Nonlinear temporal dynamics in magnetic fluids between undulatory parallel walls

Chaos, Solitons and Fractals 12 (2001) 2227±2245

www.elsevier.nl/locate/chaos

Nonlinear temporal dynamics in magnetic ¯uids between undulatory parallel walls Kadry Zakaria Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt Accepted 3 July 2000

Abstract The e€ects of undulatory parallel walls and a normal magnetic ®eld on the stability of weakly nonlinear waves at a horizontal interface of two magnetic inviscid ¯uids are investigated. We assumed that the walls have a weak sinusoidal undulation. The frequency of the main waves is similar to a problem having smooth boundaries. The breaker surface tension and the breaker magnetic ®eld are obtained. The stability analysis concerns the interaction of two propagation wave numbers satisfying the resonance condition imposed by the periodicity of the sinusoidal walls. The ®rst-resonance case occurs whenever the wall wave number is nearly equal to twice the propagation wave number while the second-resonance case occurs whenever the two kinds of wave numbers are nearly equal. When the wave number of the undulation is far from the propagation wave number, the sinusoidal walls have the same e€ect of the smooth walls on the stability criterion. The stability conditions and the transition curves in the two resonance cases are treated away from the critical state. The existence conditions and stability of Stokes waves near the critical state are discussed. Numerous illustrations and graphs amplify the work. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction The magnetic ¯uids are made in the laboratory. These ¯uids consist of ordinary nonconducting liquids in which very ®ne small particles of ferromagnetic material are suspended freely. The very small size of these particles is distributed uniformly and homogeneously in the ¯uid and prevents coagulation. The magnetic ¯uids are assumed to be nonconducting, and the only forces involved are due to polarization. These ¯uids di€er from magnetohydrodynamic ¯uids since no electric current ¯ows in these ¯uids. The introduction of a magnetic ®eld does not cause the separation of the magnetic particles from the liquid. Cowley and Rosensweig [2] have demonstrated that an instability sets in when the applied normal magnetic ®eld is slightly greater than the critical magnetic ®eld. The remarkable feature of their experiments is that such an instability is found in the appearance of the regular hexagonal cells on the ¯uid surface. This experimental observation has been theoretically studied by Gailitis [6] using the energy method, Twombly and Thomas [21] using bifurcation analysis, and other authors. Gailitis has demonstrated the existence of hard excitation of the steady waves, and has shown that for certain values of the magnetic ®eld strength, the hexagonal cells are replaced by square cells with possible hysteresis behavior for the sub-critical values of the applied magnetic ®eld. Twombly and Thomas have also obtained similar results for results of Gailitis. Zelazo and Melcher [23] considered the propagation of plane waves on the interface between two ferro¯uids in the presence of a tangential magnetic ®eld. Their analysis, based on the linear theory, revealed that the tangential magnetic ®eld has a stabilizing e€ect. This is in contrast with the results when the magnetic ®eld is normal to the interface. The Rayleigh±Taylor instability of magnetic ¯uids drew the attention of recent authors. For example, Chakraborty [1], Davalos-Oraozco and Aguilar-Rosas [3] and Shivamoggi [20] studied the stability of 0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 1 5 9 - 4

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K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

stellar and planetary and the wide range of important industrial applications. The nonlinear surface instability of two superposed semi-in®nite magnetic ¯uids in the presence of a normal and tangential magnetic ®eld is investigated by Malik and Singh [9,10] using the method of multiple scales. In the case of the normal ®eld, they found that the growth rate of the instability amplitude is governed by a nonlinear Klein±Gordon equation, which leads to the derivation of bell-shaped solutions and kink solutions as special cases. For a tangential magnetic ®eld, they obtained the nonlinear Schr odinger equation and showed that the wave train solution of constant amplitude is unstable against modulation if the product of the group velocity rate and the nonlinear interaction coecient is negative. Malik and Singh [11,12] investigated the nonlinear evolution of the wave packets on the free surface of semi-in®nite magnetic ¯uids. It was proved that there exist di€erent regions of instability and that magnetic permeability has varying e€ects in these regions of instability. Elhefnawy [4,5] investigated the criterion of the Rayleigh±Taylor instability of waves at the interface of two superposed magnetic ¯uids, each of ®nite stationary depth, in the presence of a uniform tangential and normal magnetic ®eld. He found that the ®nite stationary thickness of the ¯uids plays an important role in the nonlinear stability criterion of the considered system. Mahr and Rehberg [8] studied the nonlinear dynamics of a single ¯uid peak in an oscillating magnetic ®eld. They found that the instability is sub-critical and leads to peaks of a characteristic shape. They investigated the neighborhood of this instability experimentally under the in¯uence of a temporal modulation of magnetic ®eld. A small vessel, where only one peak arises, was used. It is found that the modulation can either be stabilizing or destabilizing, depending on the frequency and amplitude. Also, they proposed a minimal model involving a cuto€ condition, which captures the essence of the experimental observations. Zakaria [22] studied the case of the normal ®eld. He found that the normal ®eld has a destabilizing e€ect on the waves. The nonlinearity has a destabilizing in¯uence and the e€ect of the ratio between the permeabilities depends on the thicknesses of the ¯uids. Also, he discussed the relation between the ®eld and the solitons in the instability case. In a class of dissipative systems, spatially periodic structures arise as a result of a primary instability that develops above a critical value of control parameters. Such fully nonlinear structures are themselves subjected to a secondary instability which may take the form of phase modulations of the primary patterns, as documented both experimentally and theoretically. There has been considerable interest in periodicboundary perturbed problems. This is a consequence of technological advances in the ®elds of surface acoustic-wave science and water waves between periodic boundaries. Ever since the pioneering work of Rayleigh [19], the theory of wave propagation in periodic structures has received a great deal of attention. Mathematically, the problem is characterized by the Mathieu and Hill di€erential equations. The classic approach to the analysis of periodic-structure problems is Floquet theory. This approach, though useful in some simple problems, can be quite cumbersome when numerical solutions of Hill's determinant are involved. Another approach which is most useful when the periodicity in the medium of propagation is a small perturbation of one of the parameters or boundary conditions is the coupled modes approach [16]. A re®ned version of the coupled modes approach that makes use of systematic perturbation techniques is the application of the method of multiple scales. The advantages of the latter method are evident when one tries to ®nd higher-order approximations or treat the problem of propagation of transients or wave packets in a periodic medium by a simple natural extension of the multiple-space scales to multiple-time scales as well. This combination of the multiple-space and multiple-time scales is well known and is widely used in the treatment of nonlinear waves. Nayfeh and Asfer [13] investigated a uniformly valid asymptotic expansion for the propagation of an electromagnetic wave between sinusoidally perturbed walls. Their analysis shows that resonance occurs whenever the wave number of the wall distortion kw is equal to the di€erence between the wave numbers of two propagation modes. Nayfeh [14] studied the propagation of acoustic waves in a hard-wall duct with sinusoidally perturbed walls. He obtained similar results for results of Nayfeh and Asfar [13]. Pouliquen et al. [18] studied the spatio-temporal evolution of the vortex sheet separating two ®nite depth layers of immiscible ¯uids when spatially periodic forcing is imposed on one of the horizontal boundaries. They studied their model in the Boussinesq approximation, where it reduces to a forced Klein± Gordon equation. The secondary Benjamin±Feir (BF) instability of Stokes waves is analyzed in the absence of forcing. When spatial forcing is reintroduced, they studied one-dimensional propagation of Sine±Gordon phase solitons.

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

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The aim of this work is to study the e€ects of sinusoidally perturbed two boundaries and a normal magnetic ®eld on the behavior of weakly nonlinear waves at the interface of two superposed magnetic ¯uids within these boundaries. 2. The problem de®nition and a linear analysis Consider the ®nite amplitude of two-dimensional capillary-gravity waves on the interface y ˆ 0, which separates two inviscid incompressible magnetic ¯uids (see Fig. 1). The ¯uid with the density q1 and permeability l1 has the thickness y ˆ h1 ‡ d cos kw x; d  h1 : The ¯uid of density q2 and permeability l2 has the thickness y ˆ h2 ‡ d cos kw x; d  h2 : Both the ¯uids are subjected to normal uniform constant magnetic ®elds H1 and H2 . The motion is assumed to be irrotational under the gravity g and the surface tension T. We shall assume that there are no free currents at the surface of separation in the equilibrium state, and therefore, the magnetic induction is continuous at the interface, i.e., l1 H1 ˆ l2 H2 : The basic equations that govern the perturbed velocity potential uj …vj ˆ ruj † are r2 /1 ˆ 0 at

h1 ‡ d cos kw x < y < g…x; t†;

2

r /2 ˆ 0 at g…x; t† < y < h2 ‡ d cos kw x;

…1†

where y ˆ g…x; t† is the elevation of the interface. The solutions for /j …j ˆ 1; 2† have to satisfy the condition at y ˆ … 1†j hj ‡ d cos kw x;

N j
r/j ˆ 0

j ˆ 1; 2;

…2†

where Nj ˆ

rfy jrfy

j

… 1† hj … 1†j hj

d cos kw xg : d cos kw xgj

The condition that the interface y ˆ g…x; t† is moving with the ¯uids leads to ot g

oy /j ‡ ox gox /j ˆ 0

at y ˆ g…x; t†:

…3†

We assume that the magneto-quasi-static approximation is valid and we introduce the magnetic potential wj such that r2 w1 ˆ 0 at

h1 ‡ d cos kw x < y < g…x; t†;

2

r w2 ˆ 0 at g…x; t† < y < h2 ‡ d cos kw x; along with the boundary conditions:

Fig. 1. Schematic of the problem.

…4†

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K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245 j

N j ^ rwj ˆ 0

at y ˆ … 1† hj ‡ d cos kw x;

ox w1 ‡ ox goy w1 l1 foy w1

…H1

j ˆ 1; 2;

H2 †ox g ˆ ox w2 ‡ ox goy w2

ox gox w1 g ˆ l2 foy w2

ox gox w2 g

…5†

at y ˆ g…x; t†;

…6†

at y ˆ g…x; t†:

…7†

The conservation of momentum balance is 2

kqj fot /j ‡ 0:5…r/j † ‡ ggg 2

‡ 0:5lj f2Hj …ox g† ‰2oy wj ‡ 4Hj ox gox wj

Hj Š

2

2Hj oy wj

…ox wj † ‡ …oy wj †

2

2

4ox gox wj oy wj gk ‡ T o2x gf1

1:5…ox g† g ˆ 0 at y ˆ g…x; t†;

…8†

where kfj k ˆ f2 f1 : To investigate the nonlinear interactions of small but ®nite amplitude wave, we use the method of multiple scales. To that end, let xn ˆ en x;

tn ˆ en t;

f …y; x; t; e† ˆ

3 X

n ˆ 0; 1; 2;

…9†

en f …n† …y; x0 ; x1 ; x2 ; t0 ; t1 ; t2 † ‡ O…e4 †;

…10†

n

where f can be any of the physical quantities /j ; wj and g. The parameter e represents a small dimensionless parameter characterizing the steepness ratio of the wave. We assume the solid boundaries to have weak sinusoidal undulations. So, without loosing the generality, we can say that d ˆ me since m is a small dimensional number. To apply the method of solution, we express all the boundary conditions of the interface y ˆ g…x; t† about y ˆ 0 using Maclaurin's series. By the same method of Maclaurin, we express the boundary conditions of the solid boundaries about y ˆ … 1†j hj ; j ˆ 1; 2: Substituting Eqs. (9) and (10) into Eqs. (2), (3) and (5)±(8), and equating the coecients of equal powers in e, we obtain the linear as well as the successive higher-order equations. The equations of the ®rst-order, second-order and third-order are given in Appendix A. As the sinusoidal undulations are small, their e€ects cannot appear in the linear step of the analysis. So, the linear problem is the linear Rayleigh±Taylor instability of magnetic ¯uids in the presence of a normal magnetic ®eld. The solution of the ®rst-order problem in the form of progressive waves with respect to the lower scales is obtained as g…1† ˆ n…x0 ; x1 ; x2 ; t1 ; t2 †e

ixt0

‡ cc;

…11†

where cc denotes the complex conjugate and x is the frequency of the disturbance. The solutions /…1† j and w…1† j are /j…1† ˆ wj…1† ˆ

i… 1†j x ‰ne k sinh khj …H1

ixt0

‡ ccŠ cosh k…y

H2 †lj‡1 cosh khj‡1 ‰ne v1

ixt0

j

… 1† hj †;

‡ ccŠ sinh k…y

…12†

… 1†j hj †;

j ˆ 1; 2;

…13†

where v1 ˆ l2 sinh kh1 cosh kh2 ‡ l1 sinh kh2 cosh kh1 : The unknown function n satis®es the following di€erential equation: 

x2 ‰q coth kh1 ‡ q2 coth kh2 Š ‡ g…q2 k 1

where l12 ˆ l1

l2 :

 kl1 l212 H12 o2 q1 † ‡ cosh kh1 cosh kh2 ‡ T 2 n ˆ 0; l 2 v1 ox0

…14†

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The solution of Eq. (14) takes the following form: n ˆ A…x1 ; x2 ; t1 ; t2 †eikx0 ‡ cc;

…15†

where A…x1 ; x2 ; t1 ; t2 † is an unknown slowly varying function denoting the amplitude of the propagating wave and will be determined later. For the solution given by Eqs. (11)±(15) to be nontrivial, the frequency x and the wave number k must satisfy the dispersion relation  2  x kl1 l212 H12 2 ‰q coth kh1 ‡ q2 coth kh2 Š ‡ g…q2 q1 † ‡ S1 …k; x† ˆ cosh kh1 cosh kh2 Tk ˆ 0: k 1 l2 v1 …16† The dispersion relation (16) was initially obtained by Elhefnawy [4]. Letting H1 ˆ 0 and hj ! 1; we obtain the dispersion relation of Nahfey [15]. From the dispersion relation (16), we observe that the magnetic ®eld has a destabilizing in¯uence on the wave motion. The dispersion relation does not depend on the sign of l12 but depends on …q1 q2 †; H12 , the permeabilities and the thicknesses of the ¯uids. The stability of the interface depends on whether the magnetic ®eld H12 is larger or smaller than Hc ; since Hc ˆ

f1 ; f2

…17†

where f1 ˆ

fTk 3 ‡ gk…q1 q2 †g ; fq1 coth kh1 ‡ q2 coth kh2 g

f2 ˆ

l1 l212 k 2 cosh kh1 cosh kh2 : l2 v1 fq1 coth kh1 ‡ q2 coth kh2 g

It is clear that the system is stable for H12 6 Hc : This condition shows that the surface tension and the gravity are strictly stabilizing while the ®eld has a destabilizing in¯uence on the wave motion. The normal magnetic ®eld acting upon magnetic ¯uids produces the spontaneous generation of an ordered pattern of surface protuberances when the ®eld exceeds the critical value. 3. The second-order problem Now, we substitute the ®rst-order solutions into the second-order equations (A.7)±(A.13). The uniformly valid solutions of the resulting equations depend on removing the secular terms. Equating the secular terms to zero, we obtain the solvability condition. The relation between the propagation wave number and the wave number of the solid boundary undulation gives two forms of the solvability condition. The ®rst is obtained when kw is far from the propagation wave number 2k, and this is called the nonresonance case. In the nonresonance case, the secular terms are eliminated if we have the following solvability condition: oS1 oS1 ot A ˆ ox A: ox 1 ok 1

…18†

Then, the uniformly valid second-order displacement is g…2† ˆ K1 ‡ K2 e

ixt0

‡ K3 e

2ixt0

‡ cc;

…19†

where K1 ˆ 0:5K11 AA ‡ K12 fA2 e2ikx0 ‡ ccg; mK21 mK22  i…kw Aei…kw ‡k†x0 ‡ Ae K2 ˆ Tkw …kw ‡ 2k† Tkw …kw 2k† K3 ˆ 0:5K31 AA ‡ K32 fA2 e2ikx0 ‡ ccg;

k†x0

‡ cc;

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K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

since K11 ˆ



2 g…q1

q2 †

k 2 l312 l2 v2 K12 ˆ

g…q1

x2 fq2

kqj coth2 khj kg ‡ H12

q1

K21 ˆ

l1 l212 k 2 klj‡1 sinh2 khj cosh2 khj‡1 k l2 v21

2l l k 2 2l l k 2 l1 cosh kh1 cosh kh2 ‡ 1 12 ‡ 1 12 b1 l2 l2 v1 2

1 q2 †

 2

x fq2

4k 2 T

2k 2 l1 l12 k… l2 v1

2l1 l12 k 2 k… l2 v1

2



2

q1 ‡ kqj coth khj kg ‡

l1 l212 k 2 klj‡1 sinh2 khj cosh2 khj‡1 k l2 v21 ‡



j

1† lj‡1 cosh khj‡1 k

H12

j



1† lj‡1 cosh khj‡1 k

;

2l1 l12 k 2 b1 l2 v1

l1 l312 cosh2 kh1 cosh2 kh2 l2 v21  2k 2 l1 l12 ; l2

  x2 …kw ‡ k† H12 k 2 l1 l212 2 2 kqj cosech khj k ‡ klj cosh khj k ; 2k 2v21

  x2 …kw k† H12 k 2 l1 l212 2 2 kqj cosech khj k ‡ klj cosh khj k ; K22 ˆ 2k 2v21 



l1 l12 k 2 l l2 k 2 b1 ‡ 1 12 2 klj‡1 sinh2 khj cosh2 khj‡1 k l 2 v1 2l2 v1 3 2 l1 l12 k l1 l12 k 2 l1 l12 j 2 2 cosh kh cosh kh k… 1† l cosh kh k ‡ 1 2 j‡1 j‡1 2l2 v21 l 2 v1 l2  2k 2 l1 l312 cosh kh1 cosh kh2 cosh 2kh1 cosh 2kh2 fS2 ‡ Tk 2 g; l 2 v1 v2 2

2

K31 ˆ 2 0:5x fkqj coth khj k ‡ 3q2

3q1 g ‡

H12

 j K32 ˆ 0:5x2 f3q2 3q1 k… 1† qj coth2 khj k 2kqj coth khj coth 2khj kg  l1 l12 k 2 l1 l212 k 2 l1 l312 b1 klj‡1 sinh2 khj cosh2 khj‡1 k cosh2 kh1 cosh2 kh2 ‡ H12 2 l2 v1 2l2 v1 2l2 v21 k 2 l1 l12 k 2 l1 l12 2k 2 l1 l312 j ‡ k… 1† lj‡1 cosh khj‡1 k cosh kh1 cosh kh2 cosh 2kh1 cosh 2kh2 l2 v1 l2 l2 v1 v2  k 2 l21 l12 j b1 k… 1† cosh 2khj sinh 2khj‡1 k S2 ; v1 v2  S2 ˆ

2x2 ‰q1 coth 2kh1 ‡ q2 coth 2kh2 Š ‡ g…q2 k

q1 † ‡

2kl1 l212 H12 cosh 2kh1 cosh 2kh2 l2 v2

 4Tk 2 ;

v2 ˆ l2 sinh 2kh1 cosh 2kh2 ‡ l1 sinh 2kh2 cosh 2kh1 : In the case of resonance, a detuning parameter r1 exists by kw ˆ 2k ‡ 2r1 :

…20†

Then, the solvability condition in this case is i

oS1 oS1 ot A ‡ i ox A ox 1 ok 1

mK22 Aexp‰2ir1 x0 Š ˆ 0:

…21†

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The disturbances will be broken when g…q1 q2 † 4k 2 T ˆ 0 or S2 ‡ Tk 2 ˆ 0 or S2 ˆ 0: In the breaking case, the waves will be discontinuous and we have waves in a shock case. The extreme case of breaking arises when the surface tension equals g…q1 q2 †=4k 2 or the applied ®eld is equal to one of the two quantities Hb : Hb ˆ l2 v1 v2 f‰q1 coth kh1 ‡ q2 coth kh2 Š‰g…q2 2

q1 †

4Tk 2 Š

2‰q1 coth 2kh1 ‡ q2 coth 2kh2 Š

Šg=2kl1 l212 fv1 ‰q1

 ‰g…q2 q1 † Tk coth 2kh1 ‡ q2 coth 2kh2 Š cosh 2kh1 cosh 2kh2 v2 ‰q1 coth 2kh1 ‡ q2 coth 2kh2 Š cosh kh1 cosh kh2 g or Hb ˆ l2 v1 v2 fg…q2 Tk

2

q1 †‰q1 coth kh1 ‡ q2 coth kh2 Š

Šg=2kl1 l212 fv1 ‰q1

2‰…q1 coth 2kh1 ‡ q2 coth 2kh2 Š‰g…q2

coth 2kh1 ‡ q2 coth 2kh2 Š cosh 2kh1 cosh 2kh2

q1 †

v2 ‰q1 coth 2kh1

‡ q2 coth 2kh2 Š cosh kh1 cosh kh2 g:

4. The modi®ed Schr odinger equations away from the critical state To determine the stability for the amplitude of waves, we need to investigate the solvability conditions of the third-order equations (A.14)±(A.20). We substitute the ®rst- and second-order solutions into (A.14)± (A.20). As before, the uniformly valid solutions of the resulting equations depend on removing the secular terms. To accomplish this, we distinguish three cases: (i) When kw is far from k or 2k. (ii) The case of resonance treated in the previous section …kw  2k†. (iii) A new resonance case when kw  k: For the case of kw away from 2k or k, the solvability condition is   oS1 oA oS1 oA 1 o2 S1 o2 A o2 S1 o2 A 1 o2 S1 o2 A i ‡ ‡ ‡ m2 k2 A ‡ k4 A2 A ˆ 0: …22† ‡ 2 ox2 ot12 oxok ot1 ox1 2 ok 2 ox21 ox ot2 ok ox2 For the resonance cases (ii) and (iii), the solvability conditions are, respectively,   oS1 oA oS1 oA 1 o2 S1 o2 A o2 S1 o2 A 1 o2 S1 o2 A oA 2ir1 x i ‡ ‡ ‡ imk e ‡ m2 k~2 A ‡ 1 2 2 2 ox2 ot1 oxok ot1 ox1 2 ok 2 ox1 ox1 ox ot2 ok ox2 ‡ k4 A2 A ˆ 0;  i

oS1 oA oS1 oA ‡ ox ot2 ok ox2 ‡ k4 A2 A ˆ 0;

 ‡

1 o2 S1 o2 A 2 ox2 ot12

…23†

o2 S1 o2 A 1 o2 S1 o2 A  2ir2 x0 ‡ ‡ m2 k2 A ‡ m2 k3 Ae oxok ot1 ox1 2 ok 2 ox21 …24†

where kw ˆ k ‡ r2 and k1 ; k2 ; k~2 ; k3 and k4 are given in Appendix B. Eqs. (18), (21) and (22)±(24) describe the evolution of two-dimensional wave packets near (x  0) and away from (x away from zero) the cuto€ magnetic ®eld, which separates the stable from unstable states. In this section, we analyze the second case. Using (18) and (21) and replacing xn and tn by en x and en t, we rewrite (22)±(24) as   oA oA 1 o2 A i …25† ‡ x0 ‡ x00 2 ‡ d2 q2 A ‡ q4 A2 A ˆ 0; ot ox 2 ox   oA oA 1 o2 A oA 2ir1 x e ‡ d2 q~2 A ‡ q4 A2 A ˆ 0; i ‡ x0 ‡ x00 2 ‡ idq1 ot ox 2 ox ox

…26†

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K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

  oA oA 1 o2 A  2ir2 x ‡ q4 A2 A ˆ 0; i ‡ x0 ‡ x00 2 ‡ d2 q2 A ‡ d2 q3 Ae ot ox 2 ox where x ˆ

…oS1 =ok† ; …oS1 =ox†

q2 ˆ

   oS1 k2 ; ox

0

 q1 ˆ

o2 S1 o2 S1 K22 x K ‡ 22 ox2 oxok

q3 ˆ

0

   oS1 k3 ox

and

 k1

q4 ˆ

Changing the independent variables from x and t to f ˆ x forms:

oS1 ; ox

…27†

q~2 ˆ

f0:5K222

‡ k~2 g



oS1 ; ox

   oS1 2 e k4 : ox x0 t and s ˆ t, we express (25)±(27) in the ®nal

i

oA x00 o2 A ‡ ‡ d2 q2 A ‡ q4 A2 A ˆ 0; os 2 of2

…28†

i

oA x00 o2 A oA 0 ‡ ‡ idq1 e2ir1 …f‡x s† ‡ d2 q~2 A ‡ q4 A2 A ˆ 0; 2 os of 2 of

…29†

i

oA x00 o2 A  2ir2 …f‡x0 s† ‡ q4 A2 A ˆ 0; ‡ ‡ d2 q2 A ‡ d2 q3 Ae 2 os 2 of

…30†

Eq. (28) of the nonresonance case is the usual nonlinear Schr odinger equation which is valid away from the critical state (x ! 0). Eqs. (29) and (30) are not well-known Schr odinger equations. This is due to the convergence between wave numbers of the walls and the propagation. When d ! 0, we obtain the same result of [4]. We can say that Eqs. (29) and (30) are modi®ed nonlinear Schr odinger equations.

5. Stability analysis and an application away from the critical state 5.1. The stability conditions The stability condition for the nonlinear Schr odinger equation (28) is x00 q4 6 0:

…31†

Many investigators, such as [7], [15] and [17], have examined the linear stability of a ®nite amplitude wave train propagating through the interface by linearly perturbing the solution of Schr odinger equation (28). They found that the linear stability implies the same condition (31). Although the undulation appeared in Eq. (28), there is no e€ect on the stability condition. This is due to the fact that the term d2 q2 A does not a€ect in the modulation to the solution of Schr odinger equation. It means that when the wave number of the undulation kw is far from k or 2k, the sinusoidal walls have the same e€ect as the smooth walls on the energy of the wave train. In other words, the undulation neither decreases nor increases the kinematic energy of the waves during its propagation on the interface. To obtain the stability conditions for Eqs. (29) and (30), we follow the procedure of Newton and Keller [17]. The steady-state solution of Eq. (29) is given by 0

A ˆ aeir1 …f‡x s† ;

…32†

where the amplitude a is a nonzero real constant and given by a2 ˆ fr1 x0 ‡ 0:5r21 x00

d2 q~2

r1 dq1 g=q4 :

…33†

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

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The solution becomes unbounded in ®nite time if a2 < 0 [18]. Then, to obtain a stationary bounded amplitude of the waves, we must have a2 > 0. The stability criterion of the steady state can be determined by superposing small perturbations on the solution (32) by letting 0

A ˆ …a ‡ a1 ‡ ib1 †eir1 …f‡x s† ;

…34†

where a1 and b1 are small real quantities. Substituting (34) into (29) and neglecting the nonlinear terms in a1 and b1 , we obtain oa1 o2 b1 oa1 ‡ 0:5x00 2 ‡ …r1 x00 ‡ dq1 † os of of ob1 os

0:5x00

o2 a1 ‡ …r1 x00 of2

dq1 †

ob1 of

2dr1 q1 b1 ˆ 0;

…35†

2a2 q4 a1 ˆ 0:

…36†

The solution of Eqs. (35) and (36) takes the following form: a1 ˆ a~1 ei…Kf b1 ˆ b~1 ei…Kf

Xs†

‡ cc;

Xs†

‡ cc;

…37†

since a~1 and b~1 are real constants. Substituting (37) into (35) and (36), we obtain the dispersion equation X2

2r1 x00 KX

…0:5x00 K 2 ‡ 2dr1 q1 †…0:5x00 K 2

2a2 q4 †

…d2 q21

r21 x002 †K 2 ˆ 0:

…38†

The eigenvalue X will be real, for real values of K, if we have x002 K 4 ‡ 4…d2 q21 ‡ r1 x00 dq1 q4 fr1 x0 ‡ 0:5r21 x00

d2 q~2

x00 a2 q4 †K 2

16r1 dq1 a2 q4 P 0;

…39†

r1 dq1 g > 0

and so the propagating waves on the interface will be temporally stable. The condition (39) can be rewritten to be …K 2

K^1 †…K 2

K^2 † P 0;

…40†

where K^1;2 ˆ 2fx00 a2 q4

d2 q21

r1 x00 dq1  ‰…x00 a2 q4

d2 q21

2

r1 x00 dq1 † ‡ 4r1 x002 da2 q1 q4 Š

1=2

g=x002 :

…41†

The condition (40) will be satis®ed if K 2 P K^1 and K 2 P K^2 or if K 2 6 K^1 and K 2 6 K^2 . For all disturbances, we have a stable case if K^1 and K^2 are negative. When d ˆ 0, the stability condition (40) reduces to those obtained by Nayfeh [15] and thus we obtain the stability condition (31) for all disturbances. In this case of exact resonance, kw ˆ 2k, the stability condition is a2 q4 x00 6 d2 q21 provided that q~2 q4 < 0 for all waves. Likewise, the stability condition of the disturbances in the second case of resonance kw  k is as follows: …K 2

K~1 †…K 2

K~2 † P 0;

q4 fr2 x0 ‡ 0:5r22 x00

d2 q2

…42†

d2 q3 g > 0;

where K~1;2 ˆ 2fx00 a2 q4

x00 d2 q3  ‰…x00 a2 q4

2

x00 d2 q3 † ‡ 4r2 x002 d2 a2 q3 q4 Š

1=2

g=x002 :

…43†

5.2. A numerical application The stability analysis, for the di€erent cases of the resonance, may be understood by studying the stability graphs represented by (41) and (43) in d±K 2 plane. The curves in Figs. 2 and 3 correspond to the case of kw  2k while the curves in Fig. 4 corresponds to the case of kw  k. A detailed study of the limit as d ! 0

2236

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

Fig. 2. Stability diagram in d±K 2 plane for a system having h1 ˆ h2 ˆ h, q1 ˆ 1:485 g cm 3 , q2 ˆ 0:999857 g cm 3 , l1 ˆ 1:007, l2 ˆ 1:7, g ˆ 980 cm s 2 , T ˆ 32:6 dyn cm 1 , kw ˆ 2k ‡ 2r1 and r1 ˆ 0:05 cm 1 : (a) k ˆ 1 cm 1 and H1 ˆ 0; (b) k ˆ 2 cm 1 and H1 ˆ 0.

Fig. 3. Stability diagram for the same system considered in Fig. 2: (a) k ˆ 1 cm cm 1 .

1

and H12 ˆ 6 A cm 1 ; (b) k ˆ 2 cm

1

and H12 ˆ 6 A

Fig. 4. Stability diagram for the same system considered in Fig. 2 with kw ˆ k ‡ 2r2 and r2 ˆ 0:05 cm 1 : (a) h ˆ 3 cm and k ˆ 1 cm 1 ; (b) k ˆ 2 cm 1 and H1 ˆ 0; (c) k ˆ 2 cm 1 and H12 ˆ 6 A cm 1 .

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

2237

is given by Elhefnawy [4]. Our aim of this work is to study the e€ects of the undulation and the normal magnetic ®eld on the stability criterion of the weakly nonlinear waves. We plotted the curves for K^1;2 and K~1;2 against the amplitude undulation d for constant values of k, H12 and h (h1 ˆ h2 ˆ h). In Fig. 2(a), the curves K^1 and K^2 were plotted for the di€erent values of the thickness h when H12 ˆ 0 and k ˆ 1 cm 1 . It is observed in this ®gure that the increase in the thickness leads to instability. Also, the increase in the amplitude undulation d leads to the dilation in the instability. In Fig. 2(b), we have taken k ˆ 2 cm 1 . We observe the same in¯uence of the wideness of thickness, but the in¯uence of the increase in d is weak especially in the case of the small thickness. This means that the increase in the value of k hinders the in¯uence of d. In Fig. 3(a), where H12 ˆ 6 A cm 1 and k ˆ 1 cm 1 , we observe the same e€ect as in Fig. 2(a). In Fig. 3(b), the case di€ers from Fig. 2(b) since the increase in the ®eld led to the observance of the in¯uence of the amplitude undulation on the stability criterion. Also, the increase in d has the same in¯uence for all the values of thickness. The growth of the magnetic ®eld has an unstable in¯uence especially for the larger values of thickness. In case of k ˆ 1 cm 1 and h < 3 cm we ®nd that the stability condition (42) is satis®ed for all values of K 2 since K~1 and K~2 are negative. Fig. 4(a) has been plotted when k ˆ 1 cm 1 and h ˆ 3 cm, for di€erent values of magnetic ®eld. It is observed in this ®gure that the in¯uence of d, which is unstable is quick compared to that mentioned before. Also, we ®nd that the growth in the ®eld value has the same obvious unstable e€ect. When d < 0:078 cm, it is found that the system is stable for all the values of the ®eld since K~1 and K~2 are negative in this interval of amplitude undulation. When we have taken H12 ˆ 6 A cm 1 and k ˆ 1 cm 1 , it is found that the condition (42) is satis®ed for all the values of thickness. This is an indicator that the magnetic ®eld has a dual e€ect. Figs. 4(b) and (c) were plotted when H12 ˆ 0 and k ˆ 2 cm 1 and H12 ˆ 6 A cm 1 and k ˆ 2 cm 1 , respectively. It is observed that the stability criterion in Fig. 4(b) resembles to a great extent the stability criterion in Fig. 2(b). Also, we see the same similarity between Figs. 4(c) and 3(b).

6. Stability analysis near the critical state 6.1. The modi®ed Schr odinger equations near the critical state The solvability conditions (18) and (22) can be simpli®ed and combined together to produce a single equation for the nonresonance case. This can be done by di€erentiating (18) with respect to x1 and t1 , eliminating x1 from (22), and after some manipulation and using the transformation X ˆ x and v ˆ t k 0 x, we get i

oA oX

k 00 o2 A ‡ d2 c1 A ‡ c2 A2 A ˆ 0; 2 ov2

…44†

where dk ; k ˆ dx 0

d2 k k ˆ ; dx2 00

   oS1 c 1 ˆ k2 ok

 and

c2 ˆ

 2

e k4

oS1 ok



Eq. (44) is the well-known Schr odinger equation. The solutions of this equation are valid near H12 ˆ Hc , i.e., oS1 =ox ˆ 0. By the same manner, the modi®ed Schr odinger equations which are valid near the critical state for the two resonance cases kw ! 2k and kw ! k are, respectively, !  oA k 00 o2 A oA 0 oA k e2ir1 X ‡ d2 c~1 A ‡ c2 A2 A ˆ 0; ‡ idc3 …45† i oX oX ov 2 ov2 i

oA oX

k 00 o2 A  2ir2 X ‡ c2 A2 A ˆ 0; ‡ d2 c1 A ‡ d2 c4 Ae 2 ov2

…46†

2238

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

where



c3 ˆ

k1 k 

c~1 ˆ

o2 S1 0 k K22 ok 2

0

o2 S1 K22 oxok

o2 S1 2 K oxok 22

oS1 0 k k~2 ok



oS1 0 k; ox

o2 S1 0:5 2 k 0 K222 ok



2 0

…oS1 =ok† k

and

   oS1 c4 ˆ c3 ok

6.2. The Stokes waves and their stability conditions In this section, we will study the existence and stability of the Stokes waves near the critical state in the di€erent cases of resonance. 6.2.1. The Stokes waves in the nonresonance case Finite amplitude Stokes solutions of (44) are sought in the form A…X ; v† ˆ Qei…XK

Xv†

;

…47†

where Q is a real amplitude, K the wave number and X is the frequency. By substituting (47) into (44), we have Q2 ˆ fK

d2 c 1

0:5k 00 X2 g=c2 :

…48†

The nonlinear Stokes wave train will exist [18] if Q2 > 0, and in another words if we have K d c1 0:5k 00 X2 > 0 and c2 > 0 or K d2 c1 0:5k 00 X2 < 0 and c2 < 0. The domain of existence of nonlinear ^ since plane waves is found to be bounded by X 2

^ ˆ f2‰K X

d2 c1 Š=k 00 g

1=2

…49†

in K±X plane, provided that X2 P 0. We now discuss stability of the amplitude modulation of the Stokes waves when H12 ! Hc . As before, the stability criterion of the existed Stokes waves is discussed using the modulation concept. So, we perturb the solution (47) according to A…X ; v† ˆ ‰Q ‡ b…X ; v†Šei…XK

Xv†‡ih…X ;v†

;

…50†

where b and h are real small functions. Substitute Eq. (50) into (45) and neglect the nonlinear terms in b and h to have two di€erential equations for b and h. Using the solution …b; h† / ei…X `

mv†

;

…51†

we obtain the following dispersion relation: k 002 4  ‡ f…K x 4

d2 c1 †k 00

 2 ‡ 2X`k 00 x  1:5X2 k 002 gx

`2 ˆ 0:

…52†

The nonlinear Stokes wave train of amplitude Q, wave number K and frequency X will be stable if there  of negative imaginary part. Eq. (52) is exists a sideband wave number ` with a complex growth rate x  at each point of the domain of existence of Stokes waves in K±X plane solved numerically with respect to x for all values of the perturbation wave number `. When the sideband wave number ` is zero, the frequency  can be obtained analytically, and so the Stokes waves are stable if we have k 00 > 0 with x 3X2 k 00 ‡ 2c1 d2 2K > 0 or k 00 < 0 with 3X2 k 00 ‡ 2c1 d2 2K < 0. 6.2.2. The Stokes waves in the resonance cases The steady-state solutions of Eqs. (45) and (46) are, respectively, ^ ir1 X ; A ˆ Qe

^ 2 ˆ …r1 Q

dr1 c3

d2 c~1 †=c2 ;

…53†

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

~ ir2 X ; A ˆ Qe

~ 2 ˆ …r2 Q

d2 c 4

d2 c1 †=c2 :

2239

…54†

The uniform Stokes waves will exist if we have c2 < 0; d2 c~1 ‡ dr1 c3 2

2

d c4 ‡ d c1

r1 > 0;

…55†

r2 > 0

or we have c2 > 0; d2 c~1 ‡ dr1 c3 2

2

d c1 ‡ d c4

r1 < 0;

…56†

r2 < 0:

The domain of existence of nonlinear Stokes waves is found in r1 ±d plane and r2 ±d plane to be bounded by the boundaries, respectively, 8 s9 < r 1 c3 4~ c = 4~ c d1;2 ˆ 1  1 ‡ 2 12 ; 1 ‡ 2 12 > 0; …57† 2~ c1 : r1 c3 ; r1 c3 d^ ˆ

r r2 ; c1 ‡ c4

c1 ‡ c4 > 0:

…58†

The existence conditions of Stokes waves, near the critical state, for all cases of resonance are summarized in Table 1. Fig. 5 has been plotted to illustrate the areas of the wave existence in the case of nonresonance for the values of k, d and h in the presence of the condition X2 > 0. When we compare the three parts of the ®gure with one another, we ®nd that the increase in the thickness led to the dilation in the area of the existed waves. When we change the value of k to be k ˆ 1 cm 1 , we ®nd that the existence area has been shifted to the top of the curve (49). It is observed that the location of the area depends on the value of k and its width is directly proportional to the thickness. Fig. 6 was plotted to illustrate the area of the waves in the case of the ®rst resonance. Fig. 7 was plotted to illustrate the area of the waves in the second-resonance case. In Fig. 6, we observe that the increase of thickness h led to the decline of the wave area when k ˆ 0:5 cm 1 . When k is grew to be k ˆ 1 cm 1 , the wave area has been dilated, but with the same in¯uence of the increase of thickness as in the case of k ˆ 0:5 cm 1 . This means that, for all values of k, the width of the wave area is inversely proportional to the thickness. The same observations are found in Fig. 7. The stability of the stationary solutions (53) and (54) is discussed by introducing perturbations b…X ; v† and h…X ; v† to be ^ ‡ b ‡ ih†eir1 X ; A ˆ …Q ^ ‡ b ‡ ih†eir2 X : A ˆ …Q

…59†

Table 1 For the existence conditions of Stokes waves near the critical state The nonresonance case (kw does not tend to 2k and k) ^2 k 00 > 0; c2 > 0; X2 < X ^2 k 00 < 0; c2 > 0; X2 > X ^2 k 00 > 0; c2 < 0; X2 > X ^2 k 00 < 0; c2 < 0; X2 < X

The ®rst-resonance case (kw ! 2k)

The second-resonance case (kw ! k)

c~1 c~1 c~1 c~1

c1 ‡ c4 > 0; c2 > 0; d < d^ c1 ‡ c4 < 0; c2 > 0 c1 ‡ c4 > 0; c2 < 0; d > d^ ±

> 0; < 0; < 0; < 0;

c2 c2 c2 c2

> 0; > 0; < 0; < 0;

…d …d …d …d

d1 †…d d1 †…d d1 †…d d1 †…d

d2 † < 0 d2 † > 0 d2 † > 0 d2 † < 0

2240

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

Fig. 5. The shaded regions are called the domain existance of Stokes waves at the critical state for the system considered in Fig. 2 but kw does not tend to 2k and k: (a) h ˆ 0:2 cm, d ˆ 0:01 cm, kw ˆ 5k and k ˆ 1 cm 1 ; (b) h ˆ 0:8 cm, d ˆ 0:01 cm, kw ˆ 5k and k ˆ 1 cm 1 ; (c) h ˆ 0:2 cm, d ˆ 0:01, kw ˆ 5k and k ˆ 1:5 cm 1 .

Fig. 6. The shaded regions are called the domain existance of Stokes waves at the critical state for the system considered in Fig. 5 but kw ˆ 2k ‡ 2r1 : (a) h ˆ 0:2 cm and k ˆ 0:5 cm 1 ; (b) h ˆ 0:8 cm and k ˆ 0:5 cm 1 ; (c) h ˆ 0:8 cm and k ˆ 1 cm 1 .

Substituting the solutions (59) into Eqs. (45) and (46) and putting the same solution (51), the dispersion relations of the ®rst-resonance case and the second-resonance cases are, respectively, k 002 4  2 ‡ 2d2 d23 k 0 `x   ‡ fr1 k 00 ‡ d2 c23 k 02 2dr1 c3 k 00 c~1 k 00 gx x 4 ‡ f4d2 c23 r21 ‡ 4r1 c~1 c3 ‡ d2 c23 `2 `2 4dc3 r21 g ˆ 0; k 002 4  ‡ fr2 x 4

2d2 c4

 2 ‡ f4d2 c4 c1 d2 c 1 g x

`2

4dc4 r2 g ˆ 0:

…60† …61†

The nonlinear Stokes waves, in the resonance cases, will be unstable if there exists a sideband wave number  of positive imaginary part. Eq. (60) is solved numerically with respect to x  ` with a complex growth rate x at each point of the existence domain of Stokes waves in r1 ±d plane for all values of the perturbation wave

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

2241

Fig. 7. The shaded regions are called the domain existance of Stokes waves at the critical state for the system considered in Fig. 5 but kw ˆ k ‡ 2r2 : (a) h ˆ 0:2 cm and k ˆ 0:5 cm 1 ; (b) h ˆ 0:8 cm and k ˆ 0:5 cm 1 ; (c) h ˆ 0:8 cm and k ˆ 1 cm 1 ; (d) h ˆ 0:2 cm and k ˆ 1:5 cm 1 .

number `. At the critical points, we ®nd that k 0 ˆ 0 and so Eq. (60) can be solved analytically. Then, the four roots of frequency will be real and therefore we have stable Stokes waves at the marginal state if fr1

2dr1 c3

c~1 gk 00 < 0;

4d2 c23 r21 ‡ 4r1 c~1 c3 ‡ d2 c23 `2 fr1

2dr1 c3

…62† `2

4dc3 r21 > 0;

2 c~1 g P f4d2 c23 r21 ‡ 4r1 c~1 c3 ‡ d2 c23 `2

…63† `2

4dc3 r21 g:

…64†

The dispersion relation (61) will be solved analytically at each point of the domain of existence of Stokes waves in r2 ±d plane without any condition on the state, and the waves will be stable if we have the following conditions: r2

2d2 c4

4d2 c4 c1 fr2

`2

2d2 c4

d2 c1 < 0;

…65†

4dc4 r2 > 0; d2 c 1 g

2

f4d2 c4 c1

…66† `2

4dc4 r2 gk 002 > 0:

…67†

7. Concluding remarks Our aim is to develop a weakly nonlinear theory for a wave whose wave number, frequency and amplitude are all temporal and special varying functions To facilitate this, the method of multiple scales is used to derive two partial di€erential equations, for every resonance case, describing the evolution of a twodimensional wave packet on the interface. The inviscid magnetic liquids have arbitrary densities and permeabilities taking into account a constant surface tension and the e€ect of a normal magnetic ®eld. The liquids lay between undulatory parallel conducting walls. The undulation is small and so a perturbation technique is used. The frequency of the main waves is the same as a system having smooth boundaries. This frequency is obtained because the undulation is weak. The values of surface tension and the applied magnetic ®eld which produce a breaking in the waves are obtained. This breaking means that the waves are in a shock case. Following the resonance cases, we obtained di€erential equations similar to the

2242

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

Schr odinger equations away and near the critical state. These di€erential equations are solved, using the modulation concept, to obtain the stability conditions. In the non resonance case, we obtained the stability condition of Nayfeh [15]. It means that the undulation has an in¯uence, on the stability criterion, in the case of the interaction between the wave numbers of the propagation and the undulatory parallel walls. Having obtained the results of a numerical application, away from the critical state, we found that: (1) The growth of undulation amplitude has a destabilizing e€ect. This destabilizing e€ect depends, strongly, on the wave number of the propagation and the thickness. (2) The magnetic ®eld has a destabilizing in¯uence in the linear step while it plays a dual role in the nonlinear stage. (3) For large values of the thickness, the stable waves are dominant in all the cases. These results are in agreement with the experimental results [2,8]. When the undulation amplitude tends to zero, we obtain results of Elhefnawy [4]. In the critical state, we obtain the nonlinear Schr odinger-type equations which are valid near the critical state. The existence conditions and the stability of the Stokes waves are discussed. We obtained the following results: the existence area of the Stokes waves in the nonresonance case is proportional to the thickness where the growth in the thickness tends to dilate in the area. In the ®rst-resonance case, the increase in the wave number of the main waves led to dilation in the existence area. In the second-resonance case, the same behavior of the main wave number and the thickness is found as in the ®rst-resonance case. In many cases, the increase in the detuning parameter led to dilation in the existence area. The stability conditions of the existed waves are discussed for all cases. Appendix A Order e: o2x0 f/j…1† ; wj…1† g ‡ o2y f/j…1† ; wj…1† g ˆ 0 oy /…1† j ˆ 0 oy /…1† j kox0 wj…1†

and

at y ˆ … 1† hj ;

…A:2†

at y ˆ 0;

Hj ox0 g…1† k ˆ 0

klj oy w…1† j k ˆ 0

j

ox0 w…1† j ˆ 0

ot0 g…1† ˆ 0

…A:1†

…A:3†

at y ˆ 0;

…A:4†

at y ˆ 0;

kqj fot0 /j…1† ‡ gg…1† g

…A:5†

2 …1† lj Hj oy w…1† ˆ0 j k ‡ T ox0 g

at y ˆ 0;

…A:6†

Order e2 : o2x0 f/j…2† ; wj…2† g ‡ o2y f/j…2† ; wj…2† g ˆ 2 …1† oy /…2† j ‡ m cos kw x0 oy /j

mkw sin kw x0 ox0 /j…1† ˆ 0

ox0 wj…2† ‡ m cos kw x0 ox0 oy wj…1† ot0 g…2† kox0 wj…2†

oy /…2† j ˆ

…1† 2ox0 ox1 f/…1† j ; wj g; j

at y ˆ … 1† hj ;

ot1 g…1†

g…1† o2y wj…1† gk

…A:9†

at y ˆ 0;

kox1 wj…1† ‡ g…1† ox0 oy wj…1† ‡ ox0 g…1† oy wj…1†

…2† …1† klj oy w…2† j k ˆ klj fox0 g ox0 wj

…A:8†

at y ˆ … 1†j hj ;

mkw sin kw x0 oy w…1† j ˆ 0

ox0 g…1† ox0 /j…1† ‡ g…1† o2y /j…1†

Hj ox0 g…2† k ˆ

…A:7†

at y ˆ 0;

…A:10† Hj ox1 g…1† k

at y ˆ 0;

…A:11† …A:12†

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

kqj ot0 /j…2† ‡ gqj g…2† ‡ g…1† oy ot0 /j…1† g 2T ox1 ox0 g…1†

2

2 …2† lj Hj oy w…2† ˆk j k ‡ T ox0 g 2 lj f0:5…oy w…1† j †

2243

qj fot1 /j…1† ‡ 0:5…ox0 /j…1† † ‡ 0:5…oy /…1† j †

2 0:5…ox0 w…1† j †

Hj g…1† o2y wj…1† ‡ 2Hj ox0 g…1† ox0 wj…1†

2

Hj2 …ox0 g…1† †2 gk

at y ˆ 0:

…A:13†

Order e3 : o2x0 f/j…3† ; wj…3† g ‡ o2y f/j…3† ; wj…3† g ˆ

…1† 2ox0 ox2 f/…1† j ; wj g

o2x1 f/j…1† ; wj…1† g

2ox0 ox1 f/j…2† ; w…2† j g;

…A:14†

…2† …1† 2 …2† 2 2 3 …1† oy /…3† j ‡ m cos kw x0 oy /j ‡ 0:5m cos kw x0 oy /j ‡ mkw sin kw x0 ox0 /j ‡ mkw sin kw x0 ox1 /j

‡ 0:5m2 kw sin 2kw x0 oy ox0 /…1† j ˆ 0

at y ˆ … 1†j hj ;

…1† …2† ox0 wj…3† ‡ ox1 w…2† j ‡ ox2 wj ‡ m cos kw x0 ox0 oy wj

‡ m cos kw x0 ox1 oy wj…1† oy /…3† j ˆ

ot0 g…3†

ox0 g…2† ox0 /j…1† 2

Hj ox0 g…3† k ˆ

ox0 g…1† ox0 /…2† j ot1 g…2†

mkw sin kw x0 oy wj…2†

j

at y ˆ … 1† hj ;

ox1 g…1† ox0 /j…1†

ot2 g…1†

…A:16†

…2† 2 …1† ox0 g…1† ox1 /…1† j ‡ g oy /j

g…1† ox0 g…1† ox0 oy /j…1†

at y ˆ 0;

2

Hj ox1 g…2† ‡ 0:5…g…1† † ox0 o2y w…1† j

Hj ox2 g…1†

at y ˆ 0;

…A:18†

…1† …1† …1† …2† …1† …1† …2† …1† klj oy w…3† y k ˆ klj fox1 g ox0 wj ‡ ox0 g ox1 wj ‡ ox0 g ox0 wj ‡ ox0 g ox0 wj 2

g…1† o2y wj…2† ‡ 0:5…g…1† † o3y wj…1† ‡ g…1† ox0 g…1† ox0 oy wj…1† gk kqj ot0 /j…3† ‡ gqj g…3†

…A:17†

kox1 wj…2† ‡ ox2 wj…1† ‡ g…2† ox0 oy wj…1† ‡ g…1† ox0 oy wj…2† ‡ g…1† ox1 oy wj…1†

‡ ox0 g…2† oy wj…1† ‡ ox0 g…1† oy wj…2† ‡ ox1 g…1† oy wj…1† ‡ g…1† ox0 g…1† o2y wj…1† k

0:5m2 cos2 kw x0 ox0 o2y w…1† j

0:5m2 kw sin 2kw x0 o2y wj…1† ˆ 0

‡ g…1† o2y /j…2† ‡ 0:5…g…1† † o3y /j…1† kox0 wj…3†

…A:15†

li Hj oy wj…3† k ‡ T o2x0 g…3† ˆ k

g…2† o2y w…1† j

at y ˆ 0;

…A:19†

…2† …2† …1† …2† …1† qj fot2 /…1† j ‡ ot1 /j ‡ ox0 /j ox0 /j ‡ ox0 /j ox1 /j

…2† …1† …1† 2 …1† …1† ‡ g…1† oy ot1 /j…1† ‡ g…1† oy ot0 /j…2† ‡ g…2† oy ot0 /j…1† ‡ 0:5…g…1† †2 o2y ot0 /…1† j ‡ oy /j oy /j ‡ g oy /j oy /j …1† ‡ g…1† ox0 /…1† j oy ox0 /j g

‡ g…1† oy wj…1† o2y w…1† j

lj f ox0 wj…2† ox0 w…1† j

2ox0 g…1† ox0 wj…1† oy wj…1†

…1† ox0 w…1† j ox1 wj

…1† g…1† ox0 wj…1† oy ox0 wj…1† ‡ oy w…2† j oy wj

…1† 2 3 …1† Hj ‰g…2† o2y wj…1† ‡ g…1† o2y w…2† j ‡ 0:5…g † oy wj Š 2

…2† …1† …1† …1† …1† …1† …1† …1† …1† ‡ 2Hj ‰ox0 g…1† ox1 wj…1† ‡ ox0 g…2† ox0 w…1† j ‡ ox0 g ox0 wj ‡ ox1 g ox0 wj ‡ …ox0 g † oy wj ‡ g ox0 g ox0 oy wj Š

2Hj2 ‰ox0 g…1† ox0 g…2† ‡ ox0 g…1† ox1 g…1† Šgk

T ‰ox1 ox0 g…2† ‡ ox2 ox0 g…1† ‡ o2x1 g…1† Š at y ˆ 0:

…A:20†

Appendix B The constants of Eqs. (22)±(24) are:

k1 ˆ x

2 …kw

k†…kw 2k 2

2k†

…kw k†2 qj … 2k 3 "

qj hj coth khj cosech2 khj ‡

(

l l2 H 2 …kw 2k†

2 1 12 1 ‡ qj cosech khj ‡ …kw

2k 2 2v31

k† kl1 …h2

j

1† cosech2 khj

h1 † sinh kh1 sinh kh2 cosh kh1 cosh kh2

2244

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

k ‡ …l21 h2 l2

# l22 h1 † cosh2

2

kh1 cosh kh2 ‡ kl1 …h1

h2 † cosh kh1 cosh kh2 ‡ l1 k 2 …h1 cosh2 kh2

k2 2 …l h1 cosh2 kh2 l21 h2 cosh2 kh1 † cosh kh1 cosh kh2 l2 2 ) v1 k 2 2 h2 † cosh kh1 cosh kh2 …kw k 1†…l1 cosh kh1 l2 cosh kh2 † ; l2

h2 cosh2 kh1 † sinh kh1 sinh kh2 ‡ l1 k 2 …h1

k2 ˆ x

2 …1

2kw2 † … 2k

qj cosech2 khj 2Tk ( 

1†j qj coth khj cosech2 khj 

K21 …kw2 ‡ kkw ‡ k† K22 …kw2 kkw ‡ k† ‡ kw ‡ 2k kw 2k





k 2 l1 l212 H12 2v31

kl1 l2 sinh kh1 sinh kh2 …cosh2 kh1 ‡ cosh2 kh2 † ‡ k cosh kh1 cosh kh2 …l21 cosh2 kh1 ‡ l22 cosh2 kh2 †

) " # v1 K21 K22 2 2 ‡ …l cosh kh1 l2 cosh kh2 † 2kl1 cosh kh1 cosh kh2 ; Tkw 1 kw ‡ 2k kw 2k

2

…1 2k 2 †

k 2 l l2 H 2 2 q cosech kh K …k ‡ kk ‡ k† j

21 w j j 2 1 12 1 2 w w … 1† qj coth khj cosech khj k~2 ˆ x

2k kw ‡ 2k 2Tk 2v31   kl1 l2 sinh kh1 sinh kh2 …cosh2 kh1 ‡ cosh2 kh2 † ‡ k cosh kh1 cosh kh2 …l21 cosh2 kh1 ‡ l22 cosh2 kh2 †  v1 K21 2 2 …l1 cosh kh1 l2 cosh kh2 † 2kl1 cosh kh1 cosh kh2 ; Tkw …kw ‡ 2k†

2 2

k 2 l l2 H 2 2

K q cosech kh k …k 2k † k …k 2k† 22 j

w w j j 2 1 12 1 w … 1† q coth kh cosech kh k3 ˆ x2

j j j

4k 2 2Tk…kw 2k† 4v31   kl1 l2 sinh kh1 sinh kh2 …cosh2 kh1 : ‡ cosh2 kh2 † ‡ k cosh kh1 cosh kh2 …l21 cosh2 kh1 ‡ l22 cosh2 kh2 †  v1 K22 2kl1 cosh kh1 cosh kh2 ; …l1 cosh2 kh1 l2 cosh2 kh2 † Tkw …kw 2k† 2

j

k4 ˆ x2 k k… 1† qj

j

4k… 1† qj coth 2khj coth2 khj

kqj coth2 khj …K11

K12 ‡ 3K31 ‡ 3K32 †  l1 2 2 ‡ qj …K11 ‡ K12 ‡ 3K31 ‡ 3K32 † 2qj coth 2khj coth khj K32 k ‡ k l12 H1 bc k 2 l 2 v1 1 2 l1 l12 2l1 l12 c1 cosh kh1 cosh kh2 cosh kh1 cosh kh2 …cosh 2kh2 cosh 2kh1 †…K31 ‡ K32 † kl2 v1 l2 v1 v2 2l l 4l ‡ 1 12 …sinh kh2 sinh 2kh2 cosh kh1 sinh kh1 sinh 2kh1 cosh kh2 †K32 ‡ 1 …sinh 2kh1 l2 v1 v2 l2 v2 6kl1 l312 kl21 l12 ‡ sinh 2kh2 †K31 cosh2 kh1 cosh2 kh2 cosh 2kh1 cosh kh2 b1 b2 cosh kh1 cosh kh2 2 l2 v1 v2 v21 v2 2kl1 l12 ‡ cosh kh1 cosh kh2 …l1 cosh kh1 cosh 2kh1 sinh kh2 sinh 2kh2 l2 v21 v2 2kl2 l l2 cosh kh2 cosh 2kh2 sinh kh1 sinh 2kh1 † ‡ 21 12 b21 sinh 2kh1 sinh 2kh2 v1 v2  l1 49kl1 l12 ‡ …K11 7K12 ‡ K31 7K32 † ‡ cosh kh1 cosh kh2 ; l2 2l2 v1

K. Zakaria / Chaos, Solitons and Fractals 12 (2001) 2227±2245

2245

where c1 ˆ

2k 2 l1 l2 kl 2k b1 b2 ‡ 12 cosh kh1 cosh kh2 …K11 ‡ K12 ‡ K31 ‡ K32 † ‡ …K31 ‡ K32 †k cosh 2khj k v2 v1 v2 v1 ‡

c2 ˆ

6k 2 l212 cosh kh1 cosh kh2 cosh 2kh1 cosh 2kh2 v1 v2

k2 fl l v b …K11 ‡ 3K12 ‡ K31 ‡ 3K32 † v1 v2 1 2 2 1

4:5k 2 ;

j

v1 …2K31 ‡ K32 †k… 1† lj sinh 2khj k

‡ 2kl1 l2 l12 ‰5b2 cosh kh1 cosh kh2 ‡ b1 sinh 2kh1 sinh 2kh2 Šg; j

b1 ˆ k… 1† cosh khj‡1 sinh khj k; j

b2 ˆ k… 1† cosh 2khj‡1 sinh 2khj k:

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