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20320130406011 2

  1. 1. International Journal of Advanced JOURNAL OF ADVANCED RESEARCH ISSN 0976 – INTERNATIONAL Research in Engineering and Technology (IJARET), IN 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 4, Issue 7, November - December 2013, pp. 92-100 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET ©IAEME SELF-GRAVITATING ELECTRODYNAMIC STABILITY OF ACCELERATING STREAMING FLUID CYLINDERS IN SELF-GRAVITATING TENUOUS MEDIUM Hussain E. Hussain1,2 and Hossam A. Ghany1,3 1 Mathematics Department, Faculty of Science, Taif University, Hawia(888) Taif, Saudi Arabia 2 Mathematics Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt 3 Mathematics Department, Faculty of Industrial Education, Helwan University, Cairo, Egypt ABSTRACT The stability of electrodynamic self-gravitational of accelerating streaming fluid cylinder surrounded by tenuous medium has been studied for all axisymmetric and non-axisymmetric perturbation modes. The dispersion relation is derived, discussed and some reported works are recovered as limiting cases from it. The electro-dynamic forces interior and exterior the fluid cylinder are destabilizing. The self-gravitating forces are destabilizing for small axisymmetric domain while they are stabilizing in all other domains. The accelerating (periodic) streaming has a stabilizing tendency. The self-gravitating electrodynamic stable domains are enlarged due to the resonance of the periodic streaming and could overcome the instability character of the model, if it is so strong. 1. INTRODUCTION The electrodynamic stability of cylindrical fluids has been the attention of several scientists (see Melcher et. al. (1971), and Nayyar et. al. (1970)) since the pioneering works of Kelly (1965). The effect of the electrodynamic force on the capillary force and other parameters has been examined by Radwan (1992), and Mestel (1994) & (1996). In the last decades of 20th century interests have been devoted to the self-gravitating instability of cylindrical models for their crucial applications. Chandrasekhar and Fermi (1953) were the first to discuss the stability of the full fluid cylinder under the influence of the self-gravitating force and did write about its importance with the oscillation of the spiral arm of galaxy, for more studies see also Chandrasekhar (1981), Radwan (2007) and Radwan & Hussain(2009). Hasan (2011) has discussed the stability of oscillating streaming fluid cylinder subject to combined effect of the capillary, self gravitating and electrodynamic forces for all axisymmetric and non axisymmetric perturbation modes. He (2012) studied the magnetodynamic stability of a fluid jet pervaded by 92
  2. 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME transverse varying magnetic field while its surrounding tenuous medium is penetrated by uniform magnetic field. In the present work we investigate the electrodynamic self-gravitational stability of oscillating streaming fluid cylinder surrounded by self-gravitating tenuous medium of very slow motion. 2. FORMULATION OF THE PROBLEM We consider a dielectric fluid cylinder of uniform cross-section of (radius R0 ) permittivity ε (1) (2) surrounded by tenuous medium of negligible motion of permittivity ε . The fluid model is assumed to be incompressible, self-gravitating, inviscid and pervaded by uniform electric field (1),(2) E0 = (0,0, E0 ) (1) where E0 is the intensity of the electric field in the fluid. The fluid streams with periodic streaming at time t u 0 = (0,0,U cos Ωt ) (2) where U is streaming velocity at time t = 0 while Ω is the oscillation frequency of the velocity. (1),(2) The components of E 0 and u 0 are considered along the cylindrical coordinates (r ,ϕ , z ) with the z -axis coinciding with the axis of the fluid cylinder. The cylindrical fluid model is acted by the combined effect of the pressure gradient, electrodynamic and self-gravitating forces. The required basic equations are the combination of the electro-dynamic Maxwell equations, ordinary hydro-dynamic equations and Newtonian equation concerning the self-gravitating matter. For the model under consideration, these equations are given as follows: 1  ∂u  (1) (1) + ( u.∇ ) u  = −∇p + ρ∇V (1) + ∇ε (1) E .E 2  ∂t  ( ρ ∇.u = 0 ∇ ∧ (ε E ) ∇.E , (1),(2) (1),(2) ) =0 (3) (4), (5) =0 (6) ∇ 2V (1) = −4π G ρ (1) (7) ∇2V (2) = 0 (8) This system of equations is solved in the unperturbed state with (1),(2) E0 u 0 = (0,0,U ) , = (0,0, E0 ) , and after applying the required boundary conditions at r = R0 , we get V0(1) = −π Gρ r 2 (9) 93
  3. 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME R  V0(2) = 2π G ρ R02 ln  0  − π G ρ R02  r  (10) z Ucos t r 94
  4. 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME 3. PERTURBATION STATE For small departures from the unperturbed state, every physical quantity be expressed as % Q(r ,ϕ , z; t ) = Q0 (r ) + η (t )Q1 (r ,ϕ , z ) + L Q(r ,ϕ , z; t ) may (11) Q stands for u, p,V (1) ,V (2) , E (1) , E (2) with Q0 (r ) is the value of Q in the unperturbed state with Q1 is small increment of Q due to perturbation. Based on the expansion (11), the where deformation in the cylindrical interface is given by % r = R0 + η (12) % η = η (t )exp ( i ( kz + mϕ ) ) (13) with is the elevation of the surface wave measured from the unperturbed level. By using the expansion (11) in the fundamental equations (3)-(8), the relevant perturbation equations are given as follows (1) (1) ε (1)  ∂u  ∇ ( 2E 0 .E 1 ) ρ  1 + (u 0 .∇ )u 1  = −∇p1 + ρ∇V 1 + 2  ∂t  (1),( 2 ) ∇.u1 = 0 ∇ ∧ ( ε E1 ) ∇.E1 , (1),(2) =0 (14) (15), (16) =0 (17) (1) ∇ 2V1 = 0 (18) ∇2V1(2) = 0 (19) Since the fluid is irrotational, we have u 1 = −∇φ1 (20) By inserting this in (16), the velocity potential φ1 satisfies ∇ 2φ1 = 0 (21) and based on the expansion (11)-(13), the solution of (21) give 95
  5. 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME (1) φ1 = A B (t )I m (kr )exp i ( kz + mϕ )    (22) with u1r = ∂η + ikUη cos Ωt ∂t (23) From equation (14), we get ∂u  ∂u 1r + U cos Ωt 1r ∂z  ∂t ρ ( ∂Π1  =− ∂r  (1) (1) π 1 = p1 − ρV1(1) − ε (1) E 0 .E1 (24) ) (25) So ρ ∂  ∂η ∂Π   ∂η  + ikU η cos Ωt  + ikU ρ  + ikU η cos Ωt  = − 1  ∂t  ∂t ∂r   ∂t  Taking into account ∇2Π1 = 0 , so, we have Π1 = C 1 (t )η (t )I m (kr )exp i ( kz + mϕ )    (26) Hence ∂ 2η ∂η ρ 2 + 2i ρ kU cos Ωt − i ρ kUη sin Ωt − ρ k 2U 2 cos 2 Ωt ∂t ∂t ′ = −C1η kI m ( x) (27) and Π1 = − ρ I m (x )  ∂ 2η ∂η  2 2 2  2 + 2ikU cos Ωt  − ikU η sin Ωt − k U cos Ωt (28) ′ k I m (x )  ∂t ∂t  (1),(2) Now, upon using equation (17) concerning E1 (1),(2) , we have E1 = ∇ψ 1(1),(2) so, ∇2ψ 1(1),(2) = 0 , therefore (1) ψ 1 = C2 (t ) I m (kr )exp i ( kz + mϕ )    (30) 96
  6. 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME (2) ψ 1 = C3 (t ) K m ( kr ) exp i ( kz + mϕ )    (31) We have to apply the appropriate boundary conditions at r = R0 to get C2 (t ) and C3 (t ) :which are ψ1 =ψ 2 and n0 .ε (1) (1) (1) (2) (2) E1 + n1.ε (1) E 0 = n0 .ε (2) E1 + n1.ε (2) E1 from which, we get  K ( x)  C2 (t ) = C3 (t )  m ,  I m ( x)  C3 (t ) = x = kR0 (32) −iE0 ( ε (1) − ε (2) ) I m ( x) (ε (1) ′ ′ K m ( x) I m ( x ) − ε (2) K m ( x) I m ( x ) ) (33) Therefore, E (1) 1  −(ik )(−iE0 ) ( ε (1) − ε (2) ) K m ( x )  = ∇  (1) I m ( kr )exp ( i (kz + mϕ ) )  (34) (2) ′ ′  (ε K m ( x ) I m ( x) − ε K m ( x) I m ( x ) )    E (2) 1  −(ik )(−iE0 ) ( ε (1) − ε (2) ) I m ( x)  = ∇  (1) K m ( kr ) exp ( i ( kz + mϕ ) )  (35) (2) ′ ′  (ε K m ( x ) I m ( x) − ε K m ( x ) I m ( x) )    By solving equations (18) and (19), we get V1(1) = B1 (t ) I m (kr )exp i(kz + mϕ ) (36) V1(2) = B2 (t ) Km (kr )exp i(kz + mϕ ) (37) The coefficients B1 (t ) and B2 (t ) are determined upon using the boundary conditions:     at r = R0    ∂V0(1) ∂V (2) = V1(2) + η 0 ∂r ∂r (1) 2 (1) (2) ∂V1 ∂ V0 ∂V1 ∂ 2V0(2) +η = +η ∂r ∂r 2 ∂r ∂r 2 V1(1) + η 97 (38)
  7. 7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME and then upon utilizing the Wronskian relation ( cf.Abramowitz and Stegun) ′ ′ I m ( x) Km ( x) − I m ( x) Km ( x) = − x −1 (39) we get, V1(1) = 4π Gρ R0η Km ( x) I m (kr )exp i(kz + mϕ ) (40) V1(2) = 4π Gρ R0η I m ( x) Km (kr )exp i(kz + mϕ ) (41) Now, we have to apply the continuity of the normal component of the total stress tensor at r = R0 : ∂ ε (1) ε (2) (1) (1) (2) (2) (1) Π 0 + ρV 0(1) ) + 2E 0 .E 1 = 2E 0 .E 1 ( ∂r 2 2 with noting that Π 0 = const . ( Π1 + ρV 1(1) + η ) ( ) (42) We finally obtain, the second order integro-differential equation ∂ 2η ∂η = −2ikU cos Ωt + ikU Ωη sin Ωt + k 2U 2η cos Ωt 2 ∂t ∂t xI ′ ( x )  1 I m ( x ) K m ( x ) − η + 4π G ρ m I m ( x)  2  (43) 2  ε (2)  ′ x 1 − (1)  I m ( x ) K m ( x )  ε  2 ε (1) E02 − ρ R02   ε (2) ′ ( x) K m ( x ) − (1) K m ( x) I m ( x)  ′  Im ε   η 4. DISCUSSION Equation (43) is the second order integro-differential equation in the amplitude η (t ) of the perturbation. It relates η with the modified Bessel functions I m ( x) and K m ( x) , the wave numbers x and m , the permittivity ε (1) and ε (2) of the fluid and tenuous medium and with the parameters  ε E2  ρ , R0 , E0 ,V and Ω of the problem. It contains  02   ρ R0  (time)-1. If we assume that η yields − 1 2 as well as ( 4π G ρ ) − 1 2 each of unit exp(σ t ) where σ is the growth rate of the perturbation, equation (43), 98
  8. 8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME σ 2 = ( −2ikU cos Ωt )σ + ikU Ω sin Ωt + k 2U 2 cos Ωt + 4π G ρ ′ xI m ( x )  1  I m ( x) K m ( x) − 2  I m ( x)   (44) 2  ε (2)  ′ x 1 − (1)  I m ( x) K m ( x)  ε  2 ε (1) E02 − ρ R02   ε (2) ′ ′ I m ( x ) K m ( x) − (1) K m ( x ) I m ( x )   ε   As t = 0 and the fluid streams uniformly u 0 = (0,0,U ) equation (43) reduces to (σ + ikU ) 2 = 4π G ρ ′ xI m ( x)  1  I m ( x) K m ( x) − 2  I m ( x)   2  ε (2)  ′ x 1 − (1)  I m ( x) K m ( x)  ε  2 (1) − 2 0 2 0 ε E ρR (45) ε (2) ′ ′ I m ( x) K m ( x ) − (1) K m ( x ) I m ( x) ε The relation (45), as U = 0 is derived by Radwan (1992) in examining the effect of the electric field on the self-gravitating fluid cylinder. As u 0 = 0 and E 0 = 0 , the relation (45) gives the self-gravitating dispersion relation of a fluid cylinder surrounded by self-gravitational tenuous medium, viz., σ 2 = 4π G ρ ′ xI m ( x)  1 I m ( x) K m ( x) −  I m ( x)  2  (46) This relation has been early derived by Chandrasekhar (1981). We know that (cf. Abramowitz and Stegun (1970)) ′ ′ I m ( x) > 0, K m ( x) > 0, I m ( x) > 0, and K m ( x) < 0 (47) In view of the inequalities (47), the analytical and numerical discussions of the relation (46) reveal to the following results. When m = 0 : σ2 σ2 > 0 as 0 < x < 1.0667 while ≤ 0 as 1.0667 ≤ x < ∞ . 4πρG 4πρG When m ≥ 0 : we have 0 < x < ∞ . 99
  9. 9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME This means that the model is self-gravitating unstable in the axisymmetric perturbation m = 0 in the domain 0 < x < 1.0667 . While it is stable in the neighboring axisymmetric domain 1.0667 ≤ x < ∞ and in the non-axisymmetric domain 0 < x < ∞ . Therefore, we conclude that the fluid cylinder is self-gravitating unstable in small axisymmetric domain and stable otherwise. As U = 0 and we neglect the self-gravitating force effect, the relation (45), reduces to 2  ε (2)  ′ x 1 − (1)  I m ( x) K m ( x)  ε  2 ε (1) E02 σ = ρ R02 ε (2) ′ ′ K ( x) I m ( x) − I m ( x ) K m ( x ) ε (1) m 2 (48) The discussions of the relation (48), in view of the inequalities (47) show that the electrodynamic forces acting on the cylindrical fluid and the surrounding tenuous medium have strong stabilizing influence. As G = 0 and E0 = 0 , the general relation (43), shows that the periodic oscillating stream has stabilizing tendency. However, the uniform stream is destabilizing. Therefore, in our present case the electrodynamic self-gravitational stable domains are enlarged and could suppress the unstable domains and consequently the model will be completely stable as the oscillating stream of the fluid is so strong. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Abramowitz, M. and Stegun, I. "Handbook of Mathematical Functions", (Dover Publ., New York, 1970). Chandrasekhar, S. and Fermi, E., Astrophys. J., 116 (1953), 118. Chandrasekhar, S. "Hydrodynamic and Hydromagnetic Stability", (Dover Publ., New York, 1981). Kelly, R. E., J. Fluid Mech., 22 (1965), 547. Melcher, J. R. Warren, E. P., J. Fluid Mech., 47, (1971), 127. Mestel, A. J., J. Fluid Mech. 274 (1994) 93 and 312 (1996) 311. Mohammed, A. A. and Nayyar, A. K., J. Phys. A, 3(1970) 296. Radwan, A. E., Acta Phys. Polonica A, 82 (1992) 451. Radwan, A. E., Phys. Scrpt., 76 (2007)510. Radwan, A. E. and Hussain, E.H., Int.J. Maths. & Comput., 3(2009) 91. Hasan, A. A., Journal of Physica B, 406(2), 234(2011). Hasan A. A., Boundary Value Problems, (31), 1(2011). Hasan A. A., Journal of Applied Mechanics Transactions ASME, 79(2), 1(2011). Hasan A. A., Mathematical Problems in Engineering, 2012, 1(2012). M. M.Izam, E. K.Makama and M.S.Ojo, “The Gravitational Potential of a Diatomic System”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 2, 2013, pp. 206 - 209, ISSN Print: 0976-6480, ISSN Online: 0976-6499. Hany L. S. Ibrahim and Elsayed Esam M. Khaled, “Light Scattering from a Cluster Consists of Different Axisymmetric Objects”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 6, 2013, pp. 203 - 215, ISSN Print: 0976-6480, ISSN Online: 0976-6499. 100

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