research article thermal boundary layer in flow due...

Research ArticleThermal Boundary Layer in Flow due to an ExponentiallyStretching Surface with an Exponentially Moving Free Stream

Krishnendu Bhattacharyya and G C Layek

Department of Mathematics The University of Burdwan Burdwan West Bengal 713104 India

Correspondence should be addressed to Krishnendu Bhattacharyya krishmathyahoocom

Received 24 January 2014 Revised 11 April 2014 Accepted 14 April 2014 Published 8 May 2014

Academic Editor ShengKai Yu

Copyright copy 2014 K Bhattacharyya and G C Layek This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A numerical investigation is made to study the thermal boundary layer for flow of incompressible Newtonian fluid overan exponentially stretching sheet with an exponentially moving free stream The governing partial differential equations aretransformed into self-similar ordinary differential equations using similarity transformations in exponential forms Then thoseare solved numerically by shooting technique using Runge-Kutta method The study reveals that the momentum boundary layerthickness for this flow is considerably smaller than the linear stagnation point flow past a linearly stretching sheet The momentumand thermal boundary layer thicknesses reduce when the velocity ratio parameter increases For the temperature distribution inaddition to the heat transfer from the sheet the heat absorption at the sheet also occurs in certain situations and both heat transferand absorption increase with the velocity ratio parameter and the Prandtl number The temperature inside the boundary layersignificantly decreases with higher values of velocity ratio parameter and the Prandtl number

1 Introduction

The viscous fluid flow due to a stretching sheet is very signif-icant problem in fluid dynamics due to its huge applicationsinmanymanufacturing processes for example the cooling ofan infinitemetallic plate in a cooling bath the boundary layeralong material handling conveyers the aerodynamic extru-sion of plastic sheets the boundary layer along a liquid filmin the condensation processes hot rolling paper productionmetal spinning glass-fiber production and drawing of plasticfilmsThe heat transfer from a stretching surface is of interestin polymer extrusion processeswhere the object after passingthrough a die enters the fluid for cooling below a certaintemperature During the processesmechanical properties aregreatly dependent upon the rate of cooling and the rate atwhich such objects are cooled has an important bearing onthe properties of the final product So the quality of thefinal product depends on the rate of heat transfer from thestretching surfaceThe suitable choice of cooling liquid is verymuch crucial as it has a direct impact on rate of cooling of thesurface

Crane [1] first investigated the steady boundary layer flowof an incompressible viscous fluid over a linearly stretchingplate and gave an exact similarity solution in closed analyticalform Cranersquos work was extended by many researchers suchas P S Gupta and A S Gupta [2] Chen and Char [3]Pavlov [4] and Ali [5] considering the effects of heat andmass transfer and magnetic field under various physicalconditions Later Ali [6] discussed the thermal boundarylayer on a surface stretching with power law velocity Themixed convection for the flow due to linearly stretchingsheet with mass suctioninjection was studied by Ali andAl-Yousef [7] Ali [8] also investigated the buoyancy effectson the boundary layers induced by a rapidly stretchingsurface Tsai et al [9] discussed the effect of nonuniform heatsourcesink on the flow and heat transfer from an unsteadystretching sheet through a quiescent fluid medium extendingto infinity In two important papers Bhattacharyya andLayek [10 11] explained the behaviour of chemically reactivesolute distribution in MHD flow over a stretching sheet andin slip flow towards a vertical stretching sheet RecentlyBhattacharyya [12 13] presented effect of heat sourcesink

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2014 Article ID 785049 9 pageshttpdxdoiorg1011552014785049

2 Modelling and Simulation in Engineering

on the MHD boundary layer flow over steady and unsteadyshrinkingstretching sheet and Bhattacharyya et al [14]found the analytic solution of MHD flow of non-NewtonianCasson fluid flow induced due to stretchingshrinking sheetwith wall mass transfer

On the other hand Hiemenz [15] first studied the steadyflow in the neighbourhood of a stagnation point Chiam[16] investigated a problem which is a combination of theworks of Hiemenz [15] and Crane [1] that is the linearstagnation point flow towards a linear stretching sheet whenthe stretching rate of the plate is equal to the strain rateof the stagnation point flow and he found no boundarylayer structure near the plate After few years Mahapatraand Gupta [17] reinvestigated the same stagnation pointflow towards a stretching sheet with different stretching andstraining velocities and they found two kinds of boundarylayers near the sheet depending on the ratio of the stretchingand straining velocity ratesWang [18] demonstrated the stag-nation point flow towards a shrinking sheet Furthermore thebehaviour of stagnation point flow over stretchingshrinkingsheet under different physical aspects was discussed by manyresearchers [19ndash35]

In last few decades in almost all investigations of the flowover a stretching sheet the flow occurs because of linearvariation of stretching velocity of the flat sheet with thedistance from the origin So the boundary layer flow inducedby an exponentially stretching sheet is not studied muchthough it is very important and practical flow frequentlyappeared in many engineering processes In 1999 Magyariand Keller [36] first consider the boundary layer flow dueto an exponentially stretching sheet and he also studiedthe heat transfer in the flow taking exponentially variedwall temperature After that Elbashbeshy [37] numericallyexamined the flow and heat transfer over an exponentiallystretching surface considering wall mass suction Khan andSanjayanand [38] investigated the flow of viscoelastic fluidand heat transfer over an exponentially stretching sheetwith viscous dissipation effects Partha et al [39] presenteda similarity solution for mixed convection flow past anexponentially stretching surface by taking into account theinfluence of viscous dissipation on the convective transportSanjayanand and Khan [40] discussed the effects of heatand mass transfer on the boundary layer flow of viscoelasticfluid taking 119899th order chemical reaction Al-Odat et al [41]investigated the effect magnetic field on thermal boundarylayer on an exponentially stretching continuous surface withan exponential temperature distribution Later Sajid andHayat [42] showed the influence of thermal radiation onthe boundary layer flow past an exponentially stretchingsheet and they reported series solutions for velocity andtemperature using homotopy analysis method (HAM) andBidin and Nazar [43] obtained a numerical solution of thesame problem Ishak [44] studied the MHD boundary layerflow over an exponentially stretching sheet in presence ofthermal radiationMandal andMukhopadhyay [45] analyzedthe heat transfer in the flowdue to an exponentially stretchingporous sheet with surface heat flux embedded in a porousmedium Bhattacharyya [46] discussed the boundary layerflow and heat transfer past an exponentially shrinking sheet

and Bhattacharyya and Pop [47] presented the effect of mag-netic field on that flow Recently The MHD boundary layernanofluid flow due to an exponentially stretching permeablesheet was demonstrated by Bhattacharyya and Layek [48]

Inspired by the above investigations in the present paperthe thermal boundary layer in incompressible flow over anexponentially stretching sheet in an exponentially movingfree stream is studied The wall temperature distribution istaken variable in exponential form Using similarity trans-formations the governing partial differential equations aretransformed into self-similar ordinary differential equationsThen the transformed equations are solved by standardshooting technique using fourth order Runge-Kutta methodThe numerical computations are presented through somefigures and the various characteristics of flow and heattransfer are thoroughly describedThis type of flow is possiblein reality in some special situations The motive of theinvestigation is to study new type of flow dynamics and theheat transfer in the flow field

2 Analysis of Motion and Heat Transfer

Consider the steady two-dimensional laminar boundarylayer flow of a viscous incompressible fluid and heat transferover an exponentially stretching sheet (of velocity119880

119908)with an

exponentially moving free stream (of velocity119880infin)The sheet

coincides with the plane119910 = 0 and the flow confined to119910 gt 0The governing equations for the flow and the temperature arewritten in usual notation as

120597119906

120597119909

+

120597V120597119910

= 0 (1)

119906

120597119906

120597119909

+ V120597119906

120597119910

= 119880infin

119889119880infin

119889119909

+ 120592

1205972

119906

1205971199102 (2)

119906

120597119879

120597119909

+ V120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102 (3)

where 119906 and V are the velocity components in 119909 and 119910

directions respectively119880infinis the fluid velocity in free stream

120592(= 120583120588) is the kinematic fluid viscosity 120588 is the fluid density120583 is the coefficient of fluid viscosity 119879 is the temperature 120581 isthe fluid thermal conductivity and 119888

119901is the specific heat The

diagram of the physical problem is given in Figure 1The boundary conditions corresponding to the velocity

components and the temperature are given by

119906 = 119880119908(119909) V = 0 at 119910 = 0

119906 997888rarr 119880infin(119909) as 119910 997888rarr infin

(4)

119879 = 119879119908= 119879infin

+ 1198790exp(120582119909

2119871

) at 119910 = 0

119879 997888rarr 119879infin

as 119910 997888rarr infin

(5)

where 119879119908is the variable temperature of the sheet 119879

infinis

the free stream temperature assumed to be constant 1198790is

a constant which measures the rate of temperature increase

Modelling and Simulation in Engineering 3

Twx

u

Thermal boundary layer

y

Uw

Tinfin Uinfin

Velocity boundary layer

Stretching sheet

Figure 1 A diagram of the physical problem

along the sheet 119871 denotes the reference length and 120582 is aparameter which is physically very important in controllingthe exponential increment of temperature along the sheet andit may have both positive and negative values The stretchingand free stream velocities 119880

119908and 119880

infinare respectively given

by

119880119908(119909) = 119888 exp(119909

119871

) 119880infin(119909) = 119886 exp(119909

119871

) (6)

where 119888 and 119886 are constants with 119888 gt 0 and 119886 ge 0 (for withoutany free stream 119886 = 0)

Now the stream function 120595(119909 119910) is given by

119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(7)

For relations in (7) the continuity equation (1) is identicallysatisfied and the momentum equation (2) and the tempera-ture equation (3) are reduced to the following forms

120597120595

120597119910

1205972

120595

120597119909120597119910

minus

120597120595

120597119909

1205972

120595

1205971199102= 119880infin

119889119880infin

119889119909

+ 120592

1205973

120595

1205971199103

120597120595

120597119910

120597119879

120597119909

minus

120597120595

120597119909

120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102

(8)

The boundary conditions in (4) for the flow become

120597120595

120597119910

= 119880119908(119909)

120597120595

120597119909

= 0 at 119910 = 0

120597120595

120597119910

997888rarr 119880infin(119909) as 119910 997888rarr infin

(9)

Next the dimensionless variables for 120595 and 119879 are introducedas

120595 = radic2120592119871119888119891 (120578) exp( 119909

2119871

)

119879 = 119879infin

+ (119879119908minus 119879infin) 120579 (120578)

(10)

where 120578 is the similarity variable and is defined as 120578 =

119910radic1198882120592119871 exp(1199092119871)

Using (10) finally the following nonlinear self-similarequations are obtained

119891101584010158401015840

+ 11989111989110158401015840

minus 211989110158402

+ 2(

119886

119888

)

2

= 0 (11)

12057910158401015840

+ Pr (1198911205791015840 minus 1205821198911015840

120579) = 0 (12)

where 119886119888 is the velocity ratio parameter and Pr = 120583119888119901120581 is

the Prandtl numberThe boundary conditions in (9) and (5) reduce to the

following forms

119891 (120578) = 0 1198911015840

(120578) = 1 at 120578 = 0

1198911015840

(120578) 997888rarr

119886

119888


120579 (120578) = 1 at 120578 = 0 120579 (120578) 997888rarr 0 as 120578 997888rarr infin

(13)

When 119888 = 119886 (11) gives closed form analytical solution 119891(120578) =

120578 which is the same as the linear stagnation-point flow overa stretching sheet with velocity varied linearly to the distancefrom the origin

The physical quantities of interest in this problem are thelocal skin-friction coefficient and the local Nusselt numberThe dimensionless local skin-friction coefficient is expressedas

119862119891119909

=

120591119908

1003816100381610038161003816119910=0

1205881198802

119908

(14)

where 120591119908is the shear stress at the wall and is given by 120591

119908=

120583[120597119906120597119910]119910=0

So

radic2Re119862119891119909

= 11989110158401015840

(0) (15)

where Re = 119880119908119871120592 is the Reynolds number

The local Nusselt number can be written as

Nu119909= minus

119909

119879119908minus 119879infin

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

(16)

So

Nu119909

radicRe119909

= minusradic

119909

2119871

1205791015840

(0) (17)

where Re119909= 119880119908119909120592 is the local Reynolds number

3 Numerical Method for Solution

The nonlinear coupled differential equations (11) and (12)along with the boundary conditions (13) form a two pointboundary value problem (BVP) and is solved using shootingmethod by converting it into an initial value problem (IVP)In this method it is necessary to choose a suitable finite valueof 120578 rarr infin say 120578

infin The following first-order system is set

1198911015840

= 119901 1199011015840

= 119902 1199021015840

= 21199012

minus 119891119902 minus 2(

119886

119888

)

2

1205791015840

= 119911 1199111015840

= minusPr (119891119911 minus 120582119901120579)

(18)


0

05

1

15

2

25

3

0 1 2 3 4 5

f998400 (120578)

120578

ac = 25ac = 2

ac = 02

ac = 01

ac = 05

ac = 15ac = 1

ac = 3

Figure 2 Velocity profiles 1198911015840(120578) for various values of 119886119888

0

2

4

6

8

10

12

14

0 1 2 3 4 5

f(120578)

120578

ac = 01 02 05 1 15 2 25 3

Figure 3 The variations of 119891(120578) for various values of 119886119888

with the boundary conditions

119891 (0) = 0 119901 (0) = 1 120579 (0) = 1 (19)

To solve (18) with (19) as an IVP the values for 119902(0) that is11989110158401015840

(0) and 119911(0) that is 1205791015840(0) are needed but no such valuesare given The initial guess values for 119891

10158401015840

(0) and 1205791015840

(0) arechosen and the fourth-order Runge-Kutta method is appliedto obtain the solution The calculated values of 1198911015840(120578) and120579(120578) at 120578

infin(= 25) are compared with the given boundary

conditions 1198911015840(120578infin) = 119886119888 and 120579(120578

infin) = 0 and adjust values

of 11989110158401015840

(0) and 1205791015840

(0) using ldquosecant methodrdquo to give betterapproximation for the solutionThe step size is taken as Δ120578 =

001 The process is repeated until we get the results correctup to the desired accuracy of 10minus6 level

4 Results and Discussion

The numerical computations have been carried out forvarious values of the physical parameters involved in theequations namely the velocity ratio parameter 119886119888 thePrandtl number Pr and the parameter 120582 For illustration ofthe results computed values are plotted through graphs andthe physical explanations are rendered for all cases

To ensure the accuracy of the numerical scheme wecompare the values of 11989110158401015840(0) and 119891(infin) with the published

= 05 120582 = 1

ac = 01 02 05 1 15 2 25 3

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

Pr

Figure 4 Temperature profiles 120579(120578) for various values of 119886119888

Table 1 The values of 11989110158401015840(0) and 119891(infin) for 119886119888 = 0

Magyari and Keller [36] Present studyminus11989110158401015840

(0) 1281808 12818084119891 (infin) 0905639 09056433

data by Magyari and Keller [36] in Table 1 without anyexponential free stream 119880

infin= 0 that is 119886119888 = 0 (119886 = 0) and

those are found in excellent agreement Thus we are feelingconfident that the presented results are accurate

As similar to that of stagnation point flow with linearstraining velocity towards a linearly stretching sheet in theboundary layer flow with exponentially varied free streamvelocity over exponentially stretching sheet two differentkinds of boundary layer structures have formednear the sheetdepending upon the ratio of two constants relating to thestretching and free stream velocities that is on the velocityratio parameter 119886119888 for 119886119888 gt 1 and 119886119888 lt 1 Also it isimportant to note that for 119886119888 = 1 no velocity boundarylayer is formed near the sheetThe velocity profiles for variousvalues of 119886119888 are depicted in Figure 2 Also the variations of119891(120578) and the dimensionless temperature for different valuesof 119886119888 are plotted in Figures 3 and 4 It is observed that 119891(120578)increases with increasing values of 119886119888 and the temperatureat a point decreases with 119886119888

The momentum and thermal boundary layer thicknessesare denoted by 120575 and 120575

119879 respectively and are described

by the equations 120575 = 120578120575radic2120592119871119888 exp(minus1199092119871) and 120575

119879=

120578120575119879radic2120592119871119888 exp(minus1199092119871) The dimensionless boundary layer

thicknesses 120578120575and 120578120575119879

are defined as the values of 120578 (nondi-mensional distance from the surface) at which the differenceof dimensionless velocity 119891

1015840

(120578) and the parameter 119886119888 hasbeen reduced to 0001 and the dimensionless temperature120579(120578) has been decayed to 0001 respectively Both boundarylayer thicknesses are very vital in practical and theoreticalstandpoints The values 120578

120575and 120578

120575119879are given in Table 2 for

several values of 119886119888 From the table it is seen that themomentum boundary layer thickness decreases when 119886119888

increases (both for 119886119888 gt 1 and 119886119888 lt 1) But it isworth noting that themomentumboundary layer thickness isconsiderably thinner than that of the boundary layer of linear


Table 2 Values of 120578120575and 120578

120575119879for several values of 119886119888 with Pr = 05 and 120582 = 1

119886119888 01 02 05 15 20 25 30Mahapatra and Gupta [17] 120578

120575696 591 436 mdash 262 mdash 230

Present study 120578120575

504 417 291 187 179 168 159120578120575119879

897 778 563 354 311 280 257

ac = 2 = 05

0 1 2 3 5 8120582 = minus15 minus14 minus125 minus1 minus05

0

02

04

06

08

1

12

14

0 05 1 15 2 25 3 35 4

120579(120578)

120578

Pr

Figure 5 Temperature profiles 120579(120578) for various values of 120582 with119886119888 = 2

ac = 01 Pr = 05

= minus15 minus14 minus125 minus1 050 1 2 3 5 8

0

02

04

06

08

1

0 2 4 6 8 10 12

120578

120579(120578)

120582


stagnation point flow over a linearly stretching sheet whichis obtained by Mahapatra and Gupta [17] In exponentialflow past an exponentially stretching sheet though the flowdynamics is similar to that of linear stagnation point flowthe momentum boundary layer thickness is smaller and it isphysically important Due to the fact that larger velocities ofthe free stream and the surface stretching make the nonzeroviscosity layer thinner it makes boundary layer thicknessthinner Similar to the velocity field the thermal boundarylayer thickness also decreases with increase of 119886119888

The effect of the parameter 120582 on the temperature distribu-tion is very significant because the increase of wall tempera-ture along the sheet depends upon 120582 For various values of 120582the dimensionless temperature profiles 120579(120578) are presented inFigures 5 and 6 for 119886119888 = 2 and 119886119888 = 01 respectively In boththe cases the figures exhibit the overshoot of temperatureprofiles for some negative values of 120582 Hence when the

2

ac = 2 = 05

120579998400 (120578)

120578

1

05

0

minus05

minus1

minus15

minus2

minus250 05 1 15 25 3 35 4

= minus15 minus14 minus125 minus1 minus05 0 1 2 3 5 8120582

Pr

Figure 7 Temperature gradient profiles 1205791015840(120578) for various values of120582 with 119886119888 = 2

ac = 01 = 05

120582 = minus15 minus14 minus125 minus1 minus05 0 1 2 3 5 8

0 2 4 6 8 10 12

minus15

minus1

minus05

0

120578

120579998400 (120578)

Pr


wall temperature in inverse exponential forms the thermalovershoot occurs Also it is noticed that the temperatures atparticular point and the thermal boundary layer thicknessdecrease with increase of 120582 Furthermore it is very importantto note that the overshoot pick is of higher length when freestream velocity is large compared to the stretching velocity(119886119888 = 2) The dimensionless temperature gradient profiles1205791015840

(120578) for different 120582 are depicted in Figures 7 and 8 for 119886119888 = 2

and 119886119888 = 01 respectively From these twofigures it is clearlyseen that the temperature gradient at the sheet is positivefor some values of 120582 and negative for some other valuesof 120582 So heat transfer from the sheet and heat absorptionat the sheet occur depending on the values of 120582 for bothvalues of 119886119888 Because of inverse exponential distribution ofsurface temperature (120582 gt 0) the heat absorption is found inthe thermal boundary layer Thus the parameter 120582 related towall temperature plays a vital role in controlling the cooling


= 01 02 03 05 1 2

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

ac = 2 120582 = 1

Pr

Figure 9 Temperature profiles 120579(120578) for various values of Pr with119886119888 = 2 and 120582 = 1

= 01 02 03 05 1 2

ac = 01 120582 = 1

0

02

04

06

08

1

0 5 10 15 20 25

120579(120578)

120578

Pr


process of the surface and consequently it has a great impacton the quality of the manufactured products

The temperature profiles for various values of Prandtlnumber Pr are demonstrated in Figures 9ndash12 with 119886119888 = 201 and 120582 = 1 minus15 In Figures 9 and 10 for 120582 = 1 itis observed that the temperature at a fixed point decreaseswith Pr for both values of 119886119888 Accordingly the thermalboundary layer thickness decreases with increasing values ofPr and the heat transfers from the sheet to the adjacent fluidBut when 120582 = minus15 (Figures 11 and 12) the temperatureovershoot is observed that is the sheet absorbs the heatand interestingly for both values of 119886119888 the height of theovershoot pick increases with Pr so the heat absorption atthe sheet also increases However similar to the linear case inthis situation the thermal boundary layer thickness decreaseswith increasing values of Pr For increase of Prandtl numberthe fluid thermal conductivity reduces and consequently thethermal boundary layer thickness becomes thinner

The values of 11989110158401015840(0) related to local skin-friction coeffi-cient and minus120579

1015840

(0) related to local Nusselt number are plottedin Figures 13ndash15 for various values of 119886119888 120582 and Pr The valueof 11989110158401015840(0) increases with velocity ratio parameter 119886119888 Alsothe value of minus1205791015840(0) increases with 120582 Moreover for the larger

ac = 2 120582 = minus15

= 01 02 03 05 1 2

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9

120579(120578)

120578

Pr

Figure 11 Temperature profiles 120579(120578) for various values of Pr with119886119888 = 2 and 120582 = minus15

0

02

04

06

08

1

0 5 10 15 20 25

ac = 01 120582 = minus15

= 01 02 03 05 1 2

120579(120578)

120578

Pr


negative values of 120582 the heat absorption [minus1205791015840

(0) lt 0] at thesurface is found With the increase of 119886119888 and Pr the heattransfer enhances and also heat absorption becomes larger(Figures 14 and 15)

5 Concluding Remarks

The momentum and heat transfer characteristics of thelaminar boundary layer flow induced by an exponentiallystretching sheet in an exponentially moving free streamare investigated The transformed governing equations aresolved by shooting technique using Runge-Kutta methodThe findings of this study can be summarized as follows

(a) The momentum boundary layer thickness for thisflow is significantly smaller than that of the linearstagnation point flow past a linearly stretching sheet

(b) The momentum and thermal boundary layer thick-nesses decrease with increase of velocity ratio param-eter 119886119888

(c) For positive values of 120582 and smaller negative valuesheat transfers from surface to the ambient fluidand importantly for greater negative value of 120582 heat


= 05 120582 = 1

ac

06

07

08

09

1

11

12

13

05 1 15 2 25 3

ac

05 1 15 2 25 3minus2minus101234567

minus120579998400 (0)

f998400998400(0)

Pr

Figure 13 Values of 11989110158401015840(0) and minus1205791015840

(0) for various values of 119886119888

= 05

ac = 01

ac = 2

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

120582

minus120579998400 (0)

Pr

Figure 14 Values of minus1205791015840(0) versus 120582 for various values of 119886119888 withPr = 05

= 05 1

ac = 2

ac = 15

minus2

minus1

0

1

2

3

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

minus2 0 2 4 6 8

120582

120582

= 05 1

minus120579998400 (0)

Pr

Pr

Figure 15 Values of minus1205791015840(0) versus 120582 for various values of 119886119888 and Pr

transfers from the ambient fluid to the surface theheat absorption occurs

(d) Temperature and thermal boundary layer thicknessesdecrease with 120582 a parameter associated with walltemperature

(e) The heat transfer from the sheet and the heat absorp-tion at the sheet are enhanced with the increase inboth velocity ratio parameter and Prandtl number

Nomenclature

119886 Constant related to free stream velocity119888 Constant related to stretching velocity119886119888 Velocity ratio parameter119888119901 Specific heat

119891 Dimensionless stream function119871 Reference lengthPr Prandtl number119879 Temperature119879119908 Variable temperature of the sheet

119879infin Free stream temperature

119906 velocity component in 119909 directionV Velocity component in 119910 direction119880119908 Stretching velocity

119880infin Free stream velocity

120578 Similarity variable120581 Fluid thermal conductivity120582 Parameter120583 Coefficient of fluid viscosity120592 Kinematic viscosity of fluid120588 Density of fluid120595 Stream function120579 Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Oneof the authors K Bhattacharyya gratefully acknowledgesthe financial support ofNational Board for Higher Mathemat-ics (NBHM) Department of Atomic Energy Government ofIndia for pursuing this work The authors also express hissincere thanks to the esteemed referees for their constructivesuggestions

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furAngewandte Mathematik und Physik vol 21 no 4 pp 645ndash6471970

[2] P S Gupta and A S Gupta ldquoHeat and mass transfer ona stretching sheet with suction and blowingrdquo The CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash746 1977


[3] C-K Chen and M-I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[4] K B Pavlov ldquoMagnetohydrodynamic flow of an incompressibleviscous fluid caused by the deformation of a plane surfacerdquoMagnitnaya Gidrodinamika vol 10 pp 146ndash148 1974

[5] M E Ali ldquoHeat transfer characteristics of a continuous stretch-ing surfacerdquoHeat andMass Transfer vol 29 no 4 pp 227ndash2341994

[6] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[8] M E Ali ldquoThe buoyancy effects on the boundary layers inducedby continuous surfaces stretchedwith rapidly decreasing veloci-tiesrdquoHeat andMass Transfer vol 40 no 3-4 pp 285ndash291 2004

[9] R Tsai K H Huang and J S Huang ldquoFlow and heattransfer over an unsteady stretching surface with non-uniformheat sourcerdquo International Communications in Heat and MassTransfer vol 35 no 10 pp 1340ndash1343 2008

[10] K Bhattacharyya and G C Layek ldquoChemically reactive solutedistribution in MHD boundary layer flow over a permeablestretching sheet with suction or blowingrdquoChemical EngineeringCommunications vol 197 no 12 pp 1527ndash1540 2010

[11] K Bhattacharyya and G C Layek ldquoSlip effect on diffusionof chemically reactive species in boundary layer flow over avertical stretching sheet with suction or blowingrdquo ChemicalEngineering Communications vol 198 no 11 pp 1354ndash13652011

[12] K Bhattacharyya ldquoEffects of heat sourcesink on mhd flowand heat transfer over a shrinking sheet with mass suctionrdquoChemical Engineering Research Bulletin vol 15 no 1 pp 12ndash172011

[13] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[14] K Bhattacharyya T Hayat and A Alsaedi ldquoAnalytic solutionfor magnetohydrodynamic boundary layer flow of Casson fluidover a stretchingshrinking sheet with wall mass transferrdquoChinese Physics B vol 22 no 2 Article ID 024702 2013

[15] K Hiemenz ldquoDie Grenzschicht an einem in den gleichformin-gen Flussigkeits-strom einge-tauchten graden KreiszylinderrdquoDinglers Polytechnic Journal vol 326 pp 321ndash324 1911

[16] T C Chiam ldquoStagnation-point flow towards a stretching platerdquoJournal of the Physical Society of Japan vol 63 no 6 pp 2443ndash2444 1994

[17] T R Mahapatra and A S Gupta ldquoMagnetohydrodynamicstagnation-point flow towards a stretching sheetrdquoActaMechan-ica vol 152 no 1ndash4 pp 191ndash196 2001

[18] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[19] T R Mahapatra and A S Gupta ldquoHeat transfer in stagnation-point flow towards a stretching sheetrdquo Heat and Mass Transfervol 38 no 6 pp 517ndash521 2002

[20] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[21] A Ishak R Nazar and I Pop ldquoMixed convection boundarylayers in the stagnation-point flow toward a stretching verticalsheetrdquoMeccanica vol 41 no 5 pp 509ndash518 2006

[22] G C Layek S Mukhopadhyay and S A Samad ldquoHeatand mass transfer analysis for boundary layer stagnation-point flow towards a heated porous stretching sheet withheat absorptiongeneration and suctionblowingrdquo InternationalCommunications in Heat and Mass Transfer vol 34 no 3 pp347ndash356 2007

[23] T Hayat T Javed and Z Abbas ldquoMHD flow of a micropolarfluid near a stagnation-point towards a non-linear stretchingsurfacerdquoNonlinear Analysis RealWorld Applications vol 10 no3 pp 1514ndash1526 2009

[24] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on an unsteady boundary layer stagnation-point flowand heat transfer towards a stretching sheetrdquo Chinese PhysicsLetters vol 28 no 9 Article ID 094702 2011

[25] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[26] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 302ndash307 2011

[27] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[28] T RMahapatra S K Nandy andA S Gupta ldquoMomentum andheat transfer in MHD stagnation-point flow over a shrinkingsheetrdquo Journal of Applied Mechanics Transactions ASME vol78 no 2 Article ID 021015 2011

[29] K Bhattacharyya and K Vajravelu ldquoStagnation-point flow andheat transfer over an exponentially shrinking sheetrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 2728ndash2734 2012

[30] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics B Fluids vol 30 no 1 pp 120ndash128 2011

[31] K Bhattacharyya ldquoDual solutions in boundary layerstagnation-point flow and mass transfer with chemicalreaction past a stretchingshrinking sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 7 pp917ndash922 2011

[32] K Bhattacharyya ldquoDual solutions in unsteady stagnation-pointflow over a shrinking sheetrdquo Chinese Physics Letters vol 28 no8 Article ID 084702 2011

[33] K Bhattacharyya G Arif and W A Pramanik ldquoMHDboundary layer stagnation-point flow and mass transfer over apermeable shrinking sheet with suctionblowing and chemicalreactionrdquo Acta Technica vol 57 pp 1ndash15 2012

[34] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoReac-tive solute transfer in magnetohydrodynamic boundary layerstagnation-point flow over a stretching sheet with suc-tionblowingrdquo Chemical Engineering Communications vol 199no 3 pp 368ndash383 2012


[35] K Bhattacharyya ldquoHeat transfer in unsteady boundary layerstagnation-point flow towards a shrinking sheetrdquo Ain ShamsEngineering Journal vol 4 no 2 pp 259ndash264 2013

[36] E Magyari and B Keller ldquoHeat and mass transfer in theboundary layers on an exponentially stretching continuoussurfacerdquo Journal of Physics D Applied Physics vol 32 no 5 pp577ndash585 1999

[37] E M A Elbashbeshy ldquoHeat transfer over an exponen-tially stretching continuous surface with suctionrdquo Archives ofMechanics vol 53 no 6 pp 643ndash651 2001

[38] S K Khan and E Sanjayanand ldquoViscoelastic boundary layerflow and heat transfer over an exponential stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 48 no 8pp 1534ndash1542 2005

[39] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[40] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[41] M Q Al-Odat R A Damseh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[42] M Sajid and T Hayat ldquoInfluence of thermal radiation on theboundary layer flow due to an exponentially stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol35 no 3 pp 347ndash356 2008

[43] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[44] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[45] I C Mandal and S Mukhopadhyay ldquoHeat transfer analysis forfluid flow over an exponentially stretching porous sheet withsurface heat flux in porous mediumrdquo Ain Shams EngineeringJournal vol 4 no 1 pp 103ndash110 2013

[46] K Bhattacharyya ldquoBoundary layer flow and heat transfer overan exponentially shrinking sheetrdquo Chinese Physics Letters vol28 no 7 Article ID 074701 2011

[47] K Bhattacharyya and I Pop ldquoMHDBoundary layer flow due toan exponentially shrinking sheetrdquoMagnetohydrodynamics vol47 pp 337ndash344 2011

[48] K Bhattacharyya and G C Layek ldquoMagnetohydrodynamicboundary layer flow of nanofluid over an exponentially stretch-ing permeable sheetrdquo Physics Research International vol 2014Article ID 592536 12 pages 2014

International Journal of

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Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



on the MHD boundary layer flow over steady and unsteadyshrinkingstretching sheet and Bhattacharyya et al [14]found the analytic solution of MHD flow of non-NewtonianCasson fluid flow induced due to stretchingshrinking sheetwith wall mass transfer

On the other hand Hiemenz [15] first studied the steadyflow in the neighbourhood of a stagnation point Chiam[16] investigated a problem which is a combination of theworks of Hiemenz [15] and Crane [1] that is the linearstagnation point flow towards a linear stretching sheet whenthe stretching rate of the plate is equal to the strain rateof the stagnation point flow and he found no boundarylayer structure near the plate After few years Mahapatraand Gupta [17] reinvestigated the same stagnation pointflow towards a stretching sheet with different stretching andstraining velocities and they found two kinds of boundarylayers near the sheet depending on the ratio of the stretchingand straining velocity ratesWang [18] demonstrated the stag-nation point flow towards a shrinking sheet Furthermore thebehaviour of stagnation point flow over stretchingshrinkingsheet under different physical aspects was discussed by manyresearchers [19ndash35]

In last few decades in almost all investigations of the flowover a stretching sheet the flow occurs because of linearvariation of stretching velocity of the flat sheet with thedistance from the origin So the boundary layer flow inducedby an exponentially stretching sheet is not studied muchthough it is very important and practical flow frequentlyappeared in many engineering processes In 1999 Magyariand Keller [36] first consider the boundary layer flow dueto an exponentially stretching sheet and he also studiedthe heat transfer in the flow taking exponentially variedwall temperature After that Elbashbeshy [37] numericallyexamined the flow and heat transfer over an exponentiallystretching surface considering wall mass suction Khan andSanjayanand [38] investigated the flow of viscoelastic fluidand heat transfer over an exponentially stretching sheetwith viscous dissipation effects Partha et al [39] presenteda similarity solution for mixed convection flow past anexponentially stretching surface by taking into account theinfluence of viscous dissipation on the convective transportSanjayanand and Khan [40] discussed the effects of heatand mass transfer on the boundary layer flow of viscoelasticfluid taking 119899th order chemical reaction Al-Odat et al [41]investigated the effect magnetic field on thermal boundarylayer on an exponentially stretching continuous surface withan exponential temperature distribution Later Sajid andHayat [42] showed the influence of thermal radiation onthe boundary layer flow past an exponentially stretchingsheet and they reported series solutions for velocity andtemperature using homotopy analysis method (HAM) andBidin and Nazar [43] obtained a numerical solution of thesame problem Ishak [44] studied the MHD boundary layerflow over an exponentially stretching sheet in presence ofthermal radiationMandal andMukhopadhyay [45] analyzedthe heat transfer in the flowdue to an exponentially stretchingporous sheet with surface heat flux embedded in a porousmedium Bhattacharyya [46] discussed the boundary layerflow and heat transfer past an exponentially shrinking sheet

and Bhattacharyya and Pop [47] presented the effect of mag-netic field on that flow Recently The MHD boundary layernanofluid flow due to an exponentially stretching permeablesheet was demonstrated by Bhattacharyya and Layek [48]

Inspired by the above investigations in the present paperthe thermal boundary layer in incompressible flow over anexponentially stretching sheet in an exponentially movingfree stream is studied The wall temperature distribution istaken variable in exponential form Using similarity trans-formations the governing partial differential equations aretransformed into self-similar ordinary differential equationsThen the transformed equations are solved by standardshooting technique using fourth order Runge-Kutta methodThe numerical computations are presented through somefigures and the various characteristics of flow and heattransfer are thoroughly describedThis type of flow is possiblein reality in some special situations The motive of theinvestigation is to study new type of flow dynamics and theheat transfer in the flow field

2 Analysis of Motion and Heat Transfer

Consider the steady two-dimensional laminar boundarylayer flow of a viscous incompressible fluid and heat transferover an exponentially stretching sheet (of velocity119880

119908)with an

exponentially moving free stream (of velocity119880infin)The sheet

coincides with the plane119910 = 0 and the flow confined to119910 gt 0The governing equations for the flow and the temperature arewritten in usual notation as

120597119906

120597119909

+

120597V120597119910

= 0 (1)

119906

120597119906

120597119909

+ V120597119906

120597119910

= 119880infin

119889119880infin

119889119909

+ 120592

1205972

119906

1205971199102 (2)

119906

120597119879

120597119909

+ V120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102 (3)

where 119906 and V are the velocity components in 119909 and 119910

directions respectively119880infinis the fluid velocity in free stream

120592(= 120583120588) is the kinematic fluid viscosity 120588 is the fluid density120583 is the coefficient of fluid viscosity 119879 is the temperature 120581 isthe fluid thermal conductivity and 119888

119901is the specific heat The

diagram of the physical problem is given in Figure 1The boundary conditions corresponding to the velocity

components and the temperature are given by

119906 = 119880119908(119909) V = 0 at 119910 = 0

119906 997888rarr 119880infin(119909) as 119910 997888rarr infin

(4)

119879 = 119879119908= 119879infin

+ 1198790exp(120582119909

2119871

) at 119910 = 0

119879 997888rarr 119879infin


(5)

where 119879119908is the variable temperature of the sheet 119879

infinis

the free stream temperature assumed to be constant 1198790is

a constant which measures the rate of temperature increase


Twx

u


y

Uw

Tinfin Uinfin


Stretching sheet



119908and 119880


by

119880119908(119909) = 119888 exp(119909

119871

) 119880infin(119909) = 119886 exp(119909

119871

) (6)



119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(7)


120597120595

120597119910

1205972

120595

120597119909120597119910

minus

120597120595

120597119909

1205972

120595

1205971199102= 119880infin

119889119880infin

119889119909

+ 120592

1205973

120595

1205971199103

120597120595

120597119910

120597119879

120597119909

minus

120597120595

120597119909

120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102

(8)


120597120595

120597119910

= 119880119908(119909)

120597120595

120597119909

= 0 at 119910 = 0

120597120595

120597119910


(9)


120595 = radic2120592119871119888119891 (120578) exp( 119909

2119871

)

119879 = 119879infin

+ (119879119908minus 119879infin) 120579 (120578)

(10)


119910radic1198882120592119871 exp(1199092119871)


119891101584010158401015840

+ 11989111989110158401015840

minus 211989110158402

+ 2(

119886

119888

)

2

= 0 (11)

12057910158401015840

+ Pr (1198911205791015840 minus 1205821198911015840

120579) = 0 (12)



following forms

119891 (120578) = 0 1198911015840

(120578) = 1 at 120578 = 0

1198911015840

(120578) 997888rarr

119886

119888



(13)




119862119891119909

=

120591119908

1003816100381610038161003816119910=0

1205881198802

119908

(14)


119908=

120583[120597119906120597119910]119910=0

So

radic2Re119862119891119909

= 11989110158401015840

(0) (15)



Nu119909= minus

119909

119879119908minus 119879infin

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

(16)

So

Nu119909

radicRe119909

= minusradic

119909

2119871

1205791015840

(0) (17)





1198911015840

= 119901 1199011015840

= 119902 1199021015840

= 21199012

minus 119891119902 minus 2(

119886

119888

)

2

1205791015840

= 119911 1199111015840

= minusPr (119891119911 minus 120582119901120579)

(18)


0

05

1

15

2

25

3

0 1 2 3 4 5

f998400 (120578)

120578

ac = 25ac = 2

ac = 02

ac = 01

ac = 05

ac = 15ac = 1

ac = 3


0

2

4

6

8

10

12

14

0 1 2 3 4 5

f(120578)

120578

ac = 01 02 05 1 15 2 25 3



119891 (0) = 0 119901 (0) = 1 120579 (0) = 1 (19)



10158401015840

(0) and 1205791015840





of 11989110158401015840

(0) and 1205791015840






= 05 120582 = 1

ac = 01 02 05 1 15 2 25 3

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

Pr




(0) 1281808 12818084119891 (infin) 0905639 09056433


infin= 0 that is 119886119888 = 0 (119886 = 0) and






119879=


thicknesses 120578120575and 120578120575119879


1015840


120575and 120578








120575696 591 436 mdash 262 mdash 230


504 417 291 187 179 168 159120578120575119879

897 778 563 354 311 280 257

ac = 2 = 05


0

02

04

06

08

1

12

14

0 05 1 15 2 25 3 35 4

120579(120578)

120578

Pr


ac = 01 Pr = 05


0

02

04

06

08

1

0 2 4 6 8 10 12

120578

120579(120578)

120582




2

ac = 2 = 05

120579998400 (120578)

120578

1

05

0

minus05

minus1

minus15

minus2

minus250 05 1 15 25 3 35 4


Pr


ac = 01 = 05


0 2 4 6 8 10 12

minus15

minus1

minus05

0

120578

120579998400 (120578)

Pr






= 01 02 03 05 1 2

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

ac = 2 120582 = 1

Pr


= 01 02 03 05 1 2

ac = 01 120582 = 1

0

02

04

06

08

1

0 5 10 15 20 25

120579(120578)

120578

Pr





1015840


ac = 2 120582 = minus15

= 01 02 03 05 1 2

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9

120579(120578)

120578

Pr


0

02

04

06

08

1

0 5 10 15 20 25

ac = 01 120582 = minus15

= 01 02 03 05 1 2

120579(120578)

120578

Pr










= 05 120582 = 1

ac

06

07

08

09

1

11

12

13

05 1 15 2 25 3

ac

05 1 15 2 25 3minus2minus101234567

minus120579998400 (0)

f998400998400(0)

Pr



= 05

ac = 01

ac = 2

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

120582

minus120579998400 (0)

Pr


= 05 1

ac = 2

ac = 15

minus2

minus1

0

1

2

3

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

minus2 0 2 4 6 8

120582

120582

= 05 1

minus120579998400 (0)

Pr

Pr





Nomenclature









Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Twx

u


y

Uw

Tinfin Uinfin


Stretching sheet



119908and 119880


by

119880119908(119909) = 119888 exp(119909

119871

) 119880infin(119909) = 119886 exp(119909

119871

) (6)



119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(7)


120597120595

120597119910

1205972

120595

120597119909120597119910

minus

120597120595

120597119909

1205972

120595

1205971199102= 119880infin

119889119880infin

119889119909

+ 120592

1205973

120595

1205971199103

120597120595

120597119910

120597119879

120597119909

minus

120597120595

120597119909

120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102

(8)


120597120595

120597119910

= 119880119908(119909)

120597120595

120597119909

= 0 at 119910 = 0

120597120595

120597119910


(9)


120595 = radic2120592119871119888119891 (120578) exp( 119909

2119871

)

119879 = 119879infin

+ (119879119908minus 119879infin) 120579 (120578)

(10)


119910radic1198882120592119871 exp(1199092119871)


119891101584010158401015840

+ 11989111989110158401015840

minus 211989110158402

+ 2(

119886

119888

)

2

= 0 (11)

12057910158401015840

+ Pr (1198911205791015840 minus 1205821198911015840

120579) = 0 (12)



following forms

119891 (120578) = 0 1198911015840

(120578) = 1 at 120578 = 0

1198911015840

(120578) 997888rarr

119886

119888



(13)




119862119891119909

=

120591119908

1003816100381610038161003816119910=0

1205881198802

119908

(14)


119908=

120583[120597119906120597119910]119910=0

So

radic2Re119862119891119909

= 11989110158401015840

(0) (15)



Nu119909= minus

119909

119879119908minus 119879infin

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

(16)

So

Nu119909

radicRe119909

= minusradic

119909

2119871

1205791015840

(0) (17)





1198911015840

= 119901 1199011015840

= 119902 1199021015840

= 21199012

minus 119891119902 minus 2(

119886

119888

)

2

1205791015840

= 119911 1199111015840

= minusPr (119891119911 minus 120582119901120579)

(18)


0

05

1

15

2

25

3

0 1 2 3 4 5

f998400 (120578)

120578

ac = 25ac = 2

ac = 02

ac = 01

ac = 05

ac = 15ac = 1

ac = 3


0

2

4

6

8

10

12

14

0 1 2 3 4 5

f(120578)

120578

ac = 01 02 05 1 15 2 25 3



119891 (0) = 0 119901 (0) = 1 120579 (0) = 1 (19)



10158401015840

(0) and 1205791015840





of 11989110158401015840

(0) and 1205791015840






= 05 120582 = 1

ac = 01 02 05 1 15 2 25 3

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

Pr




(0) 1281808 12818084119891 (infin) 0905639 09056433


infin= 0 that is 119886119888 = 0 (119886 = 0) and






119879=


thicknesses 120578120575and 120578120575119879


1015840


120575and 120578








120575696 591 436 mdash 262 mdash 230


504 417 291 187 179 168 159120578120575119879

897 778 563 354 311 280 257

ac = 2 = 05


0

02

04

06

08

1

12

14

0 05 1 15 2 25 3 35 4

120579(120578)

120578

Pr


ac = 01 Pr = 05


0

02

04

06

08

1

0 2 4 6 8 10 12

120578

120579(120578)

120582




2

ac = 2 = 05

120579998400 (120578)

120578

1

05

0

minus05

minus1

minus15

minus2

minus250 05 1 15 25 3 35 4


Pr


ac = 01 = 05


0 2 4 6 8 10 12

minus15

minus1

minus05

0

120578

120579998400 (120578)

Pr






= 01 02 03 05 1 2

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

ac = 2 120582 = 1

Pr


= 01 02 03 05 1 2

ac = 01 120582 = 1

0

02

04

06

08

1

0 5 10 15 20 25

120579(120578)

120578

Pr





1015840


ac = 2 120582 = minus15

= 01 02 03 05 1 2

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9

120579(120578)

120578

Pr


0

02

04

06

08

1

0 5 10 15 20 25

ac = 01 120582 = minus15

= 01 02 03 05 1 2

120579(120578)

120578

Pr










= 05 120582 = 1

ac

06

07

08

09

1

11

12

13

05 1 15 2 25 3

ac

05 1 15 2 25 3minus2minus101234567

minus120579998400 (0)

f998400998400(0)

Pr



= 05

ac = 01

ac = 2

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

120582

minus120579998400 (0)

Pr


= 05 1

ac = 2

ac = 15

minus2

minus1

0

1

2

3

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

minus2 0 2 4 6 8

120582

120582

= 05 1

minus120579998400 (0)

Pr

Pr





Nomenclature









Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

05

1

15

2

25

3

0 1 2 3 4 5

f998400 (120578)

120578

ac = 25ac = 2

ac = 02

ac = 01

ac = 05

ac = 15ac = 1

ac = 3


0

2

4

6

8

10

12

14

0 1 2 3 4 5

f(120578)

120578

ac = 01 02 05 1 15 2 25 3



119891 (0) = 0 119901 (0) = 1 120579 (0) = 1 (19)



10158401015840

(0) and 1205791015840





of 11989110158401015840

(0) and 1205791015840






= 05 120582 = 1

ac = 01 02 05 1 15 2 25 3

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

Pr




(0) 1281808 12818084119891 (infin) 0905639 09056433


infin= 0 that is 119886119888 = 0 (119886 = 0) and






119879=


thicknesses 120578120575and 120578120575119879


1015840


120575and 120578








120575696 591 436 mdash 262 mdash 230


504 417 291 187 179 168 159120578120575119879

897 778 563 354 311 280 257

ac = 2 = 05


0

02

04

06

08

1

12

14

0 05 1 15 2 25 3 35 4

120579(120578)

120578

Pr


ac = 01 Pr = 05


0

02

04

06

08

1

0 2 4 6 8 10 12

120578

120579(120578)

120582




2

ac = 2 = 05

120579998400 (120578)

120578

1

05

0

minus05

minus1

minus15

minus2

minus250 05 1 15 25 3 35 4


Pr


ac = 01 = 05


0 2 4 6 8 10 12

minus15

minus1

minus05

0

120578

120579998400 (120578)

Pr






= 01 02 03 05 1 2

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

ac = 2 120582 = 1

Pr


= 01 02 03 05 1 2

ac = 01 120582 = 1

0

02

04

06

08

1

0 5 10 15 20 25

120579(120578)

120578

Pr





1015840


ac = 2 120582 = minus15

= 01 02 03 05 1 2

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9

120579(120578)

120578

Pr


0

02

04

06

08

1

0 5 10 15 20 25

ac = 01 120582 = minus15

= 01 02 03 05 1 2

120579(120578)

120578

Pr










= 05 120582 = 1

ac

06

07

08

09

1

11

12

13

05 1 15 2 25 3

ac

05 1 15 2 25 3minus2minus101234567

minus120579998400 (0)

f998400998400(0)

Pr



= 05

ac = 01

ac = 2

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

120582

minus120579998400 (0)

Pr


= 05 1

ac = 2

ac = 15

minus2

minus1

0

1

2

3

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

minus2 0 2 4 6 8

120582

120582

= 05 1

minus120579998400 (0)

Pr

Pr





Nomenclature









Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120575696 591 436 mdash 262 mdash 230


504 417 291 187 179 168 159120578120575119879

897 778 563 354 311 280 257

ac = 2 = 05


0

02

04

06

08

1

12

14

0 05 1 15 2 25 3 35 4

120579(120578)

120578

Pr


ac = 01 Pr = 05


0

02

04

06

08

1

0 2 4 6 8 10 12

120578

120579(120578)

120582




2

ac = 2 = 05

120579998400 (120578)

120578

1

05

0

minus05

minus1

minus15

minus2

minus250 05 1 15 25 3 35 4


Pr


ac = 01 = 05


0 2 4 6 8 10 12

minus15

minus1

minus05

0

120578

120579998400 (120578)

Pr






= 01 02 03 05 1 2

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

ac = 2 120582 = 1

Pr


= 01 02 03 05 1 2

ac = 01 120582 = 1

0

02

04

06

08

1

0 5 10 15 20 25

120579(120578)

120578

Pr





1015840


ac = 2 120582 = minus15

= 01 02 03 05 1 2

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9

120579(120578)

120578

Pr


0

02

04

06

08

1

0 5 10 15 20 25

ac = 01 120582 = minus15

= 01 02 03 05 1 2

120579(120578)

120578

Pr










= 05 120582 = 1

ac

06

07

08

09

1

11

12

13

05 1 15 2 25 3

ac

05 1 15 2 25 3minus2minus101234567

minus120579998400 (0)

f998400998400(0)

Pr



= 05

ac = 01

ac = 2

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

120582

minus120579998400 (0)

Pr


= 05 1

ac = 2

ac = 15

minus2

minus1

0

1

2

3

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

minus2 0 2 4 6 8

120582

120582

= 05 1

minus120579998400 (0)

Pr

Pr





Nomenclature









Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










= 01 02 03 05 1 2

0

02

04

06

08

1

0 1 2 3 4 5 6 7

120579(120578)

120578

ac = 2 120582 = 1

Pr


= 01 02 03 05 1 2

ac = 01 120582 = 1

0

02

04

06

08

1

0 5 10 15 20 25

120579(120578)

120578

Pr





1015840


ac = 2 120582 = minus15

= 01 02 03 05 1 2

0

02

04

06

08

1

12

14

0 1 2 3 4 5 6 7 8 9

120579(120578)

120578

Pr


0

02

04

06

08

1

0 5 10 15 20 25

ac = 01 120582 = minus15

= 01 02 03 05 1 2

120579(120578)

120578

Pr










= 05 120582 = 1

ac

06

07

08

09

1

11

12

13

05 1 15 2 25 3

ac

05 1 15 2 25 3minus2minus101234567

minus120579998400 (0)

f998400998400(0)

Pr



= 05

ac = 01

ac = 2

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

120582

minus120579998400 (0)

Pr


= 05 1

ac = 2

ac = 15

minus2

minus1

0

1

2

3

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

minus2 0 2 4 6 8

120582

120582

= 05 1

minus120579998400 (0)

Pr

Pr





Nomenclature









Acknowledgments


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










= 05 120582 = 1

ac

06

07

08

09

1

11

12

13

05 1 15 2 25 3

ac

05 1 15 2 25 3minus2minus101234567

minus120579998400 (0)

f998400998400(0)

Pr



= 05

ac = 01

ac = 2

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

120582

minus120579998400 (0)

Pr


= 05 1

ac = 2

ac = 15

minus2

minus1

0

1

2

3

minus2

minus1

0

1

2

3

minus2 0 2 4 6 8

minus2 0 2 4 6 8

120582

120582

= 05 1

minus120579998400 (0)

Pr

Pr





Nomenclature









Acknowledgments


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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