On Temporal Instability of Electrically Forced Axisymmetric Jets with Variable Applied Field and Nonzero Basic State Velocity

Transcription

1 Availale at Appl. Appl. Math. ISSN: Vol. 6, Issue 1 (June 11) pp. 7 (Previously, Vol. 6, Issue 11, pp ) Applications and Applied Mathematics: An International Journal (AAM) On Temporal Instaility of lectrically Forced Axisymmetric Jets with Variale Applied Field and Nonzero Basic State Velocity amaru Bhatta, Sayantan as and aniel N. Riahi epartment of Mathematics The University of Texas-Pan American dinurg, Texas USA hattad@utpa.edu, sdas@roncs.utpa.edu, Riahid@aol.com Received: June 8, 1; Accepted: Feruary 18, 11 Astract The prolem of instaility of electrically forced axisymmetric jets with respect to temporally growing disturances is investigated computationally. We derive a dispersion relation ased on the relevant approximated versions of the equations of the electro-hydrodynamics for an electrically forced jet flow. For temporal instaility, we find in the realistic cases of the non-zero asic state velocity that the growth rate of the unstale mode is unaffected y the value of the asic state velocity. However, the non-zero value of the asic state velocity affects significantly the period of the unstale mode in the sense that it decreases the period, and the rate of increase of the frequency with respect to the axial wave numer increases with the asic state velocity. It is also oserved from numerical investigations that there are two modes of instaility for small values of the wavenumer. Keywords: Axisymmetric, electrospinning, electric field, jet flow, temporal instaility MSC 1 No.: 5Q5, 5Q6 1. Introduction Our main contriution to the present wor has een to implement a numerical procedure to determine computational results aout the effects of non-zero asic flow velocity in 7

2 8 amaru Bhatta et al. the electrically driven jet flow system. Temporal instaility of a cylindrical jet of fluid with a static charge density in the presence of an external constant as well as variale electric field is considered in this paper. The investigation of electrically forced jets is gaining importance in applications such as those of electrospraying Baily (1988) and electrospinning Hohman et al. (1a, 1). lectrospraying uses electric field to produce and control sprays of very small drops that are uniform in size. lectrospinning process uses electric fields to produce and control thin, uniform, high quality fiers. In the asence of electrical effects, it had een oserved that temporal growing disturances can destailize the free shear flows which include the jet flows razin and Reid (1981). Since then several authors Hohman et al. (1a, 1), Saville (1971), Reneer et al. (), Shadov and Shutov (1) and Fridrih et al. () have done theoretical studies on temporal instaility of the electrically forced jets in the presence of electrical effects. Hohman et al. (1a, 1) developed a theoretical understanding of temporal instailities for an electrically forced jet with a static charge density. The equations for the dependent variales of the disturances were ased on the long wavelength and asymptotic approximations of the governing electro-hydrodynamic equations. They found that the dominance of the instailities depends on the surface charge density and the radius of the jet. Saville (1971) studied interactions etween electrical tractions at the interface of an electrically driven liquid jet and the linear temporal instaility phenomena. It was found, in particular, that when viscous effects are small, sufficiently small strength of the electric fields tends to decrease the growth rate of a temporally growing axisymmetric mode. However, when viscous effects predominate, then the only unstale disturance is that due to the axisymmetric mode regardless of the magnitude of the field s strength. Other investigations of electrically driven jets with applications in electrospinning of nanofier are reported in Yarin et al. (1), Sun et al. (), Li and Xia () and Yu et al. (). Spatial instaility of axisymmetric electrically forced jets with variale applied field under idealistic conditions of zero or infinite electrical conductivity was studied analytically y Riahi (9). He reported two spatial modes of instaility each of which was enhanced with increasing the strength of the externally applied electric field. In this paper, we follow an approach similar to that of Hohman et al. (1a) to otain a mathematical model for the electrically driven jets. We consider the prolem of instaility of electrically forced axisymmetric jets with respect to temporally growing disturances. We derive a dispersion relation ased on the relevant approximated versions of the equations of the electrohydrodynamics for an electrically forced jet flow. The approximations include the assumptions that the length scale along the axial direction of the jet is much larger than that in the radial direction of the jet and the disturances are axisymmetric and infinitesimal in amplitude. Indeed it is an extension of the wor of Homan et al. (1a) in the sense that our model also incorporates non-zero asic velocity and non-zero surface charge density. We then determine the dispersion relation, etween the growth rate of the spatially growing disturances and the wave numer in the axial direction, the frequency and the non-dimensional parameters of the model. We found a numer of interesting results. In particular, the growth rate of the temporally growing disturances is found to e independent of the asic state velocity, while the frequency and equivalently the period of the growing disturances are found to depend notaly on the asic state velocity. It is also oserved from numerical investigations that there are two modes of instaility for small values of the wavenumer.

3 AAM: Intern. J., Vol. 6, Issue 1 (June 11) [Previously, Vol. 6, Issue 11, pp ] 9. Formulation The present theoretical investigation for the mathematical modeling of the electrically driven jets is ased on the original governing electro-hydrodynamic equations Melcher and Taylor (1969) for the mass conservation, momentum, charge conservation and the electric potential. The system is given y u, t u P u q t q t K,, (1a) (1) (1c) and, (1d) where u is the total derivative with is used to denote dot product. Here u is the t t velocity vector, P is the pressure, is the electric field vector, is the electric potential, q is the free charge density, is the fluid density, is the dynamic viscosity, K is electric conductivity and t is the time variale. The geometry we use is shown in Figure 1. Figure 1. lectrospinning model The internal pressure in the jet can e found y taing into consideration the alances across the free oundary of the jet etween the pressure, viscous forces, capillary forces and the electric

4 amaru Bhatta et al. energy density plus the radial self-repulsion of the free charges on the free oundary Melcher and Taylor (1969). Assuming the amient air to e motionless and passive, this yield the following expression for the pressure P in the jet ~ ) / 8 / ~ / P ~, () ( where is the surface tension, is twice the mean curvature of the interface, /() is the permittivity constant in the jet, ~ / is the permittivity constant in the air and is the surface free charge. Following the previous investigation Hohman et al. (1a), we consider a cylindrical fluid jet moving axially. The fluid of air is considered as the external fluid, and the internal fluid of jet is assumed to e Newtonian and incompressile. We use the governing equations (1) in the cylindrical coordinate system with origin at the center of nozzle exit section, where the jet flow is emitted with axial z-axis along the axis of the jet. We consider the axisymmetric form of the dependent variales in the sense that the azimuthal velocity is zero and there are no variations of the dependent variales with respect to the azimuthal variale. Following approximations carried out in Hohman et al. (1a) for a long and slender jet in the axial direction, we consider length scale in the axial direction to e large in comparison to that in the radial direction and use a perturation expansion in the small jet aspect ratio. We expand the dependent variales in a Taylor series in the radial variale r. Then such expansions are used in the full axisymmetric system and eep only the leading terms. These lead to relatively simple equations for the dependent variales as functions of t and z only. Following the method of approach in Hohman et al. (1a), we employ (1d) and Coulom s integral equation to arrive at an equation for the electric field, which is essentially as the one derived y Hohman et al. (1a) and will not e repeated here. quations presented in (1) are the general governing equations applicale to general cases for electrically forced jet systems, and for this generality aspect no oundary conditions in needed to e provided. quations descried y () are simple modeling one dimensional linear equations admitting periodic solutions for perturations in the axial direction. As is nown from experimental and theoretical wor y Hohman et al. (1a, 1), such periodic form in axial direction is reasonale, provided the axial domain should e away from the jet inlet section, which is assumed in our paper. We non-dimensionalize the resulting equations using r (radius of the cross sectional area of the nozzle exit at z=), r ~, r ~ t r, and as scales for length, electric t r field, time, velocity and surface charge, respectively. The resulting non-dimensional equations are then t z h h v, 1 h hv h K, t z z (a) ()

5 AAM: Intern. J., Vol. 6, Issue 1 (June 11) [Previously, Vol. 6, Issue 11, pp ] 1 v v v t z z h h 1 z h z h 8 1 z v h, h h z z (c) and ( z) ln( ) h h, z z (d) where v is the axial velocity, h(z, t) is the radius of the jet cross-section at the axial location z, (z, t) is the surface charge, (z, t) is the electric field, the conductivity K is assumed to e a function of z in the form K = K K(z), where K is a constant dimensional conductivity and K(z) is a non-dimensional variale function. Also, non-dimensional conductivity is given y K K r ~ and ~ 1. Also r is the non-dimensional viscosity parameter, (z) is an applied electric field and 1 is the local aspect ratio, which is assumed to e small. Next, we determine the electrostatic equilirium solution, which is referred to here as the asic state solution, to the equations (a-d). The asic state solutions for the dependent variales, which are designated with a suscript, are given elow h 1, v v (1-z), (a-d),, where oth v, and are constant quantities, and = 8 /() is assumed to e a small parameter ( << 1), under which the asic state solutions given y (a-d) were found to satisfy the modeling equations Riahi (9). represents a composite parameter proportional to the asic state surface charge. It is assumed that is sufficiently small so that to leading terms eqns () do not contain z-dependent coefficients and so the present method of approach can e implemented. Here, is referred to as the acground free charge density. We consider each dependent variale as sum of its asic state solution plus a small perturation, which is assumed to e oscillatory in time and in axial variale. Thus, we write hv h v h v,,,,,,,,,, (5a) where the perturation quantities, designated y the suscript 1, are given y

6 amaru Bhatta et al ,,,,,,. h v h v e t iz (5) Here, v h ~ ~, ~,, ~ are constants which are assumed to e small, i is the imaginary unit, is the complex growth rate, and is the axial wave numer. We need four equations for the variales: v h ~ ~, ~,, ~. Using ()-(5) in (), we linearize with respect to the amplitude of perturation, consider a series expansion in powers of for all the dependent variale and only retain the lowest leading order terms, and then divide each equation y the exponential function exp [t+iz]. We then otain linear algeraic equations for the unnown constants v h ~ ~, ~,, ~. To otain non-trivial (non-zero) values of these constants, the x determinant of the coefficients of these unnowns must e zero, which yields the following dispersion relation: 1 T T T (6) where K iv T 1 (7) K T K iv iv 8 6 (8) 1 1 i K T K v i iv K v (9) with, ), ln(, To derive and compute our results for, * K we divided each term y K * and then we set 1/K * >. Thus the form of the eqn for, * K will contain only K * terms of eqn (6), other terms will vanish. As, * K the dispersion relation is given y

7 AAM: Intern. J., Vol. 6, Issue 1 (June 11) [Previously, Vol. 6, Issue 11, pp ] 1 1 iv i i v v. (1). Results and iscussion The dispersion relation (6) which presents the temporal ehavior of the system is investigated for several parameters. For all our computational purpose, we use Hohman et al. (1A). Our aim here is to present the positive real part and the imaginary part of the solution of the equation (6), which is called, respectively, the growth rate and the frequency of the unstale mode, and these contriutes to our understanding of the temporal instaility. From our computational results, it is oserved that the positive real part of is independent of the asic state velocity v, only the imaginary part depends on v. All the parameters we choose yielded negative imaginary part for nonzero asic state velocity. We use JMSL lirary to compute the complex zeros of the (6) whose coefficients are complex. Method Compute Zeros of the class Zero Polynomial of JMSL is used. Figure through present results for constant applied field and for various values of. Here, we consider four values.,.97, 1.9 and.9. Results in Figure are for infinite conductivity case, i.e., K. Others are parameters chosen as v 1,,. Figure. Positive real part of as a function with K,, and v v 1 for various ). ( As can e seen from the Figure, the instaility is reduced with increasing the magnitude of the applied field. The results indicate presence of the electrically analog of the so-called Rayleigh mode of instaility, razin and Reid (1981). The results presented in the Figure are also in

8 amaru Bhatta et al. qualitative agreement with those reported in Hohman et al. (1) for a perfect conducting fluid case and zero asic state velocity. Results for zero conductivity, i.e., ( K ), and K 19. cases are presented in Figure and Figure, respectively, eeping other parameters same as in the Figure. Figure. Positive real part of as a function with K,, and v 1 for various As in the case of perfect conducting fluid, the results for the perfect dielectric fluid cases shown in the Figure indicate that the flow instaility is reduced with increasing the magnitude of the applied field, which is again a property of the Rayleigh type mode of jet instaility. The results presented in the Figure for a finite conducting case are qualitatively similar to those presented in the Figures - and indicate those results for Rayleigh type mode of instaility. Figure. Positive real part of as a function with K 19.,, and v 1 for various

9 AAM: Intern. J., Vol. 6, Issue 1 (June 11) [Previously, Vol. 6, Issue 11, pp ] 5 Figure 5 presents some results for the constant applied field and compares the positive real solutions for three different conductivity cases, namely, zero, infinite and finite (19.) for.9,, and v 1. It is noticed that the results are closed to each other for K 19., and K. It is seen from the Figure 5 the staility effect due to conductivity of the fluid. Figure 5. Positive real part of as a function with.9,, and v 1 for various K The effect of is shown in Figure 6 for zero and finite conductivities. Other parameters used are.9, and v 1. Again the case of constant applied field is considered here. It can e seen from the results presented in the Figure 6 that oth viscosity and conductivity reduce the instaility of the unstale mode. Figure 6. Positive real solutions of as a function with and v 1 for various and K

10 6 amaru Bhatta et al. The effect of surface charge is presented next. Figure 7 displays the effect of for zero conductivity with.9, and v 1. It is seen from the results shown in the Figure 7 that surface charge density enhances the instaility of the unstale mode for the axial wave numer not too close to zero; while the opposite is true if the wave numer is sufficiently small. Figure 7. Positive real solutions of as a function with K,,.9 and v 1 for various Figure 8 presents results for variale applied field, finite and nonzero viscosity and conductivity and for different values of the strength of the applied field. It can e seen from the results shown in the Figure 8 that the instaility mode favors intermediate values of the axial wave numer (not too close to zero or one values). Figure 8. Positive real solutions of as a function with K 19.,.,.1 and v 1 for various

11 AAM: Intern. J., Vol. 6, Issue 1 (June 11) [Previously, Vol. 6, Issue 11, pp ] 7 It is oserved from numerical investigations that there are two modes of instaility for small values of. The primary mode dominates the secondary mode. The secondary mode exists only for small values of. The secondary mode is also independent of asic state velocity, i.e., real part of does not depend on the asic state velocity which is the case for primary mode also. The secondary modes are presented in Figure 9 with K 19.,,.1, v 1 for various. A comparison of these two modes are shown in Figure 1 for K 19.,,.1, v 1 and. 9. For very small values of, the secondary mode exists whereas for larger, this mode does not exist. For larger values of, only one mode (namely, primary mode) exists. Figure 9. Secondary mode: positive real solutions of as a function with K 19.,,.1, and v 1 for various Figure 1. Primary and Second modes with K 19.,,.1, v 1 and. 9

12 8 amaru Bhatta et al. Figure 11 through Figure 1 presents the imaginary part of the frequency of the unstale mode versus the axial wave numer for zero conductivity and finite conductivity. Other parameters varied are viscosity and asic state velocity. It can e seen from the Figure 11 that the period of the unstale mode is smaller for larger value of the asic state velocity and decreases with increasing the axial wave numer of the unstale mode. Also, rate of increase of the frequency of the unstale mode with respect to the axial wave numer increases with the asic state velocity. Figure 11. Imaginary part of whose real part is positive with K,,.1,. 9 It can e seen from the Figure 1 that the imaginary part of of the unstale mode for finite conductivity and zero viscosity case has sudden change in magnitude when is close to.68. Figure 1. Imaginary part of whose real part is positive with K 19.,.,.1,. 9

13 AAM: Intern. J., Vol. 6, Issue 1 (June 11) [Previously, Vol. 6, Issue 11, pp ] 9 Figure 1 presents the results for imaginary part of with K 19.,.,.1, and.9 for various asic state velocities. These is no unstale mode for approximately igger than.5. Figure 1. Imaginary part of whose real part is positive with K 19.,.,.1,. 9. Conclusions We conclude that in the realistic cases of the non-zero asic state velocity, the growth rate of the unstale mode is unaffected y the value of the asic state velocity. However, the non-zero value of the asic state velocity affects significantly the period of the unstale mode in the sense that it decreases the period, and the rate of increase of the frequency with respect to the axial wave numer increases with the asic state velocity. In all the cases that we investigated we found that the presence of the variale applied field is destailizing, while the finite values of either viscosity or conductivity are stailizing. It is also noticed for the zero conductivity case that the imaginary part of is zero if asic state velocity is zero and the imaginary part of is nonzero if asic state velocity is nonzero. It is also oserved from numerical investigations that there are two modes of instaility for small values of the wavenumer. The primary mode dominates the secondary mode. The secondary mode exists only for small values of. The secondary mode also independent of asic state velocity, i.e., real part of does not depend on the asic state velocity which is the case for primary mode also. Acnowledgement Authors would lie to than the referees for their constructive comments.

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