research article unsteady hydromagnetic rotating flow...
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 508431 8 pageshttpdxdoiorg1011552013508431
Research ArticleUnsteady Hydromagnetic Rotating Flow through an OscillatingPorous Plate Embedded in a Porous Medium
I Khan12 A Khan1 A Farhad1 M Qasim3 and S Sharidan1
1 Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia 81310 Skudai Malaysia2 College of Engineering Majmaah University PO Box 66 Majmaah 11952 Saudi Arabia3 Department of Mathematics COMSATS Institute of Information Technology Park Road Chak Shahzad Islamabad 44000 Pakistan
Correspondence should be addressed to S Sharidan sharidanutmmy
Received 23 July 2013 Revised 2 October 2013 Accepted 9 October 2013
Academic Editor Tirivanhu Chinyoka
Copyright copy 2013 I Khan et al 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 investigates unsteady hydromagnetic flow of a viscous fluid in a rotating frame The fluid is bounded by an oscillatingporous plate embedded in a porous mediumThe Laplace transform and Fourier sine transform methods are employed to find theexact solutionsThey satisfy all imposed initial and boundary conditions and as special cases are reduced to some published resultsfrom the literature The graphical results are plotted for different values of pertinent parameters and some interesting conclusionsare made
1 Introduction
Stokesrsquo problem for the flow of an incompressible viscousfluid over an oscillating plane serves as a benchmark in theliterature of fluid dynamics [1] It admits an exact analyticalsolution The Stokesrsquo problem not only is of fundamentaltheoretical interest but also occurs in many applied problems[2]The transient solutions in terms of tabulated functions forthe flowof viscous fluid due to oscillating boundary have beenexpressed by Panton [3] Later in 2000 Erdogan [4] studiedthe unsteady flow of Newtonian fluid past an oscillating hor-izontal plane wall and obtained the exact solutions Fetecauet al [5] presented new exact solutions corresponding toStokesrsquo second problem for both small and large timesErdogan and Imrak [6] made comparative study for the solu-tion of Stokesrsquo second problem by employing two differenttransform methods They expressed transient solutions interms of tabulated functions
On the other hand flow through porous media is veryprevalent in nature and therefore has become a principalinterest of researchers in many scientific and engineeringstudies For example one can refer to the books of Pop andIngham [7] Ingham et al [8] Ingham and Pop [9] Vafai [10]and Nield and Bejan [11] for the detailed literature on thistopic Moreover the rotating flow through porous media has
been the subject ofmany studies in the last few decades due totheir wide range of applications in cosmological and geophys-ical fluid dynamics astrophysics meteorology petroleumand hydrology to study the movement of underground water(Jana et al [12]) According to Hayat et al [13] the analysisof the effects of rotation and magnetic field in fluid flows hasbeen an active area of research because of its geophysical andtechnological applications such as theMHD power generatorand boundary layer flow control Based on this motivationDas et al studied in [14] the simultaneous effects of rotationandmagnetic field on the flowof a second grade fluid betweentwo parallel plates Bearing in mind the importance of MHDand porosity in a rotating fluid Hayat et al [15 16] Abelmanet al [17] Sahoo et al [18] Seth et al [19] and Farhad et al[20] studied the hydromagnetic flow of rotating fluids inporous media Recently Jana et al [21] studied the unsteadyflow of viscous fluid through a porous medium bounded by aporous plate in a rotating system
In order to further discuss the work of Jana et al [21]and to make closer relations between this study and practicalengineering we study in this work hydromagnetic rotatingflow of a viscous fluid bounded by an oscillating porous plateembedded in a porousmediumThe governingmathematicalproblem is solved by using the Laplace transform and Fouriersine transformmethodsThe expressions for velocity and skin
2 Mathematical Problems in Engineering
friction in case of cosine and sine oscillations of the plate areobtainedThe analytical results are plotted and discussedTheresults of Jana et al [21] are recovered as a special case fromour obtained solutions The present study is useful in astron-omy to study the rotating motion of astrophysical objectssuch as magnetic stars
2 Mathematical Formulation of the Problem
Consider the unsteady flow of an incompressible viscousfluid occupying the upper porous half-space of the (119909 119910)plane The fluid is bounded by a porous plate at 119911 = 0 suchthat the positive 119911-axis is taken normal to the plate and the 119909-axis is taken parallel to the plate We consider the hydromag-netic flow induced in the fluid in the presence of a uniformmagnetic field of strength 119861
0applied in a direction normal to
the plate by means of the plate oscillations in its plane (Hayatet al [13]) The electric field due to polarization of charges isneglected Both of the fluid and plate are in a state of rigidbody rotation with constant angular velocity Ω = Ωk k isa unit vector in the 119911-directionThe geometry of the problemis shown in Figure 1 Initially for 119905 le 0 both of the fluid andplate are at rest At time 119905 = 0+ the lower plate suddenlystarts to move in its own plane with oscillating velocity inthe flow direction Under these assumptions the equations ofmomentum in a rotating frame along the 119909- and 119910-directionsare [18ndash20]
120597119906
120597119905minus 1199080
120597119906
120597119911minus 2ΩV = ]
1205972119906
1205971199112minus]120601119896119906 minus1205901198612
0
120588119906
119911 119905 gt 0
120597V120597119905minus 1199080
120597V120597119911+ 2Ω119906 = ]
1205972V1205971199112minus]120601119896V minus1205901198612
0
120588V
119911 119905 gt 0
(1)
where 119906 and V are the velocity components in 119909- and 119910-directions 119908
0is the suction velocity at the plate ] is the kine-
matic viscosity 119905 is the time 120590 is the electrical conductivity1198610is the applied magnetic field 120588 is the fluid density 120601 is the
porosity and 119896 is the permeability of the porous mediumThe corresponding initial and boundary conditions are
119906 = V = 0 119905 le 0 (2)
119906 = 119880119867 (119905) cos (1205960119905) or 119880 sin (120596
0119905)
V = 0 at 119911 = 0 119905 gt 0(3)
119906 997888rarr 0 V 997888rarr 0 as 119911 997888rarr infin 119905 gt 0 (4)
in which 119880 is the constant plate velocity 1205960is the frequency
of oscillation of the plate and119867(119905) is the Heaviside functionDefining119865 = 119906+119894V the above systemof equations reduces
to
120597119865
120597119905minus 1199080
120597119865
120597119911+ 2119894Ω119865 = ]
1205972119865
1205971199112minus]120601119896119865 minus1205901198612
0
120588119865
119911 119905 gt 0
Ox
B0
Magnetic fieldΩ
z
Porous medium z gt 0
y
u(0 t) = UH(t) cos(1205960t)
u(0 t) = Usin(1205960t)
(0 t) = 0 t gt 0
Figure 1 Geometry of the problem
119865 (119911 0) = 0 119905 le 0
119865 (0 119905) = 119880119867 (119905) cos (1205960119905) or 119880 sin (1205960119905) 119905 gt 0
119865 997888rarr 0 as 119911 997888rarr infin 119905 gt 0(5)
Introducing the following dimensionless variables (see Janaet al [21])
119866 =119865
119880 120585 =
119911119880
] 120591 =
1198802119905
] 120596 =
1205960]1198802 (6)
the dimensionless problem takes the following form
120597119866
120597120591minus 119878120597119866
120597120585+ 2119894120578119866 =
1205972119866
1205971205852minus1198722119866 minus
1
119870119866 (7)
119866 (120585 0) = 0 120591 le 0 (8)
119866 (0 120591) = 119867 (120591) cos (120596120591) or sin (120596120591) 120591 gt 0 (9)
119866 (infin 120591) = 0 120591 gt 0 (10)
1198722=12059011986120]
1205881198802
1
119870=
]21206011198961198802 120578 =
Ω]1198802 119878 =
1199080
119880
(11)
where 119872 119870 120578 and 119878 are the magnetic permeability rota-tion and suction parameters respectively
3 Solution of the Problem byUsing Laplace Transform
Taking the Laplace transform of (7) and using the initialcondition (8) we get the following transformed differentialequation
1198892119866 (120585 119902)
1198891205852+ 119878119889119866 (120585 119902)
119889120585
minus [119902 +1198722+1
119870+ 2119894120578]119866 (120585 119902) = 0
(12)
Mathematical Problems in Engineering 3
where 119902 is the transform parameter In view of the boundaryconditions (9) and (10) the Laplace transform 119866(120585 119902) of119866(120585 120591) has to satisfy the following conditions
119866 (0 119902) =119902
1199022 + 1205962or 119866 (0 119902) =
120596
1199022 + 1205962 (13)
119866 (0 119902) = 0 (14)
Solution of (12) under the boundary conditions (13) and (14)yields
119866119888(120585 119902) =
1
2119890minus12058511987821
119902 + 119894120596119890minus120585radic119902+119886
+1
119902 minus 119894120596119890minus120585radic119902+119886
(15)
respectively
119866119904(120585 119902) =
1
2119894119890minus1205851198782minus
1
119902 + 119894120596119890minus120585radic119902+119886
+1
119902 minus 119894120596119890minus120585radic119902+119886
(16)
where the subscripts 119888 and 119904 denote the cosine and sine oscil-lations of the plate and
119886 =1198782
4+1
119870+1198722+ 2119894120578 (17)
Now in order to find the inverse Laplace transform of (15)and (16) we use formula (A4) from the Appendix of Farhadet al [20] hence we get the following solutions
119866119888(120585 120591)
=119867 (120591) 119890
minus1205851198782
4
times [119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 minus 119894120596) 120591)
+ 119890120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 minus 119894120596) 120591)]
+119867 (120591) 119890
minus1205851198782
4
times [119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 + 119894120596) 120591)
+ 119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 + 119894120596) 120591)]
(18)
respectively
119866119904(120585 120591)
=119890minus1205851198782
4119894
times [minus119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 minus 119894120596) 120591)
+ 119890120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 minus 119894120596) 120591)]
+119890minus1205851198782
4119894[119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 + 119894120596) 120591)
+ 119890120585radic119886+119894120596 erf 119888 ( 120585
2radic120591+radic(119886+ 119894120596) 120591)]
(19)
4 Solution of the Problem byUsing Fourier Sine Transform
Now applying Fourier sine transform to (12) and usingboundary conditions (13) and (14) we obtain
119866119888(120577 119902) =
radic2 (120577 + 120577119878) 119902
(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (20)
respectively
119866119904(120577 119902) =
radic2 (120577 + 120577119878) 120596
(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (21)
where 119866(120577 119902) denotes Fourier sine transform of 119866(120585 119902)Now we take the inverse Fourier sine transform of (20)
which yields
119866119888(120585 119902) =
2radic2119902
120587 (1199022 + 1205962)intinfin
0
(120577 + 120577119878)
(1205772 + 1205772119878 + 119902 + 119886)sin (120577120585) 119889120577
(22)
Taking the inverse Laplace transform of (22) we obtain
119866119888(120585 120591) =
2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120577120585) 119889120577
minus2
120587intinfin
0
120577119878 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120577120585) 119889120577
4 Mathematical Problems in Engineering
1
08
06
04
02
01080604020 12
G
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
01
008
006
004
002
01080604020 12
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
minusiG
(b)
Figure 2 Profiles of velocity for different values of119872
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772+ 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577(23)
Similarly for the sine oscillation of the boundary we get from(21) the following
119866119904 (120585 120591) =
2
120587intinfin
0
120577120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577120596
((1205772+1205772119878+119886)2+ 1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)120596
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577119878120596
((1205772+1205772119878+119886)2+1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772 + 1205772119878 + 119886)
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
(24)
The nondimensional shear stresses 120591119888and 120591119904at the plate 120585 =
0 due to cosine and sine oscillations of the plate are obtainedfrom (18) and (19) as follows
120591119888= minus (120591
119909+ 119894120591119910)
= minus119867 (120591) 119890
minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
Mathematical Problems in Engineering 5
1
08
06
04
02
0151050 2
G
120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
15 251050 32120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
0035
003
0025
002
0015
001
0005
0
minusiG
(b)
Figure 3 Profiles of velocity for different values of 119870
120591119904= minus (120591
119909+ 119894120591119910)
=119894119890minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
(25)
5 Special Cases
In order to verify the correctness of our obtained solutionsit is important to note that (18) and (19) satisfy the governingequation and the imposed initial and boundary conditionsFurther these solutions are more general comparing to theexisting solutions in the literature In this section we derivesome special cases of these solutions The solution given by(18) for hydrodynamic fluid (119872 = 0) over an impulsivelymoved plate (120596 = 0) reduces to the following form
119866 (120585 120591) =119867 (120591) 119890
minus1205851198782
2
times[[[
[
119890minus120585radic119878
24+1119870+2119894120578
times erf 119888( 120585
2radic120591minus radic(
1198782
4+1
119870+ 2119894120578) 120591)
+ 119890120585radic11987824+1119870+2119894120578
times erf 119888( 120585
2radic120591+ radic(
1198782
4+1
119870+ 2119894120578) 120591)
]]]
]
(26)
which is identical to the solution given by (13) obtained byJana et al [21] Further the present solution given by (18) forsuddenly moved plate (120596 = 0) reduces to the solution (19)obtained by Farhad et al [20] in the absence of Hall currentand slip boundary condition Hence this provides a usefulmathematical check to our calculi
6 Results and Discussion
The exact solutions for the unsteady hydromagnetic flow ofviscous fluid bounded by a porous plate in a porous mediumare obtained The analytical results are displayed for variousvalues of emerging parameters such as Hartmann number119872 permeability parameter 119870 rotation parameter 120578 suctionparameter 119878 and phase angle 120596120591 Figures 2ndash6 are plotted forthe cosine velocity given by (18) In these figures (a) and (b)show the real and imaginary parts of velocity
Figure 2 is prepared to show the variation of velocity fordifferent values of Hartmann number119872 It is found that thereal part of velocity and boundary layer thickness decreaseswith increasing values of Hartmann number It is due to thefact that the application of transverse magnetic field results aresistive type force (called Lorentz force) similar to drag forceand upon increasing the values of119872 the drag force increaseswhich leads to the deceleration of the flow However it isobserved that this behavior is quite opposite for the imaginarypart of velocity Figure 3 reveals that the presence of porousmedium increases the resistance to flow thus reducing its
6 Mathematical Problems in Engineering
0075
01
005
0025
0
151050 2
G
120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
minus0025
minus005
minus0075
(a)
151050 2120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
03
025
02
015
01
005
0
minusiG
(b)
Figure 4 Profiles of velocity for different values of 120578
08
08
06
06
04
04
02
02
1
1 120
0
G
120585
S = 00
S = 03
S = 06
S = 09
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
(a)
05 151 20
0120585
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
S = 00
S = 03
S = 06
S = 09
006
005
004
003
002
001
minusiG
(b)
Figure 5 Profiles of velocity for different values of 119878
velocity Itmay also be expected due to the fact that increasingvalues of119870 reduces the friction forces which assists the fluidconsiderably to move fast and increases the boundary layerthicknessThe graphs for the rotation parameter are plotted inFigure 4 It is observed that the real part of velocity increaseswhen 120578 is increased However when observed carefully it isfound from Figure 4(b) that the behavior of imaginary part ofvelocity is more oscillatory in nature The velocity increasesfirst and then decreases when 120578 is increased
Figure 5 shows the effect of suction parameter 119878 on theflow through a porous medium in a rotating frame Clearlythe absolute value of velocity and boundary layer thicknessfor both real and imaginary parts of velocity decrease with
an increase in suction parameterThe variation of velocity fordifferent values of phase angle 120596120591 is shown in Figure 6 It isfound that the real part of velocity decreases with increasingphase angle 120596120591 Two different values of the phase angle 120596120591 =0120587 and 1205872 are chosen It is interesting to note that when120596120591 = 0120587 the real part of velocity is 1 which corresponds tothe impulsivemotion of the plateMoreover for120596120591 = 1205872 thereal part of velocity is 0 The absolute value of the imaginarypart of velocity is increasing with increasing phase angle 120596120591Furthermore for large values of 120585 that is 120585 rarr infin both ofthe real and imaginary parts of velocity are approaching zeroThe graphical behavior in this figure is in accordance withthe imposed boundary conditions (3) and (4) Moreover it
Mathematical Problems in Engineering 7
08
08
06
06
04
04
02
02
1
10
0
G
120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
(a)
107505 17515025 1250120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
0
minusiG
005
004
003
002
001
(b)
Figure 6 Profiles of velocity for different values of 120596120591
is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results
7 Conclusions
The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are
plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system
References
[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982
[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968
[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968
[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000
[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008
[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009
[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001
[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004
[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005
[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005
[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012
[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004
[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009
[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008
[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008
[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
friction in case of cosine and sine oscillations of the plate areobtainedThe analytical results are plotted and discussedTheresults of Jana et al [21] are recovered as a special case fromour obtained solutions The present study is useful in astron-omy to study the rotating motion of astrophysical objectssuch as magnetic stars
2 Mathematical Formulation of the Problem
Consider the unsteady flow of an incompressible viscousfluid occupying the upper porous half-space of the (119909 119910)plane The fluid is bounded by a porous plate at 119911 = 0 suchthat the positive 119911-axis is taken normal to the plate and the 119909-axis is taken parallel to the plate We consider the hydromag-netic flow induced in the fluid in the presence of a uniformmagnetic field of strength 119861
0applied in a direction normal to
the plate by means of the plate oscillations in its plane (Hayatet al [13]) The electric field due to polarization of charges isneglected Both of the fluid and plate are in a state of rigidbody rotation with constant angular velocity Ω = Ωk k isa unit vector in the 119911-directionThe geometry of the problemis shown in Figure 1 Initially for 119905 le 0 both of the fluid andplate are at rest At time 119905 = 0+ the lower plate suddenlystarts to move in its own plane with oscillating velocity inthe flow direction Under these assumptions the equations ofmomentum in a rotating frame along the 119909- and 119910-directionsare [18ndash20]
120597119906
120597119905minus 1199080
120597119906
120597119911minus 2ΩV = ]
1205972119906
1205971199112minus]120601119896119906 minus1205901198612
0
120588119906
119911 119905 gt 0
120597V120597119905minus 1199080
120597V120597119911+ 2Ω119906 = ]
1205972V1205971199112minus]120601119896V minus1205901198612
0
120588V
119911 119905 gt 0
(1)
where 119906 and V are the velocity components in 119909- and 119910-directions 119908
0is the suction velocity at the plate ] is the kine-
matic viscosity 119905 is the time 120590 is the electrical conductivity1198610is the applied magnetic field 120588 is the fluid density 120601 is the
porosity and 119896 is the permeability of the porous mediumThe corresponding initial and boundary conditions are
119906 = V = 0 119905 le 0 (2)
119906 = 119880119867 (119905) cos (1205960119905) or 119880 sin (120596
0119905)
V = 0 at 119911 = 0 119905 gt 0(3)
119906 997888rarr 0 V 997888rarr 0 as 119911 997888rarr infin 119905 gt 0 (4)
in which 119880 is the constant plate velocity 1205960is the frequency
of oscillation of the plate and119867(119905) is the Heaviside functionDefining119865 = 119906+119894V the above systemof equations reduces
to
120597119865
120597119905minus 1199080
120597119865
120597119911+ 2119894Ω119865 = ]
1205972119865
1205971199112minus]120601119896119865 minus1205901198612
0
120588119865
119911 119905 gt 0
Ox
B0
Magnetic fieldΩ
z
Porous medium z gt 0
y
u(0 t) = UH(t) cos(1205960t)
u(0 t) = Usin(1205960t)
(0 t) = 0 t gt 0
Figure 1 Geometry of the problem
119865 (119911 0) = 0 119905 le 0
119865 (0 119905) = 119880119867 (119905) cos (1205960119905) or 119880 sin (1205960119905) 119905 gt 0
119865 997888rarr 0 as 119911 997888rarr infin 119905 gt 0(5)
Introducing the following dimensionless variables (see Janaet al [21])
119866 =119865
119880 120585 =
119911119880
] 120591 =
1198802119905
] 120596 =
1205960]1198802 (6)
the dimensionless problem takes the following form
120597119866
120597120591minus 119878120597119866
120597120585+ 2119894120578119866 =
1205972119866
1205971205852minus1198722119866 minus
1
119870119866 (7)
119866 (120585 0) = 0 120591 le 0 (8)
119866 (0 120591) = 119867 (120591) cos (120596120591) or sin (120596120591) 120591 gt 0 (9)
119866 (infin 120591) = 0 120591 gt 0 (10)
1198722=12059011986120]
1205881198802
1
119870=
]21206011198961198802 120578 =
Ω]1198802 119878 =
1199080
119880
(11)
where 119872 119870 120578 and 119878 are the magnetic permeability rota-tion and suction parameters respectively
3 Solution of the Problem byUsing Laplace Transform
Taking the Laplace transform of (7) and using the initialcondition (8) we get the following transformed differentialequation
1198892119866 (120585 119902)
1198891205852+ 119878119889119866 (120585 119902)
119889120585
minus [119902 +1198722+1
119870+ 2119894120578]119866 (120585 119902) = 0
(12)
Mathematical Problems in Engineering 3
where 119902 is the transform parameter In view of the boundaryconditions (9) and (10) the Laplace transform 119866(120585 119902) of119866(120585 120591) has to satisfy the following conditions
119866 (0 119902) =119902
1199022 + 1205962or 119866 (0 119902) =
120596
1199022 + 1205962 (13)
119866 (0 119902) = 0 (14)
Solution of (12) under the boundary conditions (13) and (14)yields
119866119888(120585 119902) =
1
2119890minus12058511987821
119902 + 119894120596119890minus120585radic119902+119886
+1
119902 minus 119894120596119890minus120585radic119902+119886
(15)
respectively
119866119904(120585 119902) =
1
2119894119890minus1205851198782minus
1
119902 + 119894120596119890minus120585radic119902+119886
+1
119902 minus 119894120596119890minus120585radic119902+119886
(16)
where the subscripts 119888 and 119904 denote the cosine and sine oscil-lations of the plate and
119886 =1198782
4+1
119870+1198722+ 2119894120578 (17)
Now in order to find the inverse Laplace transform of (15)and (16) we use formula (A4) from the Appendix of Farhadet al [20] hence we get the following solutions
119866119888(120585 120591)
=119867 (120591) 119890
minus1205851198782
4
times [119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 minus 119894120596) 120591)
+ 119890120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 minus 119894120596) 120591)]
+119867 (120591) 119890
minus1205851198782
4
times [119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 + 119894120596) 120591)
+ 119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 + 119894120596) 120591)]
(18)
respectively
119866119904(120585 120591)
=119890minus1205851198782
4119894
times [minus119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 minus 119894120596) 120591)
+ 119890120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 minus 119894120596) 120591)]
+119890minus1205851198782
4119894[119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 + 119894120596) 120591)
+ 119890120585radic119886+119894120596 erf 119888 ( 120585
2radic120591+radic(119886+ 119894120596) 120591)]
(19)
4 Solution of the Problem byUsing Fourier Sine Transform
Now applying Fourier sine transform to (12) and usingboundary conditions (13) and (14) we obtain
119866119888(120577 119902) =
radic2 (120577 + 120577119878) 119902
(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (20)
respectively
119866119904(120577 119902) =
radic2 (120577 + 120577119878) 120596
(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (21)
where 119866(120577 119902) denotes Fourier sine transform of 119866(120585 119902)Now we take the inverse Fourier sine transform of (20)
which yields
119866119888(120585 119902) =
2radic2119902
120587 (1199022 + 1205962)intinfin
0
(120577 + 120577119878)
(1205772 + 1205772119878 + 119902 + 119886)sin (120577120585) 119889120577
(22)
Taking the inverse Laplace transform of (22) we obtain
119866119888(120585 120591) =
2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120577120585) 119889120577
minus2
120587intinfin
0
120577119878 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120577120585) 119889120577
4 Mathematical Problems in Engineering
1
08
06
04
02
01080604020 12
G
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
01
008
006
004
002
01080604020 12
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
minusiG
(b)
Figure 2 Profiles of velocity for different values of119872
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772+ 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577(23)
Similarly for the sine oscillation of the boundary we get from(21) the following
119866119904 (120585 120591) =
2
120587intinfin
0
120577120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577120596
((1205772+1205772119878+119886)2+ 1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)120596
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577119878120596
((1205772+1205772119878+119886)2+1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772 + 1205772119878 + 119886)
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
(24)
The nondimensional shear stresses 120591119888and 120591119904at the plate 120585 =
0 due to cosine and sine oscillations of the plate are obtainedfrom (18) and (19) as follows
120591119888= minus (120591
119909+ 119894120591119910)
= minus119867 (120591) 119890
minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
Mathematical Problems in Engineering 5
1
08
06
04
02
0151050 2
G
120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
15 251050 32120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
0035
003
0025
002
0015
001
0005
0
minusiG
(b)
Figure 3 Profiles of velocity for different values of 119870
120591119904= minus (120591
119909+ 119894120591119910)
=119894119890minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
(25)
5 Special Cases
In order to verify the correctness of our obtained solutionsit is important to note that (18) and (19) satisfy the governingequation and the imposed initial and boundary conditionsFurther these solutions are more general comparing to theexisting solutions in the literature In this section we derivesome special cases of these solutions The solution given by(18) for hydrodynamic fluid (119872 = 0) over an impulsivelymoved plate (120596 = 0) reduces to the following form
119866 (120585 120591) =119867 (120591) 119890
minus1205851198782
2
times[[[
[
119890minus120585radic119878
24+1119870+2119894120578
times erf 119888( 120585
2radic120591minus radic(
1198782
4+1
119870+ 2119894120578) 120591)
+ 119890120585radic11987824+1119870+2119894120578
times erf 119888( 120585
2radic120591+ radic(
1198782
4+1
119870+ 2119894120578) 120591)
]]]
]
(26)
which is identical to the solution given by (13) obtained byJana et al [21] Further the present solution given by (18) forsuddenly moved plate (120596 = 0) reduces to the solution (19)obtained by Farhad et al [20] in the absence of Hall currentand slip boundary condition Hence this provides a usefulmathematical check to our calculi
6 Results and Discussion
The exact solutions for the unsteady hydromagnetic flow ofviscous fluid bounded by a porous plate in a porous mediumare obtained The analytical results are displayed for variousvalues of emerging parameters such as Hartmann number119872 permeability parameter 119870 rotation parameter 120578 suctionparameter 119878 and phase angle 120596120591 Figures 2ndash6 are plotted forthe cosine velocity given by (18) In these figures (a) and (b)show the real and imaginary parts of velocity
Figure 2 is prepared to show the variation of velocity fordifferent values of Hartmann number119872 It is found that thereal part of velocity and boundary layer thickness decreaseswith increasing values of Hartmann number It is due to thefact that the application of transverse magnetic field results aresistive type force (called Lorentz force) similar to drag forceand upon increasing the values of119872 the drag force increaseswhich leads to the deceleration of the flow However it isobserved that this behavior is quite opposite for the imaginarypart of velocity Figure 3 reveals that the presence of porousmedium increases the resistance to flow thus reducing its
6 Mathematical Problems in Engineering
0075
01
005
0025
0
151050 2
G
120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
minus0025
minus005
minus0075
(a)
151050 2120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
03
025
02
015
01
005
0
minusiG
(b)
Figure 4 Profiles of velocity for different values of 120578
08
08
06
06
04
04
02
02
1
1 120
0
G
120585
S = 00
S = 03
S = 06
S = 09
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
(a)
05 151 20
0120585
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
S = 00
S = 03
S = 06
S = 09
006
005
004
003
002
001
minusiG
(b)
Figure 5 Profiles of velocity for different values of 119878
velocity Itmay also be expected due to the fact that increasingvalues of119870 reduces the friction forces which assists the fluidconsiderably to move fast and increases the boundary layerthicknessThe graphs for the rotation parameter are plotted inFigure 4 It is observed that the real part of velocity increaseswhen 120578 is increased However when observed carefully it isfound from Figure 4(b) that the behavior of imaginary part ofvelocity is more oscillatory in nature The velocity increasesfirst and then decreases when 120578 is increased
Figure 5 shows the effect of suction parameter 119878 on theflow through a porous medium in a rotating frame Clearlythe absolute value of velocity and boundary layer thicknessfor both real and imaginary parts of velocity decrease with
an increase in suction parameterThe variation of velocity fordifferent values of phase angle 120596120591 is shown in Figure 6 It isfound that the real part of velocity decreases with increasingphase angle 120596120591 Two different values of the phase angle 120596120591 =0120587 and 1205872 are chosen It is interesting to note that when120596120591 = 0120587 the real part of velocity is 1 which corresponds tothe impulsivemotion of the plateMoreover for120596120591 = 1205872 thereal part of velocity is 0 The absolute value of the imaginarypart of velocity is increasing with increasing phase angle 120596120591Furthermore for large values of 120585 that is 120585 rarr infin both ofthe real and imaginary parts of velocity are approaching zeroThe graphical behavior in this figure is in accordance withthe imposed boundary conditions (3) and (4) Moreover it
Mathematical Problems in Engineering 7
08
08
06
06
04
04
02
02
1
10
0
G
120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
(a)
107505 17515025 1250120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
0
minusiG
005
004
003
002
001
(b)
Figure 6 Profiles of velocity for different values of 120596120591
is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results
7 Conclusions
The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are
plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system
References
[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982
[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968
[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968
[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000
[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008
[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009
[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001
[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004
[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005
[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005
[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012
[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004
[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009
[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008
[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008
[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
where 119902 is the transform parameter In view of the boundaryconditions (9) and (10) the Laplace transform 119866(120585 119902) of119866(120585 120591) has to satisfy the following conditions
119866 (0 119902) =119902
1199022 + 1205962or 119866 (0 119902) =
120596
1199022 + 1205962 (13)
119866 (0 119902) = 0 (14)
Solution of (12) under the boundary conditions (13) and (14)yields
119866119888(120585 119902) =
1
2119890minus12058511987821
119902 + 119894120596119890minus120585radic119902+119886
+1
119902 minus 119894120596119890minus120585radic119902+119886
(15)
respectively
119866119904(120585 119902) =
1
2119894119890minus1205851198782minus
1
119902 + 119894120596119890minus120585radic119902+119886
+1
119902 minus 119894120596119890minus120585radic119902+119886
(16)
where the subscripts 119888 and 119904 denote the cosine and sine oscil-lations of the plate and
119886 =1198782
4+1
119870+1198722+ 2119894120578 (17)
Now in order to find the inverse Laplace transform of (15)and (16) we use formula (A4) from the Appendix of Farhadet al [20] hence we get the following solutions
119866119888(120585 120591)
=119867 (120591) 119890
minus1205851198782
4
times [119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 minus 119894120596) 120591)
+ 119890120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 minus 119894120596) 120591)]
+119867 (120591) 119890
minus1205851198782
4
times [119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 + 119894120596) 120591)
+ 119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 + 119894120596) 120591)]
(18)
respectively
119866119904(120585 120591)
=119890minus1205851198782
4119894
times [minus119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 minus 119894120596) 120591)
+ 119890120585radic119886minus119894120596 erf 119888 ( 120585
2radic120591+ radic(119886 minus 119894120596) 120591)]
+119890minus1205851198782
4119894[119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585
2radic120591minus radic(119886 + 119894120596) 120591)
+ 119890120585radic119886+119894120596 erf 119888 ( 120585
2radic120591+radic(119886+ 119894120596) 120591)]
(19)
4 Solution of the Problem byUsing Fourier Sine Transform
Now applying Fourier sine transform to (12) and usingboundary conditions (13) and (14) we obtain
119866119888(120577 119902) =
radic2 (120577 + 120577119878) 119902
(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (20)
respectively
119866119904(120577 119902) =
radic2 (120577 + 120577119878) 120596
(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (21)
where 119866(120577 119902) denotes Fourier sine transform of 119866(120585 119902)Now we take the inverse Fourier sine transform of (20)
which yields
119866119888(120585 119902) =
2radic2119902
120587 (1199022 + 1205962)intinfin
0
(120577 + 120577119878)
(1205772 + 1205772119878 + 119902 + 119886)sin (120577120585) 119889120577
(22)
Taking the inverse Laplace transform of (22) we obtain
119866119888(120585 120591) =
2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120577120585) 119889120577
minus2
120587intinfin
0
120577119878 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120577120585) 119889120577
4 Mathematical Problems in Engineering
1
08
06
04
02
01080604020 12
G
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
01
008
006
004
002
01080604020 12
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
minusiG
(b)
Figure 2 Profiles of velocity for different values of119872
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772+ 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577(23)
Similarly for the sine oscillation of the boundary we get from(21) the following
119866119904 (120585 120591) =
2
120587intinfin
0
120577120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577120596
((1205772+1205772119878+119886)2+ 1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)120596
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577119878120596
((1205772+1205772119878+119886)2+1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772 + 1205772119878 + 119886)
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
(24)
The nondimensional shear stresses 120591119888and 120591119904at the plate 120585 =
0 due to cosine and sine oscillations of the plate are obtainedfrom (18) and (19) as follows
120591119888= minus (120591
119909+ 119894120591119910)
= minus119867 (120591) 119890
minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
Mathematical Problems in Engineering 5
1
08
06
04
02
0151050 2
G
120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
15 251050 32120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
0035
003
0025
002
0015
001
0005
0
minusiG
(b)
Figure 3 Profiles of velocity for different values of 119870
120591119904= minus (120591
119909+ 119894120591119910)
=119894119890minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
(25)
5 Special Cases
In order to verify the correctness of our obtained solutionsit is important to note that (18) and (19) satisfy the governingequation and the imposed initial and boundary conditionsFurther these solutions are more general comparing to theexisting solutions in the literature In this section we derivesome special cases of these solutions The solution given by(18) for hydrodynamic fluid (119872 = 0) over an impulsivelymoved plate (120596 = 0) reduces to the following form
119866 (120585 120591) =119867 (120591) 119890
minus1205851198782
2
times[[[
[
119890minus120585radic119878
24+1119870+2119894120578
times erf 119888( 120585
2radic120591minus radic(
1198782
4+1
119870+ 2119894120578) 120591)
+ 119890120585radic11987824+1119870+2119894120578
times erf 119888( 120585
2radic120591+ radic(
1198782
4+1
119870+ 2119894120578) 120591)
]]]
]
(26)
which is identical to the solution given by (13) obtained byJana et al [21] Further the present solution given by (18) forsuddenly moved plate (120596 = 0) reduces to the solution (19)obtained by Farhad et al [20] in the absence of Hall currentand slip boundary condition Hence this provides a usefulmathematical check to our calculi
6 Results and Discussion
The exact solutions for the unsteady hydromagnetic flow ofviscous fluid bounded by a porous plate in a porous mediumare obtained The analytical results are displayed for variousvalues of emerging parameters such as Hartmann number119872 permeability parameter 119870 rotation parameter 120578 suctionparameter 119878 and phase angle 120596120591 Figures 2ndash6 are plotted forthe cosine velocity given by (18) In these figures (a) and (b)show the real and imaginary parts of velocity
Figure 2 is prepared to show the variation of velocity fordifferent values of Hartmann number119872 It is found that thereal part of velocity and boundary layer thickness decreaseswith increasing values of Hartmann number It is due to thefact that the application of transverse magnetic field results aresistive type force (called Lorentz force) similar to drag forceand upon increasing the values of119872 the drag force increaseswhich leads to the deceleration of the flow However it isobserved that this behavior is quite opposite for the imaginarypart of velocity Figure 3 reveals that the presence of porousmedium increases the resistance to flow thus reducing its
6 Mathematical Problems in Engineering
0075
01
005
0025
0
151050 2
G
120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
minus0025
minus005
minus0075
(a)
151050 2120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
03
025
02
015
01
005
0
minusiG
(b)
Figure 4 Profiles of velocity for different values of 120578
08
08
06
06
04
04
02
02
1
1 120
0
G
120585
S = 00
S = 03
S = 06
S = 09
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
(a)
05 151 20
0120585
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
S = 00
S = 03
S = 06
S = 09
006
005
004
003
002
001
minusiG
(b)
Figure 5 Profiles of velocity for different values of 119878
velocity Itmay also be expected due to the fact that increasingvalues of119870 reduces the friction forces which assists the fluidconsiderably to move fast and increases the boundary layerthicknessThe graphs for the rotation parameter are plotted inFigure 4 It is observed that the real part of velocity increaseswhen 120578 is increased However when observed carefully it isfound from Figure 4(b) that the behavior of imaginary part ofvelocity is more oscillatory in nature The velocity increasesfirst and then decreases when 120578 is increased
Figure 5 shows the effect of suction parameter 119878 on theflow through a porous medium in a rotating frame Clearlythe absolute value of velocity and boundary layer thicknessfor both real and imaginary parts of velocity decrease with
an increase in suction parameterThe variation of velocity fordifferent values of phase angle 120596120591 is shown in Figure 6 It isfound that the real part of velocity decreases with increasingphase angle 120596120591 Two different values of the phase angle 120596120591 =0120587 and 1205872 are chosen It is interesting to note that when120596120591 = 0120587 the real part of velocity is 1 which corresponds tothe impulsivemotion of the plateMoreover for120596120591 = 1205872 thereal part of velocity is 0 The absolute value of the imaginarypart of velocity is increasing with increasing phase angle 120596120591Furthermore for large values of 120585 that is 120585 rarr infin both ofthe real and imaginary parts of velocity are approaching zeroThe graphical behavior in this figure is in accordance withthe imposed boundary conditions (3) and (4) Moreover it
Mathematical Problems in Engineering 7
08
08
06
06
04
04
02
02
1
10
0
G
120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
(a)
107505 17515025 1250120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
0
minusiG
005
004
003
002
001
(b)
Figure 6 Profiles of velocity for different values of 120596120591
is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results
7 Conclusions
The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are
plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system
References
[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982
[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968
[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968
[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000
[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008
[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009
[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001
[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004
[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005
[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005
[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012
[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004
[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009
[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008
[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008
[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
1
08
06
04
02
01080604020 12
G
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
01
008
006
004
002
01080604020 12
120585
M = 0
M = 3M = 6
M = 9
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
minusiG
(b)
Figure 2 Profiles of velocity for different values of119872
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772+ 1205772119878 + 119886)
((1205772 + 1205772119878 + 119886)2+ 1205962)
times cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596
((1205772 + 1205772119878 + 119886)2+ 1205962)
times sin (120596120591) sin (120577120585) 119889120577(23)
Similarly for the sine oscillation of the boundary we get from(21) the following
119866119904 (120585 120591) =
2
120587intinfin
0
120577120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577120596
((1205772+1205772119878+119886)2+ 1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577 (1205772 + 1205772119878 + 119886)120596
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878120596119890minus(1205772+1205772119878+119886)120591
((1205772 + 1205772119878 + 119886)2+ 1205962)
sin (120577120585) 119889120577
minus2
120587intinfin
0
120577119878120596
((1205772+1205772119878+119886)2+1205962)
cos (120596120591) sin (120577120585) 119889120577
+2
120587intinfin
0
120577119878 (1205772 + 1205772119878 + 119886)
((1205772+1205772119878+119886)2+1205962)
sin (120596120591) sin (120577120585) 119889120577
(24)
The nondimensional shear stresses 120591119888and 120591119904at the plate 120585 =
0 due to cosine and sine oscillations of the plate are obtainedfrom (18) and (19) as follows
120591119888= minus (120591
119909+ 119894120591119910)
= minus119867 (120591) 119890
minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
Mathematical Problems in Engineering 5
1
08
06
04
02
0151050 2
G
120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
15 251050 32120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
0035
003
0025
002
0015
001
0005
0
minusiG
(b)
Figure 3 Profiles of velocity for different values of 119870
120591119904= minus (120591
119909+ 119894120591119910)
=119894119890minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
(25)
5 Special Cases
In order to verify the correctness of our obtained solutionsit is important to note that (18) and (19) satisfy the governingequation and the imposed initial and boundary conditionsFurther these solutions are more general comparing to theexisting solutions in the literature In this section we derivesome special cases of these solutions The solution given by(18) for hydrodynamic fluid (119872 = 0) over an impulsivelymoved plate (120596 = 0) reduces to the following form
119866 (120585 120591) =119867 (120591) 119890
minus1205851198782
2
times[[[
[
119890minus120585radic119878
24+1119870+2119894120578
times erf 119888( 120585
2radic120591minus radic(
1198782
4+1
119870+ 2119894120578) 120591)
+ 119890120585radic11987824+1119870+2119894120578
times erf 119888( 120585
2radic120591+ radic(
1198782
4+1
119870+ 2119894120578) 120591)
]]]
]
(26)
which is identical to the solution given by (13) obtained byJana et al [21] Further the present solution given by (18) forsuddenly moved plate (120596 = 0) reduces to the solution (19)obtained by Farhad et al [20] in the absence of Hall currentand slip boundary condition Hence this provides a usefulmathematical check to our calculi
6 Results and Discussion
The exact solutions for the unsteady hydromagnetic flow ofviscous fluid bounded by a porous plate in a porous mediumare obtained The analytical results are displayed for variousvalues of emerging parameters such as Hartmann number119872 permeability parameter 119870 rotation parameter 120578 suctionparameter 119878 and phase angle 120596120591 Figures 2ndash6 are plotted forthe cosine velocity given by (18) In these figures (a) and (b)show the real and imaginary parts of velocity
Figure 2 is prepared to show the variation of velocity fordifferent values of Hartmann number119872 It is found that thereal part of velocity and boundary layer thickness decreaseswith increasing values of Hartmann number It is due to thefact that the application of transverse magnetic field results aresistive type force (called Lorentz force) similar to drag forceand upon increasing the values of119872 the drag force increaseswhich leads to the deceleration of the flow However it isobserved that this behavior is quite opposite for the imaginarypart of velocity Figure 3 reveals that the presence of porousmedium increases the resistance to flow thus reducing its
6 Mathematical Problems in Engineering
0075
01
005
0025
0
151050 2
G
120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
minus0025
minus005
minus0075
(a)
151050 2120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
03
025
02
015
01
005
0
minusiG
(b)
Figure 4 Profiles of velocity for different values of 120578
08
08
06
06
04
04
02
02
1
1 120
0
G
120585
S = 00
S = 03
S = 06
S = 09
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
(a)
05 151 20
0120585
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
S = 00
S = 03
S = 06
S = 09
006
005
004
003
002
001
minusiG
(b)
Figure 5 Profiles of velocity for different values of 119878
velocity Itmay also be expected due to the fact that increasingvalues of119870 reduces the friction forces which assists the fluidconsiderably to move fast and increases the boundary layerthicknessThe graphs for the rotation parameter are plotted inFigure 4 It is observed that the real part of velocity increaseswhen 120578 is increased However when observed carefully it isfound from Figure 4(b) that the behavior of imaginary part ofvelocity is more oscillatory in nature The velocity increasesfirst and then decreases when 120578 is increased
Figure 5 shows the effect of suction parameter 119878 on theflow through a porous medium in a rotating frame Clearlythe absolute value of velocity and boundary layer thicknessfor both real and imaginary parts of velocity decrease with
an increase in suction parameterThe variation of velocity fordifferent values of phase angle 120596120591 is shown in Figure 6 It isfound that the real part of velocity decreases with increasingphase angle 120596120591 Two different values of the phase angle 120596120591 =0120587 and 1205872 are chosen It is interesting to note that when120596120591 = 0120587 the real part of velocity is 1 which corresponds tothe impulsivemotion of the plateMoreover for120596120591 = 1205872 thereal part of velocity is 0 The absolute value of the imaginarypart of velocity is increasing with increasing phase angle 120596120591Furthermore for large values of 120585 that is 120585 rarr infin both ofthe real and imaginary parts of velocity are approaching zeroThe graphical behavior in this figure is in accordance withthe imposed boundary conditions (3) and (4) Moreover it
Mathematical Problems in Engineering 7
08
08
06
06
04
04
02
02
1
10
0
G
120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
(a)
107505 17515025 1250120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
0
minusiG
005
004
003
002
001
(b)
Figure 6 Profiles of velocity for different values of 120596120591
is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results
7 Conclusions
The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are
plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system
References
[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982
[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968
[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968
[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000
[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008
[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009
[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001
[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004
[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005
[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005
[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012
[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004
[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009
[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008
[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008
[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
1
08
06
04
02
0151050 2
G
120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
(a)
15 251050 32120585
K = 02
K = 04
K = 06
K = 08
120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
0035
003
0025
002
0015
001
0005
0
minusiG
(b)
Figure 3 Profiles of velocity for different values of 119870
120591119904= minus (120591
119909+ 119894120591119910)
=119894119890minus119886119905minus119894120596120591
4radic120587120591
times [(minus4119890119894120596120591minus 119878radic120587120591119890
119886120591minus 119878radic120587120591119890
119886120591+2119894120596120591
minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890
119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591
+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)
+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]
(25)
5 Special Cases
In order to verify the correctness of our obtained solutionsit is important to note that (18) and (19) satisfy the governingequation and the imposed initial and boundary conditionsFurther these solutions are more general comparing to theexisting solutions in the literature In this section we derivesome special cases of these solutions The solution given by(18) for hydrodynamic fluid (119872 = 0) over an impulsivelymoved plate (120596 = 0) reduces to the following form
119866 (120585 120591) =119867 (120591) 119890
minus1205851198782
2
times[[[
[
119890minus120585radic119878
24+1119870+2119894120578
times erf 119888( 120585
2radic120591minus radic(
1198782
4+1
119870+ 2119894120578) 120591)
+ 119890120585radic11987824+1119870+2119894120578
times erf 119888( 120585
2radic120591+ radic(
1198782
4+1
119870+ 2119894120578) 120591)
]]]
]
(26)
which is identical to the solution given by (13) obtained byJana et al [21] Further the present solution given by (18) forsuddenly moved plate (120596 = 0) reduces to the solution (19)obtained by Farhad et al [20] in the absence of Hall currentand slip boundary condition Hence this provides a usefulmathematical check to our calculi
6 Results and Discussion
The exact solutions for the unsteady hydromagnetic flow ofviscous fluid bounded by a porous plate in a porous mediumare obtained The analytical results are displayed for variousvalues of emerging parameters such as Hartmann number119872 permeability parameter 119870 rotation parameter 120578 suctionparameter 119878 and phase angle 120596120591 Figures 2ndash6 are plotted forthe cosine velocity given by (18) In these figures (a) and (b)show the real and imaginary parts of velocity
Figure 2 is prepared to show the variation of velocity fordifferent values of Hartmann number119872 It is found that thereal part of velocity and boundary layer thickness decreaseswith increasing values of Hartmann number It is due to thefact that the application of transverse magnetic field results aresistive type force (called Lorentz force) similar to drag forceand upon increasing the values of119872 the drag force increaseswhich leads to the deceleration of the flow However it isobserved that this behavior is quite opposite for the imaginarypart of velocity Figure 3 reveals that the presence of porousmedium increases the resistance to flow thus reducing its
6 Mathematical Problems in Engineering
0075
01
005
0025
0
151050 2
G
120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
minus0025
minus005
minus0075
(a)
151050 2120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
03
025
02
015
01
005
0
minusiG
(b)
Figure 4 Profiles of velocity for different values of 120578
08
08
06
06
04
04
02
02
1
1 120
0
G
120585
S = 00
S = 03
S = 06
S = 09
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
(a)
05 151 20
0120585
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
S = 00
S = 03
S = 06
S = 09
006
005
004
003
002
001
minusiG
(b)
Figure 5 Profiles of velocity for different values of 119878
velocity Itmay also be expected due to the fact that increasingvalues of119870 reduces the friction forces which assists the fluidconsiderably to move fast and increases the boundary layerthicknessThe graphs for the rotation parameter are plotted inFigure 4 It is observed that the real part of velocity increaseswhen 120578 is increased However when observed carefully it isfound from Figure 4(b) that the behavior of imaginary part ofvelocity is more oscillatory in nature The velocity increasesfirst and then decreases when 120578 is increased
Figure 5 shows the effect of suction parameter 119878 on theflow through a porous medium in a rotating frame Clearlythe absolute value of velocity and boundary layer thicknessfor both real and imaginary parts of velocity decrease with
an increase in suction parameterThe variation of velocity fordifferent values of phase angle 120596120591 is shown in Figure 6 It isfound that the real part of velocity decreases with increasingphase angle 120596120591 Two different values of the phase angle 120596120591 =0120587 and 1205872 are chosen It is interesting to note that when120596120591 = 0120587 the real part of velocity is 1 which corresponds tothe impulsivemotion of the plateMoreover for120596120591 = 1205872 thereal part of velocity is 0 The absolute value of the imaginarypart of velocity is increasing with increasing phase angle 120596120591Furthermore for large values of 120585 that is 120585 rarr infin both ofthe real and imaginary parts of velocity are approaching zeroThe graphical behavior in this figure is in accordance withthe imposed boundary conditions (3) and (4) Moreover it
Mathematical Problems in Engineering 7
08
08
06
06
04
04
02
02
1
10
0
G
120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
(a)
107505 17515025 1250120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
0
minusiG
005
004
003
002
001
(b)
Figure 6 Profiles of velocity for different values of 120596120591
is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results
7 Conclusions
The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are
plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system
References
[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982
[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968
[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968
[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000
[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008
[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009
[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001
[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004
[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005
[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005
[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012
[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004
[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009
[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008
[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008
[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
0075
01
005
0025
0
151050 2
G
120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
minus0025
minus005
minus0075
(a)
151050 2120585
120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0
120578 = 0
120578 = 2
120578 = 4
120578 = 6
03
025
02
015
01
005
0
minusiG
(b)
Figure 4 Profiles of velocity for different values of 120578
08
08
06
06
04
04
02
02
1
1 120
0
G
120585
S = 00
S = 03
S = 06
S = 09
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
(a)
05 151 20
0120585
120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0
S = 00
S = 03
S = 06
S = 09
006
005
004
003
002
001
minusiG
(b)
Figure 5 Profiles of velocity for different values of 119878
velocity Itmay also be expected due to the fact that increasingvalues of119870 reduces the friction forces which assists the fluidconsiderably to move fast and increases the boundary layerthicknessThe graphs for the rotation parameter are plotted inFigure 4 It is observed that the real part of velocity increaseswhen 120578 is increased However when observed carefully it isfound from Figure 4(b) that the behavior of imaginary part ofvelocity is more oscillatory in nature The velocity increasesfirst and then decreases when 120578 is increased
Figure 5 shows the effect of suction parameter 119878 on theflow through a porous medium in a rotating frame Clearlythe absolute value of velocity and boundary layer thicknessfor both real and imaginary parts of velocity decrease with
an increase in suction parameterThe variation of velocity fordifferent values of phase angle 120596120591 is shown in Figure 6 It isfound that the real part of velocity decreases with increasingphase angle 120596120591 Two different values of the phase angle 120596120591 =0120587 and 1205872 are chosen It is interesting to note that when120596120591 = 0120587 the real part of velocity is 1 which corresponds tothe impulsivemotion of the plateMoreover for120596120591 = 1205872 thereal part of velocity is 0 The absolute value of the imaginarypart of velocity is increasing with increasing phase angle 120596120591Furthermore for large values of 120585 that is 120585 rarr infin both ofthe real and imaginary parts of velocity are approaching zeroThe graphical behavior in this figure is in accordance withthe imposed boundary conditions (3) and (4) Moreover it
Mathematical Problems in Engineering 7
08
08
06
06
04
04
02
02
1
10
0
G
120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
(a)
107505 17515025 1250120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
0
minusiG
005
004
003
002
001
(b)
Figure 6 Profiles of velocity for different values of 120596120591
is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results
7 Conclusions
The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are
plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system
References
[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982
[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968
[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968
[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000
[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008
[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009
[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001
[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004
[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005
[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005
[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012
[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004
[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009
[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008
[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008
[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
08
08
06
06
04
04
02
02
1
10
0
G
120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
(a)
107505 17515025 1250120585
120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0
120596120591 = 0120587
120596120591 =120587
2
0
minusiG
005
004
003
002
001
(b)
Figure 6 Profiles of velocity for different values of 120596120591
is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results
7 Conclusions
The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are
plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system
References
[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982
[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968
[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968
[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000
[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008
[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009
[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001
[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004
[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005
[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005
[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012
[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004
[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009
[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008
[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008
[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010
[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010
[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012
[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of