Asymptotic analysis of compound, hollow, annular liquid jets

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1 Asymptotic analysis of compound, hollow, annular liquid jets J.I. Ramos E.T.S. Ingenieros Industriales e Ma/m/o, Abstract Asymptotic methods based on the slenderness ratio are used to obtain the leading-order equations which govern the fluid dynamics of compound, hollow jets such as those employed in the manufacture of textile fibers, composite fibers and optical fibers. These fibers consist of an inner material which is an annular jet and which, in turn, is surrounded by another annular jet in contact with ambient air. The one-dimensional leading-order equations for the fluid dynamics of annular liquid jets and compound jets at low Reynolds numbers are also derived and some sample results are shown. 1 Introduction Although there has been quite a lot research on the development of onedimensional, mathematical models for the analysis of single-component filaments and jets under both isothermal and non-isothermal conditions at low Reynolds numbers [1], compound fibers such as those used in reinforced materials and optical fibers (which are manufactured in coextrusion processes) have received very little attention despite the fact that the combination of two or more different materials with different properties may result in composite fibers which highly desireable properties. For example, in the manufacture of optical fibers, the core is surrounded by a sheath of cladding material. The objective of this paper is to perform an asymptotic analysis of compound, annular, isothermal jets of large aspect ratio at low Reynolds numbers, by means of perturbation techniques based on the smallness of the aspect ratio. This analysis is performed for both hollow and solid, compound

2 456 Advances in Fluid Mechanics III jets consisting of a Newtonian fluid surrounded by another Newtonian fluid, and for annular jets. If the inner fluid encloses some gases, the resulting compound jet is referred to as a hollow, compound jet, whereas, if it is a solid or round jet, it is referred to as a compound jet. In either case, the outer fluid is an annular liquid jet. Annular liquid jets are used in film blowing processes which are methods of producing thin sheets of thermoplastics rather more rapidly and economically than is possible with casting processes [1]; they are also used in spray applications, in protection systems of inertial-confinement laser fusion reactors [2], and in ink-jet printing and particle sorting [3] where the Reynolds numbers involved are so large that the flows may be considered as inviscid. When surface tension effects are important, annular liquid jets may acquire closed tulip shapes which may be used as chemical reactors [4]. Pearson and Petrie [5] studied the steady axisymmetric flow of a thin tubular liquid film at low Reynolds numbers in streamline coordinates using as a perturbation parameter the ratio between a characteristic thickness and a characteristic mean radius. They also performed a phase plane analysis for very low Reynolds numbers in the absence of gravity and surface tension [6]. Yeow [7] analyzed the linear stability of the model proposed by Pearson and Petrie [6] when the disturbances are axisymmetric and the flow is isothermal. Park [8] used perturbation methods based on the slenderness ratio and the smallness of the Deborah number in his studies of steady, isothermal, two-phase or compound fibers consisting of a Newtonian core layer surrounded by a sheath of non-newtonian layer with a Maxwell rheology. His studies resulted in a system of ordinary differential equations for both the axial velocity component and the radii of the two-phase fiber which is more manageable to analyze than the two-dimensional conservation equations of mass and linear momentum from which it was derived. The focus of this paper is on the modelling of both hollow and solid, compound liquid jets at low Reynolds numbers under isothermal conditions. Non-isothermal effects associated with the cooling used in the manufacture of thermoplastics and non-newtonian rheologies are not considered. The objective of the paper is, therefore, to consider the fluid dynamics of compound or two-phase liquid jets and the development of a set of onedimensional equations from the original partial differential equations. The paper has been divided in three main sections. Sections 2 and 3 contain the asymptotic analysis of the fluid dynamics of both hollow and round (or solid), compound liquid jets which has been carried out up to secondorder in the slenderness ratio. Section 4 contains the one-dimensional equations for annular jets at low Reynolds numbers. Some sample results are presented in Section 5, and a summary of the main results obtained in the paper is presented in the last section.

$isothermal and Newtonian. The inner and outer annular jets correspond to #i(f,z) < r < #(2,z) and #(f,z) < r < #2(6,%), respectively, with R\(t,x) / 0.$

3 Advances in Fluid Mechanics III Formulation Consider an axisymmetric, compound, hollow, annular liquid jet consisting of two immiscible, incompressible (constant density) fluids which are isothermal and Newtonian. The inner and outer annular jets correspond to #i(f,z) < r < #(2,z) and #(f,z) < r < #2(6,%), respectively, with R\(t,x) / 0. The fluid dynamics of the compound, hollow jet are governed by the conservation equations of mass and linear momentum which are subjected to kinematic and dynamic boundary conditions at the jet's interfaces, RI (, x), R(t, x) and #2(2, %), where R corresponds to the interface between the inner and outer annular jets. The kinematic conditions establish that the compound jet's interfaces are material surfaces where the shear stress is continuous, and the jump in normal stresses across the interface is balanced by surface tension. At the interface between the two liquid jets, i.e., at r R(t,x), the kinematic condition implies that that (v2- YI) -n = 0, i.e., that the normal velocity at the outer-inner jet interface is continuous, where n denotes the unit vector normal to r = R(t,x), the subscripts 1 and 2 refer to the inner and outer, annular jets, respectively, and v denotes the velocity vector. For viscous fluids, (v2 vi) t = 0 at r = R(t,x) where t is the unit vector tangent to r R(t, x) and the tangential velocity, i.e., u, components at this interface are continuous, i.e., ui(r,x,t) = u>2(r,x,t). Moreover, the normal stress condition at the inner jet's inner interface and at the outer jet's outer interface include the velocity gradients and the pressure of the gases gases surrounding (pe) and enclosed by (pi), respectively, the compound, hollow liquid jet; these gases are assumed to be dynamically passive since, in general, they have smaller density and dynamic viscosity than those of liquids. This implies that the gases surrounding the liquid may not introduce strong velocity variations along each cross section of the jet, although they may affect its dynamics. The surface tension at the inner jet's inner surface and at the outer jet's outer surface are denoted by a\ and <T2, respectively, whereas a is the surface tension at the interface between the inner and outer jets. In addition to the above boundary conditions in the radial direction, conditions in the axial direction must also be provided. If the compound. hollow jet emerges from a nozzle, there is a stress singularity at the nozzlejet's interfaces due to the relaxation of the velocity profile from no-slip conditions at the nozzle walls to the free-surface flow away from the nozzle. This relaxation may result in a jet contraction or swelling which implies that the radial velocity component is of importance near the nozzle. Moreover. the stress singularity at the nozzle exit and the jet contraction or swelling near the nozzle may result in a relatively important radial pressure gradient near the nozzle; therefore, an accurate analysis of the flow near the nozzle requires a full solution of the Navier-Stokes equations within the nozzle and in the free surface flow, and requires the use of numerical methods.

4 458 Advances in Fluid Mechanics III In this paper, a long wavelength approximation is used to reduce the conservation equations of mass and linear momentum and the kinematic and dynamic boundary conditions at the jet's interfaces at low Reynolds numbers to a one-dimensional set of equations as follows. For slender jets at low Reynolds number, it is convenient to nondimensionalize the radial (r) and axial (x) coordinates, the time (t), the axial (u) and radial (v) velocity components and the pressure (p) with respect to RQ, A, A/WO, %o, VQ and /mo /A, respectively, where RQ and A denote a characteristic radius and a characteristic wave length in the axial direction, respectively, UQ is a characteristic (constant) axial velocity component, VQ = RQUQ/\, and // is a reference viscosity. Here, we will take the reference viscosity that of the outer jet, i.e., //2, so that the nondimensional governing equations and boundary conditions can be written as (i = 1, 2) («i)* + (ViT)r/T = 0, (1) trel(u>) = -c*(p*/pi)(pi)x + (e*(«i)** + (r(«i)r)r/r) + Re/F, (2) \iipi erel(vi) = -(p2/pi)(pi)r + (HiPz/HiPi) (M^ + (( i)r /r),.), (3) Vi(Ri,x,t) = (Ri)t + Ui(Ri,x,t)(Ri)x, (4) Vi(R,x,t) = Rt+Ui(R,x,t)Rx, (5) 2e*(vi)r - (ui)*(ri)x + (u,)r + S(vi)t) (1 - t*(ri)l) =0, r = B,,, (6) + ((ui)r+(?(vi)x)(l-f?rz), r = R, (7) vi)r - 2 (Cm),, + e*(vi),) (Ri)* (e(ri)*)*) = (Wi/WzC)Ji, r = &, (8) )r - 2 ((Wz)r + e^(u2)«) (^)x r = ^2, (9) where -W/i2) (2c*(Ui)»^ + 2(ui)r - 2 ((Ui)r + e^(«l)x) ^) = (a/e(t->c)j, r = P., (10) ^),^/%-^(^J^J(l + ^((%)^,)i, (11) and ^e = denote the Reynolds, Froude and capillary numbers, respectively, and the same symbols have been used for dimensional and dimensionaless quantities for the sake of brevity. J is given by cqn. (11) with % replaced by 7?. For small Reynolds numbers, Re = tr with R = O(l), and, depending on the magnitude of the Fronde and capillary numbers, several flow

5 Advances in Fluid Mechanics III 459 regimes can be identified. Here, we consider F F/t and C C /t which correspond to low gravitational fields and small surface tension. Substitution of these values and expansion of the dependent variables as ^ = <^+e^2 + O(e4), (14) where </> denotes dependent variables, in the governing equations and boundary conditions, together with the expansion of the boundary conditions around R$(t,x], Rio(t,x) and R>2o(t,x) yield asymptotic expansions which at leading order, i.e., at O(e^), have the following solutions T^o = B^,z), pio = Df(f,z), ^o = Q/r-(Bi)zr/2, (15) corresponding to eqns. (l)-(3), (6) and (7), where Q = QK%) can be determined from eqns. (4) and (5) as 2Ci = (R$>)t + (BiR?o)*, (16) 2Ci = (Rftt + (BiRfrx, 2C2 = (Rl)t + (BzR~).,;, (17) and, from eqns. (8)-(10), one can easily obtain the following equations which can be written as - 2 (C2/4o + 0%),/2) ^ (^) ) - 2(m/m) (Q/^o + (BiL/2), (19) o). (20) Equations (16)-(20) contain nine unknowns, i.e., D\, Do, B\, &, d, C>2, RO, RIO and %o and, therefore, do not form a closed system. However, by imposing the conditions that the jets are viscous, i.e., uio(ro,x,t) = u>2q(ro,x,t) at leading order, one can easily show that BI B'2 B and C\ (7?, i.e., the leading-order axial velocity component of the inner and outer liquid jets is the same. Moreover, eqns. (16) and (17) can be written as Ms), + (B^)z = 0, Mi), 4- (fui)% = 0, (21) which correspond to the conservation of mass at leading order, where,4i = (#g - 7%))/2 and /L = (^ " #o)/2, and 2Ci=2C2 = (^), + (B^)_ (22) Equations (18)-(22) contain seven unknowns; therefore, these equations are not a closed system. In order to close this system of equations, it is necessary to go to higher orders in the asymptotic expansion. To O(e"). eqn. (2) yields %,2 - Qif"/4 + M; In r + /V,, (23)

6 460 Advances in Fluid Mechanics III where MI and Ni are functions of (x, ), and *-B^. (24) The shear stress conditions at the jet's interfaces (cf. eqns. (6) and (7)) yield to O(e^) % 4- (B^ - %)^/2, (25) (B_-Qi)^/2+ = - (B,, - Q2)4o/2 + M2, (26) which provide three more equations for two additional unknowns, i.e., MI and M2. These three equations together with eqns. (18), (19) and (22) constitute a closed system of one-dimensional equations for the fluid dynamics of compound, immiscible, isothermal, incompressible, hollow, annular liquid jets. By substituting M2 and MI from eqn. (25) into eqn. (26) and using eqn. (24) in the resulting equation, one can easily obtain the following equation for the leading-order axial velocity where ),^ 4- (%o)z/%o) + 2^2(^2/^0), + 2(/2iAi/^)(Ci/^o)^ (27) o/l 1, Mi / 1 1_\\ "lag B^"^//2l% ^^ Equations (21), (22), (27) and (28) constitute a system of four onedimensional equations for B, RQ, RW and %o which is much easier to solve than the time-dependent Navier-Stokes equations in cylindrical polar coordinates. These equations also show the coupling between thefluiddynamics of the compound, hollow annular jet and the densities, viscosities and surface tension of the liquids, and the pressure of the gases that surround the inner and outer, annular jets. 3 Compound liquid jets at low Reynolds numbers The formulation presented in the previous section can also be applied to analyze the fluid dynamics of slender compound jets. For these jets, #i(t,z) = 0, and eqns. (l)-(3), (5), (7), (9) and (10), eqns. (4) and (6) are only valid for i = 2, i.e., for the outer jet, and eqn. (8) has to be replaced by 'Ui(0,a;,f) = 0, (ui)r(0,z,f) = 0 and (pi)?.(0,%,f) = 0. Moreover, using the

7 Advances in Fluid Mechanics III 461 same nondimensionalization and parameters than in the previous section, one can easily show that C\ C>2 0, and, = 0, (29) (3(A2 4- m^i/^2)bj^ + (l/2c)((%o), + (c7/^2)(%),), (30) where Ai = Ag/2 and /^ = (^20 " ^o)/2- Equations (29) and (30) constitute a system of three one-dimensional equations for B, RQ and &o- Note that, if ^ \i\, p\ p2 and a 0, eqn. (30) reduces to that of round jet at low Reynolds numbers [1]. 4 Annular liquid jets at low Reynolds numbers The equations governing single-component, annular liquid jets can be readily obtained from those presented in Section 2 by simply disregarding the boundary condition at R(x,t), and can be written as 4-(B%)z, (31) where,.,,^,\ /_ //^ _ \ _^_ Equations (31)-(33) provide the values of RIQ, R-2o> and B. These values can then be used to determine the leading-order geometry, axial and radial velocity components, and pressure of the annular liquid jet, i.e., 5 Results Po =Pi - (7i/(f2C#io) - 2 (Ci/7^o 4- B%/2) = p, + l/(c%o) - 2 (Ci/B^ 4- BJ2). (34) Some sample results for compound, hollow jets, obtained with the leadingorder equations are shown in Figures 1 and 2. These figures illustrate the jet's geometry, axial velocity, axial traction in the inner jet and the ratio of the axial traction in the inner jet to the that in the outer one as functions of the axial coordinate. Note that the scaling of the axial coordinate is based on the wave length which is much larger than the jet's mean radius at the nozzle exit. Figure 1 indicates that the initial jet contraction increases as the ratio of the viscosity of the inner jet to that of the outer one increases; however. the radii of the jet at the take-up point increases as the viscosity ratio is increased. The axial velocity clearly exhibits a boundary layer behaviour near the take-up point where it increases substantially; the thickness of this

8 462 Advances in Fluid Mechanics III Figure 1: Geometry (a), axial velocity (b), axial traction (c)_ and axial traction in the outer jet to axial traction in the inner one (d). (R = R/F = C = Pl/p2 = 0"i/CT2 = CT/CT2 = AiBi = A2% = 1, Pi = Pe = 0; (solid line), 10 (dashed line) and 0.5 (dashed-dotted line)). boundary layer increases as the viscosity ratio p,i/p>2 is increased. The axial traction in the inner jet also increases as the viscosity ratio is increased and also exhibits a boundary layer behaviour at the downstream boundary. The results presented in Figure 1 also show that the ratio of the axial traction in the inner jet to that in the outer one decreases as the viscosity ratio is increased, but it decreases downstream. Figure 2 exhibits similar trends to those presented in Figure 1 and indicates that the ratio of the axial traction in the inner jet to that in the outer one increases as the Reynolds number is increased, and the thickness of the boundary layers of both the axial velocity component and the inner jet's axial traction decreases as the Reynolds number is increased. However, the jet contraction decreases as the Reynolds number is increased, and the ratio of the axial traction in the inner jet to that in the outer one decreases from the nozzle exit to the take-up point. Figure 2 also shows that there is a large difference between the axial traction in the inner jet to that in the outer one at the nozzle exit, especially at the higher Reynolds numbers considered in this paper. Although not shown here, it has been found that the ratio of the axial

9 Advances in Fluid Mechanics Figure 2: Geometry (a), axial velocity (b), axial traction (c) and axial traction in the outer jet to axial traction in the inner one (d). (1^1/^2 E/F = C = pi/^2 = (fi/oro = cr/^2 = -4iBi =,42% = 1, p, = pe = 0; R 1 (solid line), 0.1 (dashed line) and 2 (dashed-dotted line)). traction in the inner jet to that of the outer one can reach values approximately equal to three at the nozzle exit when pi/p2 5 for the same parameters as those of Figures 1 and 2. These large tractions might cause the breakup of the fiber if this were cooled. The formulation presented here can also be used to study thefluiddynamics offiberspinning processes when there is solidification by accounting for the temperature distribution in the fiber and the dependence of the viscosity on temperature. In this case, the leading-order equations derived here must be modified to account for the temperature distribution, but, at the low Reynolds numbers of typical fiber spinning processes, the thickness of the thermal boundary layer at the jet's interfaces is of the same order of magnitude or even smaller than that of the linear momentum, so that the thermal problem can be characterized by a core with thin boundary layers at the jet's interfaces. For compound jets, it has been found that the jet's contraction decreases, whereas the the ratio of the axial traction in the inner jet to that of the outer one increases as p\/ pi is increased, but it decreases downstream to almost 1 at the take-up point, The ratio of axial forces at the nozzle was almost equal to 4 for ^1/^2 = 0.5.

10 464 Advances in Fluid Mechanics III 6 Conclusions Asymptotic methods based on the slenderness ratio have been used to study thefluiddynamics of compound jets at low Reynolds numbers under isothermal conditions and for Newtonian rheologies. These compound jets may be hollow or solid depending on wheter the inner jet encloses or does not enclose gases. In either case, the outer jet is an annular one, and compound jets have many applications in the textile industry and manufacture of composite fibers and optical fibers. It has been shown that, in order to obtain the governing equations for the leading order quantities in the asymptotic expansion, is necessary to consider the original equations up to second-order in the perturbation parameter, i.e., the slenderness ratio, and that the leading-order equations are one-dimensional and, therefore, more amenable to numerical analysis. For hollow, compound jets, it has been shown that the radii of the jet's inner interface, common interface and outer interface must satisfy an equation which includes the difference between the pressure of the gases enclosed by the inner jet and that of the gases that surround the outer one. Acknowledgements The research reported in this paper was supported by Project PB from the D.G.E.S. of Spain. References [1] Pearson, J. R. A. Mec/mmcs o/ Former Processing. Elsevier Applied Science Publishers, Ltd., New York, [2] Ramos, J. I. Inviscid, slender, annular liquid jets, Chemical Engineering Science 51, pp , [3] Hertz, C. H. & Hermanrud, B. A liquid compound jet, Journal of Fluid AWwmcs 131, pp , [4] Ramos, J. I. Annular liquid jets: Formulation and steady state analysis, Zeitschrift fur Angewandte Mathematik und Me.cho.nik (ZAMM) 72, pp , [5] Pearson, J. R. A. & Petrie, C. J. S. The flow of a tubular film. Part 1. Formal mathematical representation, Journal of Fluid Mechanics 40, pp. 1-19, [6] Pearson, J. R. A. & Petrie, C. J. S. The flow of a tubular film. Part 2. Interpretation of the model and discussion of solutions, Journal of FMd Mec/mmcs 42, pp , [7] Yeow, Y. L. Stability of tubular film flow: a model of thefilm-blowingprocess, Journal of Fluid Mechanics 75, pp , [8] Park, C.-W. Exterisional flow of a two-phasefiber,aiche Journal 36, pp ,1990.

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