research article heat and mass transfer effect on mhd flow...

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2013 Article ID 380679 8 pageshttpdxdoiorg1011552013380679

Research ArticleHeat and Mass Transfer Effect on MHD Flow of a ViscoelasticFluid through a Porous Medium Bounded by an OscillatingPorous Plate in Slip Flow Regime

S N Sahoo

Department of Mathematics Institute of Technical Education and Research Siksha ldquoOrdquo Anusandhan University KhandagiriBhubaneswar Odisha 751030 India

Correspondence should be addressed to S N Sahoo sachisahooyahoocom

Received 30 March 2013 Accepted 18 June 2013

Academic Editor Jose C Merchuk

Copyright copy 2013 S N Sahoo This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Unsteady flow of an eclectically conducting and incompressible viscoelastic liquid of the Walter 1198611015840 model with simultaneous heatand mass transfer near an oscillating porous plate in slip flow regime under the influence of a transverse magnetic field of uniformstrength is presented The governing equations of the flow field are solved by a regular perturbation method for small elasticparameter and the expressions for the velocity temperature concentration skin friction 119862

119891 the heat flux in terms of the Nusselt

number N119906 and the rate of mass transfer in terms of the Sherwood number S

ℎare obtained The effects of the important flow

parameters on the dynamics are discussed Findings of the study reveal that the rarefaction parameter accelerates the fluid particlesin the flow domain Elastic parameter contributes to sudden fall of the velocity near the plate Magnetic force contributes to greaterskin friction as the time elapses Destructive reaction reduces whereas generative reaction enhances the concentration distribution

1 Introduction

Free convective flow in presence of heat source has beena subject of interest of many researchers because of itspossible application to geophysical sciences astrophysicalsciences and in cosmical studies Such flows arise eitherdue to unsteady motion of the boundary or the boundarytemperature The study of fluctuating flow is important inthe paper industry and many other technological fieldsThereforemany researchers have paid their attention towardsthe fluctuating flow of viscous incompressible fluid past aninfinite plate Singh et al [1] have analyzed the heat and masstransfer in MHD flow of a viscous fluids past a vertical plateunder oscillatory suction velocity Sharma and Singh [2] havereported the unsteady MHD-free convective flow and heattransfer along a vertical porous plate with variable suctionand internal heat generationThe problem of slip flow regimeis very important in this era of modern science technologyand vast ranging industrialization In many practical applica-tions the particle adjacent to a solid surface no longer takesthe velocity of the surface The particle at the surface has a

finite tangential velocity it slips along the surface The flowregime is called the slip flow regime and its effect cannotbe neglected The fluid slippage phenomenon at the solidboundaries appear in many applications such as microchan-nels or nanochannels and in application where a thin filmof light oils is attached to the moving plates or when thesurface is coated with special coating such as thickmonolayerof hydrophobic octadecyltrichlosilane that is lubrication ofmechanical device where a thin filmof lubricant is attached tothe surface slipping over one another or when the surfaces arecoated with special coating to minimize the friction betweenthem Singh and Gupta [3] have discussed the MHD-freeconvective flow of a viscous fluid through a porous mediumbounded by an oscillating porous plate in slip flow regimewith mass transfer Khandelwal and Jain [4] have analyzedthe unsteady MHD flow of a stratified fluid through porousmedium over a moving plate in slip flow regime Das et al[5] have studied the magnetohydrodynamic unsteady flowof a viscous stratified fluid through a porous medium pasta porous flat moving plate in the slip flow regime with

2 International Journal of Chemical Engineering

heat source The study of heat and mass transfer problemswith chemical reaction is of great practical importance toengineers and scientists because of their almost universaloccurrence in many branches of science and engineeringA few representative fields of interest in which combinedheat and mass transfer along with chemical reaction play animportant role are chemical process industries such as foodprocessing and polymer productionMahapatra et al [6] havestudied effects of chemical reaction on free convection flowthrough a porous medium bounded by a vertical surfaceMuthucumaraswamy [7] has studied effects of chemicalreaction on amoving isothermal vertical surfacewith suctionAl-Odat and Al-Azab [8] have studied influence of chemicalreaction on a transient MHD-free convection flow over amoving vertical plate

Viscoelastic fluid flow through porous media hasattracted the attention of scientists and engineers because ofits importance notably in the flow of the oil through porousrocks the extraction of energy from geothermal region andthe filtration of solids from liquids and drug permeationthrough human skin The flow through porous mediaalso occurs in the ground water hydrology irrigation anddrainage problems absorption and filtration processes in theground water hydrology irrigation and drainage problemsabsorption and filtration processes in chemical engineeringand soil erosion and tile drainage Chaudhary and Jain[9] have investigated the effects of the Hall current andradiation on MHD mixed convection flow of a viscoelasticfluid past and infinite vertical plate Sahoo et al [10] havediscussed the unsteady two-dimensional MHD flow andheat transfer of an elastic-viscous liquid past an infinite hotvertical porous surface bounded by porous medium withsourcesink Kumar and Chand [11] have studied the effect ofslip conditions and the Hall current on unsteady MHD flowof a viscoelastic fluid past an infinite vertical porous platethrough porous medium

The objective of the present study is to consider theviscoelastic fluid past a vertical plate in the slip flow regimepacked with uniform porous matrix in the presence of atransverse magnetic field In this paper we consider theproblem of Singh and Gupta [3] in viscoelastic fluid ofWalterrsquos1198611015840model taking into account the effect of heat sourceparameter and chemical reaction parameter

2 Formulation and Solution of the Problem

The physical configuration consists of an unsteady flow ofan electrically conducting and incompressible viscoelasticliquid of Walterrsquos 1198611015840 model with simultaneous heat and masstransfer near an oscillating infinite porous plate in slip flowregime with heat source and chemical reaction under theinfluence of a transverse magnetic field of uniform strengthThe 119910-axis is taken along the plate in vertical direction and119909-axis is perpendicular to it A uniform magnetic field ofstrength119861

0is applied in the direction of 119910-axis Let 119906 and V be

the velocity components in x- and y-directions respectivelyAs the plate is of infinite length all the variables in theproblem are functions of 119910 and 119905 Initially the plate and

fluid are at rest and then the plate is set to an oscillatorymotion The Reynolds number is assumed to be very smalland the induced magnetic field due to the flow is neglectedwith respect to the applied magnetic field The pressure 119901 inthe fluid is assumed to be constant If V

0represents the suction

or injection velocity at the plate the equation of continuity is

120597V

120597119910= 0 (1)

Under the condition 119910 = 0 V = minusV0everywhere We have

y

x998400

B0

x

Flow Diagram

Now the governing boundary layer equations of the flow fieldare

120597119906

120597119905minus V0

120597119906

120597119910= 120592

1205972119906

1205971199102minus 120590

1198612

0119906

120588minus120592119906

119870+ 119892120573 (119879 minus 119879

infin)

+ 119892120573 (119862 minus 119862infin) minus

1198700

120588(1205973119906

120597119905120597119910+ V0

1205973119906

1205971199103)

120597119879

120597119905minus V0

120597119879

120597119910= 120572

1205972119879

1205971199102minus 119878 (119879 minus 119879

infin)

120597119862

120597119905minus V0

120597119862

120597119910= 119863

1205972119862

1205971199102minus 119870119897(119862 minus 119862

infin)

(2)

The first order velocity slip boundary conditions of theproblemwhen the plate executes linear harmonic oscillationsin its own plane are given by

119910 = 0 119906 = 1198800119890int+ 1198711

120597119906

120597119910 119879 = 119879

infin 119862 = 119862

infin

119910 997888rarr infin 119906 997888rarr 0 119879 997888rarr 119879infin 119862 997888rarr 119862

infin

(3)

where 1198711= (2minus119898

1)(119871119898

1) 119871 = 120583(1205872119901120588)

12 is themean freepath and119898

1is Maxwellrsquos reflection coefficient

International Journal of Chemical Engineering 3

On introducing the following nondimensional quantities

119910lowast=1198800

120592119910 119906

lowast=

119906

1198800

119905lowast=1198802

0

120592119905

120579lowast=

119879 minus 119879infin

119879119908minus 119879infin

120601lowast=


119862119908minus 119862infin

V0

lowast=

V0

1198800

119899lowast=

120592

1198802

0

119899

119870119901=1198701198802

0

120592 119877 =

11987111198800

120592 119878

lowast=120592119878

1198802

0

119870119888=120592119870119897

1198802

0

119872 =1205901198612

0120592

1205881198802

0

119866119903=119892120573120592 (119879

119908minus 119879infin)

1198803

0

119866119898=119892120573120592 (119862


1198803

0

119877119888=1198802

01198700

1205881205922

Pr = 120592

120572 119878

119888=

120592

119863

(4)

in (2) and dropping the asterisks we have

120597119906

120597119905minus V0

120597119906

120597119910=1205972119906

1205971199102minus 119877119888(

1205973119906

1205971199051205971199103+ V0

1205973119906

1205971199103)

minus (1198722+

1

119870119901

)119906 + 119866119903120579 + 119866119898120601

(5)

Pr(120597120579120597119905

minus V0

120597120579

120597119910) =

1205972120579

1205971199102minus Pr119878120579 (6)

119878119888(120597120601

120597119905minus V0

120597120601

120597119910) =

1205972120601

1205971199102minus 119870119888119878119888120601 (7)

with boundary conditions

119910 = 0 119906 = 119890int+ 119877

120597119906

120597119910 120579 = 1 120601 = 1

119910 997888rarr infin 119906 997888rarr 0 120579 997888rarr 0 120601 997888rarr 0

(8)

Equation (5) is of third order and two boundary condi-tions are available Due to inadequate boundary conditiona perturbation method has been applied with 119877

119888lt 1 as the

perturbation parameter This assumption is quite consistentas the model under consideration is valid only for slightlyelastic fluid

Consider the following

119906 = 1199060+ 1198771198881199061+ 119874(119877

119888)2

120579 = 1205790+ 1198771198881205791+ 119874(119877

119888)2

120601 = 1206010+ 1198771198881206011+ 119874(119877

119888)2

(9)

Substituting (9) in (5)ndash(7) and equating like powers of 119877119888 we

get the following

Zeroth order equations

1205971199060

120597119905minus V0

1205971199060

120597119910=12059721199060

1205971199102minus (119872

2+

1

119870119901

)1199060+ 1198661199031205790+ 1198661198981206010

Pr(1205971205790

120597119905minus V0

1205971205790

120597119910) =

12059721205790

1205971199102minus Pr119878120579

0

119878119888(1205971206010

120597119905minus V0

1205971206010

120597119910) =

12059721206010

1205971199102minus 1198701198881198781198881206010

(10)

first order equations

1205971199061

120597119905minus V0

1205971199061

120597119910=12059721199061

1205971199102minus (119872

2+

1

119870119901

)1199061

+ 1198661199031205791+ 1198661198981206011minus V0

12059731199061

1205971199103minus

12059731199060

1205971199051205971199102

Pr(1205971205791120597119905

minus V0

1205971205791

120597119910) =

12059721205791

1205971199102minus Pr119878120579

1

119878119888(1205971206011

120597119905minus V0

1205971206011

120597119910) =

12059721206011

1205971199102minus 1198701198881198781198881206011

(11)

The corresponding boundary conditions are

119910 = 0 1199060= 119890

int+ 119877

1205971199060

120597119910 119906

1= 0 120579

0= 1

1205791= 1 120601

0= 1 120601

1= 1

119910 997888rarr infin 1199060997888rarr 0 119906

1997888rarr 0 120579

0997888rarr 0

1205791997888rarr 0 120601

0997888rarr 0 120601

1997888rarr 0

(12)

In order to reduce the system of partial differential equations(10)ndash(11) to a system of ordinary differential equations wefurther introduce

1199060(119910 119905) = 119906

00(119910) + 119906

01(119910) 119890

int

1199061(119910 119905) = 119906

10(119910) + 119906

11(119910) 119890

int

1205790(119910 119905) = 120579

00(119910) + 120579

01(119910) 119890

int

1205791(119910 119905) = 120579

10(119910) + 120579

11(119910) 119890

int

1206010(119910 119905) = 120601

00(119910) + 120601

01(119910) 119890

int

1206011(119910 119905) = 120601

10(119910) + 120601

11(119910) 119890

int

(13)


Substituting (13) into (10)ndash(11) and equating the harmonicand nonharmonic terms we obtain

11990610158401015840

00+ V01199061015840

00minus (119872

2+

1

119870119901

)11990600= minus11986611990312057900minus 11986611989812060100

11990610158401015840

01+ V01199061015840

01minus (119872

2+

1

119870119901

+ 119894119899) 11990601= minus11986611990312057901minus 11986611989812060101

11990610158401015840

10+ V01199061015840

10minus (119872

2+

1

119870119901

)11990610= minus11986611990312057910minus 11986611989812060110+ V0119906101584010158401015840

00

11990610158401015840

11+ V01199061015840

11minus (119872

2+

1

119870119901

+ 119894119899) 11990611

= minus11986611990312057911minus 11986611989812060111+ V0119906101584010158401015840

01+ 11989411989911990610158401015840

01

12057910158401015840

00+ V0Pr120579101584000minus Pr119878120579

00= 0

12057910158401015840

01+ V0Pr120579101584001minus (119894119899Pr + Pr119878) 120579

01= 0

12057910158401015840

10+ V0Pr120579101584010minus Pr119878120579

10= 0

12057910158401015840

11+ V0Pr120579101584011minus (119894119899Pr + Pr119878) 120579

11= 0

12060110158401015840

00+ V01198781198881206011015840

00minus 11987011988811987811988812060100= 0

12060110158401015840

01+ V01198781198881206011015840

01minus (119894119899119878

119888+ 119870119888119878119888) 12060101= 0

12060110158401015840

10+ V01198781198881206011015840

10minus 11987011988811987811988812060110= 0

12060110158401015840

11+ V01198781198881206011015840

11minus (119894119899119878

119888+ 119870119888119878119888) 12060111= 0

(14)


119910 = 0 11990600= 119877

12059711990600

120597119910 119906

01= 1 + 119877

12059711990601

120597119910

11990610= 0 119906

11= 0 120579

00= 1

12057901= 0 120579

10= 1 120579

11= 0

12060100= 1 120601

01= 0 120601

10= 0 120601

11= 0

119910 997888rarr infin 11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0

11990611997888rarr 0 120579

00997888rarr 0 120579

01997888rarr 0

12057910997888rarr 0 120579

11997888rarr 0 120601

00997888rarr 0

12060101997888rarr 0 120601

10997888rarr 0 120601

11997888rarr 0

(15)

The solutions of (14) applying boundary conditions (15) are

11990600= 1198601119890minus1198861119910+ 1198602119890minus1198862119910+ 1198603119890minus1198863119910

11990601= 1198604119890minus1198864119910


11990611= 1198608119890minus1198864119910

12057900= 119890minus1198861119910

12057901= 12057910= 12057911= 0

12060100= 119890minus1198862119910

12060101= 12060110= 12060111= 0

(16)

Hence the velocity temperature and concentration of theflow field are

119906 = 1198601119890minus1198861119910+ 1198602119890minus1198862119910+ 1198603119890minus1198863119910

+ 1198604119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)

+ 1198771198881198605119890minus1198861119910+ 1198606119890minus1198862119910+ 1198607119890minus1198863119910

+1198608119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)

120579 = 119890minus1198861119910

120601 = 119890minus1198862119910

(17)

The skin friction at the plate is given by

119862119891=

120591119909119910

1198802

0

=120597119906

120597119910minus 119877119888(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102)

100381610038161003816100381610038161003816100381610038161003816119910=0

(18)

where

120591119909119910=120597119906

120597119910minus1198700

120588(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102) (19)

The rate of heat transfer that is the heat flux at the plate interms of the Nusselt number is given by

119873119906= minus

120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198861 (20)

The rate ofmass transfer at the plate in terms of the Sherwoodnumber is given by

119878ℎ= minus

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198862 (21)

3 Results and Discussion

The problem of unsteady flow of an electrically conductingand incompressible viscoelastic liquid of Walterrsquos 1198611015840 modelwith heat andmass transfer near an oscillating infinite porous


plate in slip flow regime with heat source and chemicalreaction parameter under the influence of a transversemagnetic field of uniform strength has been considered Theeffects of the flow parameters such as the Prandtl numberPr porosity parameter 119870

119901 magnetic parameter M elastic

parameter 119877119888 heat source parameter S chemical reaction

parameter119870119888 thermal Grashof number119866

119903 the mass Grashof

number 119866119898 the Scmidt number 119878

119888 suction parameter V

0

and rarefaction parameter 119877 on the velocity field have beenstudied analytically and presented with the help of Figures1ndash4 The effects of the flow parameters on the temperaturefield and concentration distribution have been presented inFigures 5 and 6 respectively Further the effects of the flowparameters on the skin friction heat flux and rate of masstransfer have been discussed with the help of Tables 1ndash3 Fornumerical computation the values of 119866

119903are taken positive

This indicates that the study has been carried out under theinfluence of the cooling of the plate Also we have takennt = 1205872 The interesting aspect of the problem is to studythe combined effect of the flow parameters with that of thefirst order velocity slip boundary condition when the plateexecutes linear harmonic oscillation in its own plane

Figure 1 shows the effect of the Prandtl number (Pr)permeability parameter (119870

119901) andmagnetic parameter (M) on

velocity profile For this figure we have taken that S = 1 119866119903

= 5 119866119898= 5 119878

119888= 024 119877

119888= 05 119870

119888= 2 R = 02 and V

0=

2 It is observed that the increase in the Prandtl number aswell as permeability parameter decreases the velocity of theflow field whereas increase in magnetic parameter increasesit Since Prandtl number is the ratio of kinematic viscosity tothermal diffusivity so as Pr increases the kinematic viscosityof the fluid dominates the thermal diffusivity of the fluidwhich leads to decreasing the velocity of the flow field Theapplication of transverse magnetic field sets up the Lorentzforce which enhances the fluid velocity

Figure 2 shows the effect of elastic parameter (119877119888) heat

source parameter (S) and chemical reaction (119870119888) parameter

on velocity frofile For this figure we have taken that Pr =071 119870

119901= 1M = 2 119866

119903= 5 119866

119898 = 5 119878

119888= 024 R = 02 and V

0

= 2 It is observed that the velocity of the flow field decreasesdue to the presence of elastic parameter chemical reactionparameter and heat source parameter For 119877

119888= S = 119870

119888= 0

the present work agrees with the work of Singh and Gupta[3]

Figure 3 depicts the effect of the thermal Grashof number(119866119903) the mass Grashof number (119866

119898) and the Schmidt

number (119878119888) on velocity frofile For this figure we have taken

that Pr = 071 S = 1 119870119901= 1 M = 2 119877

119888= 05 119870

119888= 2 R = 02

and V0= 2 It is observed that for the heavier species that is

with increasing 119878119888 the velocity decreases The velocity of the

flowfield decreases due to the increase in the thermalGrashofnumberMoreover buoyancy effect (119866

119898) due tomass transfer

enhances the velocityFigure 4 depicts the effect of suction parameter (V

0) and

rarefaction parameter (R) on velocity frofile For this figurewe have taken that Pr = 071 S = 1 119870

119901= 1 M = 2 119877

119888= 05

119870119888= 2 119866

119903= 5 119866

119898= 5 and 119878

119888= 024 It is observed that the

velocity of the flow field decreases due the presence of suction

0 05 1 15 2 25 3 35 4 45 5

1

0

minus1

minus2

minus3

minus4

minus5

u

y

Pr = 031Kp = 02M = 05

Pr = 071Kp = 02M = 1

Pr = 071Kp = 02M = 05

Pr = 071Kp = 04M = 05

Figure 1 Effect of Pr119870119901and119872 on velocity profile

0 05 1 15 2 25 3 35 4 45 5

05

0

minus05

minus1

minus15

minus2

u

y

Rc = 02 S = 05Kc = 1

Rc = 02 S = 05Kc = 2

Rc = 02 S = 1Kc = 1

Rc = 05 S = 05Kc = 1

Figure 2 Effect of 119877119888 S and 119870

119888on velocity profile

12

1

08

06

04

02

0

minus020 05 1 15 2 25 3 35 4 45 5

u

y

Gr = 25 Gm = 2 2

Gr = 2 Gm = 2 25

Sc =

Sc =

Gr = 2 Gm = 2 2Sc =

Gr = 2 Gm = 25 2Sc =

Figure 3 Effect of 119866119903 119866119898 and 119878



06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

0 = 15 R = 07

0 = 15 R = 05

0 = 2 R = 05

u

0 05 1 15 2 25 3 35 4 45 5y

Figure 4 Effect of V0and 119877 on velocity profile

parameter but the reverse effect is observed due the presenceof the rarefaction parameter

Figure 5 shows the effect of the Prandtl number heatsource parameter and suction parameter on the temperatureof the flow field It is observed that the temperature ofthe flow field diminishes as the Prandtl number increasesThis is consistent with the fact that the thermal boundarylayer thickness decreases with increasing Prandtl numberPresence of heat source reduces the temperature of the flowfield This may happen due the elastic property of the fluidIt is observed that temperature of the flow field diminishes asthe suction parameter increases

Figure 6 depicts the effect of the Schmidt number chemi-cal reaction parameter and suction parameter on concentra-tion distributionThe concentration distribution decreases atall points of the flow field with the increase in the Schmidtnumber This shows that the heavier diffusing species havea greater retarding effect on the concentration distributionof the flow field It is observed that a destructive reaction(119870119888gt 0) reduces the concentration distribution whereas

a generative reaction (119870119888

= 0) enhances it Also it isobserved that presence of suction parameter diminishes theconcentration distribution

The skin friction is an important phenomenon whichcharacterizes the frictional drag at the solid surface FromTable 1 it is observed that the skin friction decreases with theincrease in all the forcing forces but it is interesting to notethat the skin friction increases with the increase in magneticparameter

From Table 2 it is to note that all the entries are positiveIt is seen that the Prandtl number (Pr) heat source (S) andsuction parameter (V

0) increase the rate of heat transfer at the

surface of the plateFrom Table 3 it is to note that all the entries are positive

It is observed that Schmidt number (119878119888) chemical reaction

parameter (119870119888) and suction parameter (V

0) increase the rate

of mass transfer at the surface of the plate

1

09

08

07

06

05

04

03

02

01

0

Pr = 05 S = 05 0 = 02

Pr = 071 S = 05 0 = 02

Pr = 071 S = 05 0 = 05

Pr = 071 S = 1 0 = 02

0 05 1 15 2 25 3 35 4 45 5y

Pr = 071 S = 0 0 = 02

120579

Figure 5 Effect of Pr 119878 and V0on temperature profile

1

09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

y

078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =

078Kc = 05 0 = 02Sc =

078Kc = 05 0 = 05Sc =

078Kc = 1 0 = 02Sc =120601

Figure 6 Effect of 119878119888 119870119888 and V

0on concentration profile

4 Conclusion

A theoretical study of unsteady MHD incompressible vis-coelastic liquid of Walterrsquos 1198611015840 model with heat and masstransfer near an oscillating infinite porous plate in slip flowregime under the influence of a transverse magnetic fieldof uniform strength is considered Some of the importantfindings of the problem are given in the following

(i) Presence of the Prandtl number decreases the velocityof the flow field whereas presence magnetic fieldincreases it

(ii) The velocity of the flow field decreases suddenly nearthe plate due to the presence of elastic parameter

(iii) The velocity of the flow field decreases due to theincrease in the thermal Grashof number


Table 1 Skin friction (119862119891)

Pr 119870119901

119872 119877119888

119878 119870119888

119866119903

119866119898

119878119888

119877 V0

119862119891

030 1 2 05 1 2 5 5 024 02 2 183921050 1 2 05 1 2 5 5 024 02 2 108793050 15 2 05 1 2 5 5 024 02 2 71877050 1 25 05 1 2 5 5 024 02 2 304390050 1 2 02 1 2 5 5 024 02 2 130904050 1 2 05 05 2 5 5 024 02 2 179390050 1 2 05 1 15 5 5 024 02 2 116350050 1 2 05 1 2 45 5 024 02 2 131659050 1 2 05 1 2 5 45 024 02 2 118927050 1 2 05 1 2 5 5 030 02 2 91301050 1 2 05 1 2 5 5 024 05 2 275040050 1 2 05 1 2 5 5 024 02 15 140484

Table 2 The Nusselt number (119873119906)

Pr 119878 V0

119873119906

050 05 02 075887071 05 02 078166071 10 02 107351071 05 05 090654

Table 3 The Sherwood number (119878ℎ)

119878119888

119870119888

V0

119878ℎ

030 05 02 041845078 05 02 070735078 10 02 096461078 05 05 084923

(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number

(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution

Appendix


1198861=1

2[V0Pr + radic(V

0Pr)2 + 4119878Pr]

1198862=1

2[V0119878119888+ radic(V

0119878119888)2

+ 4119870119888119878119888]

1198863=1

2[V20+ radicV20+ 4119876] 119876 = 119872

2+

1

119870119901

1198864=1

2[V20+ radicV20+ 4 (119876 + 119894119899)]

1198601=

minus119866119903

1198862

1minus 1198861V0minus 119876

1198602=

minus119866119903

1198862

2minus 1198862V0minus 119876

1198603=

minus1

1 + 1198863119877[(1198861119877 + 1)119860

1+ (1198862119877 + 1)119860

2]

1198604=

1

1 + 1198771198864

1198605=

1198863

11198601V0

1198862

1minus 1198861V0minus 119876

1198606=

1198863

21198602V0

1198862

2minus 1198862V0minus 119876

1198607= 1198605+ 1198606minusV01198863

31198603119910

V0minus 21198863

1198608=

(minusV01198863

41198604+ 1198941198991198862

41198604) 119910

V0minus 21198864

(A1)

Nomenclature

119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity

119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number

119866119898 The mass Grashof number

119878119888 The Schmidt number

119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter

119870119888 Nondimensional chemical reaction parameter

119879119908 Temperature at the wall

119879infin Temperature far away from the wall

119862119908 Concentration at the wall

119862infin Concentration far away from the wall

120590 Electric conductivity1198610 Uniform magnetic field

V0 Suctioninjection velocity

119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter

1198700 Dimensional elastic parameter

119877119888 Nondimensional elastic parameter

119899 Frequency of oscillation


Acknowledgment

The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work

References

[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003

[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008

[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005

[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005

[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011

[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010

[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002

[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007

[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006

[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011

[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



heat source The study of heat and mass transfer problemswith chemical reaction is of great practical importance toengineers and scientists because of their almost universaloccurrence in many branches of science and engineeringA few representative fields of interest in which combinedheat and mass transfer along with chemical reaction play animportant role are chemical process industries such as foodprocessing and polymer productionMahapatra et al [6] havestudied effects of chemical reaction on free convection flowthrough a porous medium bounded by a vertical surfaceMuthucumaraswamy [7] has studied effects of chemicalreaction on amoving isothermal vertical surfacewith suctionAl-Odat and Al-Azab [8] have studied influence of chemicalreaction on a transient MHD-free convection flow over amoving vertical plate

Viscoelastic fluid flow through porous media hasattracted the attention of scientists and engineers because ofits importance notably in the flow of the oil through porousrocks the extraction of energy from geothermal region andthe filtration of solids from liquids and drug permeationthrough human skin The flow through porous mediaalso occurs in the ground water hydrology irrigation anddrainage problems absorption and filtration processes in theground water hydrology irrigation and drainage problemsabsorption and filtration processes in chemical engineeringand soil erosion and tile drainage Chaudhary and Jain[9] have investigated the effects of the Hall current andradiation on MHD mixed convection flow of a viscoelasticfluid past and infinite vertical plate Sahoo et al [10] havediscussed the unsteady two-dimensional MHD flow andheat transfer of an elastic-viscous liquid past an infinite hotvertical porous surface bounded by porous medium withsourcesink Kumar and Chand [11] have studied the effect ofslip conditions and the Hall current on unsteady MHD flowof a viscoelastic fluid past an infinite vertical porous platethrough porous medium

The objective of the present study is to consider theviscoelastic fluid past a vertical plate in the slip flow regimepacked with uniform porous matrix in the presence of atransverse magnetic field In this paper we consider theproblem of Singh and Gupta [3] in viscoelastic fluid ofWalterrsquos1198611015840model taking into account the effect of heat sourceparameter and chemical reaction parameter

2 Formulation and Solution of the Problem

The physical configuration consists of an unsteady flow ofan electrically conducting and incompressible viscoelasticliquid of Walterrsquos 1198611015840 model with simultaneous heat and masstransfer near an oscillating infinite porous plate in slip flowregime with heat source and chemical reaction under theinfluence of a transverse magnetic field of uniform strengthThe 119910-axis is taken along the plate in vertical direction and119909-axis is perpendicular to it A uniform magnetic field ofstrength119861

0is applied in the direction of 119910-axis Let 119906 and V be

the velocity components in x- and y-directions respectivelyAs the plate is of infinite length all the variables in theproblem are functions of 119910 and 119905 Initially the plate and

fluid are at rest and then the plate is set to an oscillatorymotion The Reynolds number is assumed to be very smalland the induced magnetic field due to the flow is neglectedwith respect to the applied magnetic field The pressure 119901 inthe fluid is assumed to be constant If V

0represents the suction

or injection velocity at the plate the equation of continuity is

120597V

120597119910= 0 (1)

Under the condition 119910 = 0 V = minusV0everywhere We have

y

x998400

B0

x

Flow Diagram

Now the governing boundary layer equations of the flow fieldare

120597119906

120597119905minus V0

120597119906

120597119910= 120592

1205972119906

1205971199102minus 120590

1198612

0119906

120588minus120592119906

119870+ 119892120573 (119879 minus 119879

infin)

+ 119892120573 (119862 minus 119862infin) minus

1198700

120588(1205973119906

120597119905120597119910+ V0

1205973119906

1205971199103)

120597119879

120597119905minus V0

120597119879

120597119910= 120572

1205972119879

1205971199102minus 119878 (119879 minus 119879

infin)

120597119862

120597119905minus V0

120597119862

120597119910= 119863

1205972119862

1205971199102minus 119870119897(119862 minus 119862

infin)

(2)

The first order velocity slip boundary conditions of theproblemwhen the plate executes linear harmonic oscillationsin its own plane are given by

119910 = 0 119906 = 1198800119890int+ 1198711

120597119906

120597119910 119879 = 119879

infin 119862 = 119862

infin

119910 997888rarr infin 119906 997888rarr 0 119879 997888rarr 119879infin 119862 997888rarr 119862

infin

(3)

where 1198711= (2minus119898

1)(119871119898

1) 119871 = 120583(1205872119901120588)

12 is themean freepath and119898

1is Maxwellrsquos reflection coefficient



119910lowast=1198800

120592119910 119906

lowast=

119906

1198800

119905lowast=1198802

0

120592119905

120579lowast=


119879119908minus 119879infin

120601lowast=


119862119908minus 119862infin

V0

lowast=

V0

1198800

119899lowast=

120592

1198802

0

119899

119870119901=1198701198802

0

120592 119877 =

11987111198800

120592 119878

lowast=120592119878

1198802

0

119870119888=120592119870119897

1198802

0

119872 =1205901198612

0120592

1205881198802

0

119866119903=119892120573120592 (119879


1198803

0

119866119898=119892120573120592 (119862


1198803

0

119877119888=1198802

01198700

1205881205922

Pr = 120592

120572 119878

119888=

120592

119863

(4)


120597119906

120597119905minus V0

120597119906

120597119910=1205972119906

1205971199102minus 119877119888(

1205973119906

1205971199051205971199103+ V0

1205973119906

1205971199103)

minus (1198722+

1

119870119901

)119906 + 119866119903120579 + 119866119898120601

(5)

Pr(120597120579120597119905

minus V0

120597120579

120597119910) =

1205972120579

1205971199102minus Pr119878120579 (6)

119878119888(120597120601

120597119905minus V0

120597120601

120597119910) =

1205972120601

1205971199102minus 119870119888119878119888120601 (7)


119910 = 0 119906 = 119890int+ 119877

120597119906

120597119910 120579 = 1 120601 = 1


(8)


119888lt 1 as the



119906 = 1199060+ 1198771198881199061+ 119874(119877

119888)2

120579 = 1205790+ 1198771198881205791+ 119874(119877

119888)2

120601 = 1206010+ 1198771198881206011+ 119874(119877

119888)2

(9)


get the following


1205971199060

120597119905minus V0

1205971199060

120597119910=12059721199060

1205971199102minus (119872

2+

1

119870119901

)1199060+ 1198661199031205790+ 1198661198981206010

Pr(1205971205790

120597119905minus V0

1205971205790

120597119910) =

12059721205790

1205971199102minus Pr119878120579

0

119878119888(1205971206010

120597119905minus V0

1205971206010

120597119910) =

12059721206010

1205971199102minus 1198701198881198781198881206010

(10)


1205971199061

120597119905minus V0

1205971199061

120597119910=12059721199061

1205971199102minus (119872

2+

1

119870119901

)1199061

+ 1198661199031205791+ 1198661198981206011minus V0

12059731199061

1205971199103minus

12059731199060

1205971199051205971199102

Pr(1205971205791120597119905

minus V0

1205971205791

120597119910) =

12059721205791

1205971199102minus Pr119878120579

1

119878119888(1205971206011

120597119905minus V0

1205971206011

120597119910) =

12059721206011

1205971199102minus 1198701198881198781198881206011

(11)


119910 = 0 1199060= 119890

int+ 119877

1205971199060

120597119910 119906

1= 0 120579

0= 1

1205791= 1 120601

0= 1 120601

1= 1

119910 997888rarr infin 1199060997888rarr 0 119906

1997888rarr 0 120579

0997888rarr 0

1205791997888rarr 0 120601

0997888rarr 0 120601

1997888rarr 0

(12)


1199060(119910 119905) = 119906

00(119910) + 119906

01(119910) 119890

int

1199061(119910 119905) = 119906

10(119910) + 119906

11(119910) 119890

int

1205790(119910 119905) = 120579

00(119910) + 120579

01(119910) 119890

int

1205791(119910 119905) = 120579

10(119910) + 120579

11(119910) 119890

int

1206010(119910 119905) = 120601

00(119910) + 120601

01(119910) 119890

int

1206011(119910 119905) = 120601

10(119910) + 120601

11(119910) 119890

int

(13)



11990610158401015840

00+ V01199061015840

00minus (119872

2+

1

119870119901

)11990600= minus11986611990312057900minus 11986611989812060100

11990610158401015840

01+ V01199061015840

01minus (119872

2+

1

119870119901

+ 119894119899) 11990601= minus11986611990312057901minus 11986611989812060101

11990610158401015840

10+ V01199061015840

10minus (119872

2+

1

119870119901

)11990610= minus11986611990312057910minus 11986611989812060110+ V0119906101584010158401015840

00

11990610158401015840

11+ V01199061015840

11minus (119872

2+

1

119870119901

+ 119894119899) 11990611

= minus11986611990312057911minus 11986611989812060111+ V0119906101584010158401015840

01+ 11989411989911990610158401015840

01

12057910158401015840

00+ V0Pr120579101584000minus Pr119878120579

00= 0

12057910158401015840

01+ V0Pr120579101584001minus (119894119899Pr + Pr119878) 120579

01= 0

12057910158401015840

10+ V0Pr120579101584010minus Pr119878120579

10= 0

12057910158401015840

11+ V0Pr120579101584011minus (119894119899Pr + Pr119878) 120579

11= 0

12060110158401015840

00+ V01198781198881206011015840

00minus 11987011988811987811988812060100= 0

12060110158401015840

01+ V01198781198881206011015840

01minus (119894119899119878

119888+ 119870119888119878119888) 12060101= 0

12060110158401015840

10+ V01198781198881206011015840

10minus 11987011988811987811988812060110= 0

12060110158401015840

11+ V01198781198881206011015840

11minus (119894119899119878

119888+ 119870119888119878119888) 12060111= 0

(14)


119910 = 0 11990600= 119877

12059711990600

120597119910 119906

01= 1 + 119877

12059711990601

120597119910

11990610= 0 119906

11= 0 120579

00= 1

12057901= 0 120579

10= 1 120579

11= 0

12060100= 1 120601

01= 0 120601

10= 0 120601

11= 0

119910 997888rarr infin 11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0

11990611997888rarr 0 120579

00997888rarr 0 120579

01997888rarr 0

12057910997888rarr 0 120579

11997888rarr 0 120601

00997888rarr 0

12060101997888rarr 0 120601

10997888rarr 0 120601

11997888rarr 0

(15)



11990601= 1198604119890minus1198864119910


11990611= 1198608119890minus1198864119910

12057900= 119890minus1198861119910

12057901= 12057910= 12057911= 0

12060100= 119890minus1198862119910

12060101= 12060110= 12060111= 0

(16)



+ 1198604119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)


+1198608119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)

120579 = 119890minus1198861119910

120601 = 119890minus1198862119910

(17)


119862119891=

120591119909119910

1198802

0

=120597119906

120597119910minus 119877119888(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102)

100381610038161003816100381610038161003816100381610038161003816119910=0

(18)

where

120591119909119910=120597119906

120597119910minus1198700

120588(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102) (19)


119873119906= minus

120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198861 (20)


119878ℎ= minus

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198862 (21)











0







= 5 119866119898= 5 119878

119888= 024 119877

119888= 05 119870

119888= 2 R = 02 and V

0=





119901= 1M = 2 119866

119903= 5 119866

119898 = 5 119878

119888= 024 R = 02 and V

0


119888= S = 119870

119888= 0





that Pr = 071 S = 1 119870119901= 1 M = 2 119877

119888= 05 119870

119888= 2 R = 02






0) and


119901= 1 M = 2 119877

119888= 05

119870119888= 2 119866

119903= 5 119866

119898= 5 and 119878



0 05 1 15 2 25 3 35 4 45 5

1

0

minus1

minus2

minus3

minus4

minus5

u

y

Pr = 031Kp = 02M = 05

Pr = 071Kp = 02M = 1

Pr = 071Kp = 02M = 05

Pr = 071Kp = 04M = 05


0 05 1 15 2 25 3 35 4 45 5

05

0

minus05

minus1

minus15

minus2

u

y

Rc = 02 S = 05Kc = 1

Rc = 02 S = 05Kc = 2

Rc = 02 S = 1Kc = 1

Rc = 05 S = 05Kc = 1



12

1

08

06

04

02

0

minus020 05 1 15 2 25 3 35 4 45 5

u

y

Gr = 25 Gm = 2 2

Gr = 2 Gm = 2 25

Sc =

Sc =

Gr = 2 Gm = 2 2Sc =

Gr = 2 Gm = 25 2Sc =




06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

0 = 15 R = 07

0 = 15 R = 05

0 = 2 R = 05

u

0 05 1 15 2 25 3 35 4 45 5y















1

09

08

07

06

05

04

03

02

01

0

Pr = 05 S = 05 0 = 02

Pr = 071 S = 05 0 = 02

Pr = 071 S = 05 0 = 05

Pr = 071 S = 1 0 = 02

0 05 1 15 2 25 3 35 4 45 5y

Pr = 071 S = 0 0 = 02

120579


1

09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

y

078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =

078Kc = 05 0 = 02Sc =

078Kc = 05 0 = 05Sc =

078Kc = 1 0 = 02Sc =120601



4 Conclusion







Pr 119870119901

119872 119877119888

119878 119870119888

119866119903

119866119898

119878119888

119877 V0

119862119891

030 1 2 05 1 2 5 5 024 02 2 183921050 1 2 05 1 2 5 5 024 02 2 108793050 15 2 05 1 2 5 5 024 02 2 71877050 1 25 05 1 2 5 5 024 02 2 304390050 1 2 02 1 2 5 5 024 02 2 130904050 1 2 05 05 2 5 5 024 02 2 179390050 1 2 05 1 15 5 5 024 02 2 116350050 1 2 05 1 2 45 5 024 02 2 131659050 1 2 05 1 2 5 45 024 02 2 118927050 1 2 05 1 2 5 5 030 02 2 91301050 1 2 05 1 2 5 5 024 05 2 275040050 1 2 05 1 2 5 5 024 02 15 140484


Pr 119878 V0

119873119906

050 05 02 075887071 05 02 078166071 10 02 107351071 05 05 090654


119878119888

119870119888

V0

119878ℎ

030 05 02 041845078 05 02 070735078 10 02 096461078 05 05 084923



Appendix


1198861=1

2[V0Pr + radic(V

0Pr)2 + 4119878Pr]

1198862=1

2[V0119878119888+ radic(V

0119878119888)2

+ 4119870119888119878119888]

1198863=1

2[V20+ radicV20+ 4119876] 119876 = 119872

2+

1

119870119901

1198864=1

2[V20+ radicV20+ 4 (119876 + 119894119899)]

1198601=

minus119866119903

1198862

1minus 1198861V0minus 119876

1198602=

minus119866119903

1198862

2minus 1198862V0minus 119876

1198603=

minus1

1 + 1198863119877[(1198861119877 + 1)119860

1+ (1198862119877 + 1)119860

2]

1198604=

1

1 + 1198771198864

1198605=

1198863

11198601V0

1198862

1minus 1198861V0minus 119876

1198606=

1198863

21198602V0

1198862

2minus 1198862V0minus 119876

1198607= 1198605+ 1198606minusV01198863

31198603119910

V0minus 21198863

1198608=

(minusV01198863

41198604+ 1198941198991198862

41198604) 119910

V0minus 21198864

(A1)

Nomenclature


















Acknowledgment


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119910lowast=1198800

120592119910 119906

lowast=

119906

1198800

119905lowast=1198802

0

120592119905

120579lowast=


119879119908minus 119879infin

120601lowast=


119862119908minus 119862infin

V0

lowast=

V0

1198800

119899lowast=

120592

1198802

0

119899

119870119901=1198701198802

0

120592 119877 =

11987111198800

120592 119878

lowast=120592119878

1198802

0

119870119888=120592119870119897

1198802

0

119872 =1205901198612

0120592

1205881198802

0

119866119903=119892120573120592 (119879


1198803

0

119866119898=119892120573120592 (119862


1198803

0

119877119888=1198802

01198700

1205881205922

Pr = 120592

120572 119878

119888=

120592

119863

(4)


120597119906

120597119905minus V0

120597119906

120597119910=1205972119906

1205971199102minus 119877119888(

1205973119906

1205971199051205971199103+ V0

1205973119906

1205971199103)

minus (1198722+

1

119870119901

)119906 + 119866119903120579 + 119866119898120601

(5)

Pr(120597120579120597119905

minus V0

120597120579

120597119910) =

1205972120579

1205971199102minus Pr119878120579 (6)

119878119888(120597120601

120597119905minus V0

120597120601

120597119910) =

1205972120601

1205971199102minus 119870119888119878119888120601 (7)


119910 = 0 119906 = 119890int+ 119877

120597119906

120597119910 120579 = 1 120601 = 1


(8)


119888lt 1 as the



119906 = 1199060+ 1198771198881199061+ 119874(119877

119888)2

120579 = 1205790+ 1198771198881205791+ 119874(119877

119888)2

120601 = 1206010+ 1198771198881206011+ 119874(119877

119888)2

(9)


get the following


1205971199060

120597119905minus V0

1205971199060

120597119910=12059721199060

1205971199102minus (119872

2+

1

119870119901

)1199060+ 1198661199031205790+ 1198661198981206010

Pr(1205971205790

120597119905minus V0

1205971205790

120597119910) =

12059721205790

1205971199102minus Pr119878120579

0

119878119888(1205971206010

120597119905minus V0

1205971206010

120597119910) =

12059721206010

1205971199102minus 1198701198881198781198881206010

(10)


1205971199061

120597119905minus V0

1205971199061

120597119910=12059721199061

1205971199102minus (119872

2+

1

119870119901

)1199061

+ 1198661199031205791+ 1198661198981206011minus V0

12059731199061

1205971199103minus

12059731199060

1205971199051205971199102

Pr(1205971205791120597119905

minus V0

1205971205791

120597119910) =

12059721205791

1205971199102minus Pr119878120579

1

119878119888(1205971206011

120597119905minus V0

1205971206011

120597119910) =

12059721206011

1205971199102minus 1198701198881198781198881206011

(11)


119910 = 0 1199060= 119890

int+ 119877

1205971199060

120597119910 119906

1= 0 120579

0= 1

1205791= 1 120601

0= 1 120601

1= 1

119910 997888rarr infin 1199060997888rarr 0 119906

1997888rarr 0 120579

0997888rarr 0

1205791997888rarr 0 120601

0997888rarr 0 120601

1997888rarr 0

(12)


1199060(119910 119905) = 119906

00(119910) + 119906

01(119910) 119890

int

1199061(119910 119905) = 119906

10(119910) + 119906

11(119910) 119890

int

1205790(119910 119905) = 120579

00(119910) + 120579

01(119910) 119890

int

1205791(119910 119905) = 120579

10(119910) + 120579

11(119910) 119890

int

1206010(119910 119905) = 120601

00(119910) + 120601

01(119910) 119890

int

1206011(119910 119905) = 120601

10(119910) + 120601

11(119910) 119890

int

(13)



11990610158401015840

00+ V01199061015840

00minus (119872

2+

1

119870119901

)11990600= minus11986611990312057900minus 11986611989812060100

11990610158401015840

01+ V01199061015840

01minus (119872

2+

1

119870119901

+ 119894119899) 11990601= minus11986611990312057901minus 11986611989812060101

11990610158401015840

10+ V01199061015840

10minus (119872

2+

1

119870119901

)11990610= minus11986611990312057910minus 11986611989812060110+ V0119906101584010158401015840

00

11990610158401015840

11+ V01199061015840

11minus (119872

2+

1

119870119901

+ 119894119899) 11990611

= minus11986611990312057911minus 11986611989812060111+ V0119906101584010158401015840

01+ 11989411989911990610158401015840

01

12057910158401015840

00+ V0Pr120579101584000minus Pr119878120579

00= 0

12057910158401015840

01+ V0Pr120579101584001minus (119894119899Pr + Pr119878) 120579

01= 0

12057910158401015840

10+ V0Pr120579101584010minus Pr119878120579

10= 0

12057910158401015840

11+ V0Pr120579101584011minus (119894119899Pr + Pr119878) 120579

11= 0

12060110158401015840

00+ V01198781198881206011015840

00minus 11987011988811987811988812060100= 0

12060110158401015840

01+ V01198781198881206011015840

01minus (119894119899119878

119888+ 119870119888119878119888) 12060101= 0

12060110158401015840

10+ V01198781198881206011015840

10minus 11987011988811987811988812060110= 0

12060110158401015840

11+ V01198781198881206011015840

11minus (119894119899119878

119888+ 119870119888119878119888) 12060111= 0

(14)


119910 = 0 11990600= 119877

12059711990600

120597119910 119906

01= 1 + 119877

12059711990601

120597119910

11990610= 0 119906

11= 0 120579

00= 1

12057901= 0 120579

10= 1 120579

11= 0

12060100= 1 120601

01= 0 120601

10= 0 120601

11= 0

119910 997888rarr infin 11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0

11990611997888rarr 0 120579

00997888rarr 0 120579

01997888rarr 0

12057910997888rarr 0 120579

11997888rarr 0 120601

00997888rarr 0

12060101997888rarr 0 120601

10997888rarr 0 120601

11997888rarr 0

(15)



11990601= 1198604119890minus1198864119910


11990611= 1198608119890minus1198864119910

12057900= 119890minus1198861119910

12057901= 12057910= 12057911= 0

12060100= 119890minus1198862119910

12060101= 12060110= 12060111= 0

(16)



+ 1198604119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)


+1198608119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)

120579 = 119890minus1198861119910

120601 = 119890minus1198862119910

(17)


119862119891=

120591119909119910

1198802

0

=120597119906

120597119910minus 119877119888(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102)

100381610038161003816100381610038161003816100381610038161003816119910=0

(18)

where

120591119909119910=120597119906

120597119910minus1198700

120588(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102) (19)


119873119906= minus

120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198861 (20)


119878ℎ= minus

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198862 (21)











0







= 5 119866119898= 5 119878

119888= 024 119877

119888= 05 119870

119888= 2 R = 02 and V

0=





119901= 1M = 2 119866

119903= 5 119866

119898 = 5 119878

119888= 024 R = 02 and V

0


119888= S = 119870

119888= 0





that Pr = 071 S = 1 119870119901= 1 M = 2 119877

119888= 05 119870

119888= 2 R = 02






0) and


119901= 1 M = 2 119877

119888= 05

119870119888= 2 119866

119903= 5 119866

119898= 5 and 119878



0 05 1 15 2 25 3 35 4 45 5

1

0

minus1

minus2

minus3

minus4

minus5

u

y

Pr = 031Kp = 02M = 05

Pr = 071Kp = 02M = 1

Pr = 071Kp = 02M = 05

Pr = 071Kp = 04M = 05


0 05 1 15 2 25 3 35 4 45 5

05

0

minus05

minus1

minus15

minus2

u

y

Rc = 02 S = 05Kc = 1

Rc = 02 S = 05Kc = 2

Rc = 02 S = 1Kc = 1

Rc = 05 S = 05Kc = 1



12

1

08

06

04

02

0

minus020 05 1 15 2 25 3 35 4 45 5

u

y

Gr = 25 Gm = 2 2

Gr = 2 Gm = 2 25

Sc =

Sc =

Gr = 2 Gm = 2 2Sc =

Gr = 2 Gm = 25 2Sc =




06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

0 = 15 R = 07

0 = 15 R = 05

0 = 2 R = 05

u

0 05 1 15 2 25 3 35 4 45 5y















1

09

08

07

06

05

04

03

02

01

0

Pr = 05 S = 05 0 = 02

Pr = 071 S = 05 0 = 02

Pr = 071 S = 05 0 = 05

Pr = 071 S = 1 0 = 02

0 05 1 15 2 25 3 35 4 45 5y

Pr = 071 S = 0 0 = 02

120579


1

09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

y

078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =

078Kc = 05 0 = 02Sc =

078Kc = 05 0 = 05Sc =

078Kc = 1 0 = 02Sc =120601



4 Conclusion







Pr 119870119901

119872 119877119888

119878 119870119888

119866119903

119866119898

119878119888

119877 V0

119862119891

030 1 2 05 1 2 5 5 024 02 2 183921050 1 2 05 1 2 5 5 024 02 2 108793050 15 2 05 1 2 5 5 024 02 2 71877050 1 25 05 1 2 5 5 024 02 2 304390050 1 2 02 1 2 5 5 024 02 2 130904050 1 2 05 05 2 5 5 024 02 2 179390050 1 2 05 1 15 5 5 024 02 2 116350050 1 2 05 1 2 45 5 024 02 2 131659050 1 2 05 1 2 5 45 024 02 2 118927050 1 2 05 1 2 5 5 030 02 2 91301050 1 2 05 1 2 5 5 024 05 2 275040050 1 2 05 1 2 5 5 024 02 15 140484


Pr 119878 V0

119873119906

050 05 02 075887071 05 02 078166071 10 02 107351071 05 05 090654


119878119888

119870119888

V0

119878ℎ

030 05 02 041845078 05 02 070735078 10 02 096461078 05 05 084923



Appendix


1198861=1

2[V0Pr + radic(V

0Pr)2 + 4119878Pr]

1198862=1

2[V0119878119888+ radic(V

0119878119888)2

+ 4119870119888119878119888]

1198863=1

2[V20+ radicV20+ 4119876] 119876 = 119872

2+

1

119870119901

1198864=1

2[V20+ radicV20+ 4 (119876 + 119894119899)]

1198601=

minus119866119903

1198862

1minus 1198861V0minus 119876

1198602=

minus119866119903

1198862

2minus 1198862V0minus 119876

1198603=

minus1

1 + 1198863119877[(1198861119877 + 1)119860

1+ (1198862119877 + 1)119860

2]

1198604=

1

1 + 1198771198864

1198605=

1198863

11198601V0

1198862

1minus 1198861V0minus 119876

1198606=

1198863

21198602V0

1198862

2minus 1198862V0minus 119876

1198607= 1198605+ 1198606minusV01198863

31198603119910

V0minus 21198863

1198608=

(minusV01198863

41198604+ 1198941198991198862

41198604) 119910

V0minus 21198864

(A1)

Nomenclature


















Acknowledgment


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











11990610158401015840

00+ V01199061015840

00minus (119872

2+

1

119870119901

)11990600= minus11986611990312057900minus 11986611989812060100

11990610158401015840

01+ V01199061015840

01minus (119872

2+

1

119870119901

+ 119894119899) 11990601= minus11986611990312057901minus 11986611989812060101

11990610158401015840

10+ V01199061015840

10minus (119872

2+

1

119870119901

)11990610= minus11986611990312057910minus 11986611989812060110+ V0119906101584010158401015840

00

11990610158401015840

11+ V01199061015840

11minus (119872

2+

1

119870119901

+ 119894119899) 11990611

= minus11986611990312057911minus 11986611989812060111+ V0119906101584010158401015840

01+ 11989411989911990610158401015840

01

12057910158401015840

00+ V0Pr120579101584000minus Pr119878120579

00= 0

12057910158401015840

01+ V0Pr120579101584001minus (119894119899Pr + Pr119878) 120579

01= 0

12057910158401015840

10+ V0Pr120579101584010minus Pr119878120579

10= 0

12057910158401015840

11+ V0Pr120579101584011minus (119894119899Pr + Pr119878) 120579

11= 0

12060110158401015840

00+ V01198781198881206011015840

00minus 11987011988811987811988812060100= 0

12060110158401015840

01+ V01198781198881206011015840

01minus (119894119899119878

119888+ 119870119888119878119888) 12060101= 0

12060110158401015840

10+ V01198781198881206011015840

10minus 11987011988811987811988812060110= 0

12060110158401015840

11+ V01198781198881206011015840

11minus (119894119899119878

119888+ 119870119888119878119888) 12060111= 0

(14)


119910 = 0 11990600= 119877

12059711990600

120597119910 119906

01= 1 + 119877

12059711990601

120597119910

11990610= 0 119906

11= 0 120579

00= 1

12057901= 0 120579

10= 1 120579

11= 0

12060100= 1 120601

01= 0 120601

10= 0 120601

11= 0

119910 997888rarr infin 11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0

11990611997888rarr 0 120579

00997888rarr 0 120579

01997888rarr 0

12057910997888rarr 0 120579

11997888rarr 0 120601

00997888rarr 0

12060101997888rarr 0 120601

10997888rarr 0 120601

11997888rarr 0

(15)



11990601= 1198604119890minus1198864119910


11990611= 1198608119890minus1198864119910

12057900= 119890minus1198861119910

12057901= 12057910= 12057911= 0

12060100= 119890minus1198862119910

12060101= 12060110= 12060111= 0

(16)



+ 1198604119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)


+1198608119890minus1198864119910

(cos 119899119905 + 119894 sin 119899119905)

120579 = 119890minus1198861119910

120601 = 119890minus1198862119910

(17)


119862119891=

120591119909119910

1198802

0

=120597119906

120597119910minus 119877119888(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102)

100381610038161003816100381610038161003816100381610038161003816119910=0

(18)

where

120591119909119910=120597119906

120597119910minus1198700

120588(1205972119906

120597119905120597119910+ V0

1205972119906

1205971199102) (19)


119873119906= minus

120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198861 (20)


119878ℎ= minus

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198862 (21)











0







= 5 119866119898= 5 119878

119888= 024 119877

119888= 05 119870

119888= 2 R = 02 and V

0=





119901= 1M = 2 119866

119903= 5 119866

119898 = 5 119878

119888= 024 R = 02 and V

0


119888= S = 119870

119888= 0





that Pr = 071 S = 1 119870119901= 1 M = 2 119877

119888= 05 119870

119888= 2 R = 02






0) and


119901= 1 M = 2 119877

119888= 05

119870119888= 2 119866

119903= 5 119866

119898= 5 and 119878



0 05 1 15 2 25 3 35 4 45 5

1

0

minus1

minus2

minus3

minus4

minus5

u

y

Pr = 031Kp = 02M = 05

Pr = 071Kp = 02M = 1

Pr = 071Kp = 02M = 05

Pr = 071Kp = 04M = 05


0 05 1 15 2 25 3 35 4 45 5

05

0

minus05

minus1

minus15

minus2

u

y

Rc = 02 S = 05Kc = 1

Rc = 02 S = 05Kc = 2

Rc = 02 S = 1Kc = 1

Rc = 05 S = 05Kc = 1



12

1

08

06

04

02

0

minus020 05 1 15 2 25 3 35 4 45 5

u

y

Gr = 25 Gm = 2 2

Gr = 2 Gm = 2 25

Sc =

Sc =

Gr = 2 Gm = 2 2Sc =

Gr = 2 Gm = 25 2Sc =




06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

0 = 15 R = 07

0 = 15 R = 05

0 = 2 R = 05

u

0 05 1 15 2 25 3 35 4 45 5y















1

09

08

07

06

05

04

03

02

01

0

Pr = 05 S = 05 0 = 02

Pr = 071 S = 05 0 = 02

Pr = 071 S = 05 0 = 05

Pr = 071 S = 1 0 = 02

0 05 1 15 2 25 3 35 4 45 5y

Pr = 071 S = 0 0 = 02

120579


1

09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

y

078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =

078Kc = 05 0 = 02Sc =

078Kc = 05 0 = 05Sc =

078Kc = 1 0 = 02Sc =120601



4 Conclusion







Pr 119870119901

119872 119877119888

119878 119870119888

119866119903

119866119898

119878119888

119877 V0

119862119891

030 1 2 05 1 2 5 5 024 02 2 183921050 1 2 05 1 2 5 5 024 02 2 108793050 15 2 05 1 2 5 5 024 02 2 71877050 1 25 05 1 2 5 5 024 02 2 304390050 1 2 02 1 2 5 5 024 02 2 130904050 1 2 05 05 2 5 5 024 02 2 179390050 1 2 05 1 15 5 5 024 02 2 116350050 1 2 05 1 2 45 5 024 02 2 131659050 1 2 05 1 2 5 45 024 02 2 118927050 1 2 05 1 2 5 5 030 02 2 91301050 1 2 05 1 2 5 5 024 05 2 275040050 1 2 05 1 2 5 5 024 02 15 140484


Pr 119878 V0

119873119906

050 05 02 075887071 05 02 078166071 10 02 107351071 05 05 090654


119878119888

119870119888

V0

119878ℎ

030 05 02 041845078 05 02 070735078 10 02 096461078 05 05 084923



Appendix


1198861=1

2[V0Pr + radic(V

0Pr)2 + 4119878Pr]

1198862=1

2[V0119878119888+ radic(V

0119878119888)2

+ 4119870119888119878119888]

1198863=1

2[V20+ radicV20+ 4119876] 119876 = 119872

2+

1

119870119901

1198864=1

2[V20+ radicV20+ 4 (119876 + 119894119899)]

1198601=

minus119866119903

1198862

1minus 1198861V0minus 119876

1198602=

minus119866119903

1198862

2minus 1198862V0minus 119876

1198603=

minus1

1 + 1198863119877[(1198861119877 + 1)119860

1+ (1198862119877 + 1)119860

2]

1198604=

1

1 + 1198771198864

1198605=

1198863

11198601V0

1198862

1minus 1198861V0minus 119876

1198606=

1198863

21198602V0

1198862

2minus 1198862V0minus 119876

1198607= 1198605+ 1198606minusV01198863

31198603119910

V0minus 21198863

1198608=

(minusV01198863

41198604+ 1198941198991198862

41198604) 119910

V0minus 21198864

(A1)

Nomenclature


















Acknowledgment


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















0







= 5 119866119898= 5 119878

119888= 024 119877

119888= 05 119870

119888= 2 R = 02 and V

0=





119901= 1M = 2 119866

119903= 5 119866

119898 = 5 119878

119888= 024 R = 02 and V

0


119888= S = 119870

119888= 0





that Pr = 071 S = 1 119870119901= 1 M = 2 119877

119888= 05 119870

119888= 2 R = 02






0) and


119901= 1 M = 2 119877

119888= 05

119870119888= 2 119866

119903= 5 119866

119898= 5 and 119878



0 05 1 15 2 25 3 35 4 45 5

1

0

minus1

minus2

minus3

minus4

minus5

u

y

Pr = 031Kp = 02M = 05

Pr = 071Kp = 02M = 1

Pr = 071Kp = 02M = 05

Pr = 071Kp = 04M = 05


0 05 1 15 2 25 3 35 4 45 5

05

0

minus05

minus1

minus15

minus2

u

y

Rc = 02 S = 05Kc = 1

Rc = 02 S = 05Kc = 2

Rc = 02 S = 1Kc = 1

Rc = 05 S = 05Kc = 1



12

1

08

06

04

02

0

minus020 05 1 15 2 25 3 35 4 45 5

u

y

Gr = 25 Gm = 2 2

Gr = 2 Gm = 2 25

Sc =

Sc =

Gr = 2 Gm = 2 2Sc =

Gr = 2 Gm = 25 2Sc =




06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

0 = 15 R = 07

0 = 15 R = 05

0 = 2 R = 05

u

0 05 1 15 2 25 3 35 4 45 5y















1

09

08

07

06

05

04

03

02

01

0

Pr = 05 S = 05 0 = 02

Pr = 071 S = 05 0 = 02

Pr = 071 S = 05 0 = 05

Pr = 071 S = 1 0 = 02

0 05 1 15 2 25 3 35 4 45 5y

Pr = 071 S = 0 0 = 02

120579


1

09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

y

078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =

078Kc = 05 0 = 02Sc =

078Kc = 05 0 = 05Sc =

078Kc = 1 0 = 02Sc =120601



4 Conclusion







Pr 119870119901

119872 119877119888

119878 119870119888

119866119903

119866119898

119878119888

119877 V0

119862119891

030 1 2 05 1 2 5 5 024 02 2 183921050 1 2 05 1 2 5 5 024 02 2 108793050 15 2 05 1 2 5 5 024 02 2 71877050 1 25 05 1 2 5 5 024 02 2 304390050 1 2 02 1 2 5 5 024 02 2 130904050 1 2 05 05 2 5 5 024 02 2 179390050 1 2 05 1 15 5 5 024 02 2 116350050 1 2 05 1 2 45 5 024 02 2 131659050 1 2 05 1 2 5 45 024 02 2 118927050 1 2 05 1 2 5 5 030 02 2 91301050 1 2 05 1 2 5 5 024 05 2 275040050 1 2 05 1 2 5 5 024 02 15 140484


Pr 119878 V0

119873119906

050 05 02 075887071 05 02 078166071 10 02 107351071 05 05 090654


119878119888

119870119888

V0

119878ℎ

030 05 02 041845078 05 02 070735078 10 02 096461078 05 05 084923



Appendix


1198861=1

2[V0Pr + radic(V

0Pr)2 + 4119878Pr]

1198862=1

2[V0119878119888+ radic(V

0119878119888)2

+ 4119870119888119878119888]

1198863=1

2[V20+ radicV20+ 4119876] 119876 = 119872

2+

1

119870119901

1198864=1

2[V20+ radicV20+ 4 (119876 + 119894119899)]

1198601=

minus119866119903

1198862

1minus 1198861V0minus 119876

1198602=

minus119866119903

1198862

2minus 1198862V0minus 119876

1198603=

minus1

1 + 1198863119877[(1198861119877 + 1)119860

1+ (1198862119877 + 1)119860

2]

1198604=

1

1 + 1198771198864

1198605=

1198863

11198601V0

1198862

1minus 1198861V0minus 119876

1198606=

1198863

21198602V0

1198862

2minus 1198862V0minus 119876

1198607= 1198605+ 1198606minusV01198863

31198603119910

V0minus 21198863

1198608=

(minusV01198863

41198604+ 1198941198991198862

41198604) 119910

V0minus 21198864

(A1)

Nomenclature


















Acknowledgment


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

0 = 15 R = 07

0 = 15 R = 05

0 = 2 R = 05

u

0 05 1 15 2 25 3 35 4 45 5y















1

09

08

07

06

05

04

03

02

01

0

Pr = 05 S = 05 0 = 02

Pr = 071 S = 05 0 = 02

Pr = 071 S = 05 0 = 05

Pr = 071 S = 1 0 = 02

0 05 1 15 2 25 3 35 4 45 5y

Pr = 071 S = 0 0 = 02

120579


1

09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

y

078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =

078Kc = 05 0 = 02Sc =

078Kc = 05 0 = 05Sc =

078Kc = 1 0 = 02Sc =120601



4 Conclusion







Pr 119870119901

119872 119877119888

119878 119870119888

119866119903

119866119898

119878119888

119877 V0

119862119891

030 1 2 05 1 2 5 5 024 02 2 183921050 1 2 05 1 2 5 5 024 02 2 108793050 15 2 05 1 2 5 5 024 02 2 71877050 1 25 05 1 2 5 5 024 02 2 304390050 1 2 02 1 2 5 5 024 02 2 130904050 1 2 05 05 2 5 5 024 02 2 179390050 1 2 05 1 15 5 5 024 02 2 116350050 1 2 05 1 2 45 5 024 02 2 131659050 1 2 05 1 2 5 45 024 02 2 118927050 1 2 05 1 2 5 5 030 02 2 91301050 1 2 05 1 2 5 5 024 05 2 275040050 1 2 05 1 2 5 5 024 02 15 140484


Pr 119878 V0

119873119906

050 05 02 075887071 05 02 078166071 10 02 107351071 05 05 090654


119878119888

119870119888

V0

119878ℎ

030 05 02 041845078 05 02 070735078 10 02 096461078 05 05 084923



Appendix


1198861=1

2[V0Pr + radic(V

0Pr)2 + 4119878Pr]

1198862=1

2[V0119878119888+ radic(V

0119878119888)2

+ 4119870119888119878119888]

1198863=1

2[V20+ radicV20+ 4119876] 119876 = 119872

2+

1

119870119901

1198864=1

2[V20+ radicV20+ 4 (119876 + 119894119899)]

1198601=

minus119866119903

1198862

1minus 1198861V0minus 119876

1198602=

minus119866119903

1198862

2minus 1198862V0minus 119876

1198603=

minus1

1 + 1198863119877[(1198861119877 + 1)119860

1+ (1198862119877 + 1)119860

2]

1198604=

1

1 + 1198771198864

1198605=

1198863

11198601V0

1198862

1minus 1198861V0minus 119876

1198606=

1198863

21198602V0

1198862

2minus 1198862V0minus 119876

1198607= 1198605+ 1198606minusV01198863

31198603119910

V0minus 21198863

1198608=

(minusV01198863

41198604+ 1198941198991198862

41198604) 119910

V0minus 21198864

(A1)

Nomenclature


















Acknowledgment


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Pr 119870119901

119872 119877119888

119878 119870119888

119866119903

119866119898

119878119888

119877 V0

119862119891

030 1 2 05 1 2 5 5 024 02 2 183921050 1 2 05 1 2 5 5 024 02 2 108793050 15 2 05 1 2 5 5 024 02 2 71877050 1 25 05 1 2 5 5 024 02 2 304390050 1 2 02 1 2 5 5 024 02 2 130904050 1 2 05 05 2 5 5 024 02 2 179390050 1 2 05 1 15 5 5 024 02 2 116350050 1 2 05 1 2 45 5 024 02 2 131659050 1 2 05 1 2 5 45 024 02 2 118927050 1 2 05 1 2 5 5 030 02 2 91301050 1 2 05 1 2 5 5 024 05 2 275040050 1 2 05 1 2 5 5 024 02 15 140484


Pr 119878 V0

119873119906

050 05 02 075887071 05 02 078166071 10 02 107351071 05 05 090654


119878119888

119870119888

V0

119878ℎ

030 05 02 041845078 05 02 070735078 10 02 096461078 05 05 084923



Appendix


1198861=1

2[V0Pr + radic(V

0Pr)2 + 4119878Pr]

1198862=1

2[V0119878119888+ radic(V

0119878119888)2

+ 4119870119888119878119888]

1198863=1

2[V20+ radicV20+ 4119876] 119876 = 119872

2+

1

119870119901

1198864=1

2[V20+ radicV20+ 4 (119876 + 119894119899)]

1198601=

minus119866119903

1198862

1minus 1198861V0minus 119876

1198602=

minus119866119903

1198862

2minus 1198862V0minus 119876

1198603=

minus1

1 + 1198863119877[(1198861119877 + 1)119860

1+ (1198862119877 + 1)119860

2]

1198604=

1

1 + 1198771198864

1198605=

1198863

11198601V0

1198862

1minus 1198861V0minus 119876

1198606=

1198863

21198602V0

1198862

2minus 1198862V0minus 119876

1198607= 1198605+ 1198606minusV01198863

31198603119910

V0minus 21198863

1198608=

(minusV01198863

41198604+ 1198941198991198862

41198604) 119910

V0minus 21198864

(A1)

Nomenclature


















Acknowledgment


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgment


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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