Nonaxisymmetrtc Instability of a Streaming

NONAXISYMMETRTC INSTABILITY OF A STREAMING ANNULAR FLUID JET SURROUNDING A TAR CYLINDER UNDER GENERAL TENUOUS VARYING MAGNETIC FIELDS

Ahmed E. Radwan, Sahar M. Abdel-Gaied Mathematics Department, Faculty of Science, Ain-Shams University,Cairo,Egypt Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt

Abstract:-The magnetodynamic stability of streaming annular jet surrounding a tar mantle under inertia and electromagnetic forces with varying magnetic fields, is developed. A general eigenvalue relation is derived and discussed. The axial interior and exterior fields have strong stabilizing influences for symmetric m = 0 and asymmetric m  0 modes. The azimuthal varying field is purely destabilizing for m = 0 but in the m  0 it is stabilizing or destabilizing according to restrictions. The streaming has a strong destabilizing influence in all modes for all wavelengths. Its influence increases the MFD unstable domains and decreases those of stability. As the tenuous azimuthal magnetic field influence is superior to those of axial fields, the MFD unstable domains are increasing with increasing q (the tar cylinder radius normalized with respect to that of the fluid) values and vice versa. If the unperturbed fluid velocity is smaller than the Alfven wave velocity, the model destabilizing character is suppressed and stability arises.

1. INTRODUCTION The instability of a full fluid jet in the found that the model is only unstable hydrodynamic and hydromagnetic for small asymmetric disturbances and versions has been documented and that the area under the instability mastered through numerous curves are decreasing with increasing q investigations [1] to [3]. That is due to values, but the capillary instability will its practical applications in several never be suppressed. Recently Radwan domains such as fuel atomization, [10] examined the influence of a spinning of synthetic fibers, the spray constant magnetic field on the forgoing drying, inkjet printer … etc. study [8]. It is found that the magnetic The stability of an annular fluid jet is field is stabilizing and could suppress also interesting to be discussed. For its the capillary instability. crucial applications in the astrophysics, The eneavours of the present Kendall [4] has recently explained and work is to investigate the magneto- made very neat experiments with dynamic instability and oscillation of a modern equipments to deduce and streaming annular fluid jet surrounding study the capillary stability of that a tar mantle under general varying model which is a gas jet coaxial with a tenuous magnetic fields for all possible liquid jet. Radwan has discussed the (symmetric and asymmetric) modes of stability of a viscous [5] and a perturbations. streaming [7] hollow jet. The capillary For the importance of the instability of an annular fluid jet annular fluid jet stability discussions, around a solid axis has been elaborated in general, in several domains of byRadwan [8]on utilizing the applications we may refer to Kendall lagrangian principle of energy. He [8] [4].

-1- Here u and p are the fluid velocity vector and kinetic pressure,  is the 3. FORMULATION OF THE coefficient of the magnetic PROBLEM permeability and H is the magnetic consider a circular fluid jet of field intensity. Equations (3) and (5) uniform density  , radius a and are the conservation of the magnetic streams with uiform velocity (0, 0, U); fluxies interior and exterior the annular coaxial and concentric with a tar jet. Equation (4) is that of evolution cylinder of radius q a where 0  q  1 . of the magnetic field and equation (2) This model is ambient with is the equation of continuity for a tenuous medium of negligible inertia. incompressible fluid. The fluid is assumed to be non- The unperturbed state is studied and as viscous, incompressible and perfectly a result conducting. The model is penetrated by       p  H . H   0 the magnetic field 0, 0, Ho  and  o    o o   (7)   2   embedded in the general tenuous Since the acting forces in the initial varying magnetic field state would be balanced at r = a, the Hs 0, H a r -1, H H o    o  o  where o is unperturbed fluid pressure is being the intensity of the magnetic field   H 2   o  2 2 interior the annular jet and  ,  are po       - 1 (8)  2  parameters satisfy certain restrictions with the restrictions (see equation 9). The components of 2 2 the streaming velocity and the     1 (9) magnetic fields are taken along where the equality occurs as a limiting utilized cylindrical polar coordinates case of zero fluid pressure. (r,  , z) -system with the z-axis 3. PERTURBATION ANALYSIS The relevant perturbation coinciding with the axis of the annular equations derived from equations (1) to jet. The model is acting upon the (6) due to small departures from the inertia and electromagnetic forces. initial state are accomplished by The basic equations to be used substituting the expansions here are the combination of the Q( r ,  , z ; t)  Q (r)   Q (r ,  , z ; t) ordinary fluid dynamic equations with o o 1 those of Maxwell concern the and retaining first order terms only n electromagnetic theory. For the small fluctuating variable problem at hand these equations are Q1 (r ,  , z ; t) . Here  o is the given by in the fluid amplitude of the perturbation at time t   u  =0 while Q stands for     u .  u  -  p     H   H (1) s   t  u  u r , u , u z  , p, H , H and the  . u  0 (2) radial distance of the fluid jet. For a single Fourier term, the latter is  . H  0 (3) described by

r  a   o  (11)  H     u  H  (4) with  t   exp i (k z  m  - k c t )  (12) in the tenuous Here  is the elevation of the surface s  . H  0 (5) wave measured from the unperturbed   Hs  0 (6) position. k (any real number) is the longitudinal wave-number, m (an

-2- integer) is the azimuthal wavenumber s s s H1   1 , 1  C K m (k r) and k c is the oscillation frequency at exp i (k z  m  - k c t ) time t. In magnetodynamic (22) approximation, the relevant where A, B and C are arbitrary perturbation equations are given as : constants of integrations, Im (k r) and in the fluid K m (k r) are modified Bessel functions   u  u  of the first and second kind of order m;   1  U 1     t  z  and er , e , ez are three unit vectors in (r, , z) -  p1     H1   H o    H o   H1  the cylindrical . (13) 4. EIGENVALUE RELATION  . u  0 (14) The solution of the perturbation 1 equation (13) to (18) represented by  . H  0 1 (15) (19) to (22) must satisfy appropriate  H1 conditions. Under the present    u1  H1     u o  H1   t circumstances these boundary (16) conditions are the following : in the tenuous (i) The kinematic boundary conditoin s  . H1  0 (17) states that the normal component of the s velocity vector u (i. e. N . u ) must be   H1  0 (18) Consequent to the deformation compatible with the velocity of the (11) and based the (, z ;t) (fluid-tenuous) surface across the interface (11) at r = a. This reads dependence (cf. equation (12)) and  r according to the linear perturbation N o . u1  N1 . u o  at r  a (23) technique, as usual for the stability  t problems of cylindrical models, every with perturbed quantity can be expressed as N  N o   o N1  (1, 0, 0)  exp i (k z  m  - k c t times an  i m  (24)  o 0 , - , - i k  amplitude function of r. Thence the  a  linearized system (13) to (18) is is the outward unit vector normal to the simplified and solved. Apart from the perturbed interface (11). Therefore singular solutions we have obtained A Im (k a)  B Km (k a)  i (U - c) (25)  H o  (ii) The normal component of the H1    u1r e r  u1 e  u1z e z   (U - c)  velocity N . u must vanish at r=qa. (19) This condition yields p1  i k  (c - U)  A I m (k r)  B K m (k r)  A Im (q k a)  B Km (q k a)  0 exp i (k z  m  - k c t) (20) (26) Equations (25) and (26) give u1r  k A Im (k r)  B Km (k r) i (U - c) Km (y) exp i (k z  m  - k c t) A  m (27) L i m xy u1  k A I m (k r)  B K m (k r) i (U - c) Im (y) r B  m (28) L exp i (k z  m  - k c t) xy with u  i k A I (k r)  B K (k r) 1z m m Lm  Im (x) K (y) - I (y) K (x) (29) exp i (k z  m  - k c t) xy m m m where x ( = k a) and y ( = q x) are the (21) dimensionless longitudinal wave- and numbers.

-3- (iii) The normal component of the geometric factor q which is the radius magnetic field must be continuous of the tar cylinder normalized with across the fluid-tenuous interface (11) respect to a, the compound functions m m at r = a; Fm (x, y) , L xy and L y and with the N o . H1  N1 . H o  parameters  , U ,  , a and Ho of the s N o . H1  N1 . H o  at r  a problem. One has to refer to the fact (30) that the fundamental natural quantity 1 from which we get 2   H 2  (m    x)  o  has a unit of time, so the C  i H  2  o (31)   a  x Km (x) (iv) The normal component of the total dimensionless growth rate and stress tensor must also be continuous oscillation frequency are given by   across the disturbed interface (11) at r=a 1 1 2 2 2 and 2 2 2 resp  p o      H o .H o    Ho  a   Ho  a  p1      2 H o . H1     r  2    r  ectively. s s (32) Before we are going to discuss     s s  H .H   2 H . H    o o   at r  a the eigenvalue relation (33) in its    0 1   2    r  general form, we intend to involve This condition, taking into account the ourselves in some limiting cases.If we solutions in the present section and in let q tends to zero i.e. y  0 , in such the preceding sections, degenerates to a case the model of a streaming full following eigenvalue relation fluid jet ambient with the tenuous medium pervaded by a general varying  H 2 k 2 (U - c) 2  o magnetic field and acting upon it the  a 2 intertia and electromagnetic forces. The corresponding eigenvalues relation  2  2 2 K m (x)   x    m    x Fm (x, y) can be obtained from (33) by utilizing      x K m (x)   the asymptotic behaviour of the (33) cylindrical functions appearing in it. with As q tends to zero, using the m asymptotic behaviour of modified L xy Fm (x, y)  x m , x  k a , y  q x (34) Bessel functions L y lim Im (y)  0 m q  0 (37) Lxy  Im (x) Km (y) - Im (y) Km (x) (35) lim Km (y)   m q  0 (38) L y  Im (x) Km (y) - Im (y) K m (x) (36) we obtain 2 2 2  5. DISCUSSIONS AND k (U - c) 2  x I m (x) 2  x - -  Ho CONCLUSION I m (x)  a 2  The desired magnetodynamic I (x) K (x) dispersion relation (33) relates the (m   x) 2 m m oscillation frequency   k c or I m (x) Km (x) rather the temporal amplification (39)   - i k c with the longitudinal In absence of the streaming character, wavenumber x and y normalized with the relation (39) recovers our recent respect to the scale length a of the result [8] if we neglect the contribution problem, the azimuthal wavenumber of the surface tension there. For detail m, the parameters  ,  of the discussions we may refer to the tenuous varying magnetic field, the

-4- analytical and numerical analysis of model is stable for all  and q  1 ref. [8]. values for all (short and long) If we consider a very thin fluid wavelength. This is physically layer surrounding a very thick tar plausible because the longitudinal mantle i.e. we let q  1 from below, uniform fields give a measure of the associated eigenvalue relation can rigidity to the conducting fluid and that be identified from (33) by using the of course the main basic reason for the asymptotic behaviour of modified stabilizing character. Bessel functions and their derivatives (ii) If the model is very thin cylindrical as q  1 . Using a series development fluid shells having very thick tar mantle and penetrated by the uniform for I , I , K , K around the value q m m m m axial field H e and ambient with the x and neglecting terms of second order o z of (1 - q), the eigenvalue relation (33) azimuthal tenuous varying magnetic in the rotationally axisymmetric mode s    a H o   field Ho     e  then degenerates to   r   In such a case the temporal 2 2   0 . k (U - c) 2 amplification ( - ikc) is depleted as q 2  x   Ho tends to unity from below depending  a 2  on the value of  . For q = 0.90 we  x k  x 2 I K - IK  (1- q)   2  2 0  1 1 1 1 get high stability for small values of  K  I K - K I  0   0 1 0 1   but high instability for large values (40) of  . It is found also that for q = where the arguments of all modified Bessel functions are x. In view of the 0.99, the model is stable for all values well-known Wronskian relation of  and the stability becomes slower and slower as  is increased.  WI m (x) , K m (x)  I m (x) K m (x) - This is verified since in the last case (q -1 2 Im (x) K m (x)  - x = 0.99) we find  (1- q)  1 and if (41)  2 (1- q) equals unity we must take and the recurrence relations terms in the relation (44) of order I (x)  I (x) 2 0 1 (42) (1- q) or even more. It is worthwhile K0 (x)  - K1 (x) (43) to mention here that the increasing the equation (4) reduces to thickness of the solid mantle leads to k (U - c)2 the increasing of the stability with  x 2  2 respect to the full fluid jet for the  Ho  a 2  corresponding values of  =1.0,

  2 2 x K 0 (x)  20.0.  1 - (1 - q)   -   (iii) The third subclass is the general   K1 (x)  case in which  0 ,   0 and the (44) we have here the following three model is a very this fluid shells different subclasses. surrounding a very thick tar axis. This (i) If the model is very thin cylindrical model is stable for all  ,  and fluid shells and surrounding a very x  0 values if the restriction 1 thick mantle and pervaded by and 2     x K 0 (x)  embedded with axial magnetic fields      is satisfied as U  a  K (x) then   0 . The discussion of (44) in  1  = 0. this case shows, as U = 0, that the

-5- Form the discussions of the In view of these results, following above mentioned foregoing three cases inequalities (note that y is less than x) : and their results, we may conclude that the instability states and those of Im (x)  Im (y) (47) stability are depending on  ,  and q   values whether the deformation is K m (x)  K m (y) (48) axisymmetric or / and asymmetric. In contrast to the same problem acting upon it through capillary force only as Im (x) Km (y)  Im (y) Km (x) (49) we have obtained [6] where it is found that the considered system is unstable are satisfied for every non-zero real only for small wavenumbers in the values of x and y. therefore the axisymmetric mode m = 0. Moreover m m compound I and L are being the area under the instability curves are xy y m found to be decreasing with increasing Lxy  0 (50) q values (in the range 0  q  1 ) but m the instability is never be suppressed L y  0 (51) whatever is the largest value of q. for every non-zero real values of x and Now, let us return to the y for all possible (m = 0 and m  0 ) general eigenvalue relation (33) which modes of perturbation. From the is valid to all possible (m = 0 and foregoing discussions we may deduce m  0 ) modes of perturbation. that The influence of the longitudinal magnetic field pervaded Fm (x, y)  0 (52) interior the annular jet is represented and never changes sign for every non-   H 2  zero real values of x and y and for 2  o  by the term x following  2  . every integer value of m.   a  By the use of inequality (52) and the It has a strong stabilizing effect for all  K (x)  (short and long) wavelengths whether m fact that   is always negative the perturbation is non-axisymmetric  Km (x)  m  0 or / and axisymmetric one m = (see (46)) : the analytical discussions 0 and that is independent of q values. of the relation (33) show that the In order to identify the tenuous longitudinal tenuous magnetic field is varying magnetic field influence we strongly stabilizing for all q, x and m have to determine the behaviour of the values. The azimuthal tenuous varying different cylindrical functions magnetic field is strongly destabilizing occurring in the relation (33). As is in the symmetric mode m = 0 which in well known, for every non-zero real the asymmetric m  0 , it is values of x that Im (x) is positive and stabilizing if the restrictions monotonic increasing while K m (x) is m K m (x)  x Km (x) (53) monotonic decreasing but never are satisfied and vice versa. Therefore negative. Using that together with the the azimuthal tenuous magnetic field is recurrence relations destabilizing or stabilizing according to

2 Im (x)  Im-1 (x)  Im1 (x) (45) some restrictions. The general tenuous varying magnetic 2 Km (x)  - K m-1(x) - K m1(x) (46) field is absolutely destabilizing if m    x  0 . The latter restriction is one can show that Im (x) is always satisfied if the coefficient of the positive while Km (x) is never positive.

-6- perturbed tenuous magnetic field instability of an annular jet having a tar vanishes, see equation (31). Also the mantle becomes more worse as fluid is destabilizing influence of the azimuthal uniformly streaming. field is largest, when the perturbed 1. order to verify and confirm tenuous field is orthogonal to the the foregoing analytical results : to s clarify the influence of q and unperturbed basic tenuous field Ho at   the boundary surface r = a. This can be   * - i k shown as follows : U  1  on the 2 From the obtained solution we have   Ho 2     a 2   established, apart from the singular magnetodynamic instability of the solutions and the value of C, that annular jet, the eigenvalue relation (33) Hs  0 ,  a H r -1 ,  H o  o o  has been analysed numerically. s H1  C  K m (k r) exp i (k z  m  - i k c t)  The criterion (33) has been from which the direction numbers at calculated in computer in order to r=a of Hs are (0,  , ) and those of obtain the values of o  (  - ikc) or  ( k c) s s in the natural H1 are H1r , m , x . Therefore, we 1 2 2 s   Ho  have Ho .H1  is proportional to unit  , corresponding to the   a 2  m    x , which equals zero if   instability states or those of stability Hs is orthogonal to Hs . o 1 respectively, for various values of x. To sum up the non-streaming This is carried out for different values jet having a tar mantle pervaded by of (,  ) and q for numerous values uniform field and ambient which * general varying tenuous magnetic field of U . The numerical data are is not absolutely stable not only in the collected and tabulated and presented symmetric mode m = 0 but also in the graphically, there are many features in asymmetric modes m  0 of these numerical presentations. perturbations. In contrast to the same (I) In the rotationally axisymmetric problem subject to surface tension only mode m = 0 [6], it is found that the model is The numerical data are absolutely stable in all asymmetric presented graphically for different modes m  0 while it is unstable to pairs of values of (,  ) = (1, 1), (2, small wavenumbers in the 1), (2, 2) and (10, 1). For every pair axisymmetric disturbance m = 0. (,  ) values, several values of U* From the analytical discussions are considered and at the same time of the eigenvalue relation (33), it is numerous fluid cylindrical shells (of found that the streaming has a strong different thickness) surrounding the tar destabilizing influence for all q values mantle are assumed. The values of for all (short and long wavelengths U* are taken to be 0.2, 0.4 and 0.7 whether the perturbation is symmetric while the geometric factor q (the ratio or / and asymmetric modes. This is between the radii of the solid cylinder also true even for other cylindrical and the fluid jet) values are 0.1, 0.5, configurations and other acting forces, 0.9 and 0.99. from these numerical see refs. [7] and [9]. Therefore, the data we deduce the following : streaming has the influence of     1 enlarging the instability regions but As and 2 for a * decreasing the stable domains. This given set values of U , the unstable means that the magneto-dynamic domain are found to be decreasing with

-7- increasing q values while the stable The numerical data are domains are extensively increasing. presented numerically for different This means that as the fluid cylindrical values of (,  ) as (,  ) =(1, 1), shell is thin and surrounding a very (1, 2) and (5, 5). Then for every pair of thick tar mantle, it is more these values numerous values of U* mangetodynamic stable than the other and also of q are considered. Similar cases as long as    whatever is discussions can be carried out here for the value of U* . the present case m  ∓ 1 as in the case As    for a given values of of m = 0. However it is remarkable that U* , it is found that the domains of there are very little domains of instability are increasing with instabilities while those of stability are increasing q values. Therefore, a very very wide and very high, in particular thin shell surrounding a very thick tar for very thin cylindrical fluid shells mantle is magnetodynamic less stable surrounding very thick tar mantles. The domains of instability, for than the other cases as long as    the same values of  and  , are * whatever is the value of U . We quickly decreased with increasing q predict that the motion is not slowly values. While they are increasing with and experimentally the observations increasing U* values. This means that are not easy in comparison with the q has a stabilizing influence while U* case of MHD full fluid jet (as q = 0). is strongly destabilizing. This confirms Moreover, from the above the analytical discussions. discussions we may conclude also that As    , the domains of the Lorentz force is not purely instabilities are much larger than those   0 stabilizing, in particular as , of the cases in which    . This is but it is stabilizing or destabilizing physically plausible because the according to some restrictions. This is tenuous azimuthal varying magnetic due to the fact that the azimuthal field is destabilizing whether the tenuous magnetic field is not uniform disturbance modes are axisymmetric and consequently due to the influence (sausage) m = 0 or / and non- of that field the magnetodynamic axisymmetric modes m  0 . unstable domains are never be suppressed. REFERENCES For given value of q and (,  ) , it is found that the streaming 1. G.Bateman,Magnetohydrodynami has a strong destabilizing influence. c, Instabilities. The M. I. T. Press, This is very clear from the numerical 1978. representation where it is found that 2. S.Chandrasekhar,Hydrodynamic the unstable domains are increasing and Hydromagnetic Stability, while those of stable are decreasing Dover, New York, 1981. with increasing U* values. This result 3. P. G. Drazin and W. H. Reid, is true whether the components of the Hydrodynamic Stability. s tenuous magnetic field Ho are equal Cambridge Univ. Press, London, or not and also whether the cylindrical 1981. shells are thick or thin. 4. J. M. Kendall, Experiments on the Capillary Instability of an Annular (II) In the non-axisymmetric disturbances m  ∓ 1 Liquid Jet, Phys. Fluids 29,1986, 2086.

-8- 5. A. E. Radwan and S. S. Elazab, Simon Stevin, 61,1987, 293. 6. A. E. Radwan Indian Pure & Appl. Maths. 19, 1988, 1105. 7. A. E. Radwan, Instability of a Hollow Jet with Effects of Surface Tension and Fluid Inertia, J. Phys. Soc. Japan, 58 ,1989, 1225. 8. A. E. Radwan and S. S. Elazab, Nuovo Cimento, 117, 2002, 257. 9. A. E. Radwan J. Appl. Maths. & Comput. 147 ,2003, 191. Fig. 3 The numerical results for m = 0.0, 10. A. E. Radwan, Hydromagnetic (,   (1,1)) ,and U*= 0.7 Stability of a Streaming Annular Liquid Jet Having a Solid Rod as a Mantle Under Longitudinal Magnetic Fields, Simon Stevin. 64, 1990, 365. 11. J. W. Rayleigh, The Theory of Sound, Dover, New York, 1 and 2, 1945.

Fig. 1 The numerical results for m = ,   (1,1) 0.0, , and U*= 0.2

Fig. 2 The numerical results for m = 0.0, , (,   (1,1)) and U*= 0.4