The physical situation considered is that of quasi-state hydro magnetic flow of a viscous, incompressible and electrically conducting fluid between two parallel plates distance™ apart, when the lower plate is set in sinusoidal motion and the upper stationary .Neglecting the Magnetic Prandtl Number, Solution for Quasi-study state when the lower plate moves with cos ( and the corresponding Skin friction at the lower plate have been obtained. Discussion has been made of these two features for the two cases when the magnetic lines of force have been fixed relative to the fluid and the moving plate respectively

Rani, CH. Baby

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ISSN 2347-1921 2083 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4    Quasi-Steady State Solutions of MHD Oscillatory Flow between two Parallel Plates in a Rotating System Dr.Ch. Baby Rani Associate professor, Department of Mathematics V. R. Siddhartha Engineering College Vijayawada, Andhra Pradesh, India. Pin: 520 007. Email: ranicbr1@yahoo.com ABSTRACT The physical situation considered is that of quasi-state hydro magnetic flow of a viscous, incompressible and electrically conducting fluid between two parallel plates distance ‘2L’ apart, when the lower plate is set in sinusoidal motion and the upper stationary .Neglecting the Magnetic Prandtl Number, Solution for Quasi-steady state when the lower plate moves with 𝑈0 cos (𝜔𝑡) and the corresponding Skin –friction at the lower plate have been obtained. Discussion has been made of these two features for the two cases when the magnetic lines of force have been fixed relative to the fluid and the moving plate respectively. Keywords: Quasi-state; Porous plates; Velocity Distribution; Skin friction.                               Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics Vol 9, No 3 editor@cirjam.org www.cirjam.com  ISSN 2347-1921 2084 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4    1. INTRODUCTION Ramana Rao and Vidyanidhi (1970) studied the unsteady hydromagnetic flow between two parallel porous plates of which the lower is stationary and the upper moved with uniform velocity A 𝑡𝑛  (n = 0, 1) for the two cases when (i) the magnetic lines of force are fixed relative to the fluid and (ii) the magnetic lines of force are fixed relative to the upper plate. They however assumed the magnetic Prandtl number 𝑃𝑚   0 i.e, neglecting the induced magnetic field. Singh, Sacheti and Chandran (1994) brought out  the transient effects of couette flow of an electrically conducting fluid subject to rotation and magnetic field, when one of the plates has been set into uniformly accelerated motion for the above two cases. Sutton and Sherman (1965) studied the hydromagnetic transient couette flow when magnetic Prandtl number 𝑃𝑚  is unity. They obtained exact solutions for the velocity and the induced magnetic field assuming the upper plate is in uniform motion and the lower plate is stationary. This work holds good when the magnetic lines of force are fixed relative to the fluid. Ramesh (1996) presented the work of Sutton and Sherman assuming that the magnetic lines of force are fixed relative to the upper plate. Here we consider the hydromagnetic flow of a viscous, incompressible and electrically conducting fluid between two parallel plates, distant 2L apart, when the lower plate is set in sinusoidal motion and upper stationary .Neglecting the magnetic Prandtl number solution for quasi steady state when the  lower plate moves with 0 cosU t  and the corresponding skin-friction at the lower plate  have been obtained. Discussion has been made of these two features for the two cases when the Magnetic lines of force have been fixed relative to the fluid and the moving plate respectively. 2. MATHEMATICAL FORMULATION The physical situation considered is that of quasi-state hydromagnetic flow of a viscous, incompressible and electrically conducting fluid bounded by two infinite parallel plates, distant 2L apart, when 𝑍1-axis is taken normal to the plates. It is assumed that the plates are electrically non conducting and an applied uniform magnetic field𝐻0is acting parallel to the 𝑍1-axis. When 𝑡1 > 0, the lower plate 1LZ    moves sinusoidallyi.e,  1 10cos( )tU   or1 10sin( )tU  . The upper plate 1LZ   is stationary for 𝑡1 > 0. The governing equations of continuity and motion and Maxwell’s equations are: .V 0           (1.1) 211( . ) ( )V J BV V p Vt                   (1.2) 0B J           (1.3) 1BEt            (1.4) 0B           (1.5) ( )J E V B              (1.6) Where 1( ,0,0)V uthe velocity vector, p is the pressure,  is the density,   is the kinematic viscosity, J is the current density, Bis the magnetic induction vector, E is the electric field vector, 0 is the magnetic permeability and   is the electric conductivity. We assume the magnetic Reynolds number is so small, in others, mP  the magnetic Prandtl number is almost zero being the ratio of magnetic Reynolds number and the Reynolds number, that the induced magnetic field can be neglected in comparison with applied one (1962), so that  0(0,0, )B B          (1.7) where 0B is a constant. It is also assumed that no applied and polarization voltage exists (i. e, 0E ).This then corresponds to the case when no energy is added or extracted from the fluid by the electric field. Science the plates are infinite in extent, all physical variables (except pressure) are functions of 1Z and 1t . Equation (1.1) is automatically satisfied. Now, the equation for the conservation of electric charge, 0J  leads to 1zJ   constant, where1 1 1(J , , )X Y ZJ J J . As in the case of vertical velocity, we immediately see that 1 0ZJ   ISSN 2347-1921 2085 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4    Equation (1.6) thus yields 1 0XJ   , 1 11 10 0(u )i tYJ U e       (1.8) Noting that when the magnetic lines of force are fixed relative to the lower plate, moving sinusoidally equation (1.6) is replaced by  1 110ˆ( ( )i tJ E V u U e i B              (1.9) where iˆ is the unit vector in 1X - direction In view of the above considerations equation (1.2) can be written in the component form as in (1986). 1 121 2 1101 1( )i tu um u KU et z           (1.10) Where 20Bm  K=      0             𝑖𝑓 𝐵0 𝑖𝑠 𝑓𝑖𝑥𝑒𝑑 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡𝑜 𝑡𝑕𝑒 𝑓𝑙𝑢𝑖𝑑1 𝑖𝑓 𝐵0 𝑖𝑠 𝑓𝑖𝑥𝑒𝑑 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡𝑜 𝑡𝑕𝑒 𝑠𝑖𝑛𝑢𝑠𝑜𝑖𝑑𝑎𝑙𝑙𝑦 𝑚𝑜𝑣𝑖𝑛𝑔 𝑙𝑜𝑤𝑒𝑟 𝑝𝑙𝑎𝑡𝑒  Since we seek the quasi-state solution for 𝑢1 the boundary conditions are  1 110i tu U e  at 1Z L   and 1 0u   at 1Z L  for 1t > 0     (1.11) The real and imaginary parts of1u , when obtained corresponds to the motion of the lower plate either with 1 10 cosU t  or 1 10 sinU t  respectively. 3. METHOD OF SOLUTION Substituting: 1 11 10 ( )i tu U e f Z         (1.12) In equation (1.10) we get 111 m i mkf f              (1.13) which is to be solved subject to the conditions of equation (1.11) namely ( ) 1f L   , ( ) 0f L          (1.14) And dash denotes the differentiation with respect to 1Z  In terms of the non-dimensional quantities 1 11 10, , , ,2 2u m zV L L Z T tU L            (1.15) 14 4 2 2( )P      , 14 4 2 2( )Q           (1.16) We get ISSN 2347-1921 2086 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4    2 2 2 244 42 24 444 42 24 41(Z)cos sin1( )(Ch cos cos sin sin )2(Sh sin cos cos sin )1{ (Sh sin cos cos sin )2( cos cofCh P Q Sh P QKPZ QZChP Q ShPZ QZShP QKPZ QZChP Q ChPZ QZShP QKi PZ QZchPQ Q ChP QZShP QKChPZ QZChP                2 2 2 24 2 24 4 4 4s sin sin )}12( cos sincos cos sin sinsin cos sinQ ShPZ QZShP QSh P Q Ch P QShPZ QZShP Q ChPZ QZChP Qi ChPZ QZShP Q ShPZcosQZChP QK Ki                              (1.17) The boundary condition 1 1 10 cosu U t  of equation (1.11) gives the real part of 1 11 10 ( )i tu U e f Z i.e, 2 2 2 244 42 24 444 42 24 41cos sin1{( )(Ch cos cos sin sin )2(Sh sin cos cos sin )}cosT1{ (Sh sin cos cos sin )2( cosVCh P Q Sh P QKZP ZQChP Q ShZP ZQShP QKZP ZQChP Q ChZP ZQShP QKZP ZQchP Q ChZP ZQShP QKChZP ZQChP                2 2 2 24 2 24 4 4 4cos sin sin )}sinT12( cos sin( cos cos sin sin )cossin cos sin sincos sinQ ShZP ZQShP QSh P Q Ch P QShZP ZQShP Q ChZP ZQChP Q TChZP ZQShP Q ShZPcosZQChP Q TK KT T                              (1.18) Where in equations (1.17-1.18) Ch and Sh stands for cos hyperbolic and sine hyperbolic respectively. The corresponding skin-friction   at the lower plate 1zVZ    is given by ISSN 2347-1921 2087 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4    2 2 2 24 2 24 4 4 44 2 24 4 4 42 2 2 212( cos sin )1{( )( PSh 2 sin 2 ) ( sin 2 2 )}cosT21{( )( sin 2 2 ) ( 2 sin 2 )}sin214( cos sin )( 2Ch P Q Sh P QK KP Q Q P Q QSh PK KP Q QSh P Psh P Q Q TSh P Q Ch P QPSh P Q                                    sin 2 )cosT ( Psin 2 2 )sinQ Q QSh P T     (1.19) The boundary condition 1 1 10 cosU U t    of equation (1.11) gives the imaginary part of 1 11 10 ( )i tU U e f Zand its corresponding skin- friction at the lower plate can be determined but these two expressions are not obtained here. We have computed the expressions for the velocity and skin-friction as given by equations (1.18) and (1.19). The velocity distribution has been entered in Table I-VI and skin-friction in table VII. Figures 1 and 2 show tables I, III and IV respectively. 4. DISCUSSION In the non-magnetic case i.e., α = 0, shown in Fig .1the effect of increasing T(𝑤1𝑡1) is to reduce the velocity at any point of the channel independent of K( K does not rise) for fixed 𝜎, the penetration depth. In the magnetic case shown in Fig.2, the effect of increasing T also reduces the velocity at any point of the channel for fixed 𝜎 in both the cases when the magnetic lines of force are fixed relative to the fluid  K = 0 shown  by          and the lower plate K = 1 shown by-------. The effect of increasing K as shown in Fig.2 is to increase the velocity at any point of the channel, its width taken as unity. From Table II, in non-magnetic case, for fixed T the effect of increasing 𝜎 is to increase the velocity but very feebly. This also seen in the magnetic case from tables V and VI, corresponding to K=0 and 1 respectively. From these tables, as K increases the velocity increases. From Table VII ,it observed that the skin-friction at the lower plate (i) when 𝛼 = 0, i.e, in the non-magnetic case is the same independent of K( K does not arise) , (ii) in magnetic case decreases as K increases , (iii)  decreases as both T and 𝜎 increase in both non-magnetic and magnetic cases, (iv) increases as 𝛼 increases when K = 0 and decreases when K = 1.                                                                                TABLE-I            T 0.5  1.0   5.0 Z             -1.0   .87758260  .54030230  .28366220  -0.8   .79399850  .49382050  .24649410 -0.6   .70905900  .44493130  .21209510 -0.4   .62293530  .39391520  .18016350 -0.2   .53579140  .34104820  .15039520 0   .44778500  .28660250  .12248450 0.2   .35906820  .23084650  .09612405 0.4   .26978870  .17404580  .07100530 0.6   .18009040  .11646370  .04681866 0.8   .09011438  .05836169  .02325379 1.0   0      0        0             Velocity distribution when 𝛼 = 0, 𝜎 = 0.2, K = 0 and 1 ISSN 2347-1921 2088 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4                               TABLE-II              𝜎 0.2      0.4     0.6 Z            -1.0   .87758260  .87758260  .87758260 -0.8   .79399850  .80403510  .81233530 -0.6   .70905900  .72538010  .73679440 -0.4   .62293530  .64252720  .65386150 -0.2   .53579140  .55626710  .56582870 0   .44778500  .46728540  .47444590 0.2   .35906820  .37617670  .38099470 0.4   .26978870  .28345900  .28636310 0.6   .18009040  .18958880  .19112330 0.8   .09011438  .09497619  .09561023 1.0      0     0     0              Velocity distribution when 𝛼 = 0, T= 0.5, K = 0 and 1                                                                                           TABLE-III               T 0.5  1.0  5.0 Z             -1.0   .87758260  .54030230  .28366220  -0.8   .66153110  .40951220  .20928790 -0.6   .49759850  .30965790  .15410920 -0.4   .37288020  .23321390  .11309990 -0.2   .27755530  .17441380  .08251508 0   .20411540  .12881780  .05955396 0.2   .14677330  .09298015  .04210772 0.4   .10100920  .06418966  .02856892 0.6   .06321429  .04026641  .01768596 0.8   .03040743  .01939742  .00844937 1.0   0  0  0              Velocity distribution when 𝛼 = 1, 𝜎 = 0.2, K = 0        ISSN 2347-1921 2089 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4                                                                              TABLE-IV               T      0.5  1.0  5.0 Z             -1.0   .87758260  .54030230  .28366220  -0.8   .85117060  .52804790  .26695550 -0.6   .82116040  .51220150  .25189280 -0.4   .78520610  .49159570  .23715110 -0.2   .74048600  .46469190  .22141350 0   .68348220  .42946620  .20325330 0.2   .60970400  .38325820  .18100620 0.4   .51333510  .32257140  .15262010 0.6   .38677640  .24281010  .11546960 0.8   .22004700  .13793330  .06611700 1.0   0                  0                  0              Velocity distribution when 𝛼 = 1, 𝜎 = 0.2, K = 1               TABLE-V               𝜎                 0.2  0.4  0.6 Z             -1.0   .87758260  .87758260  .87758260  -0.8   .66153110  .66679070  .67387630 -0.6   .49759850  .50520470  .51485320 -0.4   .37288020  .38107450  .39082780 -0.2   .27755530  .28532100  .29397460 0   .20411540  .21090040  .21797770 0.2   .14677330  .15230640  .15772380 0.4   .10100920  .10517840  .10903610 0.6   .06321429  .06598935  .06844366 0.8   .03040743  .03179109  .03297934 1.0   0  0  0              Velocity distribution when 𝛼 = 1, 𝑇 = 0.K = 0            ISSN 2347-1921 2090 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4                      TABLE-VI               𝜎 0.2  0.4  0.6 Z             -1.0   .87758260  .87758260  .87758260  -0.8   .85117060  .86034430  .87178670 -0.6   .82116040  .83608260  .85381200 -0.4   .78520610  .80329900  .82390430 -0.2   .74048600  .75975080  .78094540 0   .68348220  .70230730  .72249130 0.2   .60970400  .62673620  .64469460 0.4   .51333510  .52740290  .54211260 0.6   .38677640  .39686730  .40740250 0.8   .22004700  .22534470  .23088980 1.0   0  0  0              Velocity distribution when 𝛼 = 1, 𝑇 = 0.5, K = 1                                                                       TABLE-VII              𝛼  𝜎  T   K   𝜏              0  0.2  0.5   0  0.41423600        1  0.41423600     1   0  0.22591600 1  0.22591600 0.4              0.5  0   0  0.35329120 1  0.35329120 1  0.2  0.5   0  1.23688100 1  0.12656620     1   0  0.74656890 0.4     1  0.05386135   0.5   0  1.20067900      1  0.07028925             Values of skin-friction 𝜏 at the lower plate ISSN 2347-1921 2091 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4                                             ISSN 2347-1921 2092 | P a g e                                                        A u g u s t  2 5 ,  2 0 1 4    REFERENCES: [1] S.I. Pai, Magnetogas dynamics and plasma dynamics, Springer, Berlin, 1962. [2] V.V. Ramana Rao and V. Vidyanidhi , Proc. Ind Acad.Sci. Vol. LXXII, No.3, Sec.A, P.272, 1970. [3] A.Raptis and A.K. Singh, Acta Phys. Hung, Vol.60, P.221,1986. [4] A.K. Singh, N.C. Sacheti  and P. Chandran, Int. J. Engg Science, Vol.32, P.133, 1994. [5] A. Sherman and G.W.Sutton, Engineering Magnetohydromics,Mc Graw-Hill, New York, 1965. [6] Ch. Ramesh, M.Phill. Dissertation , Andhra University, 1996. [7] A. N. Kozlov, Journal of Applied Mechanics and Technical Physics, Vol. 50, No.3, 2009. [8] T. Linga Raju and P. S. R Murthy, Bulletin of Pure and Applied Sciences,( International Research Journal of Sciences) , Vol. 24, No. 2, 2005. [9] T. Linga Raju and P. S. R Murthy, J. Indian Academy of Mathematics, Vol. 28, No. 2, 2006.            

Quasi-Steady State Solutions of MHD Oscillatory Flow Between two Parallel Plates in a Rotating System

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Quasi-Steady State Solutions of MHD Oscillatory Flow Between two Parallel Plates in a Rotating System

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