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Magnetohydrodynamic flow due to non-coaxial rotations of a porous oscillating disk and a fluid at infinity
International Journal of Engineering Science 41 (2003) 1177–1196 www.elsevier.com/locate/ijengsci
Magnetohydrodynamic flow due to non-coaxial rotations of a porous oscillating disk and a fluid at infinity T. Hayat
a,*
, M. Zamurad a, S. Asghar a, A.M. Siddiqui
b
a
b
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Department of Mathematics, Pennsylvania State University York Campus, York, PA 17403, USA Received 16 September 2002; accepted 4 October 2002
Abstract An exact solution of the Navier–Stokes equations is constructed for the case of flow due to non-coaxial rotations of a porous disk and a fluid at infinity. The disk executes oscillations in its own plane and is nonconducting. The viscous fluid is incompressible and electrically conducting. Analytical solution is established by the method of Laplace transform. The velocity fields are obtained for the cases when the angular velocity is greater than, smaller than or equal to the frequency of oscillations. The structure of the steady and the unsteady velocity fields are investigated. The difficulty of the hydrodynamic steady solution associated with the case of resonant frequency is resolved in the present analysis. Ó 2003 Published by Elsevier Science Ltd. Keywords: Magnetohydrodynamic effects; Porous disk; Laplace transform; Time-dependent equations; Resonance
1. Introduction The increasing number of technical applications using magnetohydrodynamic (MHD) effects has made it desirable to extend many of the available hydrodynamic solutions to include the effects of magnetic field for those cases when the viscous fluid is electrically conducting. For example, liquid–metal MHDs take their roots in conventional hydrodynamics of incompressible media, which become important in the metallurgical industry, nuclear reactor, sodium cooling system, energy storage and electrical power generation [1–3]. The greatest advantage of inductiontype pumps over other types of MHD devices is the absence of electrodes [4]. Induction pumps have been used to pump coolants in nuclear reactors. These designs are also being considered in *
Corresponding author. E-mail address: [email protected] (T. Hayat).
0020-7225/03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd. doi:10.1016/S0020-7225(03)00004-1
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MHD power generation [5]. The basic equations of incompressible MHD are non-linear. But there are many interesting cases where the equations become linear in terms of the unknown quantities and may be solved without any difficulty. Linear MHD permits exact solutions and adopts the approximations that the density and transport properties be constant. No fluid is incompressible, but all may be treated as such whenever the pressure changes are small in comparison with the bulk modulus. Mention may be made to the interesting works in [6–12]. Erdogan [13] has recently made an initial value investigation for flow due to non-coaxial rotations of an oscillating disk and a fluid at infinity. He claimed that if the angular velocity is equal to the frequency of oscillations, the system resonates and steady solution does not exist. It is apparent from physical considerations that suction and blowing have opposite effects. Indeed, the suction prevents the imposed non-torsional oscillations from spreading away from the disk by viscous diffusion for all values of the frequency. On the contrary, the blowing promotes the spreading of the oscillations far away from the disk and hence one boundary layer tends to be infinitely thick when the disk is forced to oscillate with resonant frequency. In other words, in the case of blowing and resonance, the oscillatory boundary layer flows are no longer possible. This kind of resonance effect is found to be inherent in present problem if an attempt is made to force oscillations with a frequency which is equal to the angular frequency of rotation. Thus, it remains to answer the question of finding a meaningful steady solution for the case of blowing and the resonant frequency. An attempt is made to answer this question by posing a MHD problem. This paper is concerned with the unsteady MHD flow in the semi-infinite expanse of an electrically conducting fluid bounded by an infinite non-conducting porous, oscillating disk with uniform suction or blowing in the presence of a transverse uniform magnetic field. The disk and the fluid exhibit non-coaxial rotation. By using the Laplace transform method, the structure of the steady and the associated boundary layers are investigated including the case of blowing and resonance. It is found that unlike the hydrodynamic situation for the case of blowing and resonance, the present steady state solution satisfies the boundary condition at infinity. Effects of suction or blowing and the external magnetic field are examined. It is shown that the difficulty of the problem considered by Erdogan [13] associated with resonance is automatically resolved in the present analysis. 2. Mathematical formulation We consider the unsteady flow induced in the semi-infinite expanse of an electrically conducting viscous fluid bounded by an infinite non-conducting porous disk at z ¼ 0 subject to uniform suction or blowing in the presence of a transverse uniform magnetic field B0 normal to the disk. Additionally, disk is oscillating with frequency n in its own plane at time t > 0. The axes of rotation, of both the disk and the fluid, are assumed to be in the plane x ¼ 0, with the distance between the axes being l. The disk and the fluid are initially rotating about z-axis and suddenly sets in motion; the disk rotating about z-axis and fluid about z0 -axis. The common angular velocity of the disk and the fluid is taken as X. The boundary and initial conditions can be written in the following form: u ¼ Xy þ U cos nt or u ¼ Xy þ U sin nt; v ¼ Xx u ¼ Xðy lÞ; v ¼ Xx as z ! 1 for all t; u ¼ Xðy lÞ; v ¼ Xx at t ¼ 0 for z > 0:
at z ¼ 0 for t > 0; ð1Þ
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For the motion under consideration, the velocity field has the form u ¼ Xy þ f ðz; tÞ;
v ¼ Xx þ gðz; tÞ;
ð2Þ
where u and v are the components of the velocity in the directions x and y respectively. The unsteady motion of the conducting fluid in the Cartesian coordinate system is governed by the conservation laws of momentum and of mass which are oV 1 1 þ ðV:$ÞV ¼ $p þ tr2 V þ j B; ot q q
ð3Þ
div V ¼ 0;
ð4Þ
where V ¼ ðu; v; wÞ is the velocity vector, p is the pressure, q is the density, j is the electric current density, B is the total magnetic field so that B ¼ B0 þ b, b is the induced magnetic field, and t is the kinematic viscosity. Neglecting the displacement currents, the Maxwell equations and the generalized OhmÕs law are div B ¼ 0;
curl B ¼ lj;
curl E ¼
oB ; ot
j ¼ rðE þ V BÞ;
ð5Þ ð6Þ
where l is the magnetic permeability, E is the electric field and r is the electric conductivity of the fluid. We make the following assumptions: (i) The quantities q, t, l and r are all constants throughout the flow field. (ii) The magnetic field B is perpendicular to the velocity field V, and the induced magnetic field is negligible compared with the imposed field so that the magnetic Reynolds number is small. (iii) The electric field is assumed to be zero. In view of the above assumptions, the electromagnetic body force involved in Eq. (3) takes the following form 1 r r ðj BÞ ¼ ½ðV BÞ B ¼ ½B0 ðV:B0 Þ VðB0 :B0 Þ ¼ N V; q q q
ð7Þ
where N ¼ ðr=qÞB20 has the same dimension as X. Using Eq. (2) in Eq. (4), we obtain ðow=ozÞ ¼ 0: Following Kaloni [14] we take w ¼ W0 . Obviously W0 > 0 is the suction velocity and W0 < 0 is the blowing velocity. Substituting Eqs. (2) and (7) in Eq. (3), we obtain of of 1 op^ o2 f Xg W0 ¼ þ t 2 N ðf XyÞ; ot oz q ox oz
ð8Þ
og og 1 op^ o2 g þ Xf W0 ¼ þ t 2 N ðg þ XxÞ; ot oz q oy oz
ð9Þ
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rB20 W0 ¼
op^ ; oz
ð10Þ
where p^ ¼ p
qr2 X2 ; 2
r 2 ¼ x2 þ y 2 :
The boundary and initial conditions for f and g are f ð0; tÞ ¼ U cos nt or f ð0; tÞ ¼ U sin nt; gð0; tÞ ¼ 0; f ð1; tÞ ¼ Xl; gð1; tÞ ¼ 0 f ðz; 0Þ ¼ Xl; gðz; 0Þ ¼ 0
t > 0; ð11Þ
for all t; for z > 0:
Eliminating p^ from Eqs. (8)–(10), and then using conditions (11), we have t
o2 G oG oG þ W0 ðiX þ N ÞG ¼ 0; 2 oz ot oz
ð12Þ
with Gð0; tÞ ¼
U cos nt 1 Xl
or Gð0; tÞ ¼
U sin nt 1; Xl
Gð1; tÞ ¼ 0;
Gðz; 0Þ ¼ 0;
ð13Þ
where G¼
f ig þ 1: Xl Xl
ð14Þ
Using G ¼ H eiXt :
ð15Þ
Eqs. (12) and (13) become t
o2 H oH oH þ W0 NH ¼ 0; 2 oz ot oz
H ð0; tÞ ¼
U cos nt 1 Xl
or H ð0; tÞ ¼
ð16Þ U sin nt 1; Xl
Hð1; tÞ ¼ 0;
H ðz; 0Þ ¼ 0:
ð17Þ
3. Methodology of solution The problem given by Eqs. (16) and (17) can be solved directly by the use of the Laplace transform pair [15]
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
H ðz; sÞ ¼
Z
1181
1
H ðz; tÞest dt;
ð18Þ
0
1 H ðz; tÞ ¼ 2pi
Z
kþi1
H ðz; sÞest ds;
ð19Þ
k > 0:
ki1
In view of the Laplace transform (18), the differential system can be written as sþN H ¼ 0; t 1 U 1 1 H ð0; sÞ ¼ þ þ or s iX 2Xl s þ iðn XÞ s iðn þ XÞ 1 iU 1 1 þ H ð0; sÞ ¼ ; s iX 2Xl s þ iðn XÞ s iðn þ XÞ d2 H W0 dH þ dz2 t dz
ð20Þ
H ð1; sÞ ¼ 0: For U cos nt, the solutions are given by i.e. for n > X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ 1 U 1 U 1 ðW2t0 Þ þNt þts z H ðz; sÞ ¼ ; þ þ e s iX 2pl s þ iðn XÞ 2pl s iðn þ XÞ
ð21Þ
for n < X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ 1 U 1 U 1 ðW2t0 Þ þNt þts z H ðz; sÞ ¼ : þ e þ s iX 2pl s iðX nÞ 2pl s iðn þ XÞ
ð22Þ
For U sin nt and n > X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ 1 U 1 U 1 ðW2t0 Þ þNt þts z i H ðz; sÞ ¼ ; þi e s iX 2pl s þ iðn XÞ 2pl s iðn þ XÞ
ð23Þ
for n < X H ðz; sÞ ¼
1 U 1 U 1 i þi s iX 2pl s iðX nÞ 2pl s iðn þ XÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ ðW2t0 Þ þNt þts z : e
ð24Þ
The inverse Laplace transforms have been obtained employing the procedure used in [15] and after using Eqs. (14) and (15) in the resulting expression, we arrive at
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for U cos nt, n > X
3 ffi 8 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2
W0 2 > iX 7 6 > þN > tþ t z 2t > > ðW2t0 Þ þNt þiXt tt 7 6 > pz ffiffiffi þ > > erfc e > > 2 tt 7 6 < = 7 6 U int 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e þ
W 2 ffi! > 2Xl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 >
> > 2 0 N iX W 7 6 > þtþ t > 0 þN þiX tt 2t > z z ffiffiffi ð Þ t t 2t 7 6 > p > > erfc 2 tt ; 7 6 :þe 7 6 9 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 W0 2 N iðnXÞ > > 7 6 > þ z ð Þ iðnXÞ t t W0 2t > z ffiffiffi N p > > 7 6 > e erfc þ þ tt W0 < = t t 2t 2 tt 7 f g eð 2t Þz 6 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þi ¼1þ 7; 6 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > W0 iðnXÞ N > 7 Xl Xl 2 6 >
W0 2 N iðnXÞ ð 2t Þ þ t t z > z ffiffiffi > > 7 6 > p : ; erfc þ þ e tt 7 6 t t 2t 2 tt 7 6 7 6 U int 7 6 þ 2Xl e 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 W0 2 N iðnþXÞ > 7 6 > > þ þ z ð 2t Þ t t iðnþXÞ W0 > z ffiffiffi N p > > 7 6 > erfc þ þ þ e tt < = t 2t t 2 tt 7 6 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > W0 2 N iðnþXÞ > 7 6 >
W0 2 N iðnþXÞ ð 2t Þ þ t þ t z > z > > 5 4 > :þe ; erfc 2pffiffittffi þ þ tt t 2t t 2
ð25Þ for U cos nt and n < X
3 ffi 8 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2
W0 2 iX > 7 6 > þN > tþ t z 2t ðW2t0 Þ þNt þiXt tt > > 7 6 > pz ffiffiffi þ > > erfc e > > 7 6 < 2 tt = 7 6 U int 7 6 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e þ ! 2Xl
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > 7 6 > 2
W0 > 2 N 7 iX 6 > W þtþ t > 0 þN þiX tt 2t > þ e > 7 6 > z z ffiffiffi ð Þ t t 2t > > p erfc 2 tt ; 7 6 : 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r W0 2 N iðXnÞ > 7 6 >
þtþ t z > > 2 ð Þ iðXnÞ W 2t > > z N 7 6 >e 0 > erfc 2pffiffittffi þ þtþ t tt W0 < = 7 6 2t 7 f g eð 2t Þz 6
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 7: 6 þi ¼1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > > 7 6 W iðXnÞ > 0 þN þ Xl Xl 2 6 >
z > > ð 2t Þ t t 7 iðXnÞ W0 2 N > :þe ; erfc 2pz ffiffittffi þ þ tt 7 6 > t 2t t 7 6 7 6 7 6 þ U eint 7 6 8 2Xl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > W0 2 N iðnþXÞ >
þ þ z > > 2 ð Þ 7 6 > iðnþXÞ t t 2t W0 > N pz ffiffiffi þ > tt e erfc þ þ 6 > =7 t 2t t 2 tt 7 6 < 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > > 2 W0 2 N iðnþXÞ > 7 6 > þ þ z > > ð Þ iðnþXÞ t t W 2t z N > 0 4 > :þe erfc 2pffiffittffi þtþ t tt ; 5 2t 2
ð26Þ
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For U sin nt and n > X ffi 9 3 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
W 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! > > 2 N 0 iX W þ þ > > t tz 2t > ð 2t0 Þ þNt þiXt tt >e > 7 6 > erfc 2pz ffiffittffi þ > < = 7 6 > 7 6 iU int qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e þ 7 6 !
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2Xl 2
W 2 > 7 6 > W0 > þN þiX N iX 0 > > 7 6 > t t 2t z ð 2t Þ þ t þ t tt > pz ffiffiffi > > 7 6 > erfc þ e : ; 2 tt 7 6 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 97 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 W0 2 N iðnXÞ 7 6 > > > > ð iðnXÞ 7 6 > > 2t Þ þ t t z W z N 0 > > p ffiffi ffi erfc þ þ e tt W0 < =7 t 2t t 2 tt ð 2t Þz 6 7 6 f g e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7: 6
þi ¼1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 7 > > 2 W0 N iðnXÞz > Xl Xl 2 6 þ ð Þ 7 6 > > > iðnXÞ t t W0 2t N > erfc 2pz ffiffittffi þ tt :þe ; 7 6 > 2t t t 7 6 7 6 iU int 7 6 2Xl e 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 97 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 W 2 iðnþXÞ > 7 6 > > iðnþXÞ W0 > e ð 2t0 Þ þNt þ t z erfc pz ffiffiffi þ > N 6 > > þtþ t tt =7 7 6 < 2t 2 tt 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > W iðnþXÞ 5 4 > 0 þN þ >
W0 2 N iðnþXÞ z ð Þ > > t t 2t z > erfc 2pffiffittffi þtþ t tt > :þe ; 2t ð27Þ For U sin nt and n < X 3 ffi 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 !
W 2 ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > 0 > iX W0 2 N iX þN > tþ t z 2t þ t þ t tt 7 6 > > > z ffiffiffi ð Þ 2t p > > e erfc þ 7 6 > > 2 tt < = 7 6 iU int 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ 2Xl e ! 7 6 >
W 2 ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 7 6 > > 0 iX W0 2 N iX þN > tþ t z 2t 7 6 > þ þ tt > > z ð Þ t t 2t p ffiffi ffi > > þ e erfc 7 6 : ; 2 tt 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W0 2 N iðXnÞ > 7 6 >
W0 2 N iðXnÞ > > ð 2t Þ þ t þ t z > > 7 6 z p ffiffi ffi >e > tt erfc 2 tt þ þ þ W0 < = 7 6 t t 2t ð 2t Þz 6 7 f g e q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 7: 6
þi ¼1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > > 7 6 W0 iðXnÞ > Xl Xl 2 6 >
þNt þ t z 2 > > ð Þ 7 iðXnÞ W 2t z ffiffiffi N 0 > > p : ; þe erfc 2 tt þtþ t tt 7 6 2t 7 6 7 6 iU int 7 6 e 2Xl 7 6 8 9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 7 6 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W0 2 N iðnþXÞ 7 6 > >
W0 2 N iðnþXÞ > ð 7 6 > 2t Þ þ t þ t z > > z p ffiffi ffi > > tt erfc þ þ þ e 6 < =7 t 2t t 2 tt 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 > 2 W0 N þiðnþXÞz > þ 7 6 > > > ð Þ iðnþXÞ t t W0 2t z N > > p ffiffi ffi 4 :þe erfc 2 tt þtþ t tt ; 5 2t 2
ð28Þ
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4. Non-dimensionalization Introducing the following change of variables rffiffiffiffiffi X n U rB2 n¼ ; s ¼ Xt; N1 ¼ 0 ; z; k ¼ ; e ¼ 2t X 2Xl qX
W0 S ¼ pffiffiffiffiffiffi 2 tX
ð29Þ
in Eqs. (25)–(28) we have for U cos nt, n > X 8 9 3 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis 7 6 2: nffiffiffi ðaþibÞn ; p þ e erfc ða þ ibÞ 7 6 2 8 9 7 6 2s pffiffis 7 6 n ða þib Þn < = 1 1 erfc p ffiffiffi ffiffi p e þ ða þ ib Þ 7 6 1 f g 1 2 2s 2Sn 6 iks 7; þ e þi ¼1þe pffiffis 6 n ða þib Þn : þ e 1 1 erfc pffiffiffi ða1 þ ib1 Þ ;7 Xl Xl 7 6 2s 6 8 29 7 ffiffi p 6 < eða2 þib2 Þn erfc pnffiffiffi þ ða2 þ ib2 Þ s = 7 7 6 2 2s 5 4 þ eiks ffiffi p : þ eða2 þib2 Þn erfc pnffiffiffi ða2 þ ib2 Þ s ; 2 2s 2
ð30Þ
for U cos nt, n < X 8 9 3 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffi 7 6 2: nffiffiffi ðaþibÞn p þe erfc 2s ða þ ibÞ 2s ; 7 6 8 97 6 ffiffi p 6 < eða3 þib3 Þn erfc pnffiffiffi þ ða3 þ ib3 Þ s =7 pffiffi 6 7 f g 2 2s 2Sn 6 iks 7; þ e þi ¼1þe ffiffi p 6 : þ eða3 þib3 Þn erfc pnffiffiffi ða3 þ ib3 Þ s ; 7 Xl Xl 7 6 2s 6 8 29 7 ffiffi p 6 n < eða2 þib2 Þn erfc pffiffiffi þ ða2 þ ib2 Þ s = 7 7 6 2 2s 5 4 þ eiks pffiffis n ða þib Þn : þ e 2 2 erfc pffiffiffi ða2 þ ib2 Þ ; 2
2s
ð31Þ
2
for U sin nt, n > X 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: n ðaþibÞn 7 6 þe erfc pffiffiffi ða þ ibÞ 2 2s 6 8 97 7 6 ffiffi p < eða1 þib1 Þn erfc pnffiffiffi þ ða1 þ ib1 Þ s =7 pffiffi 6 f g 2 6 2s 7 þi ¼ 1 þ e 2Sn 6 þ ieiks 7; ffiffi p n ða þib Þn s : þ e 1 1 erfc pffiffiffi ða1 þ ib Þ ;7 6 Xl Xl 1 6 2 2s 8 9 7 7 6 ffiffi p 6 < eða2 þib2 Þn erfc pnffiffiffi þ ða2 þ ib2 Þ s = 7 7 6 2 2s 5 4 ieiks ffiffi p : þ eða2 þib2 Þn erfc pnffiffiffi ða2 þ ib Þ s ; 2 2 2s 2
ð32Þ
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for U sin nt, n < X 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: nffiffiffi ðaþibÞn p 7 6 þe erfc 2s ða þ ibÞ 2 7 6 8 9 7 6 pffiffis nffiffiffi ða3 þib3 Þn < = 7 6 p pffiffi e erfc 2s þ ða3 þ ib3 Þ 2 f g 7 2Sn 6 iks þ ie þi ¼1þe 7; 6 : þ eða3 þib3 Þn erfc pnffiffiffi ða3 þ ib Þpffiffis ; 7 6 Xl Xl 3 6 2 2s 8 9 7 7 6 pffiffis n 7 6 ða þib Þn 2 2 erfc < = pffiffiffi þ ða2 þ ib Þ e 7 6 2 2 2s 5 4 ieiks : þ eða2 þib2 Þn erfc pnffiffiffi ða2 þ ib Þpffiffis ; 2 2 2s 2
ð33Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 a¼ ðS 2 þ N1 Þ þ 1 þ ðS þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b¼ ðS þ N1 Þ þ 1 ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðS 2 þ N1 Þ þ ðk 1Þ þ ðS þ N1 Þ ; a1 ¼ 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 b1 ¼ ðS þ N1 Þ þ ðk 1Þ ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS 2 þ N1 Þ2 þ ðk þ 1Þ2 þ ðS 2 þ N1 Þ ; a2 ¼
ð34Þ
12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 b2 ¼ ðS þ N1 Þ þ ðk þ 1Þ ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 a3 ¼ ðS þ N1 Þ þ ð1 kÞ þ ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ðS þ N1 Þ þ ð1 kÞ ðS þ N1 Þ : b3 ¼ The solutions for blowing can be obtained by replacing S with S1 ðS1 > 0Þ in all the results of suction.
5. Resonant suction case If the angular velocity is equal to the frequency of oscillations i.e. n ¼ X, the system resonates and solutions in this case are given by
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for U cos nt 3 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9
W 2 ffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> > 2 0 iX > þN tþ t z > > 7 2t 6 > ðW2t0 Þ þNt þiXt tt > erfc 2pz ffiffittffi þ e > > 7 6 > > > = 7 6 < 7 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 !>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2 7 6 >
W0 > > 2 N iX W0 > > 7 6 > þtþ t N þiX tt > 2t þ z z ð Þ > t t 2t 7 6 > erfc 2pffiffittffi > :þe ; 7 6 > 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r 2 W 7 6 > >
W0 2 N ð 2t0 Þ þNt z > > W0 7 6 z > > pffiffiffi þ ð 2t Þz 6 e erfc þ tt > > 7 < = t 2t f g e 2 tt 7: 6 þi ¼1þ U iXt 7 6 e þ q ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 6 2Xl Xl Xl ffi > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 > W0 2 N > > 7 6
> > þ z > > W0 2 N 7 6 : þ e ð 2t Þ t erfc pz ffiffiffi ; þ tt 2t t 7 6 2 tt 7 6 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 97 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 2 W0 2 N 2X > > 7 6 ð > > W 2t Þ þ t þi t z z N 2X 0 > > 6 erfc 2pffiffittffi þ þ t þ i t tt > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > 2 7 6 W > > 0 þN þi2Xz
> > ð 2t Þ t t 5 4 > > W0 2 N 2X :þe ; erfc 2pz ffiffittffi þ þ i tt t t 2t ð35Þ For U sin nt 3 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9
W 2 ffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> > 2 N 0 iX W > > þtþ t > 2t 7 6 > z ð 2t0 Þ þNt þiXt tt > erfc 2pz ffiffittffi þ e > > 7 6 > > > < = 7 6 7 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 !
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q > 2 7 6 >
W 2 > W0 N þiX > > 7 6 > þ N iX 0 t tz > > 2t þ þ tt z ð Þ > t t 2t 7 6 > erfc 2pffiffittffi > :þe ; 7 6 > 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r W0 2 N 7 6 > >
þtz 2 ð Þ > > W0 W 2t 7 6 z N 0 > > pffiffiffi þ ð 2t Þz 6 e erfc þ tt > > 7 < = 2t t f g e 2 tt 7: 6 iU iXt þi ¼1þ 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffi Xl Xl 2 6 þ 2Xl e > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r 7 > 2 W > > N 0 7 6
W0 2 N > > > þ e ð 2t Þ þ t z erfc pz ffiffiffi > 7 6 : ; þ t tt 2t 7 6 2 tt 7 6 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 8 9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 2 W0 2 N 2X > > 7 6 þ þi z > > W0 N 2X > e ð 2t Þ t t erfc pz ffiffiffi þ > 7 6 þ þ i tt > > < = t t 2t 2 tt 7 6 7 6 iU eiXt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 2Xl > r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 7 6 2 > > > ðW2t0 Þ þNt þi2Xtz > 5 4 > > W0 z ffiffiffi N 2X :þe ; p erfc 2 tt þ t þ i t tt 2t ð36Þ
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Making use of Eq. (29), Eqs. (35) and (36) become for U cos nt 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: n ðaþibÞn 7 6 þe erfc pffiffiffi ða þ ibÞ 2 2s 6 8 97 7 6 ffiffi p < eða4 þib4 Þn erfc pnffiffiffi þ ða4 þ ib4 Þ s =7 pffiffi 6 f g 2 6 2s 7 þi ¼ 1 þ e 2Sn 6 þ eis 7: ffiffi p n ða þib Þn s : þ e 4 4 erfc pffiffiffi ða4 þ ib Þ ;7 6 Xl Xl 4 6 2 2s 8 9 7 7 6 ffiffi p n 6 < eða5 þib5 Þn erfc pffiffiffi þ ða5 þ ib5 Þ s = 7 7 6 2 2s 5 4 þ eis ffiffi p : þ eða5 þib5 Þn erfc pnffiffiffi ða5 þ ib Þ s ; 5 2 2s 2
ð37Þ
For U sin nt 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: n ðaþibÞn ffiffiffi p 7 6 þe erfc 2s ða þ ibÞ 2 7 6 8 9 7 6 pffiffis nffiffiffi ða4 þib4 Þn < = 7 6 p pffiffi e erfc 2s þ ða4 þ ib4 Þ 2 f g 7 2Sn 6 is þi ¼1þe 7; 6 þ ie : þ eða4 þib4 Þn erfc pnffiffiffi ða4 þ ib Þpffiffis ; 7 6 Xl Xl 4 6 2 2s 8 9 7 7 6 pffiffis n 7 6 ða5 þib5 Þn < = pffiffiffi þ ða5 þ ib Þ e erfc 7 6 5 2 2s 5 4 ieis : þ eða5 þib5 Þn erfc pnffiffiffi ða5 þ ib Þpffiffis ; 5 2 2s 2
ð38Þ
where 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 a4 ¼ ðS þ N1 Þ þ ðS þ N1 Þ ; b4 ¼
12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS 2 þ N1 Þ2 ðS 2 þ N1 Þ ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 ðS þ N1 Þ þ 4 þ ðS þ N1 Þ ; a5 ¼
ð39Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b5 ¼ ðS þ N1 Þ þ 4 ðS þ N1 Þ :
6. Resonant blowing case In this section W0 ¼ W0 ðS ¼ S1 ; S1 > 0Þ and the solutions (37) and (38) become of the following form
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for U cos nt 8 9 3 pffiffi ~ s > > ~ < eða~þibÞn erfc pnffiffiffi = ~ þ a þ i b 2 2s 7 61 pffiffi 7 6 2> ~ ~ a þi b n > n s ð Þ 7 6 ~ :þe ; ~ erfc pffiffiffi a þ i b 7 6 2 2s 6 8 97 7 6 > > ða~4 þib~4 Þn erfc pnffiffiffi þ a~ þ ib~ pffiffis 7 6 < = e ffiffi p 4 4 7 6 2 f g 2s 2S1 n 6 is pffiffi 7 þi ¼1þe þ e 7: 6 ~ > þ eða~4 þib4 Þn erfc pnffiffiffi a~ þ ib~ Xl Xl s > : ; 7 6 4 4 2 2s 7 6 6 8 9 7 p ffiffi 7 6 ~ > s > ~ 7 6 < eða~5 þib5 Þn erfc pnffiffiffi = ~ þ a þ i b 5 5 2 2s 7 6 5 4 þ eis p ffiffi ~5 þib~5 Þn a > > n s ð :þe ; erfc pffiffiffi a~5 þ ib~5 2 2
ð40Þ
2s
For U sin nt 3 pffiffi 8 9 s < eða~þib~Þn erfc pnffiffiffi þ a~ þ ib~ = 2s 7 61 2 pffiffi 7 6 2: ða~þib~Þn n s ; ~ p ffiffiffi 7 6 þe erfc 2s a~ þ ib 2 7 6 8 9 7 6 pffiffis a~4 þib~4 Þn nffiffiffi ð ~ < = 7 6 p pffiffi e erfc 2s þ a~4 þ ib4 f g 2 7 2S1 n 6 is þi ¼1þe þ ie 7 6 : þ eða~4 þib~4 Þn erfc pnffiffiffi a~4 þ ib~ pffiffis ; 7 6 Xl Xl 4 6 2 2s pffiffi 8 9 7 7 6 a~5 þib~5 Þn n 7 6 s ð ~ < = pffiffiffi þ a ~ e erfc þ i b 5 7 6 5 2 2s 5 4 ieis : þ eða~5 þib~5 Þn erfc pnffiffiffi a~5 þ ib~ pffiffis ; 5 2 2s 2
ð41Þ
in which qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 a~ ¼ ðS1 þ N1 Þ þ 1 þ ðS1 þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b~ ¼ ðS1 þ N1 Þ þ 1 ðS1 þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 a~4 ¼ ðS1 þ N1 Þ þ ðS1 þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b~4 ¼ ðS1 þ N1 Þ ðS1 þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 ðS1 þ N1 Þ þ 4 þ ðS1 þ N1 Þ ; a~5 ¼ b~5 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 ðS12 þ N1 Þ2 þ 4 ðS12 þ N1 Þ :
ð42Þ
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In order to determine the steady state structure of the solution, we use the asymptotic formula for the complementary error function rffiffiffi n s p ffiffiffiffiffi ða þ ibÞ ! ð0; 2Þ as s ! 1: erfc 2 2s Evidently, in the limit s ! 1, solutions (40) and (41) reduce to for U cos nt h i pffiffi f g ~ ~ ~ þi ¼ 1 þ e 2S1 n eða~þibÞn þ 2eis eða~4 þib4 Þn þ 2eis eða~5 þib5 Þn ; Xl Xl
ð43Þ
for U sin nt h i pffiffi f g ~ ~ ~ þi ¼ 1 þ e 2S1 n eða~þibÞn þ i2eis eða~4 þib4 Þn i2eis eða~5 þib5 Þn : Xl Xl
ð44Þ
Fig. 1. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
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These results describe meaningful hydromagnetic boundary layer flows for resonant frequency. It may be observed that both the effects of rotation and electromagnetic forces are reflected in the ultimate steady velocity fields.
7. Discussion In this paper, we discuss the MHD flow due to non-coaxial rotations of an oscillating disk and a fluid at infinity. A method of Laplace transform has been used to get the solution of the problem. The effects of the various parameters s, S, k and N1 on velocity distribution have been studied and the results have been presented by several graphs. In order to study the effect of the magnetic field parameter N1 (¼0 and 2) on the velocity distribution, we have plotted ðf =XlÞ and ðg=XlÞ against n in Figs. 1 and 2 for U cos nt and in Figs. 3 and 4 for U sin nt with dotted lines in all the three cases i.e. n > X, n < X and n ¼ X. From these figures it is found that boundary layer thickness decreases with increase in N1 . The full lines in Figs. 1–4 depict the effect of suction (S ¼ 0:5)=blowing (S ¼ 0:5). In this case, we see that with
Fig. 2. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU=4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
1191
Fig. 3. The variation of ðf =XlÞ and ðg=XlÞ with n for the sine oscillation at ðU =4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
the increase in suction parameter, the boundary layer thickness decreases and with increase in blowing parameter the boundary layer thickness increases as compared to the case of suction. The case S ¼ 0 represent no suction/no blowing. The time required to attain the steady state for the cosine oscillation of the disk is shorter than that for the sine oscillation of the disk. The value of this time depends on ðn=XÞ (ratio of the frequency of oscillation to the angular velocity of the disk) and ðU =XlÞ. Fig. 5 shows the variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðn=XÞ ¼ 5, ðU=4XlÞ ¼ 1, S ¼ 0 and N1 ¼ 2 for different values of s. The times required to attain the steady state in Fig. 5(a) and (b) are s ¼ 0:4 and s ¼ 1:8 respectively. It is remarked that both values of s are less when compared with case of N1 ¼ 0 (Figs. 2 and 3 in Erdogan [13]). Further, Fig. 6(a) and (b) respectively denote the variation of ðf =XlÞ and ðg=XlÞ with n for various values of s for the cosine oscillation at ðn=XÞ ¼ 0:5, ðU =4XlÞ ¼ 1, N1 ¼ 0 and S ¼ 0:5. It is found that time required to attain the steady state for ðf =XlÞ and ðg=XlÞ are s ¼ 3 and s ¼ 4:5 respectively, which is also shorter than for S ¼ 0 (Figs. 6 and 7 in Erdogan [13]). Fig. 6(c) and (d) are prepared for ðf =XlÞ and ðg=XlÞ against n for the cosine oscillation at ðn=XÞ ¼ 0:5, ðU =4XlÞ ¼ 1, N1 ¼ 0 and S ¼ 0:5 for different s. We observe that time to attain the
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Fig. 4. The variation of ðf =XlÞ and ðg=XlÞ with n for the sine oscillation at ðU =4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
Fig. 5. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
1193
Fig. 6. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
steady state for both ðf =XlÞ and ðg=XlÞ is s ¼ 6, which is greater than for the case S ¼ 0 (Figs. 6 and 7 in Erdogan [13]). It is further observed that similar behaviors occur for time required to attain the steady state for sine oscillation. In the resonant case (n ¼ X) it is known that no steady asymptotic solution is possible for flow due to non-coaxial rotations of an oscillating disk and a fluid at infinity (i.e. the boundary condition at infinity is not satisfied). But the present analysis exhibits a striking difference between the structures of the hydrodynamic and the hydromagnetic boundary layers (Figs. 7 and 8). Fig. 7(a) and (b) indicate the variation of ðf =XlÞ and ðg=XlÞ respectively against n for the cosine oscillation at n ¼ X, ðU =4XlÞ ¼ 1, S ¼ 0:5 and N1 ¼ 2 for different s (the dotted lines denote the steady-state velocity and the full lines show the starting velocity). The respective times required for steady state for ðf =XlÞ and ðg=XlÞ are s ¼ 0:9 and s ¼ 1:6. In Fig. 7(c) and (d) we plotted ðf =XlÞ and ðg=XlÞ against n for cosine oscillation with different s at n ¼ X, ðU =4XlÞ ¼ 1, S ¼ 0:5 and N1 ¼ 2. The steady state situation for ðf =XlÞ and ðg=XlÞ are obtained at s ¼ 1:2 and s ¼ 2 which are greater than that of suction case.
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Fig. 7. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
In the case of sine oscillations, the resonant steady state for ðf =XlÞ and ðg=XlÞ in suction/ blowing cases are s ¼ 0:7 and s ¼ 1:4=s ¼ 0:9 and s ¼ 1:8 respectively (see Fig. 8).
8. Concluding remarks The most important features of Eqs. (37), (38), (40) and (41) are that unlike the hydrodynamic situation for the resonant case, Eqs. (37), (38), (40) and (41) satisfy the boundary condition at infinity. Consequently, the associated boundary layers remain bounded for all values of frequencies including the resonant frequency. In contrast to the hydrodynamic solution for the case of blowing and resonance where the blowing promotes the spreading of the oscillations far away from the disk, the solutions (37), (38), (40) and (41) represent the sensible oscillatory boundary layer flow. The physical implication of this conclusion is that for the case of resonance, the unbounded spreading of the oscillations away from the disk is controlled by the external magnetic field.
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
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Fig. 8. The variation of ðf =XlÞ and ðg=XlÞ with n for the sine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
References [1] P.C. Stangeby, A review of the status of MHD power generation technology including suggestions for a Canadian MHD research programe, UTIAS Rev. 39 (1974). [2] O.A. Lielausis, Liquid–metal magnetohydrodynamics, Atomic Energy Rev. 13 (1975) 527. [3] J.C.R. Hunt, R. Moreau, Liquid–metal magnetohydrodynamics with strong magnetic fields a report on Euromech, J. Fluid Mech. 78 (1976) 261. [4] J.M. Lyons, D.L. Turcotte, A study of MHD induction devices, AFOSRTR-1864, 1962. [5] I.B. Bernstein, J.B. Fischbeck, K.H. Lessen, W. Ness, J. Jaren, R.M. Kulsrud, An electrodeless MHD generator, in: 2nd Symposium on the engineering aspects of magnetohydrodynamics, University of Pennsylvania, Philadelphia, 1961. [6] S.N. Murthy, MHD Ekman layer on a porous plate, Nuovo Cimento 29 (1975) 296. [7] L. Debnath, On unsteady magnetohydrodynamic boundary layers in a rotating flow, ZAMM 52 (1972) 623. [8] L. Debnath, On unsteady hydromagnetic boundary layers in a rotating flow, Proc. Int. Union Theor. Appl. Mech. 2 (1971) 1397. [9] A.M. Siddiqui, T. Haroon, T. Hayat, S. Asghar, Unsteady MHD flow of a non-Newtonian fluid due to accentric rotations of a porous disk and a fluid at infinity, Acta Mech. 147 (2001) 99. [10] T. Hayat, S. Nadeem, S. Asghar, A.M. Siddiqui, MHD rotating flow of a third grade fluid on an oscillatory porous plate, Acta Mech. 152 (2001) 177.
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[11] K.R. Rajagopal, On the flow of a simple fluid in an orthogonal rheometer, Arch. Rational Mech. Anal. 79 (1982) 39. [12] K.R. Rajagopal, Flow of viscoelastic fluids between rotating disks, Theor. Comput. Fluid Dyn. 3 (1992) 185. [13] M.E. Erdogan, Flow induced by non-coaxial rotations of a disk executing non-torsional oscillations and a fluid rotating at infinity, Int. J. Eng. Sci. 38 (2000) 175. [14] P.N. Kaloni, Fluctuating flow of an elastico-viscous fluid past a porous flat plate, Phys. Fluids 10 (1967) 1344. [15] B.M. Abramovitz, I.A. Stegun, Handbook of mathematical functions, Dover, New York, 1964, p. 325.
Magnetohydrodynamic flow due to non-coaxial rotations of a porous oscillating disk and a fluid at infinity T. Hayat
a,*
, M. Zamurad a, S. Asghar a, A.M. Siddiqui
b
a
b
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Department of Mathematics, Pennsylvania State University York Campus, York, PA 17403, USA Received 16 September 2002; accepted 4 October 2002
Abstract An exact solution of the Navier–Stokes equations is constructed for the case of flow due to non-coaxial rotations of a porous disk and a fluid at infinity. The disk executes oscillations in its own plane and is nonconducting. The viscous fluid is incompressible and electrically conducting. Analytical solution is established by the method of Laplace transform. The velocity fields are obtained for the cases when the angular velocity is greater than, smaller than or equal to the frequency of oscillations. The structure of the steady and the unsteady velocity fields are investigated. The difficulty of the hydrodynamic steady solution associated with the case of resonant frequency is resolved in the present analysis. Ó 2003 Published by Elsevier Science Ltd. Keywords: Magnetohydrodynamic effects; Porous disk; Laplace transform; Time-dependent equations; Resonance
1. Introduction The increasing number of technical applications using magnetohydrodynamic (MHD) effects has made it desirable to extend many of the available hydrodynamic solutions to include the effects of magnetic field for those cases when the viscous fluid is electrically conducting. For example, liquid–metal MHDs take their roots in conventional hydrodynamics of incompressible media, which become important in the metallurgical industry, nuclear reactor, sodium cooling system, energy storage and electrical power generation [1–3]. The greatest advantage of inductiontype pumps over other types of MHD devices is the absence of electrodes [4]. Induction pumps have been used to pump coolants in nuclear reactors. These designs are also being considered in *
Corresponding author. E-mail address: [email protected] (T. Hayat).
0020-7225/03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd. doi:10.1016/S0020-7225(03)00004-1
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MHD power generation [5]. The basic equations of incompressible MHD are non-linear. But there are many interesting cases where the equations become linear in terms of the unknown quantities and may be solved without any difficulty. Linear MHD permits exact solutions and adopts the approximations that the density and transport properties be constant. No fluid is incompressible, but all may be treated as such whenever the pressure changes are small in comparison with the bulk modulus. Mention may be made to the interesting works in [6–12]. Erdogan [13] has recently made an initial value investigation for flow due to non-coaxial rotations of an oscillating disk and a fluid at infinity. He claimed that if the angular velocity is equal to the frequency of oscillations, the system resonates and steady solution does not exist. It is apparent from physical considerations that suction and blowing have opposite effects. Indeed, the suction prevents the imposed non-torsional oscillations from spreading away from the disk by viscous diffusion for all values of the frequency. On the contrary, the blowing promotes the spreading of the oscillations far away from the disk and hence one boundary layer tends to be infinitely thick when the disk is forced to oscillate with resonant frequency. In other words, in the case of blowing and resonance, the oscillatory boundary layer flows are no longer possible. This kind of resonance effect is found to be inherent in present problem if an attempt is made to force oscillations with a frequency which is equal to the angular frequency of rotation. Thus, it remains to answer the question of finding a meaningful steady solution for the case of blowing and the resonant frequency. An attempt is made to answer this question by posing a MHD problem. This paper is concerned with the unsteady MHD flow in the semi-infinite expanse of an electrically conducting fluid bounded by an infinite non-conducting porous, oscillating disk with uniform suction or blowing in the presence of a transverse uniform magnetic field. The disk and the fluid exhibit non-coaxial rotation. By using the Laplace transform method, the structure of the steady and the associated boundary layers are investigated including the case of blowing and resonance. It is found that unlike the hydrodynamic situation for the case of blowing and resonance, the present steady state solution satisfies the boundary condition at infinity. Effects of suction or blowing and the external magnetic field are examined. It is shown that the difficulty of the problem considered by Erdogan [13] associated with resonance is automatically resolved in the present analysis. 2. Mathematical formulation We consider the unsteady flow induced in the semi-infinite expanse of an electrically conducting viscous fluid bounded by an infinite non-conducting porous disk at z ¼ 0 subject to uniform suction or blowing in the presence of a transverse uniform magnetic field B0 normal to the disk. Additionally, disk is oscillating with frequency n in its own plane at time t > 0. The axes of rotation, of both the disk and the fluid, are assumed to be in the plane x ¼ 0, with the distance between the axes being l. The disk and the fluid are initially rotating about z-axis and suddenly sets in motion; the disk rotating about z-axis and fluid about z0 -axis. The common angular velocity of the disk and the fluid is taken as X. The boundary and initial conditions can be written in the following form: u ¼ Xy þ U cos nt or u ¼ Xy þ U sin nt; v ¼ Xx u ¼ Xðy lÞ; v ¼ Xx as z ! 1 for all t; u ¼ Xðy lÞ; v ¼ Xx at t ¼ 0 for z > 0:
at z ¼ 0 for t > 0; ð1Þ
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
1179
For the motion under consideration, the velocity field has the form u ¼ Xy þ f ðz; tÞ;
v ¼ Xx þ gðz; tÞ;
ð2Þ
where u and v are the components of the velocity in the directions x and y respectively. The unsteady motion of the conducting fluid in the Cartesian coordinate system is governed by the conservation laws of momentum and of mass which are oV 1 1 þ ðV:$ÞV ¼ $p þ tr2 V þ j B; ot q q
ð3Þ
div V ¼ 0;
ð4Þ
where V ¼ ðu; v; wÞ is the velocity vector, p is the pressure, q is the density, j is the electric current density, B is the total magnetic field so that B ¼ B0 þ b, b is the induced magnetic field, and t is the kinematic viscosity. Neglecting the displacement currents, the Maxwell equations and the generalized OhmÕs law are div B ¼ 0;
curl B ¼ lj;
curl E ¼
oB ; ot
j ¼ rðE þ V BÞ;
ð5Þ ð6Þ
where l is the magnetic permeability, E is the electric field and r is the electric conductivity of the fluid. We make the following assumptions: (i) The quantities q, t, l and r are all constants throughout the flow field. (ii) The magnetic field B is perpendicular to the velocity field V, and the induced magnetic field is negligible compared with the imposed field so that the magnetic Reynolds number is small. (iii) The electric field is assumed to be zero. In view of the above assumptions, the electromagnetic body force involved in Eq. (3) takes the following form 1 r r ðj BÞ ¼ ½ðV BÞ B ¼ ½B0 ðV:B0 Þ VðB0 :B0 Þ ¼ N V; q q q
ð7Þ
where N ¼ ðr=qÞB20 has the same dimension as X. Using Eq. (2) in Eq. (4), we obtain ðow=ozÞ ¼ 0: Following Kaloni [14] we take w ¼ W0 . Obviously W0 > 0 is the suction velocity and W0 < 0 is the blowing velocity. Substituting Eqs. (2) and (7) in Eq. (3), we obtain of of 1 op^ o2 f Xg W0 ¼ þ t 2 N ðf XyÞ; ot oz q ox oz
ð8Þ
og og 1 op^ o2 g þ Xf W0 ¼ þ t 2 N ðg þ XxÞ; ot oz q oy oz
ð9Þ
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rB20 W0 ¼
op^ ; oz
ð10Þ
where p^ ¼ p
qr2 X2 ; 2
r 2 ¼ x2 þ y 2 :
The boundary and initial conditions for f and g are f ð0; tÞ ¼ U cos nt or f ð0; tÞ ¼ U sin nt; gð0; tÞ ¼ 0; f ð1; tÞ ¼ Xl; gð1; tÞ ¼ 0 f ðz; 0Þ ¼ Xl; gðz; 0Þ ¼ 0
t > 0; ð11Þ
for all t; for z > 0:
Eliminating p^ from Eqs. (8)–(10), and then using conditions (11), we have t
o2 G oG oG þ W0 ðiX þ N ÞG ¼ 0; 2 oz ot oz
ð12Þ
with Gð0; tÞ ¼
U cos nt 1 Xl
or Gð0; tÞ ¼
U sin nt 1; Xl
Gð1; tÞ ¼ 0;
Gðz; 0Þ ¼ 0;
ð13Þ
where G¼
f ig þ 1: Xl Xl
ð14Þ
Using G ¼ H eiXt :
ð15Þ
Eqs. (12) and (13) become t
o2 H oH oH þ W0 NH ¼ 0; 2 oz ot oz
H ð0; tÞ ¼
U cos nt 1 Xl
or H ð0; tÞ ¼
ð16Þ U sin nt 1; Xl
Hð1; tÞ ¼ 0;
H ðz; 0Þ ¼ 0:
ð17Þ
3. Methodology of solution The problem given by Eqs. (16) and (17) can be solved directly by the use of the Laplace transform pair [15]
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
H ðz; sÞ ¼
Z
1181
1
H ðz; tÞest dt;
ð18Þ
0
1 H ðz; tÞ ¼ 2pi
Z
kþi1
H ðz; sÞest ds;
ð19Þ
k > 0:
ki1
In view of the Laplace transform (18), the differential system can be written as sþN H ¼ 0; t 1 U 1 1 H ð0; sÞ ¼ þ þ or s iX 2Xl s þ iðn XÞ s iðn þ XÞ 1 iU 1 1 þ H ð0; sÞ ¼ ; s iX 2Xl s þ iðn XÞ s iðn þ XÞ d2 H W0 dH þ dz2 t dz
ð20Þ
H ð1; sÞ ¼ 0: For U cos nt, the solutions are given by i.e. for n > X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ 1 U 1 U 1 ðW2t0 Þ þNt þts z H ðz; sÞ ¼ ; þ þ e s iX 2pl s þ iðn XÞ 2pl s iðn þ XÞ
ð21Þ
for n < X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ 1 U 1 U 1 ðW2t0 Þ þNt þts z H ðz; sÞ ¼ : þ e þ s iX 2pl s iðX nÞ 2pl s iðn þ XÞ
ð22Þ
For U sin nt and n > X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ 1 U 1 U 1 ðW2t0 Þ þNt þts z i H ðz; sÞ ¼ ; þi e s iX 2pl s þ iðn XÞ 2pl s iðn þ XÞ
ð23Þ
for n < X H ðz; sÞ ¼
1 U 1 U 1 i þi s iX 2pl s iðX nÞ 2pl s iðn þ XÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hW0 2 2t þ ðW2t0 Þ þNt þts z : e
ð24Þ
The inverse Laplace transforms have been obtained employing the procedure used in [15] and after using Eqs. (14) and (15) in the resulting expression, we arrive at
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T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
for U cos nt, n > X
3 ffi 8 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2
W0 2 > iX 7 6 > þN > tþ t z 2t > > ðW2t0 Þ þNt þiXt tt 7 6 > pz ffiffiffi þ > > erfc e > > 2 tt 7 6 < = 7 6 U int 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e þ
W 2 ffi! > 2Xl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 >
> > 2 0 N iX W 7 6 > þtþ t > 0 þN þiX tt 2t > z z ffiffiffi ð Þ t t 2t 7 6 > p > > erfc 2 tt ; 7 6 :þe 7 6 9 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 W0 2 N iðnXÞ > > 7 6 > þ z ð Þ iðnXÞ t t W0 2t > z ffiffiffi N p > > 7 6 > e erfc þ þ tt W0 < = t t 2t 2 tt 7 f g eð 2t Þz 6 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þi ¼1þ 7; 6 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > W0 iðnXÞ N > 7 Xl Xl 2 6 >
W0 2 N iðnXÞ ð 2t Þ þ t t z > z ffiffiffi > > 7 6 > p : ; erfc þ þ e tt 7 6 t t 2t 2 tt 7 6 7 6 U int 7 6 þ 2Xl e 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 W0 2 N iðnþXÞ > 7 6 > > þ þ z ð 2t Þ t t iðnþXÞ W0 > z ffiffiffi N p > > 7 6 > erfc þ þ þ e tt < = t 2t t 2 tt 7 6 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > W0 2 N iðnþXÞ > 7 6 >
W0 2 N iðnþXÞ ð 2t Þ þ t þ t z > z > > 5 4 > :þe ; erfc 2pffiffittffi þ þ tt t 2t t 2
ð25Þ for U cos nt and n < X
3 ffi 8 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2
W0 2 iX > 7 6 > þN > tþ t z 2t ðW2t0 Þ þNt þiXt tt > > 7 6 > pz ffiffiffi þ > > erfc e > > 7 6 < 2 tt = 7 6 U int 7 6 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e þ ! 2Xl
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > 7 6 > 2
W0 > 2 N 7 iX 6 > W þtþ t > 0 þN þiX tt 2t > þ e > 7 6 > z z ffiffiffi ð Þ t t 2t > > p erfc 2 tt ; 7 6 : 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r W0 2 N iðXnÞ > 7 6 >
þtþ t z > > 2 ð Þ iðXnÞ W 2t > > z N 7 6 >e 0 > erfc 2pffiffittffi þ þtþ t tt W0 < = 7 6 2t 7 f g eð 2t Þz 6
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 7: 6 þi ¼1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > > 7 6 W iðXnÞ > 0 þN þ Xl Xl 2 6 >
z > > ð 2t Þ t t 7 iðXnÞ W0 2 N > :þe ; erfc 2pz ffiffittffi þ þ tt 7 6 > t 2t t 7 6 7 6 7 6 þ U eint 7 6 8 2Xl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > W0 2 N iðnþXÞ >
þ þ z > > 2 ð Þ 7 6 > iðnþXÞ t t 2t W0 > N pz ffiffiffi þ > tt e erfc þ þ 6 > =7 t 2t t 2 tt 7 6 < 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > > 2 W0 2 N iðnþXÞ > 7 6 > þ þ z > > ð Þ iðnþXÞ t t W 2t z N > 0 4 > :þe erfc 2pffiffittffi þtþ t tt ; 5 2t 2
ð26Þ
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
1183
For U sin nt and n > X ffi 9 3 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
W 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! > > 2 N 0 iX W þ þ > > t tz 2t > ð 2t0 Þ þNt þiXt tt >e > 7 6 > erfc 2pz ffiffittffi þ > < = 7 6 > 7 6 iU int qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e þ 7 6 !
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2Xl 2
W 2 > 7 6 > W0 > þN þiX N iX 0 > > 7 6 > t t 2t z ð 2t Þ þ t þ t tt > pz ffiffiffi > > 7 6 > erfc þ e : ; 2 tt 7 6 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 97 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 W0 2 N iðnXÞ 7 6 > > > > ð iðnXÞ 7 6 > > 2t Þ þ t t z W z N 0 > > p ffiffi ffi erfc þ þ e tt W0 < =7 t 2t t 2 tt ð 2t Þz 6 7 6 f g e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7: 6
þi ¼1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 7 > > 2 W0 N iðnXÞz > Xl Xl 2 6 þ ð Þ 7 6 > > > iðnXÞ t t W0 2t N > erfc 2pz ffiffittffi þ tt :þe ; 7 6 > 2t t t 7 6 7 6 iU int 7 6 2Xl e 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 97 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 W 2 iðnþXÞ > 7 6 > > iðnþXÞ W0 > e ð 2t0 Þ þNt þ t z erfc pz ffiffiffi þ > N 6 > > þtþ t tt =7 7 6 < 2t 2 tt 7 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > W iðnþXÞ 5 4 > 0 þN þ >
W0 2 N iðnþXÞ z ð Þ > > t t 2t z > erfc 2pffiffittffi þtþ t tt > :þe ; 2t ð27Þ For U sin nt and n < X 3 ffi 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 !
W 2 ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 > 0 > iX W0 2 N iX þN > tþ t z 2t þ t þ t tt 7 6 > > > z ffiffiffi ð Þ 2t p > > e erfc þ 7 6 > > 2 tt < = 7 6 iU int 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ 2Xl e ! 7 6 >
W 2 ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 7 6 > > 0 iX W0 2 N iX þN > tþ t z 2t 7 6 > þ þ tt > > z ð Þ t t 2t p ffiffi ffi > > þ e erfc 7 6 : ; 2 tt 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W0 2 N iðXnÞ > 7 6 >
W0 2 N iðXnÞ > > ð 2t Þ þ t þ t z > > 7 6 z p ffiffi ffi >e > tt erfc 2 tt þ þ þ W0 < = 7 6 t t 2t ð 2t Þz 6 7 f g e q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 7: 6
þi ¼1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 > > 7 6 W0 iðXnÞ > Xl Xl 2 6 >
þNt þ t z 2 > > ð Þ 7 iðXnÞ W 2t z ffiffiffi N 0 > > p : ; þe erfc 2 tt þtþ t tt 7 6 2t 7 6 7 6 iU int 7 6 e 2Xl 7 6 8 9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 7 6 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W0 2 N iðnþXÞ 7 6 > >
W0 2 N iðnþXÞ > ð 7 6 > 2t Þ þ t þ t z > > z p ffiffi ffi > > tt erfc þ þ þ e 6 < =7 t 2t t 2 tt 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 > 2 > 2 W0 N þiðnþXÞz > þ 7 6 > > > ð Þ iðnþXÞ t t W0 2t z N > > p ffiffi ffi 4 :þe erfc 2 tt þtþ t tt ; 5 2t 2
ð28Þ
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4. Non-dimensionalization Introducing the following change of variables rffiffiffiffiffi X n U rB2 n¼ ; s ¼ Xt; N1 ¼ 0 ; z; k ¼ ; e ¼ 2t X 2Xl qX
W0 S ¼ pffiffiffiffiffiffi 2 tX
ð29Þ
in Eqs. (25)–(28) we have for U cos nt, n > X 8 9 3 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis 7 6 2: nffiffiffi ðaþibÞn ; p þ e erfc ða þ ibÞ 7 6 2 8 9 7 6 2s pffiffis 7 6 n ða þib Þn < = 1 1 erfc p ffiffiffi ffiffi p e þ ða þ ib Þ 7 6 1 f g 1 2 2s 2Sn 6 iks 7; þ e þi ¼1þe pffiffis 6 n ða þib Þn : þ e 1 1 erfc pffiffiffi ða1 þ ib1 Þ ;7 Xl Xl 7 6 2s 6 8 29 7 ffiffi p 6 < eða2 þib2 Þn erfc pnffiffiffi þ ða2 þ ib2 Þ s = 7 7 6 2 2s 5 4 þ eiks ffiffi p : þ eða2 þib2 Þn erfc pnffiffiffi ða2 þ ib2 Þ s ; 2 2s 2
ð30Þ
for U cos nt, n < X 8 9 3 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffi 7 6 2: nffiffiffi ðaþibÞn p þe erfc 2s ða þ ibÞ 2s ; 7 6 8 97 6 ffiffi p 6 < eða3 þib3 Þn erfc pnffiffiffi þ ða3 þ ib3 Þ s =7 pffiffi 6 7 f g 2 2s 2Sn 6 iks 7; þ e þi ¼1þe ffiffi p 6 : þ eða3 þib3 Þn erfc pnffiffiffi ða3 þ ib3 Þ s ; 7 Xl Xl 7 6 2s 6 8 29 7 ffiffi p 6 n < eða2 þib2 Þn erfc pffiffiffi þ ða2 þ ib2 Þ s = 7 7 6 2 2s 5 4 þ eiks pffiffis n ða þib Þn : þ e 2 2 erfc pffiffiffi ða2 þ ib2 Þ ; 2
2s
ð31Þ
2
for U sin nt, n > X 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: n ðaþibÞn 7 6 þe erfc pffiffiffi ða þ ibÞ 2 2s 6 8 97 7 6 ffiffi p < eða1 þib1 Þn erfc pnffiffiffi þ ða1 þ ib1 Þ s =7 pffiffi 6 f g 2 6 2s 7 þi ¼ 1 þ e 2Sn 6 þ ieiks 7; ffiffi p n ða þib Þn s : þ e 1 1 erfc pffiffiffi ða1 þ ib Þ ;7 6 Xl Xl 1 6 2 2s 8 9 7 7 6 ffiffi p 6 < eða2 þib2 Þn erfc pnffiffiffi þ ða2 þ ib2 Þ s = 7 7 6 2 2s 5 4 ieiks ffiffi p : þ eða2 þib2 Þn erfc pnffiffiffi ða2 þ ib Þ s ; 2 2 2s 2
ð32Þ
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
1185
for U sin nt, n < X 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: nffiffiffi ðaþibÞn p 7 6 þe erfc 2s ða þ ibÞ 2 7 6 8 9 7 6 pffiffis nffiffiffi ða3 þib3 Þn < = 7 6 p pffiffi e erfc 2s þ ða3 þ ib3 Þ 2 f g 7 2Sn 6 iks þ ie þi ¼1þe 7; 6 : þ eða3 þib3 Þn erfc pnffiffiffi ða3 þ ib Þpffiffis ; 7 6 Xl Xl 3 6 2 2s 8 9 7 7 6 pffiffis n 7 6 ða þib Þn 2 2 erfc < = pffiffiffi þ ða2 þ ib Þ e 7 6 2 2 2s 5 4 ieiks : þ eða2 þib2 Þn erfc pnffiffiffi ða2 þ ib Þpffiffis ; 2 2 2s 2
ð33Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 a¼ ðS 2 þ N1 Þ þ 1 þ ðS þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b¼ ðS þ N1 Þ þ 1 ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðS 2 þ N1 Þ þ ðk 1Þ þ ðS þ N1 Þ ; a1 ¼ 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 b1 ¼ ðS þ N1 Þ þ ðk 1Þ ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS 2 þ N1 Þ2 þ ðk þ 1Þ2 þ ðS 2 þ N1 Þ ; a2 ¼
ð34Þ
12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 b2 ¼ ðS þ N1 Þ þ ðk þ 1Þ ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 a3 ¼ ðS þ N1 Þ þ ð1 kÞ þ ðS þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ðS þ N1 Þ þ ð1 kÞ ðS þ N1 Þ : b3 ¼ The solutions for blowing can be obtained by replacing S with S1 ðS1 > 0Þ in all the results of suction.
5. Resonant suction case If the angular velocity is equal to the frequency of oscillations i.e. n ¼ X, the system resonates and solutions in this case are given by
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T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
for U cos nt 3 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9
W 2 ffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> > 2 0 iX > þN tþ t z > > 7 2t 6 > ðW2t0 Þ þNt þiXt tt > erfc 2pz ffiffittffi þ e > > 7 6 > > > = 7 6 < 7 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 !>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q 2 7 6 >
W0 > > 2 N iX W0 > > 7 6 > þtþ t N þiX tt > 2t þ z z ð Þ > t t 2t 7 6 > erfc 2pffiffittffi > :þe ; 7 6 > 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r 2 W 7 6 > >
W0 2 N ð 2t0 Þ þNt z > > W0 7 6 z > > pffiffiffi þ ð 2t Þz 6 e erfc þ tt > > 7 < = t 2t f g e 2 tt 7: 6 þi ¼1þ U iXt 7 6 e þ q ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 6 2Xl Xl Xl ffi > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 > W0 2 N > > 7 6
> > þ z > > W0 2 N 7 6 : þ e ð 2t Þ t erfc pz ffiffiffi ; þ tt 2t t 7 6 2 tt 7 6 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 97 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 2 W0 2 N 2X > > 7 6 ð > > W 2t Þ þ t þi t z z N 2X 0 > > 6 erfc 2pffiffittffi þ þ t þ i t tt > >
> > ð 2t Þ t t 5 4 > > W0 2 N 2X :þe ; erfc 2pz ffiffittffi þ þ i tt t t 2t ð35Þ For U sin nt 3 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9
W 2 ffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> > 2 N 0 iX W > > þtþ t > 2t 7 6 > z ð 2t0 Þ þNt þiXt tt > erfc 2pz ffiffittffi þ e > > 7 6 > > > < = 7 6 7 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 !
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q > 2 7 6 >
W 2 > W0 N þiX > > 7 6 > þ N iX 0 t tz > > 2t þ þ tt z ð Þ > t t 2t 7 6 > erfc 2pffiffittffi > :þe ; 7 6 > 7 6 7 6 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 9 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r W0 2 N 7 6 > >
þtz 2 ð Þ > > W0 W 2t 7 6 z N 0 > > pffiffiffi þ ð 2t Þz 6 e erfc þ tt > > 7 < = 2t t f g e 2 tt 7: 6 iU iXt þi ¼1þ 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffi Xl Xl 2 6 þ 2Xl e > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r 7 > 2 W > > N 0 7 6
W0 2 N > > > þ e ð 2t Þ þ t z erfc pz ffiffiffi > 7 6 : ; þ t tt 2t 7 6 2 tt 7 6 7 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 8 9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 7 6 2 W0 2 N 2X > > 7 6 þ þi z > > W0 N 2X > e ð 2t Þ t t erfc pz ffiffiffi þ > 7 6 þ þ i tt > > < = t t 2t 2 tt 7 6 7 6 iU eiXt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 2Xl > r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 7 6 2 > > > ðW2t0 Þ þNt þi2Xtz > 5 4 > > W0 z ffiffiffi N 2X :þe ; p erfc 2 tt þ t þ i t tt 2t ð36Þ
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Making use of Eq. (29), Eqs. (35) and (36) become for U cos nt 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: n ðaþibÞn 7 6 þe erfc pffiffiffi ða þ ibÞ 2 2s 6 8 97 7 6 ffiffi p < eða4 þib4 Þn erfc pnffiffiffi þ ða4 þ ib4 Þ s =7 pffiffi 6 f g 2 6 2s 7 þi ¼ 1 þ e 2Sn 6 þ eis 7: ffiffi p n ða þib Þn s : þ e 4 4 erfc pffiffiffi ða4 þ ib Þ ;7 6 Xl Xl 4 6 2 2s 8 9 7 7 6 ffiffi p n 6 < eða5 þib5 Þn erfc pffiffiffi þ ða5 þ ib5 Þ s = 7 7 6 2 2s 5 4 þ eis ffiffi p : þ eða5 þib5 Þn erfc pnffiffiffi ða5 þ ib Þ s ; 5 2 2s 2
ð37Þ
For U sin nt 3 8 9 pffiffi < eðaþibÞn erfc pnffiffiffi þ ða þ ibÞ s = 2 2s 7 61 pffiffis ; 7 6 2: n ðaþibÞn ffiffiffi p 7 6 þe erfc 2s ða þ ibÞ 2 7 6 8 9 7 6 pffiffis nffiffiffi ða4 þib4 Þn < = 7 6 p pffiffi e erfc 2s þ ða4 þ ib4 Þ 2 f g 7 2Sn 6 is þi ¼1þe 7; 6 þ ie : þ eða4 þib4 Þn erfc pnffiffiffi ða4 þ ib Þpffiffis ; 7 6 Xl Xl 4 6 2 2s 8 9 7 7 6 pffiffis n 7 6 ða5 þib5 Þn < = pffiffiffi þ ða5 þ ib Þ e erfc 7 6 5 2 2s 5 4 ieis : þ eða5 þib5 Þn erfc pnffiffiffi ða5 þ ib Þpffiffis ; 5 2 2s 2
ð38Þ
where 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 a4 ¼ ðS þ N1 Þ þ ðS þ N1 Þ ; b4 ¼
12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS 2 þ N1 Þ2 ðS 2 þ N1 Þ ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 ðS þ N1 Þ þ 4 þ ðS þ N1 Þ ; a5 ¼
ð39Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b5 ¼ ðS þ N1 Þ þ 4 ðS þ N1 Þ :
6. Resonant blowing case In this section W0 ¼ W0 ðS ¼ S1 ; S1 > 0Þ and the solutions (37) and (38) become of the following form
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for U cos nt 8 9 3 pffiffi ~ s > > ~ < eða~þibÞn erfc pnffiffiffi = ~ þ a þ i b 2 2s 7 61 pffiffi 7 6 2> ~ ~ a þi b n > n s ð Þ 7 6 ~ :þe ; ~ erfc pffiffiffi a þ i b 7 6 2 2s 6 8 97 7 6 > > ða~4 þib~4 Þn erfc pnffiffiffi þ a~ þ ib~ pffiffis 7 6 < = e ffiffi p 4 4 7 6 2 f g 2s 2S1 n 6 is pffiffi 7 þi ¼1þe þ e 7: 6 ~ > þ eða~4 þib4 Þn erfc pnffiffiffi a~ þ ib~ Xl Xl s > : ; 7 6 4 4 2 2s 7 6 6 8 9 7 p ffiffi 7 6 ~ > s > ~ 7 6 < eða~5 þib5 Þn erfc pnffiffiffi = ~ þ a þ i b 5 5 2 2s 7 6 5 4 þ eis p ffiffi ~5 þib~5 Þn a > > n s ð :þe ; erfc pffiffiffi a~5 þ ib~5 2 2
ð40Þ
2s
For U sin nt 3 pffiffi 8 9 s < eða~þib~Þn erfc pnffiffiffi þ a~ þ ib~ = 2s 7 61 2 pffiffi 7 6 2: ða~þib~Þn n s ; ~ p ffiffiffi 7 6 þe erfc 2s a~ þ ib 2 7 6 8 9 7 6 pffiffis a~4 þib~4 Þn nffiffiffi ð ~ < = 7 6 p pffiffi e erfc 2s þ a~4 þ ib4 f g 2 7 2S1 n 6 is þi ¼1þe þ ie 7 6 : þ eða~4 þib~4 Þn erfc pnffiffiffi a~4 þ ib~ pffiffis ; 7 6 Xl Xl 4 6 2 2s pffiffi 8 9 7 7 6 a~5 þib~5 Þn n 7 6 s ð ~ < = pffiffiffi þ a ~ e erfc þ i b 5 7 6 5 2 2s 5 4 ieis : þ eða~5 þib~5 Þn erfc pnffiffiffi a~5 þ ib~ pffiffis ; 5 2 2s 2
ð41Þ
in which qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 a~ ¼ ðS1 þ N1 Þ þ 1 þ ðS1 þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b~ ¼ ðS1 þ N1 Þ þ 1 ðS1 þ N1 Þ ; 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 a~4 ¼ ðS1 þ N1 Þ þ ðS1 þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 b~4 ¼ ðS1 þ N1 Þ ðS1 þ N1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 2 ðS1 þ N1 Þ þ 4 þ ðS1 þ N1 Þ ; a~5 ¼ b~5 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 ðS12 þ N1 Þ2 þ 4 ðS12 þ N1 Þ :
ð42Þ
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In order to determine the steady state structure of the solution, we use the asymptotic formula for the complementary error function rffiffiffi n s p ffiffiffiffiffi ða þ ibÞ ! ð0; 2Þ as s ! 1: erfc 2 2s Evidently, in the limit s ! 1, solutions (40) and (41) reduce to for U cos nt h i pffiffi f g ~ ~ ~ þi ¼ 1 þ e 2S1 n eða~þibÞn þ 2eis eða~4 þib4 Þn þ 2eis eða~5 þib5 Þn ; Xl Xl
ð43Þ
for U sin nt h i pffiffi f g ~ ~ ~ þi ¼ 1 þ e 2S1 n eða~þibÞn þ i2eis eða~4 þib4 Þn i2eis eða~5 þib5 Þn : Xl Xl
ð44Þ
Fig. 1. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
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These results describe meaningful hydromagnetic boundary layer flows for resonant frequency. It may be observed that both the effects of rotation and electromagnetic forces are reflected in the ultimate steady velocity fields.
7. Discussion In this paper, we discuss the MHD flow due to non-coaxial rotations of an oscillating disk and a fluid at infinity. A method of Laplace transform has been used to get the solution of the problem. The effects of the various parameters s, S, k and N1 on velocity distribution have been studied and the results have been presented by several graphs. In order to study the effect of the magnetic field parameter N1 (¼0 and 2) on the velocity distribution, we have plotted ðf =XlÞ and ðg=XlÞ against n in Figs. 1 and 2 for U cos nt and in Figs. 3 and 4 for U sin nt with dotted lines in all the three cases i.e. n > X, n < X and n ¼ X. From these figures it is found that boundary layer thickness decreases with increase in N1 . The full lines in Figs. 1–4 depict the effect of suction (S ¼ 0:5)=blowing (S ¼ 0:5). In this case, we see that with
Fig. 2. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU=4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
1191
Fig. 3. The variation of ðf =XlÞ and ðg=XlÞ with n for the sine oscillation at ðU =4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
the increase in suction parameter, the boundary layer thickness decreases and with increase in blowing parameter the boundary layer thickness increases as compared to the case of suction. The case S ¼ 0 represent no suction/no blowing. The time required to attain the steady state for the cosine oscillation of the disk is shorter than that for the sine oscillation of the disk. The value of this time depends on ðn=XÞ (ratio of the frequency of oscillation to the angular velocity of the disk) and ðU =XlÞ. Fig. 5 shows the variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðn=XÞ ¼ 5, ðU=4XlÞ ¼ 1, S ¼ 0 and N1 ¼ 2 for different values of s. The times required to attain the steady state in Fig. 5(a) and (b) are s ¼ 0:4 and s ¼ 1:8 respectively. It is remarked that both values of s are less when compared with case of N1 ¼ 0 (Figs. 2 and 3 in Erdogan [13]). Further, Fig. 6(a) and (b) respectively denote the variation of ðf =XlÞ and ðg=XlÞ with n for various values of s for the cosine oscillation at ðn=XÞ ¼ 0:5, ðU =4XlÞ ¼ 1, N1 ¼ 0 and S ¼ 0:5. It is found that time required to attain the steady state for ðf =XlÞ and ðg=XlÞ are s ¼ 3 and s ¼ 4:5 respectively, which is also shorter than for S ¼ 0 (Figs. 6 and 7 in Erdogan [13]). Fig. 6(c) and (d) are prepared for ðf =XlÞ and ðg=XlÞ against n for the cosine oscillation at ðn=XÞ ¼ 0:5, ðU =4XlÞ ¼ 1, N1 ¼ 0 and S ¼ 0:5 for different s. We observe that time to attain the
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Fig. 4. The variation of ðf =XlÞ and ðg=XlÞ with n for the sine oscillation at ðU =4XlÞ ¼ 1; (- - -) S ¼ 0, N1 ¼ 2; (––) N1 ¼ 0, for various values of S.
Fig. 5. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
1193
Fig. 6. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
steady state for both ðf =XlÞ and ðg=XlÞ is s ¼ 6, which is greater than for the case S ¼ 0 (Figs. 6 and 7 in Erdogan [13]). It is further observed that similar behaviors occur for time required to attain the steady state for sine oscillation. In the resonant case (n ¼ X) it is known that no steady asymptotic solution is possible for flow due to non-coaxial rotations of an oscillating disk and a fluid at infinity (i.e. the boundary condition at infinity is not satisfied). But the present analysis exhibits a striking difference between the structures of the hydrodynamic and the hydromagnetic boundary layers (Figs. 7 and 8). Fig. 7(a) and (b) indicate the variation of ðf =XlÞ and ðg=XlÞ respectively against n for the cosine oscillation at n ¼ X, ðU =4XlÞ ¼ 1, S ¼ 0:5 and N1 ¼ 2 for different s (the dotted lines denote the steady-state velocity and the full lines show the starting velocity). The respective times required for steady state for ðf =XlÞ and ðg=XlÞ are s ¼ 0:9 and s ¼ 1:6. In Fig. 7(c) and (d) we plotted ðf =XlÞ and ðg=XlÞ against n for cosine oscillation with different s at n ¼ X, ðU =4XlÞ ¼ 1, S ¼ 0:5 and N1 ¼ 2. The steady state situation for ðf =XlÞ and ðg=XlÞ are obtained at s ¼ 1:2 and s ¼ 2 which are greater than that of suction case.
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Fig. 7. The variation of ðf =XlÞ and ðg=XlÞ with n for the cosine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
In the case of sine oscillations, the resonant steady state for ðf =XlÞ and ðg=XlÞ in suction/ blowing cases are s ¼ 0:7 and s ¼ 1:4=s ¼ 0:9 and s ¼ 1:8 respectively (see Fig. 8).
8. Concluding remarks The most important features of Eqs. (37), (38), (40) and (41) are that unlike the hydrodynamic situation for the resonant case, Eqs. (37), (38), (40) and (41) satisfy the boundary condition at infinity. Consequently, the associated boundary layers remain bounded for all values of frequencies including the resonant frequency. In contrast to the hydrodynamic solution for the case of blowing and resonance where the blowing promotes the spreading of the oscillations far away from the disk, the solutions (37), (38), (40) and (41) represent the sensible oscillatory boundary layer flow. The physical implication of this conclusion is that for the case of resonance, the unbounded spreading of the oscillations away from the disk is controlled by the external magnetic field.
T. Hayat et al. / International Journal of Engineering Science 41 (2003) 1177–1196
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Fig. 8. The variation of ðf =XlÞ and ðg=XlÞ with n for the sine oscillation at ðU =4XlÞ ¼ 1; (- - -) steady-state velocity; (––) the starting velocity.
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[11] K.R. Rajagopal, On the flow of a simple fluid in an orthogonal rheometer, Arch. Rational Mech. Anal. 79 (1982) 39. [12] K.R. Rajagopal, Flow of viscoelastic fluids between rotating disks, Theor. Comput. Fluid Dyn. 3 (1992) 185. [13] M.E. Erdogan, Flow induced by non-coaxial rotations of a disk executing non-torsional oscillations and a fluid rotating at infinity, Int. J. Eng. Sci. 38 (2000) 175. [14] P.N. Kaloni, Fluctuating flow of an elastico-viscous fluid past a porous flat plate, Phys. Fluids 10 (1967) 1344. [15] B.M. Abramovitz, I.A. Stegun, Handbook of mathematical functions, Dover, New York, 1964, p. 325.