Transient Free Convective MHD Flow Past an Exponentially...

Research ArticleTransient Free Convective MHD Flow Past anExponentially Accelerated Vertical Porous Plate withVariable Temperature through a Porous Medium

Ashish Paul

Department of Mathematics Cotton College State University Guwahati 781001 India

Correspondence should be addressed to Ashish Paul ashpaul85gmailcom

Received 30 June 2016 Accepted 30 November 2016 Published 17 January 2017

Academic Editor Alberto Cardona

Copyright copy 2017 Ashish Paul 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

This paper is concerned with analytical solution of one-dimensional unsteady laminar boundary layer MHD flow of a viscousincompressible fluid past an exponentially accelerated infinite vertical plate in presence of transverse magnetic field The verticalplate and the medium of flow are considered to be porous The fluid is assumed to be optically thin and the magnetic Reynoldsnumber is considered small enough to neglect the induced hydromagnetic effects The governing boundary layer equations arefirst converted to dimensionless form and then solved by Laplace transform technique Numerical values of transient velocitytemperature skin friction and Nusselt number are illustrated and are presented in graphs for various sets of physical parametricvalues namely Grashof number accelerating parameter suction parameter permeability parameter radiation parameter magneticparameter and time It is found that the velocity decreases with increases of the suction parameter for both cases of cooling andheating of the porous plate whereas skin friction increases with increase of suction parameter

1 Introduction

The study of unsteady natural convective flow of viscousincompressible fluid past vertical bodies has wide engineer-ing and technological applications When free convectionflows occurs at high temperature the effects of radiation arevital importantThermal radiation is key to many fundamen-tal phenomena surrounding us from solar radiation to fireincandescent lamp and has played a major role in combus-tion and furnace design design of fins nuclear power plantscooling of towers gas turbines and various propulsion devicefor aircraft energy utilization temperature measurementsremote sensing for astronomy and space exploration

Magnetohydrodynamic (MHD) flow and heat and masstransfer processes occur in many industrial applications suchas the geothermal system aerodynamic processes chemicalcatalytic reactors and processes electromagnetic pumps andMHD power generators Many studies have been carriedout to investigate the magnetohydrodynamic transient freeconvective flow Gupta [1] first studied transient free con-vection of an electrically conducting fluid from a vertical

plate in the presence of magnetic field Free convectioneffects on flow past an exponentially accelerated verticalplate was studied by Singh and Kumar [2] Jha et al [3]analyzed mass transfer effects on exponentially acceleratedinfinite vertical plate with constant heat flux and uniformmass diffusion Muthucumaraswamy et al [4] studied theflow past an exponentially accelerated infinite vertical plate inthe presence of variable surface temperature Again Muthu-cumaraswamy et al [5] analysed the combined effects ofheat and mass transfer on exponentially accelerated verticalplate in presence of uniform magnetic field Rajesh et al[6] presented the analytic investigation of radiation andmass transfer effects on MHD free convection flow past anexponentially accelerated vertical plate in the presence ofvariable mass diffusion Rajesh and Chamkha [7] presentednumerical solution to study the unsteady convective flowpast an exponentially accelerated infinite vertical porousplate with Newtonian heating and viscous dissipation byGalerkin finite element method They have found that theskin friction coefficient increases with increases in either ofthe Eckert number Recently Srinivasa et al [8] presented

HindawiInternational Journal of Engineering MathematicsVolume 2017 Article ID 2981071 9 pageshttpsdoiorg10115520172981071

2 International Journal of Engineering Mathematics

both numerical and analytical solutions to study the effectof chemical reaction on unsteady incompressible viscousflow past an exponentially accelerated vertical plate with heatabsorption and variable temperature in a magnetic field

In recent years convective heat transfer in porous mediahas attracted considerable attention owing to its wide indus-trial and technological applications such as geothermalenergy recovery oil extraction fibre and granular insulationelectronic system cooling and porous material regenerativeheat exchangers Gupta et al [9] have studied free convectionon flow past an accelerated vertical plate in the presence ofviscous dissipative heat using perturbation method Kafou-sias and Raptis [10] extended their (Gupta et al [9]) work byincluding mass transfer effects subjected to variable suctionor injection Transient free convection flow past a plateembedded in a porous medium pioneered by Ping and Pop[11] Jha [12] presented analytic investigation of uniformtransverse magnetic field effect on the free-convection andmass-transform flow of an electrically conducting fluid pastan infinite vertical plate for uniformly accelerated motion ofthe plate through a porous medium Magyari et al [13] havepresented an analytical solution for unsteady free convectionin porousmedia Chaudhary and Jain [14ndash16] andChaudharyet al [17] analyzed MHD transient free convection effectson flow past a moving vertical plate embedded in porousmedium under different physical situations by employingLaplace transform technique Rajesh [18] presented an ana-lytic investigation of MHD free convection flow past anaccelerated vertical porous plate with variable temperaturethrough a porous medium Again Rajesh et al [19] studiedthe transient free convection MHD flow and heat transfer ofnanofluid past an impulsively started vertical porous plate inthe presence of viscous dissipation Recently Ibrahim [20]studied the effects of thermal radiation viscous dissipationand chemical reaction on unsteady two-dimensional bound-ary layer MHD flow over a nonisothermal stretching sheetembedded in porous medium

However to the best of authorsrsquo knowledge MHD freeconvective flow past an exponentially accelerated verticalporous plate with variable temperature through a porousmedium was never considered in the literatureThe objectiveof this paper is to study magnetohydrodynamic transientheat transfer flow past an exponentially accelerated infinitevertical porous plate with variable temperature and the plateis embedded in a porous medium The exact solutions ofthe dimensionless unsteady linear governing equations areobtained by Laplace transform technique

2 Mathematical Analysis

Anunsteady one-dimensional laminar free convective flow ofa viscous incompressible fluid past an infinite vertical porousplate through a porous medium with variable temperatureis considered The 119909-axis is being taken vertically upwardsalong the vertical plate and 119910-axis is taken to be normal tothe plate The physical model and coordinate system of theflow problem is shown in Figure 1 Initially it is assumed thatthe plate and fluid are at the same temperature 1198791015840infin in thestationary condition At 1199051015840 ge 0 the plate is exponentially

x

y

Bo

T998400w

g

ouoe

a998400 t998400

u998400

998400T998400infin

Figure 1 Physical model and coordinate system

accelerated with a velocity 1199061015840 = 11990610158400 exp(11988610158401199051015840) in its ownplane and the plate temperature is raised linearly with time119905 A uniform magnetic field is applied in the directionperpendicular to the plate The fluid is assumed to be slightlyconducting so that the magnetic Reynolds number is muchless than unity and hence the induced magnetic field isnegligible in comparison with the applied magnetic fieldThefluid considered here is a gray absorbingemitting radiationbut a nonscattering medium The viscous dissipation is alsoassumed to be negligible in the energy equation as themotion is due to free convection only It is also assumedthat all the fluid properties are constant except for thedensity in the buoyancy term which is given by the usualBoussinesqrsquos approximation Under these assumptions thegoverning boundary layer equations are

12059711990610158401205971199051015840 + V1015840

12059711990610158401205971199101015840 = ]

1205972119906101584012059711991010158402 + 119892120573 (1198791015840 minus 1198791015840infin)

minus 12059011986120120588 1199061015840 minus ]11990610158401198961015840

(1)

120588119862119901 (1205971198791015840

1205971199051015840 + V101584012059711987910158401205971199101015840 ) = 120581

1205972119879101584012059711991010158402 minus

1205971199021199031205971199101015840 (2)

with the following initial and boundary conditions

1199051015840 le 0 1199061015840 = 01198791015840 = 1198791015840infin forall1199101015840

1199051015840 gt 0 1199061015840 = 1199060 exp (11988610158401199051015840) 1198791015840 = 1198791015840infin (1198791015840119908 minus 1198791015840infin)1198601199051015840

at 1199101015840 = 01199061015840 997888rarr 01198791015840 997888rarr 1198791015840infin

as 119910 997888rarr infin

(3)

where 119860 = 11990620]

International Journal of Engineering Mathematics 3

The local radiant for the case of an optically thin gray gasis expressed by

1205971199021199031205971199101015840 = minus4119886lowast120590 (11987910158404infin minus 11987910158404) (4)

We assume that the temperature differences within theflow are sufficiently small such that 11987910158404 may be expressed asa linear function of the temperature This is accomplished byexpanding 11987910158404 in a Taylor series about 1198791015840infin and neglectinghigher-order terms thus

11987910158404 = minus411987910158403infin1198791015840 minus 311987910158404infin (5)

By using (4) and (5) (2) gives

120588119862119901 1205971198791015840

1205971199051015840 = 1205811205972119879101584012059711991010158402 minus 16119886lowast12059011987910158403infin (1198791015840infin minus 1198791015840) (6)

In order to write the governing equations initial andboundary conditions in dimensionless form the followingnondimensional quantities are introduced

119880 = 11990610158401199060

119884 = 11991010158401199060V

119905 = 11990510158401199060]

119879 = 1198791015840 minus 1198791015840infin1198791015840119908 minus 1198791015840infin

Pr = 120583119862119901120581

Gr = 119892120573]1198791015840119908 minus 1198791015840infin11990630

119872 = 1205901198612011990320V120588

119896 = 119896101584011990620]2

119877 = 16119886lowast]212059011987910158403infin12058111990620

119886 = 1198861015840]11990620

120574 = minus V10158401199060

(7)

In view of (7) the governing equations (1) and (6) reduce tothe following nondimensional form

120597119880120597119905 minus 120574

120597119880120597119884 =

12059721198801205971198842 + Gr119879 minus119872119880 minus

119880119896 (8)

120597119879120597119905 minus 120574

120597119879120597119884 =

1Pr12059721198791205971198842 minus

119877Pr119879 (9)

with following initial and boundary conditions

119905 le 0 119880 = 0119879 = 0

forall119884119905 gt 0 119880 = exp (119886119905)

119879 = 119905at 119884 = 0

119880 997888rarr 0119879 997888rarr 0


(10)

All physical variables and parameters are mentioned innomenclature

3 Solution Technique

To solve the unsteady equations (8) and (9) subject to initialand boundary conditions (10) we apply Laplace transformtechnique for the case unit Parndtl number

Laplace transforms of (8) and (9) give rise to

11988921198801198891198842 + 120574

119889119880119889119884 minus (119901 +119872 +

1119896)119880 + Gr119879 = 0 (11)

11988921198791198891198842 + 120574

119889119879119889119884 minus (119901 + 119877)119879 = 0 (12)

where p is the Laplace transformation parameter and 119880 amp 119879are the Laplace transform of U amp T respectively

Solutions of (11) and (12) subject to initial and boundaryconditions (10) give

119880 = 1119901 minus 119886119890minus119884((1205742)+radic119901+119889) +

Gr119871119890minus119884((1205742)+radic119901+119889)

1199012

minus Gr119871119890minus119884((1205742)+radic119901+119878)

1199012 (13)

119879 = 11199012 119890minus119884((1205742)+radic119901+119878) (14)


Inverse Laplace transforms of (13) and (14) respectively givethe velocity and temperature profiles as

119880 = 1198901198861199052 119890minus119884((1205742)+radic119886+119889)erfc( 1198842radic119905 minus radic(119886 + 119889) 119905)

+ 119890minus119884((1205742)minusradic119886+119889)erfc( 1198842radic119905 + radic(119886 + 119889) 119905)

+ Gr119871 (1199052 minus

1198844radic119889) exp(minus119884(

1205742 + radic119889))

sdot erfc( 1198842radic119905 minus radic119889119905) +Gr119871 (1199052 +

1198844radic119889)

sdot exp (minus119884(1205742 minus radic119889)) erfc(1198842radic119905 + radic119889119905)

minus Gr119871 (1199052 minus

1198844radic119878) exp(minus119884(

1205742 + radic119878)) erfc(

1198842radic119905

minus radic119878119905) minus Gr119871 (1199052 +

1198844radic119878) exp(minus119884(

1205742 minus radic119878))

sdot erfc( 1198842radic119905 + radic119878119905)

(15)

119879 = ( 1199052 minus1198844radic119878) exp(minus119884(

1205742 + radic119878)) erfc(

1198842radic119905

minus radic119878119905) + ( 1199052 +1198844radic119878) exp(minus119884(

1205742 minus radic119878))


(16)

Knowing the velocity and temperature field it is veryinteresting to study the skin friction and Nusselt number Innondimensional form the skin friction and Nusselt numberare defined respectively as follows

120591 = minus 12059711988012059711988410038161003816100381610038161003816100381610038161003816119884=0

Nu = minus 12059711987912059711988410038161003816100381610038161003816100381610038161003816119884=0

(17)

31 Skin Friction Expression of the skin-friction 120591 isobtained from (15) as

120591 = 119890minus119889119905radic120587119905 +1198901198861199052 (

1205742 + radic119886 + 119889) erfc (minusradic119905 (119886 + 119889))

+ (1205742 minus radic119886 + 119889) erfc (radic119905 (119886 + 119889))+ Gr 1199052119871 (

1205742 + radic119889) erfc (minusradic119889119905)

+ (1205742 minus radic119889) erfc (radic119889119905)+ Gr 1199052119871 (


+ (1205742 minus radic119878) erfc (radic119878119905) +Gr4119871radic119889 erfc (minusradic119889119905)

minus erfc (radic119889119905) minus Gr4119871radic119878 erfc (minusradic119878119905)

minus erfc (radic119878119905) + Grradic119905119871radic120587 exp (minus119889119905) minus exp (minus119878119905)

(18)

32 Nusselt Number Expression of Nusselt number Nu isobtained from (16) as

Nu = 1199052 (1205742 + radic119878) erfc (minusradic119878119905)

+ (1205742 minus radic119878) erfc (radic119878119905) minus14radic119878 erfc (minusradic119878119905)

minus erfc (radic119878119905) + radic 119905120587 exp (minus119878119905) (19)

where 119889 = 12057424 +119872 + 1119896 119878 = 12057424 + 119877 119871 = 119877 minus119872 minus 11198964 Results and Discussion

In order to get an insight into the physical solution of theproblem the numerical computations of velocity profiletemperature profile skin friction and Nusselt number areobtained for different values of magnetic field parameter119872 Grashof numbers Gr accelerating parameter 119886 suctionparameter 120574 permeability parameter 119896 radiation parameter119877 and time 119905 and presented graphically in Figures 2ndash14

The transient velocity profiles for different values ofGrashof number at 119886 = 06 119872 = 12 119896 = 03 119877 = 2119905 = 06 and 120574 = 04 are shown in Figure 2 The Grashofnumber signifies the relative effect of the buoyancy force tothe hydrodynamic viscous force The positive values of Grcorrespond to cooling of the plate and the negative values ofGr correspond to heating of the plate by free convection Asexpected it is found that an increase in the Grashof numberleads to increase the velocity due to enhancement in thebuoyancy force


U

00

05

10

15

05 10 15 20 25 3000Y

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

a = 06 120574 = 04M = 12

k = 03 R = 2 t = 06

Figure 2 Effect of Gr on velocity profiles at 119886 = 06119872 = 12 119896 =03 119877 = 2 119905 = 06 and 120574 = 04

0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06 120574 = 04

k = 05 R = 2 t = 06

M = 06

M = 15

M = 3

Figure 3 Effect of119872 on velocity profiles at Gr = (minus8 amp 8) 119886 = 06119896 = 05 119877 = 2 119905 = 06 and 120574 = 04

The transient velocity profiles for different values ofMagnetic parameter 119872 at Gr = minus8 amp 8 119886 = 06 119896 = 05119877 = 2 119905 = 06 and 120574 = 04 are shown in Figure 3 It isobserved from the figure that an increase in magnetic fieldleads to decrease in the velocity field for both the cases ofcooling and heating of the porous plate It is because that theapplication of transverse magnetic field will result a resistivetype force (Lorentz force) similar to drag force which tendsto resist the fluid flow and thus reducing its velocity

The effects of suction parameter 120574 on velocity profiles atGr = minus8 amp 8 119886 = 06 119896 = 05 119905 = 06 119877 = 2 and119872 = 15 are

0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06M = 15 k = 05

R = 2 t = 06

120574 = 02

120574 = 08

120574 = 16

Figure 4 Effect of 120574 on velocity profiles at Gr = (minus8 amp 8) 119886 = 06119896 = 05 119905 = 06 119877 = 2 and119872 = 15

00

06

12

18

U

05 10 15 20 25 3000Y

Gr = 5 a = 06 R = 2M = 15 120574 = 04

k = 05

k = 1

k = 21

t = 09

t = 05

t = 02

Figure 5 Effects of 119896 and 119905 on velocity profiles at Gr = 5 119886 = 06119877 = 2119872 = 15 and 120574 = 04

shown in Figure 4 It is foundhere that velocity decreaseswithincreases of the suction parameter 120574 for both cases of coolingand heating of the porous plate The effects of permeabilityparameter 119896 and time 119905 on velocity profiles at Gr = 5 119886 = 06119877 = 2119872 = 15 and 120574 = 04 are depicted in Figure 5 It canbe seen that velocity increases with increase of permeabilityparameter or time

The transient velocity profiles for different values ofaccelerating parameter 119886 and radiation parameter 119877 at Gr= 8 119896 = 05 119905 = 08 119872 = 1 and 120574 = 04 are plottedin Figure 6 It is observed that velocity decreases with anincrease in radiation parameter but decreases with increase


U

05 10 15 20 2500Y

00

04

08

12

16

20

24

28

Gr = 8 120574 = 04

M = 1 k = 05 t = 08

R = 2

R = 8

R = 20

a = 02

a = 12

a = 06


00

02

04

06

T

05 10 15 20 2500Y

R = 2

R = 4

R = 10

120574 = 08 t = 05

R = 05

Figure 7 Effect of 119877 on temperature profiles at 120574 = 08 and 119905 = 05

in accelerating parameter Effects of radiation parameter 119877suction parameter 120574 and time 119905 on temperature profiles areshown in Figures 7 8 and 9 respectively It is observed fromthese figures that temperature decreaseswith increased valuesof radiation parameter or suction parameter but increaseswith increased values of time

Effect of Grashof number Gr on skin friction is presentedin Figure 10 It is observed from the figure that skin frictiondecreases with increase of Grashof number In case ofcooling the plate skin friction decreases continuously withtime whereas in case of heating the plate though initiallyit decreases but after certain time it tends to increase withincrease of time Effect of accelerating parameter 119886 on skinfriction is presented in Figure 11 at Gr = minus5 and 5119872 = 12

00

02

04

06

T

05 10 15 20 2500Y

R = 02 t = 05

120574 = 01

120574 = 04

120574 = 1

120574 = 25


T

05 10 15 20 25 3000Y

00

02

04

06

08

10120574 = 04 R = 2

t = 04

t = 06

t = 06

t = 1


120574 = 04 119877 = 2 and 119896 = 05 It is observed that skin frictionincreases with an increase of accelerating parameter a forboth the cases of cooling and heating of the porous plate

Figure 12 shows the effects of magnetic parameter119872 andpermeability parameter 119896 at Gr = 5 120574 = 04 119886 = 03 and119877 = 2 and it is observed that skin friction increases withincrease in magnetic parameter but decreases with increasein permeability parameter Effects of radiation parameter 119877and suction parameter 120574 on skin friction at Gr = 5119872 = 12119886 = 03 and 119896 = 05 are depicted in Figure 13 and it canbe seen that skin friction increases with increase of radiationparameter or suction parameter

The Nusselt number for different values radiation param-eter 119877 and suction parameter 120574 are shown in Figure 14 The


08

120591

t

a = 03 120574 = 04 R = 2

M = 12 k = 05

Gr = minus15

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

Gr = 15

minus4

minus2

0

2

4

6

8

10

1200 04

Figure 10 Effect of Gr on skin friction at 120574 = 04 119886 = 03 119877 = 2119872 = 12 and 119896 = 05

08

t

5

1

2

3

4

6

1200 04

120591

M = 12 120574 = 04

k = 05 R = 2

Gr = minus5

Gr = 5

a = 02

a = 03

a = 04

Figure 11 Effect of 119886 on skin friction at119872 = 12 120574 = 04 119877 = 2and 119896 = 05

rate of heat transfer increases with increase of radiationparameter or suction parameter

5 Conclusions

The analytical study on unsteady one-dimensional naturalconvective MHD flow of a viscous incompressible and elec-trically conducting fluid past an exponentially acceleratedinfinite vertical porous plate through a porous medium withvariable temperature is presented The exact solutions ofthe dimensionless governing boundary layer equations are

120591

t

1200 08040

1

2

3

4

5

Gr = 5 120574 = 04a = 03 R = 2

M = 06 k = 05

M = 12 k = 05

M = 2 k = 05

M = 12 k = 1

M = 12 k = 2

Figure 12 Effect of 119872 and 119896 on skin friction at Gr = 5 120574 = 04119886 = 03 and 119877 = 2

120591

t

1200 08041

2

3

4

5 Gr = 5M = 12

a = 03 k = 05

R = 2 120574 = 04

R = 8 120574 = 04

R = 20 120574 = 04

R = 8 120574 = 08

R = 8 120574 = 15

Figure 13 Effects of 119877 and 120574 on skin friction at Gr = 5119872 = 12119886 = 03 and 119896 = 05

obtained by Laplace transform technique On the basis of theobservations results and discussions the conclusions of thepresent study are as follows

(1) Velocity increases with increase in Gr or 119896 or 119905 or 119886but decreases with increase in119872 or 120574 or 119877

(2) Temperature increaseswith increase in 119905but decreaseswith increase in 120574 or 119877

(3) Skin friction increases with increase in119872 or 120574 or 119877 or119886 but decreases with increase in Gr or 119896(4) Rate of heat transfer increases with increase in 119877 or 120574


120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t

Figure 14 Effect of 120574 and 119877 on Nusselt number

Nomenclature

1198861015840 Accelerating parameter119886 Dimensionless accelerating parameter119886lowast Absorption coefficient119862119901 Specific heat at constant pressure1198610 Transverse magnetic field strengthGr Grashof number119892 Acceleration due to gravity120581 Thermal conductivity of the fluid1198961015840 Permeability parameter119896 Dimensionless permeability parameter119872 Magnetic field parameterNu Nusselt numberPr Prandtl number119902119903 Radiative heat flux in the 119910 direction119877 Radiation parameter1199051015840 Time119905 Dimensionless time1198791015840 Temperature119879 Dimensionless temperature1198791015840119908 Temperature of the plate1198791015840infin Temperature of the fluid far away from the plate1199061015840 119909-component of velocity11990610158400 Velocity of the plate119880 Dimensionless velocityV1015840 119910-component of velocity1199101015840 Coordinate axis normal to the plate119884 Dimensionless coordinate axis normal to the plate

Greek Symbols

120573 Volumetric coefficient of thermal expansion120574 Suction parameter] Kinematic viscosity120588 Fluid density120590 Electrical conductivity of fluid

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A K Singh and N Kumar ldquoFree-convection flow past anexponentially accelerated vertical platerdquo Astrophysics and SpaceScience vol 98 no 2 pp 245ndash248 1984

[3] B K Jha R Prasad and S Rai ldquoMass transfer effects on the flowpast an exponentially accelerated vertical plate with constantheat fluxrdquoAstrophysics and Space Science vol 181 no 1 pp 125ndash134 1991

[4] R Muthucumaraswamy K Sathappan and R Natarajan ldquoHeattransfer effects on flow past an exponentially accelerated ver-tical plate with variable temperaturerdquo Theoretical and AppliedMechanics vol 35 no 4 pp 323ndash331 2008

[5] R Muthucumaraswamy K E Sathappan and R NatarajanldquoHeat and mass transfer effects on exponentially acceleratedvertical plate with uniformmagnetic fieldrdquo Journal of Engineer-ing Annals vol 6 no 3 pp 188ndash193 2008

[6] V Rajesh S Varma andV Kumar ldquoRadiation andmass transfereffects on MHD free convection flow past an exponentiallyaccelerated vertical plate with variable temperaturerdquo ARPNJournal of Engineering and Applied Sciences vol 4 no 6 pp 20ndash26 2009

[7] V Rajesh and A J Chamkha ldquoUnsteady convective flowpast an exponentially accelerated infinite vertical porous platewith Newtonian heating and viscous dissipationrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 24 no5 pp 1109ndash1123 2014

[8] R Srinivasa G Aruna S Naidu S Varma and M RashidildquoChemically reacting fluid flow induced by an exponentiallyaccelerated infinite vertical plate in a magnetic field andvariable temperature via LTT and FEMrdquoTheoretical andAppliedMechanics vol 43 no 1 pp 49ndash83 2016

[9] A S Gupta I Popp and V M Soundalgekar ldquoFree convectioneffects on the flow past an accelerated vertical plate in anincompressible dissipative fluidrdquo Revue Roumaine des SciencesTechniques Serie deMecanique Appliquee vol 24 no 4 pp 561ndash568 1979

[10] N G Kafousias and A A Raptis ldquoMass transfer and free con-vection effects on the flow past an accelerated vertical infiniteplate with variable suction or injectionrdquo Revue Roumaine desSciences TechniquesmdashSerie de Mecanique Appliquee vol 26 pp11ndash22 1981

[11] C Ping and I Pop ldquoTransient free convection about a verticalflat plate embedded in a porous mediumrdquo International Journalof Engineering Science vol 22 no 3 pp 253ndash264 1984

[12] B K Jha ldquoMHD free-convection and mass-transform flowthrough a porous mediumrdquo Astrophysics and Space Science vol175 no 2 pp 283ndash289 1991

[13] E Magyari I Pop and B Keller ldquoAnalytical solutions forunsteady free convection in porous mediardquo Journal of Engineer-ing Mathematics vol 48 no 2 pp 93ndash104 2004

[14] R C Chaudhary andA Jain ldquoCombined heat andmass transfereffects MHD free convection flow past an oscillating plate


embedded in porous mediumrdquo Romanian Journal of Physicsvol 52 pp 505ndash524 2007

[15] R C Chaudhary andA Jain ldquoMagnetohydrodynamic transientconvection flow past a vertical surface embedded in a porousmedium with oscillating temperaturerdquo Turkish Journal of Engi-neering and Environmental Sciences vol 32 no 1 pp 13ndash222008

[16] R C Chaudhary and A Jain ldquoMHD heat and mass diffusionflow by natural convection past a surface embedded in a porousmediumrdquo Theoretical and Applied Mechanics vol 36 no 1 pp1ndash27 2009

[17] R C Chaudhary A Jain and M C Goyal ldquoFree convectioneffects on MHD flow past an infinite vertical acceleratedplate embedded in porous media with constant heat fluxrdquoMatematicas Ensenanza Universitaria vol 17 no 2 pp 73ndash822009

[18] V Rajesh ldquoMHD free convection flow past an accelerated ver-tical porous plate with variable temperature through a porousmediumrdquo Acta Technica CorviniensismdashBulletin of Engineeringvol 2 pp 91ndash96 2010

[19] V Rajesh M Mallesh and O A Beg ldquoTransient MHD freeconvection flow and heat transfer of nanofluid past an impul-sively started vertical porous plate in the presence of viscousdissipationrdquo Procedia Materials Science vol 10 pp 80ndash89 2015

[20] S M Ibrahim ldquoEffects of chemical reaction on dissipativeradiative MHD flow through a porous medium over a non-isothermal stretching sheetrdquo Journal of Industrial Mathematicsvol 2014 Article ID 243148 10 pages 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


both numerical and analytical solutions to study the effectof chemical reaction on unsteady incompressible viscousflow past an exponentially accelerated vertical plate with heatabsorption and variable temperature in a magnetic field

In recent years convective heat transfer in porous mediahas attracted considerable attention owing to its wide indus-trial and technological applications such as geothermalenergy recovery oil extraction fibre and granular insulationelectronic system cooling and porous material regenerativeheat exchangers Gupta et al [9] have studied free convectionon flow past an accelerated vertical plate in the presence ofviscous dissipative heat using perturbation method Kafou-sias and Raptis [10] extended their (Gupta et al [9]) work byincluding mass transfer effects subjected to variable suctionor injection Transient free convection flow past a plateembedded in a porous medium pioneered by Ping and Pop[11] Jha [12] presented analytic investigation of uniformtransverse magnetic field effect on the free-convection andmass-transform flow of an electrically conducting fluid pastan infinite vertical plate for uniformly accelerated motion ofthe plate through a porous medium Magyari et al [13] havepresented an analytical solution for unsteady free convectionin porousmedia Chaudhary and Jain [14ndash16] andChaudharyet al [17] analyzed MHD transient free convection effectson flow past a moving vertical plate embedded in porousmedium under different physical situations by employingLaplace transform technique Rajesh [18] presented an ana-lytic investigation of MHD free convection flow past anaccelerated vertical porous plate with variable temperaturethrough a porous medium Again Rajesh et al [19] studiedthe transient free convection MHD flow and heat transfer ofnanofluid past an impulsively started vertical porous plate inthe presence of viscous dissipation Recently Ibrahim [20]studied the effects of thermal radiation viscous dissipationand chemical reaction on unsteady two-dimensional bound-ary layer MHD flow over a nonisothermal stretching sheetembedded in porous medium

However to the best of authorsrsquo knowledge MHD freeconvective flow past an exponentially accelerated verticalporous plate with variable temperature through a porousmedium was never considered in the literatureThe objectiveof this paper is to study magnetohydrodynamic transientheat transfer flow past an exponentially accelerated infinitevertical porous plate with variable temperature and the plateis embedded in a porous medium The exact solutions ofthe dimensionless unsteady linear governing equations areobtained by Laplace transform technique

2 Mathematical Analysis

Anunsteady one-dimensional laminar free convective flow ofa viscous incompressible fluid past an infinite vertical porousplate through a porous medium with variable temperatureis considered The 119909-axis is being taken vertically upwardsalong the vertical plate and 119910-axis is taken to be normal tothe plate The physical model and coordinate system of theflow problem is shown in Figure 1 Initially it is assumed thatthe plate and fluid are at the same temperature 1198791015840infin in thestationary condition At 1199051015840 ge 0 the plate is exponentially

x

y

Bo

T998400w

g

ouoe

a998400 t998400

u998400

998400T998400infin

Figure 1 Physical model and coordinate system

accelerated with a velocity 1199061015840 = 11990610158400 exp(11988610158401199051015840) in its ownplane and the plate temperature is raised linearly with time119905 A uniform magnetic field is applied in the directionperpendicular to the plate The fluid is assumed to be slightlyconducting so that the magnetic Reynolds number is muchless than unity and hence the induced magnetic field isnegligible in comparison with the applied magnetic fieldThefluid considered here is a gray absorbingemitting radiationbut a nonscattering medium The viscous dissipation is alsoassumed to be negligible in the energy equation as themotion is due to free convection only It is also assumedthat all the fluid properties are constant except for thedensity in the buoyancy term which is given by the usualBoussinesqrsquos approximation Under these assumptions thegoverning boundary layer equations are

12059711990610158401205971199051015840 + V1015840

12059711990610158401205971199101015840 = ]

1205972119906101584012059711991010158402 + 119892120573 (1198791015840 minus 1198791015840infin)

minus 12059011986120120588 1199061015840 minus ]11990610158401198961015840

(1)

120588119862119901 (1205971198791015840

1205971199051015840 + V101584012059711987910158401205971199101015840 ) = 120581

1205972119879101584012059711991010158402 minus

1205971199021199031205971199101015840 (2)

with the following initial and boundary conditions

1199051015840 le 0 1199061015840 = 01198791015840 = 1198791015840infin forall1199101015840

1199051015840 gt 0 1199061015840 = 1199060 exp (11988610158401199051015840) 1198791015840 = 1198791015840infin (1198791015840119908 minus 1198791015840infin)1198601199051015840

at 1199101015840 = 01199061015840 997888rarr 01198791015840 997888rarr 1198791015840infin


(3)

where 119860 = 11990620]







120588119862119901 1205971198791015840



119880 = 11990610158401199060

119884 = 11991010158401199060V

119905 = 11990510158401199060]


Pr = 120583119862119901120581

Gr = 119892120573]1198791015840119908 minus 1198791015840infin11990630

119872 = 1205901198612011990320V120588

119896 = 119896101584011990620]2

119877 = 16119886lowast]212059011987910158403infin12058111990620

119886 = 1198861015840]11990620

120574 = minus V10158401199060

(7)


120597119880120597119905 minus 120574

120597119880120597119884 =

12059721198801205971198842 + Gr119879 minus119872119880 minus

119880119896 (8)

120597119879120597119905 minus 120574

120597119879120597119884 =

1Pr12059721198791205971198842 minus

119877Pr119879 (9)


119905 le 0 119880 = 0119879 = 0

forall119884119905 gt 0 119880 = exp (119886119905)

119879 = 119905at 119884 = 0

119880 997888rarr 0119879 997888rarr 0


(10)





11988921198801198891198842 + 120574

119889119880119889119884 minus (119901 +119872 +

1119896)119880 + Gr119879 = 0 (11)

11988921198791198891198842 + 120574

119889119879119889119884 minus (119901 + 119877)119879 = 0 (12)




Gr119871119890minus119884((1205742)+radic119901+119889)

1199012


1199012 (13)

119879 = 11199012 119890minus119884((1205742)+radic119901+119878) (14)





+ Gr119871 (1199052 minus

1198844radic119889) exp(minus119884(

1205742 + radic119889))


1198844radic119889)



1198844radic119878) exp(minus119884(

1205742 + radic119878)) erfc(

1198842radic119905


1198844radic119878) exp(minus119884(

1205742 minus radic119878))


(15)


1205742 + radic119878)) erfc(

1198842radic119905


1205742 minus radic119878))


(16)


120591 = minus 12059711988012059711988410038161003816100381610038161003816100381610038161003816119884=0

Nu = minus 12059711987912059711988410038161003816100381610038161003816100381610038161003816119884=0

(17)


120591 = 119890minus119889119905radic120587119905 +1198901198861199052 (









(18)









U

00

05

10

15

05 10 15 20 25 3000Y

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

a = 06 120574 = 04M = 12

k = 03 R = 2 t = 06


0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06 120574 = 04

k = 05 R = 2 t = 06

M = 06

M = 15

M = 3




0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06M = 15 k = 05

R = 2 t = 06

120574 = 02

120574 = 08

120574 = 16


00

06

12

18

U

05 10 15 20 25 3000Y

Gr = 5 a = 06 R = 2M = 15 120574 = 04

k = 05

k = 1

k = 21

t = 09

t = 05

t = 02





U

05 10 15 20 2500Y

00

04

08

12

16

20

24

28

Gr = 8 120574 = 04

M = 1 k = 05 t = 08

R = 2

R = 8

R = 20

a = 02

a = 12

a = 06


00

02

04

06

T

05 10 15 20 2500Y

R = 2

R = 4

R = 10

120574 = 08 t = 05

R = 05




00

02

04

06

T

05 10 15 20 2500Y

R = 02 t = 05

120574 = 01

120574 = 04

120574 = 1

120574 = 25


T

05 10 15 20 25 3000Y

00

02

04

06

08

10120574 = 04 R = 2

t = 04

t = 06

t = 06

t = 1






08

120591

t

a = 03 120574 = 04 R = 2

M = 12 k = 05

Gr = minus15

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

Gr = 15

minus4

minus2

0

2

4

6

8

10

1200 04


08

t

5

1

2

3

4

6

1200 04

120591

M = 12 120574 = 04

k = 05 R = 2

Gr = minus5

Gr = 5

a = 02

a = 03

a = 04



5 Conclusions


120591

t

1200 08040

1

2

3

4

5

Gr = 5 120574 = 04a = 03 R = 2

M = 06 k = 05

M = 12 k = 05

M = 2 k = 05

M = 12 k = 1

M = 12 k = 2


120591

t

1200 08041

2

3

4

5 Gr = 5M = 12

a = 03 k = 05

R = 2 120574 = 04

R = 8 120574 = 04

R = 20 120574 = 04

R = 8 120574 = 08

R = 8 120574 = 15







120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t


Nomenclature


Greek Symbols


Competing Interests


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120588119862119901 1205971198791015840



119880 = 11990610158401199060

119884 = 11991010158401199060V

119905 = 11990510158401199060]


Pr = 120583119862119901120581

Gr = 119892120573]1198791015840119908 minus 1198791015840infin11990630

119872 = 1205901198612011990320V120588

119896 = 119896101584011990620]2

119877 = 16119886lowast]212059011987910158403infin12058111990620

119886 = 1198861015840]11990620

120574 = minus V10158401199060

(7)


120597119880120597119905 minus 120574

120597119880120597119884 =

12059721198801205971198842 + Gr119879 minus119872119880 minus

119880119896 (8)

120597119879120597119905 minus 120574

120597119879120597119884 =

1Pr12059721198791205971198842 minus

119877Pr119879 (9)


119905 le 0 119880 = 0119879 = 0

forall119884119905 gt 0 119880 = exp (119886119905)

119879 = 119905at 119884 = 0

119880 997888rarr 0119879 997888rarr 0


(10)





11988921198801198891198842 + 120574

119889119880119889119884 minus (119901 +119872 +

1119896)119880 + Gr119879 = 0 (11)

11988921198791198891198842 + 120574

119889119879119889119884 minus (119901 + 119877)119879 = 0 (12)




Gr119871119890minus119884((1205742)+radic119901+119889)

1199012


1199012 (13)

119879 = 11199012 119890minus119884((1205742)+radic119901+119878) (14)





+ Gr119871 (1199052 minus

1198844radic119889) exp(minus119884(

1205742 + radic119889))


1198844radic119889)



1198844radic119878) exp(minus119884(

1205742 + radic119878)) erfc(

1198842radic119905


1198844radic119878) exp(minus119884(

1205742 minus radic119878))


(15)


1205742 + radic119878)) erfc(

1198842radic119905


1205742 minus radic119878))


(16)


120591 = minus 12059711988012059711988410038161003816100381610038161003816100381610038161003816119884=0

Nu = minus 12059711987912059711988410038161003816100381610038161003816100381610038161003816119884=0

(17)


120591 = 119890minus119889119905radic120587119905 +1198901198861199052 (









(18)









U

00

05

10

15

05 10 15 20 25 3000Y

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

a = 06 120574 = 04M = 12

k = 03 R = 2 t = 06


0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06 120574 = 04

k = 05 R = 2 t = 06

M = 06

M = 15

M = 3




0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06M = 15 k = 05

R = 2 t = 06

120574 = 02

120574 = 08

120574 = 16


00

06

12

18

U

05 10 15 20 25 3000Y

Gr = 5 a = 06 R = 2M = 15 120574 = 04

k = 05

k = 1

k = 21

t = 09

t = 05

t = 02





U

05 10 15 20 2500Y

00

04

08

12

16

20

24

28

Gr = 8 120574 = 04

M = 1 k = 05 t = 08

R = 2

R = 8

R = 20

a = 02

a = 12

a = 06


00

02

04

06

T

05 10 15 20 2500Y

R = 2

R = 4

R = 10

120574 = 08 t = 05

R = 05




00

02

04

06

T

05 10 15 20 2500Y

R = 02 t = 05

120574 = 01

120574 = 04

120574 = 1

120574 = 25


T

05 10 15 20 25 3000Y

00

02

04

06

08

10120574 = 04 R = 2

t = 04

t = 06

t = 06

t = 1






08

120591

t

a = 03 120574 = 04 R = 2

M = 12 k = 05

Gr = minus15

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

Gr = 15

minus4

minus2

0

2

4

6

8

10

1200 04


08

t

5

1

2

3

4

6

1200 04

120591

M = 12 120574 = 04

k = 05 R = 2

Gr = minus5

Gr = 5

a = 02

a = 03

a = 04



5 Conclusions


120591

t

1200 08040

1

2

3

4

5

Gr = 5 120574 = 04a = 03 R = 2

M = 06 k = 05

M = 12 k = 05

M = 2 k = 05

M = 12 k = 1

M = 12 k = 2


120591

t

1200 08041

2

3

4

5 Gr = 5M = 12

a = 03 k = 05

R = 2 120574 = 04

R = 8 120574 = 04

R = 20 120574 = 04

R = 8 120574 = 08

R = 8 120574 = 15







120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t


Nomenclature


Greek Symbols


Competing Interests


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













+ Gr119871 (1199052 minus

1198844radic119889) exp(minus119884(

1205742 + radic119889))


1198844radic119889)



1198844radic119878) exp(minus119884(

1205742 + radic119878)) erfc(

1198842radic119905


1198844radic119878) exp(minus119884(

1205742 minus radic119878))


(15)


1205742 + radic119878)) erfc(

1198842radic119905


1205742 minus radic119878))


(16)


120591 = minus 12059711988012059711988410038161003816100381610038161003816100381610038161003816119884=0

Nu = minus 12059711987912059711988410038161003816100381610038161003816100381610038161003816119884=0

(17)


120591 = 119890minus119889119905radic120587119905 +1198901198861199052 (









(18)









U

00

05

10

15

05 10 15 20 25 3000Y

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

a = 06 120574 = 04M = 12

k = 03 R = 2 t = 06


0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06 120574 = 04

k = 05 R = 2 t = 06

M = 06

M = 15

M = 3




0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06M = 15 k = 05

R = 2 t = 06

120574 = 02

120574 = 08

120574 = 16


00

06

12

18

U

05 10 15 20 25 3000Y

Gr = 5 a = 06 R = 2M = 15 120574 = 04

k = 05

k = 1

k = 21

t = 09

t = 05

t = 02





U

05 10 15 20 2500Y

00

04

08

12

16

20

24

28

Gr = 8 120574 = 04

M = 1 k = 05 t = 08

R = 2

R = 8

R = 20

a = 02

a = 12

a = 06


00

02

04

06

T

05 10 15 20 2500Y

R = 2

R = 4

R = 10

120574 = 08 t = 05

R = 05




00

02

04

06

T

05 10 15 20 2500Y

R = 02 t = 05

120574 = 01

120574 = 04

120574 = 1

120574 = 25


T

05 10 15 20 25 3000Y

00

02

04

06

08

10120574 = 04 R = 2

t = 04

t = 06

t = 06

t = 1






08

120591

t

a = 03 120574 = 04 R = 2

M = 12 k = 05

Gr = minus15

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

Gr = 15

minus4

minus2

0

2

4

6

8

10

1200 04


08

t

5

1

2

3

4

6

1200 04

120591

M = 12 120574 = 04

k = 05 R = 2

Gr = minus5

Gr = 5

a = 02

a = 03

a = 04



5 Conclusions


120591

t

1200 08040

1

2

3

4

5

Gr = 5 120574 = 04a = 03 R = 2

M = 06 k = 05

M = 12 k = 05

M = 2 k = 05

M = 12 k = 1

M = 12 k = 2


120591

t

1200 08041

2

3

4

5 Gr = 5M = 12

a = 03 k = 05

R = 2 120574 = 04

R = 8 120574 = 04

R = 20 120574 = 04

R = 8 120574 = 08

R = 8 120574 = 15







120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t


Nomenclature


Greek Symbols


Competing Interests


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










U

00

05

10

15

05 10 15 20 25 3000Y

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

a = 06 120574 = 04M = 12

k = 03 R = 2 t = 06


0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06 120574 = 04

k = 05 R = 2 t = 06

M = 06

M = 15

M = 3




0 1 2 300

05

10

15

U

Y

Gr = minus8

Gr = 8

a = 06M = 15 k = 05

R = 2 t = 06

120574 = 02

120574 = 08

120574 = 16


00

06

12

18

U

05 10 15 20 25 3000Y

Gr = 5 a = 06 R = 2M = 15 120574 = 04

k = 05

k = 1

k = 21

t = 09

t = 05

t = 02





U

05 10 15 20 2500Y

00

04

08

12

16

20

24

28

Gr = 8 120574 = 04

M = 1 k = 05 t = 08

R = 2

R = 8

R = 20

a = 02

a = 12

a = 06


00

02

04

06

T

05 10 15 20 2500Y

R = 2

R = 4

R = 10

120574 = 08 t = 05

R = 05




00

02

04

06

T

05 10 15 20 2500Y

R = 02 t = 05

120574 = 01

120574 = 04

120574 = 1

120574 = 25


T

05 10 15 20 25 3000Y

00

02

04

06

08

10120574 = 04 R = 2

t = 04

t = 06

t = 06

t = 1






08

120591

t

a = 03 120574 = 04 R = 2

M = 12 k = 05

Gr = minus15

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

Gr = 15

minus4

minus2

0

2

4

6

8

10

1200 04


08

t

5

1

2

3

4

6

1200 04

120591

M = 12 120574 = 04

k = 05 R = 2

Gr = minus5

Gr = 5

a = 02

a = 03

a = 04



5 Conclusions


120591

t

1200 08040

1

2

3

4

5

Gr = 5 120574 = 04a = 03 R = 2

M = 06 k = 05

M = 12 k = 05

M = 2 k = 05

M = 12 k = 1

M = 12 k = 2


120591

t

1200 08041

2

3

4

5 Gr = 5M = 12

a = 03 k = 05

R = 2 120574 = 04

R = 8 120574 = 04

R = 20 120574 = 04

R = 8 120574 = 08

R = 8 120574 = 15







120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t


Nomenclature


Greek Symbols


Competing Interests


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










U

05 10 15 20 2500Y

00

04

08

12

16

20

24

28

Gr = 8 120574 = 04

M = 1 k = 05 t = 08

R = 2

R = 8

R = 20

a = 02

a = 12

a = 06


00

02

04

06

T

05 10 15 20 2500Y

R = 2

R = 4

R = 10

120574 = 08 t = 05

R = 05




00

02

04

06

T

05 10 15 20 2500Y

R = 02 t = 05

120574 = 01

120574 = 04

120574 = 1

120574 = 25


T

05 10 15 20 25 3000Y

00

02

04

06

08

10120574 = 04 R = 2

t = 04

t = 06

t = 06

t = 1






08

120591

t

a = 03 120574 = 04 R = 2

M = 12 k = 05

Gr = minus15

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

Gr = 15

minus4

minus2

0

2

4

6

8

10

1200 04


08

t

5

1

2

3

4

6

1200 04

120591

M = 12 120574 = 04

k = 05 R = 2

Gr = minus5

Gr = 5

a = 02

a = 03

a = 04



5 Conclusions


120591

t

1200 08040

1

2

3

4

5

Gr = 5 120574 = 04a = 03 R = 2

M = 06 k = 05

M = 12 k = 05

M = 2 k = 05

M = 12 k = 1

M = 12 k = 2


120591

t

1200 08041

2

3

4

5 Gr = 5M = 12

a = 03 k = 05

R = 2 120574 = 04

R = 8 120574 = 04

R = 20 120574 = 04

R = 8 120574 = 08

R = 8 120574 = 15







120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t


Nomenclature


Greek Symbols


Competing Interests


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08

120591

t

a = 03 120574 = 04 R = 2

M = 12 k = 05

Gr = minus15

Gr = minus10

Gr = minus5

Gr = 5

Gr = 10

Gr = 15

minus4

minus2

0

2

4

6

8

10

1200 04


08

t

5

1

2

3

4

6

1200 04

120591

M = 12 120574 = 04

k = 05 R = 2

Gr = minus5

Gr = 5

a = 02

a = 03

a = 04



5 Conclusions


120591

t

1200 08040

1

2

3

4

5

Gr = 5 120574 = 04a = 03 R = 2

M = 06 k = 05

M = 12 k = 05

M = 2 k = 05

M = 12 k = 1

M = 12 k = 2


120591

t

1200 08041

2

3

4

5 Gr = 5M = 12

a = 03 k = 05

R = 2 120574 = 04

R = 8 120574 = 04

R = 20 120574 = 04

R = 8 120574 = 08

R = 8 120574 = 15







120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t


Nomenclature


Greek Symbols


Competing Interests


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120574 = 01 R = 2

120574 = 04 R = 2

120574 = 1 R = 2

0

2

4

6

8

Nu

120574 = 04 R = 05

120574 = 04 R = 4

1 2 30

t


Nomenclature


Greek Symbols


Competing Interests


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Transient Free Convective MHD Flow Past an Exponentially...

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