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  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME 1 STABILITY OF A GAS CYLINDER IN A COMPRESSIBLE LIQUID Hussain E. Hussain1,2 1 Mathematics Department, Faculty of Science, Taif University, Hawia(888) Taif, Saudi Arabia 2 Mathematics Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt ABSTRACT The stability of a gas cylinder in a compressible liquid endowed with surface tension has been studied. The dispersion relation is derived and discussed for all axisymmetric and non-axisymmetric perturbations. The capillary force is destabilizing only in small domain of axisymmetric mode. While it is capillary stabilizing in the rest domains. The compressibility has a strong destabilizing tendency and causes collapsing the model. 1. INTRODUCTION The stability criterion of a gas jet pervaded into an incompressible liquid endowed with surface tension for axisymmetric perturbation is indicated for first time by Chandrasekhar (1981). See also Drazin and Reid (1980) (p.16) where the inertia force of the liquid is paramount over that of the gas. Cheng (1985) studied the instability of a streaming gas jet in an incompressible liquid for all axisymmetric and non-axisymmetric modes of perturbation. However, we have to mention here that the results given by Cheng (1985), in Eqs.(4) and (5) are incorrect in the third term. In fact the quantity (1-m2 -k2 R2 0) must be in the numerator as it is clear from Eq. (3) there. See also equation (30) in the present work and Drazin’s result (1980) p.16 and also Chandrasekhar’s dispersion relation P.538 and P.540 [Eqs. (147) and (155) there]. Radwan and Elazab (1987) examined the viscosity effect on the capillary instability of this model for axisymmetric perturbation. In (1989) Radwan identified the stabilizing effect of the magnetic field on the stability of this model, for other topics see Radwan (2005). The stability of different cylindrical models under the action of self gravitating force in addition to other forces has been elaborated by Radwan and Hasan (2008) and (2009). They (2008) studied the gravitational stability of a fluid cylinder under transverse time-dependent electric field for axisymmetric perturbations. 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 INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2014): 7.8273 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME 2 axisymmetric and non axisymmetric perturbation modes. Hasan (2011) studied the instability of a full fluid cylinder surrounded by self-gravitating tenuous medium pervaded by transverse varying electric field under the combined effect of the capillary, self-gravitating and electric forces for all modes of perturbations. He (2012) discussed the instability of a full fluid cylinder surrounded by selfgravitating tenuous medium pervaded by transverse varying electric field under the combined effect of the capillary, self-gravitating and electric forces for all modes of perturbations. He (2012) studied the magnetodynamic stability of a fluid jet pervaded by transverse varying magnetic field while its surrounding tenuous medium is penetrated by uniform magnetic field. In all foregoing works it is assumed that the fluid moves such that the divergence of the flow fluid vanishes. Here we, will not consider this behavior i.e. the velocity of the liquid is not solenoid anymore, investigate the capillary instability of a gas cylinder embedded into (real) compressible liquid. This phenomenon may occur in the geological drillings as a gas escapes from below oil layers in the crust. For astrophysical applications, we may refer to the experimental work of Kendall (1986). 2. FORMULATION OF THE PROBLEM Consider a gas cylinder of radius R0 pervaded into a compressible liquid.Assuming that the inertia force of the liquid is paramount over that of the gas. The basic equations are being: 0)()( =∇⋅+⋅∇+ ∂ ∂ ρρ ρ uu t (1) r P u z u r u r u t u rzr r ∂ ∂ −=            ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ϕ ρ ϕ (2) ϕϕ ρ ϕ ϕϕ ∂ ∂ −=            ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ P r u z u r u r u t u zr 1 (3) z P u z u r u r u t u zzr z ∂ ∂ −=            ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ϕ ρ ϕ (4) )( uPT z u r u r u t T C zrv ⋅∇−=            ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ϕ ρ ϕ (5) TRP c ρ= (6) )( 1 2 1 1 −− += rrSPs , Nrr ⋅∇=+ −− 1 2 1 1 (7), (8) f f N ∇ ∇ = , 0),,,( =tzrf ϕ (9), (10) Here ρ, u(=(ur,uϕ, uz)) and P are the polytropic liquid density, velocity vector and kinetic pressure. T is the temperature, Rc polytropic gas constant, Cv the specific heat at constant volume,
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME 3 S the surface tension coefficient and N the outward unit vector normal to the gas-liquid interface pointed as r does where (r, ϕ, z) are the cylindrical coordinates with the z-axis coinciding with the axis of the gas-liquid model. r1 and r2 are the principle radii of curvature of the gas-liquid interface. 3. PERTURBATION ANALYSIS For small departures from the initial state, every perturbed quantity χ(r,ϕ, z, t) may be expressed as χ(r,ϕ, z, t) = χ0 + ε(t) χ1 (r, ϕ, z) + … (11) Here χ(r, ϕ, z, t) stands for u, ρ, P, N and the radial distance of the gas cylinder. ε(t) is the amplitude of the perturbation ε(t) = ε0eσt (12) where ε0 (=ε at t=0) is the initial amplitude and σ is the growth rate. Based on the perturbation technique Q1 (r, ϕ, z) = Q1 (r) exp [i(kz + mϕ)] (13) where k is the longitudinal wavenumber and m the azimuthal wavenumber. The radial distance of the gas cylinder due to the perturbation is given by: R = R0 + ε0R1 (14) with R1 ≈ exp (i(kz + mϕ) + σt) (15) where ε0R1 is the elevation of the surface wave measured from the unperturbed position. By inserting the expansion (11) into the basic equations (1) – (10) we get two systems of partial differential equations, the unperturbed system and the perturbed one. By solving the unperturbed system of equations with [u0 = (0, 0, 0)], we obtain       −= 0 00 R S PP g (16) where g P0 is the gas constant pressure. If )( 00 RSPg < , the kinetic pressure 0P of the liquid will be negative and the model collapses. The perturbed system of equations is given by: 1 1 0 P t u −∇=      ∂ ∂ ρ , 1 2 1 ρaP = (17), (18) ( ) 01 1 =⋅∇+ ∂ ∂ u t oρ ρ (19)
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME 4       ∂ ∂ + ∂ ∂ += 2 1 2 2 2 1 2 121 z R R R R R S P o o s ϕ (20) where )( 00 ργPa = is the speed of sound in the liquid By solving equations (17)-(19), we get       −=⋅∇ 2 0 1 1 )( a P u ρ σ (21) and 110 Pu −∇=σρ (22) Combining equations (21) and (22), yields 02 1 2 1 2 =−∇ a P P σ (23) This leads, on using the space dependence (13), to 0 1 1 2 2 2 1 =      +−      P r m dr dP r dr d r ζ (24) With 2 2 22 a k σ ζ += (25) The non-singular solution of equation (24) is given by ( )[ ]ϕζ mkzirAKP m += exp)(1 (26) where Km (ζr) is the second kind of modified Bessel function of order m while A is constant of integration could be determined upon using appropriate boundary condition at r = R0: viz., t R u r ∂ ∂ = 1 1 at 0Rr = (27) From which we get )( 00 2 yyK R A m ι ρσ −= (28) where y (= ζR0) is the dimensionless longitudinal wavenumber of a compressible liquid. Also equation (20) yields
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME 5 ( ) 1 22 2 0 1 1 Rxm R S Ps −−= (29) where x(=kR0) is the ordinary dimensionless longitudinal wavenumber. By applying the balance of the pressure across the gas-liquid interface at r=R0, following dispersion relation is obtained ( ) )( )( 1 22 3 2 yK yyK xm R S m m oo ι ρ σ −− − = (30) 4. DISCUSSIONS Equation (30) is the required dispersion relation of a gas cylinder embedded into a compressible liquid endowed with surface tension. It relates the growth rate σ with the longitudinal dimensionless wavenumbers x and y, the azimuthal wavenumber m, the modified Bessel function Km(y) of second kind of order m and its derivative and with other parameters S, ρ0 and R0 of the problem. As a → ∞ we have y → x and the liquid is incompressible here, we have ( ) )( )( 1 22 3 2 xK xxK xm R S m m oo ι ρ σ −− − = (31) This relation coincides with that given by Drazin and Reid (1980) p. 16, and Radwan result (1989) as we neglect the magnetic field effect there. As a → ∞ we have y → x and suppose m = 0, we get ( ) )( )( 1 0 12 3 2 xK xxK x R S oo −= ρ σ (32) where )()( 10 xKxK −=ι . This relation coincides with that given by Chandrasekhar (1981). In discussing equations (31) and (32), it is found that               ≠∀≠< <<=> === ∞<<=< 000 100,0 1,0,0 1,0,0 )( 3 00 2 xasm xm xm xmas RS K ρ σ This means that the gas-liquid model as the liquid is incompressible is capillary unstable only for m = 0 as 0 < x < 1 while it is stable for (m = 0 as 1 < x < ∞) and (m≠0, ∀ x ≠0). These results are confirmed numerically (cf. Chandrasekhar (1981) for m=0 and Radwan (1989) and (2005) for m ≥ 0).
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME 6 Here the effect of the capillary instability on the present model could be identified via discussing equation (30) numerically in its general form. However, we see as above that the most dangerous mode of instability is m=0. In the present study since the argument of ι mK and K`m in equation (30) (is being y(=ζR0), the longitudinal wavenumber of the compressible liquid with ( )2 0 2222 Raxy σ+= ) includes σ2 (see equation (25)), it is cumbersome to identify the stability conditions and states using the well-known standard approaches. So, we have used the numerical iterative technique as m=0 for different values of a(= 0/ Ra )=1.0, 1.5, 2.0, 5.0 and 10.0. Due to the discussing results we may see (fig.1) that the unstable domains are decreasing vertically with increasing a-values while they are the same 0< x <1 horizontally. Keep in mind that the parameter "a" in the denominator of equation (25), we conclude that the compressibility has strong destabilizing tendency on the present model and causes collapsing it. It is worthwhile to mention here that through the numerical calculations, we may have )( 3 0 2 RS ρ σ > 0 or )( 3 0 2 RS ρ σ =0 or )( 3 0 2 RS ρ σ < 0 i.e. we have the cases 3 0RS ρ σ is real or zero or imaginary. As 3 0RS ρ σ is real, then the area under the curves σ/√S/(ρR3 0) (see equation (15) for time dependence) represents the ordinary unstable domains. As 3 0RS ρ σ is imaginary we have to write σ = iω (i=√-1, imaginary factor) with ω/2π is the oscillation frequency, then the area under the curves (ω/√S/ρR3 0) is being the ordinary stable domains. As 3 0RS ρ σ =0, we have neutral (marginal) stability states, from which one obtain the critical values of x(=xc) which separates stability domains from those of instability: here in our problem xc=1. REFERENCES [1] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability “(Dover Publ., New York) 1981. [2] L.Y. Cheng, Phys. Fluids, 28 (1985) 2614. [3] P.G. Drazin and W. Reid, Hydrodynamic Stability, (Cambridge University Press, London) 1980, pp 16. [4] J.M. Kendall, Phys. Fluids, 29 (1986) 2086. [5] A.E. Radwan and S.S. Elazab, Simon Stevin, 61 (1987)29. [6] A.E. Radwan, J. Phys. Soc. Japan, 58 (1989) 1225. [7] A.E. Radwan, Mechanics and Mechanical Engineering, 8 (2005)127. [8] Mohammed, A. A. and Nayyar, A. K., J. Phys. A, 3(1970) 296. [9] Radwan, A. E., Acta Phys. Polonica A, 82 (1992) 451. [10] Radwan, A. E., Phys. Scrpt., 76 (2007)510. [11] Radwan, A. E. and Hussain, E.H., Int.J. Maths. & Comput., 3(2009) 91. [12] Radwan AE and Hasan AA, Magnetohydrodynamic stability of selfgravitational fluid [13] cylinder, Appl. Mathal Modlling, 33(4)(2009) 2121. [14] Hasan, A. A., Journal of Physica B, 406(2), 234(2011). [15] Hasan A. A., Boundary Value Problems, (31), 1(2011).
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME 7 [16] Hasan A. A., Journal of Applied Mechanics Transactions ASME, 79(2), 1(2011). [17] Hasan A. A., Mathematical Problems in Engineering, 2012, 1(2012). [18] 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. [19] 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. [20] Hussain E. Hussain and Hossam A. Ghany, “Self-Gravitating Electrodynamic Stability of Accelerating Streaming Fluid Cylinders in Self-Gravitating Tenuous Medium”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 7, 2013, pp. 92 - 100, ISSN Print: 0976-6480, ISSN Online: 0976-6499.