

(4)


119862119891 = minus1205911199081205881199061199082 120591119908 = 120583(120597119906120597119910)

10038161003816100381610038161003816100381610038161003816119910=0 (5)






119891 = 1205721 + 1205722119890minus1205723120578119892 = 1205724 + 1205725119890minus1205726120578 (6)



(7)


120577 = radic1 +119872 + 1205731205741 + 120573 (8)



(9)


1198911015840 (120578) = 119890minus1205771205781198921015840 (120578) = 1205731 + 120573119890minus120577120578

minus11989110158401015840 (0) = 120577 = radic1 +119872 + 1205731205741 + 120573 (10)





M = 00 50 100

f㰀(휂)

g㰀(휂)

0

02

04

06

08

1

1 2 3 4 50휂


훽 = 01 05 10

1 2 3 4 50휂

0

02

04

06

08

1

f㰀(휂)

g㰀(휂)

(b)



120574 120573 119872 minus11989110158401015840(0)02 143759105 05 10 147196010 1527525

02 142009401 05 10 1425950

10 143178220 1741647

01 05 50 2456284100 3321646


4 Conclusion



Nomenclature



Greek Symbols




Superscripts



Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











M = 00 50 100

f㰀(휂)

g㰀(휂)

0

02

04

06

08

1

1 2 3 4 50휂


훽 = 01 05 10

1 2 3 4 50휂

0

02

04

06

08

1

f㰀(휂)

g㰀(휂)

(b)



120574 120573 119872 minus11989110158401015840(0)02 143759105 05 10 147196010 1527525

02 142009401 05 10 1425950

10 143178220 1741647

01 05 50 2456284100 3321646


4 Conclusion



Nomenclature



Greek Symbols




Superscripts



Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Superscripts



Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









an exact solution of mhd boundary layer flow of dusty

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