The role of microstructure in taming the Rayleigh capillary instability of cylindrical jets

PHYSICA ELS EV 1ER

Physica D 123 (1998) 161-182

The role of microstructure in taming the Rayleigh capillary instability of cylindrical jets M. Gregory Forest a,., Qi Wang b, 1 a Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA b Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202, USA

Abstract It is observed both in nature and in technological processes that filaments with anisotropic molecular-scale structure are less susceptible to breakup due to capillary instability than homogeneous, isotropic fluids in similar filament flows. Here we provide rigorous evidence that the strong coupling of microstructure to the hydrodynamics of cylindrical axisymmetric free surface filaments, indeed fundamentally alters the linearized stability of cylindricaljets. We extend Rayleigh's classical inviscid analysis of cylindrical jets to the three-dimensional (3D), macroscopic flow-orientation equations derived from the Doi kinetic theory for liquid crystalline polymers (LCPs). These equations assume rigid rod-like molecules and incorporate LCP effects of molecular relaxation, anisotropic drag, polymer kinetic energy, LCP density, and an intermolecular potential which couple orientation dynamics to standard free surface fluid equations. Depending on the LCP density, there are between one and three flow-independent orientation equilibria which persist in a constant-velocity, cylindrical free surface flow: an isotropic phase exists at all concentrations, whereas two anisotropic phases exist at sufficiently high LCP density. These equilibrium LCP cylindrical jets have two independent sources of instability, hydrodynamic and orientational, each identified within the coupled flow/orientation free surface equations. For this paper we restrict to equilibria free of orientational instabilities. All streamwise perturbations of wavelength greater than the jet circumference are unstable to capillary instability; only the strength of the instability and most dominant wavelength are affected by LCP microstructure. The degree to which microstructure reduces the capillary instability depends on two critical scaling parameters: an LCP capillary number Calcp (a ratio of LCP-induced surface stress to interfacial capillary stress); and the anisotropic drag/friction parameter cr. The most striking result is: for sufficiently large Calcp and highly anisotropic drag (cr ~ 0) the capillary growthrate can be uniformly lowered, arbitrarily close to zero. For sufficiently small Calcp, all capillary-dominated growthrates are reduced, but are bounded below in terms of an explicit, sharp estimate and bounded above by the Rayleigh formula. The upshot is: inviscid LCPjets are predicted to yield bigger drops which form on longer timescales than an inviscid isotropic fluid with the same surface tension. Copyright © 1998 Elsevier Science B.V. Keywords: Liquid crystals; Flow stability; Free surface flows; Capillarity; Non-Newtonian flows

1. Introduction The capillary instability of cylindrical columns of inviscid and viscous fluids was characterized by Lord Rayleigh [3,11 ]: all perturbations along the jet axis of symmetry of wavelength exceeding the jet circumference are unstable. Rayleigh derived an explicit formula for the inviscid linearized growthrate, Vinv(K), of a constant radius (r = ~b0) * Corresponding author. Tel.: (919) 962 9606; fax: 919 962 2568; e-mail: [email protected]. I Tel.: (317) 274-8144; fax: (317) 274-3460; e-mail: [email protected]. 0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reserved PH S0167-2789(98)00119-5

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jet, where K is the non-dimensional wave number (K = ~bok)of the superimposed streamwise perturbation of axial wave-number k, and the rate is normalized with respect to a characteristic timescale to: V2nv(K) = (1 - K2)KI~ (K) WIo(K) '

(1)

where W = (pfb3)/(ast~) is the Weber number measuring inertia relative to surface tension, as is the surface tension of the fluid/air interface, p is the fluid density, and Ii (K), Io(K) are the modified Bessel functions of the first kind of orders 1, 0, respectively. The dimensional instability band is 0 < k < (~b0)- I ; the cutoff, (~b0)-l, is purely geometrical, independent of surface tension. Rayleigh and later Chandrasekhar [3] showed that the unstable band of wave numbers is also independent of fluid viscosity. The order of the growthrate, Vinv (k) or Vvisc(k), however, scales with material properties as well as the jet radius. Here we extend Rayleigh's inviscid analysis to fluids with internal orientation. To do so, we adopt a Doi-type constitutive law and orientation dynamics derived for nematic liquid crystalline polymers (LCPs) in the Doi closure approximation by Bhave et al. [2] (hereafter referred to as the BMAB theory). The microstructure in this theory is represented by rod-like molecules. For this paper, we suppress the effect of solvent viscosity; the analysis is far more difficult with viscosity even for isotropic Newtonian fluids. We surmise, however, that the essential predictions of this paper remain (as Rayleigh, Weber and Chandrasekhar showed for isotropic fluids) when solvent viscosity is present. The growthrates almost surely reduce even further with solvent viscosity; this conjecture is confirmed in the longwave limit in [ 10]. We remark that the slender longwave asymptotic models in [ 10] provide a simplified analysis and clear predictions that the orientation-induced reduction of the Rayleigh instability is indeed a threedimensional (3D) effect. That is, since capillary instability is a longwave instability, slender fiber models are valid precisely in the scales where the instabilities reside. Once we observed these dramatic predictions in the simplified setting, we were encouraged to push the analysis to the full 3D equations. The results with solvent viscosity, however, still require more courage. Recall that we are coupling microstructural dynamics to hydrodynamics. The aforementioned classical analysis describes the flow decoupled from orientation. Likewise, the LCP flow/orientation equations presented below simplify greatly if the flow is suppressed, or if we posit simple flows. The orientation dynamics decouples in such situations, and one can classify all equilibrium phases and their stability if the flow coupling is ignored; those results [2,7] are summarized below. The upshot of this paper is a treatment of the interaction between these separate, quite tractable and transparent, stability phenomena; the coupling preserves both the constant velocity, cylindrical free surface jet and the nematic LCP equilibrium phases, setting the stage for an interesting dynamical interaction of these two instability phenomena. The analysis of the LCP cylindrical jet reveals linearized growthrate formulae which identify capillary-dominated and orientation-dominated growthrates and the influence of the coupling. We report elsewhere [9] on additional phenomena peculiar to unstable orientation phases, whereby new instabilities born in the flow/orientation coupling arise. The 3D axisymmetric LCP analysis contained here is analogous to the viscous generalization of Rayleigh's inviscid results. One loses a closed-form, algebraic growthrateformula in disfavor of a transcendental linearized dispersion equation, the roots of which must be, and are, graphed or tabled numerically. In special asymptotic limits, such as a longwave limit where the capillary instability resides, the dispersion equation yields explicit closed-form information on the orientation-induced corrections to the Rayleigh instability, and the hydrodynamic influence on orientational dynamics. We exhibit that our dispersion equations and graphs recover Rayleigh's exact results if we suppress all LCP effects, and likewise reproduce the exact orientation stability results if the flow is decoupled. (We recall that Rayleigh showed torsional modes are neutrally stable in the inviscid isotropic case, which rationalizes the axisymmetric assumption that we impose as well.)

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The rest of the paper is organized in three sections. First, we formulate the 3D flow/orientation free surface boundary value problem. Then, we characterize all flow/orientation equilibria for cylindrical, axisymmetric filaments. Finally, we derive the linearized dispersion equation for these equilibria in anisotropic and isotropic LCP phases, respectively, and deduce the stability results.

2. Three d i m e n s i o n a l m a t h e m a t i c a l formulation

We first present the governing equations for free surface axisymmetric LCP jet flows using the approximate BMAB theory, then discuss the free surface boundary conditions and the centerline conditions for the velocity field and the orientation variables. 2.1, Dynamic 3D equations and boundary conditions In the BMAB theory for LCPs, the macroscopic, or average, internal orientation properties of nematic liquid crystals are defined in terms of a second order orientation tensor Q, Q = (m®m)

- 1I 3 '

(2)

where ((.)) =

[

(3)

(-)f(m,x,t)dm,

Iml=l and f ( m , x, t) is the probability density function (or normalized orientation distribution function) corresponding to the probability that an arbitrary dumbbell or rod-like LCP molecule is in direction m at location x and time t. Bhave et al. [2] derived a diffusion equation for f ( m , x, t) from kinetic theory, dt = 6---~ Urn" ~mm

- O--ram' I V y . m - Vv : m ® m ® ml f - 6B T----~ ~ m

f

,

(4)

which accounts for polymers in a Newtonian solvent, subject to anisotropic drag (with drag coefficient cr 6 (0, 1]), and a polymer-polymer mean-field interaction with Maier-Saupe potential, = -3NBT(m®m-

½I) • Q,

(5)

where v is the velocity field, Vv denotes the velocity gradient, n is the number of polymer molecules per unit volume, N is a dimensionless measure o f n which characterizes the strength of the intermolecular potential, cr is a dimensionless parameter describing the anisotropic drag that a molecule experiences as it moves relative to the solution (cr = 1 indicates the friction is isotropic, whereas a = 0 corresponds to the case where the friction is hyper-anisotropic so that the molecular motion is confined only to the direction of the molecular axis), )~ is the relaxation time of the LCP molecules associated with rotation of the dumbbell molecules, B is the Boltzmann constant, T is absolute temperature, and d / d t ( . ) denotes the material derivative: O/Ot + v . V. From the kinetic equation, Bhave et al. further derived equations for the orientation tensor and the stress tensor, These equations couple to the kinetic equation for f through fourth order moments of m. A Doi-type quadratic closure approximation, that the fourth order moments of m are approximated by products of second order moments, (.) : ( m ® m ® m ® m )

= (.) : (m ® m ) ) ( m ® m),

(6)

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allows one to decouple the kinetic equation for f and obtain an approximate constitutive equation for the stress tensor which couples only the orientation tensor Q, and a self-consistent orientation tensor dynamical equation. The approximate constitutive equations along with the equation for conservation of momentum and the incompressible constraint constitute the governing equations for nematic liquid crystals in the approximate BMAB theory. Conservation of momentum: d p - ~ v = V • r + g,

(7)

where p is the density of the polymeric liquid, r is the total stress tensor, g is the external force, which will be neglected in this paper, since we are focusing on constant cylindrical filaments. Incompressibility: v.

v = 0.

(8)

Constitutive equation f o r stresses: r = - p I + ~-,

~ - = 2 ~ D + 3 n B T [ ( 1 - ½ N ) Q - N ( Q . Q ) + N(Q : Q ) ( Q + ½I)

(9)

+ 2 Z ( V v : Q)(Q + ½I)], where D is the rate of strain tensor, p is the scalar pressure and r/is the solvent viscosity. Orientation tensor equation d ~--~Q - (Vv. Q + Q . VV T) -~- F(Q) + G(Q, Vv),

F(Q) = -(or/3.){(1 - ½N)Q - N ( Q . Q) + N(Q : Q)(Q + 1I)},

(10)

G(Q, Vv) -- ~ D - 2(Vv : Q)(Q ÷ ½I). Note that F characterizes the orientation dynamics independent of flow, whereas G describes the flow-orientation interaction. We adopt cylindrical coordinates (r, 0, z), where the z-axis coincides with the axis of flow symmetry (i.e., the filament centerline). Consistent with the axisymmetry of the flow, the radial and torsional (swirling) components of the velocity vector must vanish at the fiber centerline (r = 0), i.e.,

Or(O, O, Z, t) = O, vo(O, 0, Z, t) = 0,

(11)

where v = (Vr (r, O, z, t), vo(r, O, z, t), vz(O, O, z, t)) is the velocity vector in cylindrical coordinates. The axisymmetricfree surface is parametrized by r = ~b(z, t).

(12)

The kinematic boundary condition is d ~-~(r - ~P(z, t)) = 0,

(13)

i.e., the free surface convects with the flow; the kinetic boundary condition is (7- -- - r a ) n f = - - t r s K n f ,

/¢ = ~b- l (1 + ~b2)-1/2 - (bzz(l + q~2)-3/2,

(14)

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165

where n f is the unit outward normal of the free surface (12), crs is the surface tension coefficient, x is the mean curvature of the free surface, and Ta is the ambient stress tensor. This condition indicates that the shear stress is continuous across the free surface in the tangential direction whereas the normal stress is discontinuous with jump proportional to surface tension times mean curvature. Due to the orientational contribution to stress, this boundary condition allows f o r competition between the microstrueture and interfacial tension, a balance which will reveal itself in the linearized analysis to follow. For simplicity, we assume that the ambient stress "r~ is a constant pressure, i.e., r~ = - p a I .

(15)

2.2. Representations o f nematic orientation tensors In [ 10], we introduce a biaxial representation for the orientation tensor Q: Q = s(n3 ® n 3 - ~I) +/5(n2 @n2 - 1I).

(16)

Here s and fl are two independent nematic LCP order parameters, equivalent to the eigenvalues of Q, that carry nonlinear average information about the degree of orientation that the polymer molecule m makes with the orthonormal eigenvectors ni, i ---- 1,2, 3, of Q, known as the LCP nematic directors and defined by s = ((n3 • m) 2) - ((ni -m)2),

fl = ((n2 • m) 2) - ((nl • m)2).

(17)

The range of values for (s,/5) is the closed triangular region in the (s, 15) plane with vertices (1~ 0), (0, 1) and (-1, -1). When s/5 = 0 or s = /5, (16) reduces to a uniaxial representation [10]. While the orientation tensor Q is free of the discontinuities which are apparently linked to defects, axisymmetry of the cylindrical flow field demands that the nematic liquid crystal exhibits uniaxial symmetry along the flow centerline, i.e., the z-axis. For all z, with r -- 0, we identify the particular eigenvector n3 as the one which limits to e: as r goes to zero and the order parameter/5(0, 0, z, t) = 0. If the flow is, in addition, torsionless, symmetry considerations allow the following parametrization of the orthonormal eigenvectors of Q in cylindrical coordinates [ 10]: nl = ( 0 . - - 1 , 0 ) .

n2 = ( - - c o s ~ , 0, sin~p),

n3 = (sin fit, 0, cos~p),

(18)

where ~p is the angle between the director n3 and e:. This angle must satisfy ~p(r = O, O, z, t) = 0,

(19)

which guarantees that at the centerline the LCP is uniaxial with the director n3 aligned with the axis of flow symmetry. We focus here on uniaxial equilibria, parametrized by the particular limit of our above biaxial representation in which/5 = 0; the other uniaxial limits, s = 0, s = /5, are merely relabels of this uniaxial representation and do not represent another physical solution. (These uniaxial steady states are the only equilibria when Vv = 0; more complex velocity fields are required to sustain more complex biaxial equilibria, such as an imposed, exact simple elongational flow [2,7,12] and shear flow [2,14].) For the uniaxial states, the eigenvector n3 is the distinguished optical axis or uniaxial director and s is the uniaxial order parameter. When 0 < s _< 1, the liquid crystal is said to exhibit prolate uniaxial symmetry; when - ½ _< s < 0, one infers oblate uniaxial symmetry; s ---- - ½ corresponds to all molecules aligned in the plane orthogonal to n3; s = 1 corresponds to parallel alignment of n3 and m; finally, s = 0 corresponds to an isotropic state.

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3. Steady state base flows 3.1. Flow~orientation equilibria In cylindrical coordinates the free surface boundary value problem outlined in the previous section admits the following constant solutions, which we will refer to as base flows hereafter: Ur = 0 ,

V0 = 0 ,

v~ = v 0 ,

$=$0,

s=so,

~p = 0 ,

/3 = 0 ,

(20)

where v0 is an arbitrary constant, q~0 > 0 is an arbitrary positive constant, and so c [ - 1 / 2 , 1] is a critical point of the intermolecular potential (bulk free energy), f U(s) ds, i.e., the roots of

U(s) = s(1 - ½N(1 - s)(2s + 1)) = 0,

(21)

parametrized explicitly in terms of the dimensionless LCP concentration N:

so =

0, ¼(1 -4- 3v/1 - 8/3N).

(22)

We note that these orientation equilibria are identical both without flow and with a constant velocity, cylindrical free surface flow. These steady state nematic LCP solutions exhibit uniaxial symmetry (since/3 = 0) with the uniaxial director aligned with the z-axis (since ap = 0). The number of orientation equilibria, and the corresponding degree of orientation, depend on the polymer concentration N; refer to Fig. 1. The isotropic phase exists for all N, whereas anisotropic phases are only supported at sufficiently high concentrations. For 0 < N < 38, the isotropic LCP phase, so = 0, is the unique equilibrium; for N > 8 there are three equilibrium phases whose structure changes with N. In the narrow range 8 < N < 3, the two new anisotropic equilibria emerge from the prolate state so = ¼: so = ¼(1 + 3~/1 - 8/3N) represents the most aligned prolate phase while so = ¼(1 - 3x/1 - 8/3N) represents the less aligned prolate phase. For N > 3, the highly aligned prolate and the isotropic phases remain, while so = ¼(1 - 3x/1 - 8 / 3 N ) becomes negative representing an oblate phase. As N approaches infinity, the two anisotropic phases approach 1, - 2 J, respectively, which correspond to the LCP molecules completely aligned with or orthogonal to the optical axis of symmetry, respectively.

3.2. Flow-independent stability of orientation equilibria Fig. 1 also depicts theflow-independent, pure orientational stability of these equilibrium phases. The general features of this bifurcation diagram are contained in [2]; here we provide further details from [7] as to the precise orientational degrees of freedom in which these instabilities reside, and exact formulas for the associated growthrates. Recall that this orientational stability picture couples in the full orientation/flow free surface equations to the stability results of inviscid cylindrical jet flows with surface tension, and the interactions of these two stability structures are the focus of this paper. This perspective on the dynamics of LCP cylindrical jets is appropriate because both the orientation equilibria and the cylindrical jet equilibria are preserved by the coupled flow/orientation boundary value problem. The only LCP equilibria without flow (or for constant velocity flows) are the uniaxial nematic phases shown in Fig. 1. It is valuable to note the following nested set of invariant subspaces of the general biaxial space of orientation tensors Q: the general space of biaxial Q is five-dimensional (traceless, symmetric second-order tensors); a threedimensional biaxial invariant subspace of"torsion-free" Q, parametrized by two order parameters s,/3 and an angle parameter 7t, described in Section 2.2; a two-dimensional biaxial invariant subspace, also torsionless, parametrized

M.G. Forest, Q. Wang/Physica D 123 (1998) 161-182 1

#

0.8 0.6 0.4

167

f

800.2

k

-0.2 -0.4

.........

6

~

4

g~

• ...................

+ "tb

't'z

1'4'

N Fig. 1. Uniaxial nematic LCP equilibria and stability to orientation dynamics alone, independent of flow. These equilibrium phases arise as critical points of the intermolecular potential (bulk free energy). The orientational stability/instability, absent ofbothflow and a free surface, is recalled here for each branch; solid curves indicate orientational stability, whereas dotted branches indicate orientational instability. This bifurcation diagram, which does not reflect the dimension of unstable manifold nor the degree of freedom of Q where each instability resides, is the same for the three-dimensional, torsion-free, invariant dynamical system (denoted here by M 3) and the full five-dimensional system (denoted by .Ms). The isotropic phase (s = 0) exists for all LCP concentrations N: il is stable for 0 < N < 3; three-dimensionally unstable for N > 3 within M 3 and five-dimensionally unstable for N > 3 within M s . At high enough LCP concentration, Ncl = ~, a saddle-node bifurcation within the one-dimensional special uniaxial invariant flow gives birth to two anisotropic equilibria starting at s = ¼. The upper branch consists of the more highly aligned LCP prolate phase, and is unconditionally stable. The lower branch consists of a less-aligned anisotropic phase. For ~8 < N < 3, this phase is prolate, with a one-dimensional uniaxial instability only. For N > 3, this phase becomes oblate (so < 0); the uniaxial instability is lost, but one biaxial order-parameter-instability emerges within M 3, and another torsional biaxial instability emerges within M ~ .

by s and fi, with ~ ---- 0; and a one-dimensional uniaxial subspace parametrized by s, with ~p ---- 0 and fi = 0. We have exploited this nested structure in [7] to identify new sources of pure orientational instability, as well as to generalize appealing results of Rey [ 12] on elongational-flow-induced nematic patterns and their stability. All equilibrium phases of Fig. 1 correspond to fixed points of the one-dimensional ode for the uniaxial torsionless phases. The stability of these equilibria, because of the nested structure, can be performed in sequence to isolate where the instabilities lie: within the one-dimensional space, any instabilities are uniaxial order parameter; within the two-dimensional uniaxial space, any new instabilities must be associated with a director instability; within the lwo-dimensional biaxial space, any new instabilities beyond the one-dimensional uniaxial space necessarily involve the biaxial order parameter degree of freedom; any new instabilities arising in the three-dimensional biaxial ode are then identified with director angle instability; finally as we pass to the full five-dimensional space, any new instabilities not already captured in the previous subspaces are necessarily due to torsional degrees of freedom. Again, we record this growthrate information for the various unstable LCP phases to establish points of contact in the flow/orientation analysis of the next section. The last issue of torsional instabilities is not addressed here, since it takes us too far astray of the primary focus on the torsion-free Rayleigh instability [7]. From Fig. 1 [7] we observe the stability features of the three branches: The isotropic phase is linearly stable for N < 3 within the full five-dimensional system. When N > 3, instabilities emerge in all available degrees of freedom! For example, there is a transcritical bifurcation within the uniaxial one-dimensional subspace, in which the isotropic phase transfers uniaxial stability to the oblate phase with so < 0 for N > 3. Thus the isotropic phase is five-dimensionally unstable for N > 3, with all growthrates identical and ,~iven by y = -(or/K)(1 - N/3). -

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M.G. Forest, Q. Wang/Physica D 123 (1998) 161-182

- The highly aligned prolate phase is born in a saddle-node bifurcation within the one-dimensional uniaxial subspace at N = 8. This upper branch is stable in the full five-dimensional space. - All other branches of equilibria are unstable [7]. 8 < N < 3, is one-dimensionally - The moderately aligned prolate phase, 0 < so < ¼, which exists for .~ unstable; as noted earlier the instability lies within the uniaxial one-dimensional flow, with growthrate Yuni =

- ( ~ /;~)U' (so). - The lower anisotropic phase becomes an oblate phase, so < 0, for N > 3 which is stable if one restricts to uniaxial dynamics [2], but is actually two-dimensionally unstable. In particular, there is a biaxial order parameter instability and a torsional director instability, each with exponential growthrate Fbi = --so (a N / Z ) , In the next section we restrict to torsion-free Q, so this latter unstable mode is suppressed. For N < 8, the isotropic phase is the unique equilibrium which also is stable. In the small window 8 < N < 3, the highly aligned prolate phase and the isotropic phase are co-existing stable equilibria; for all N > 3, the highly aligned prolate phase is the unique stable branch. Thus, these stable phases within the isolated orientation dynamics are candidates f o r observable phases in nontrivial, cylindrical jet flows. For this reason we focus only on these phases in the next section where flow and orientation are coupled. It is worth noting that the invariance of these orientation subspaces corresponds to special symmetries of the orientation dynamical system when there is no flow. In the presence of nontrivial flow, such symmetries are generically broken, and thus one cannot restrict to such a subspace of the full biaxial orientation tensor space. Fortunately, for axisymmetric, free surface, nonswirling (torsionless) flows, the three-dimensional space of Q given above remains invariant; i.e., this symmetry is preserved by the full orientation/flow equations. Consistent with Rayleigh's fluid analysis, we do not consider possible instabilities that arise from nonaxisymmetric and torsional degrees of freedom in v or Q. 4. Stability for L C P jet flows in the highly aligned, anisotropic phase (so > ¼)

Following Rayleigh [11], we first linearize the governing equations and boundary conditions about the above cylindrical LCP equilibrium jet solutions and then seek solutions of the linearized system in the form: (.)(r, z, t) = (')o + (')(r)e i(ct-kz),

(23)

where (')0 denotes the steady state solution, e.g., v0, ~b0,so, 7z = 0, fl = 0 and (.) denotes the r-dependent separable structure of the respective physical variables Vr, v=, s, qz, fl ; ~b of course has no r-dependent factor. The resulting governing equations consist of constant coefficient, ordinary differential equations for the r-dependent linearized eigenfunctions, together with algebraic constraints. (These equations are long and tedious and thus deleted.) The information sought is the linearized dispersion relation, c(k), which for this system is multi-valued, and from which one concludes linearized stability/instability of the equilibrium LCP jet solution. The linearized growth vs. decay of the superimposed axial disturbance of wave number k is determined from the signs of Re(ic(k)) = F(k), for each dispersion function c(k), which we refer to as the linearized growthrates. If all Re(ic) are negative for all k, we conclude linearized stability; if Re(ic (k)) = 0 we conclude neutral stability for that dispersion function at that wave number; for each dispersion function for which Re(ic(k)) > 0 there is a one-dimensional instability in that wave number mode. We identify both capillary and orientational sources of instability in distinct branches of c(k), and the influence of the coupling. We only report dispersion relations corresponding to instability or neutral stability. We draw special attention to the perturbation of the shear stress component rrz due to its prominent role in the derivation of the dispersion relation for different phases, rrz(r)

U(so) ["(2so + 1)ikvr (r) + (so - 1)v~ (r) ] s---~ [ 3-0c T ( - ~ ) - ~ o ~ s ~ "J '

(24)

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169

where (25)

ot = 3 n B T

is an LCP characteristic stress parameter, whose value relative to interracial stress (cr~/q~0) defines an LCP capillary number, introduced just below, which turns out to be a critical parameter with regard to capillary instability. When the LCP base flow is isotropic, i.e., so = 0, rrz(r) is nonzero; in contrast, rrz(r) = 0 for the anisotropic base flows (Fig. 1), where so is a nonzero root of U ( s o ) / s o = 0. This distinction leads to distinct dispersion relations for the isotropic family and the anisotropic families of steady states. Without loss of generality, we assign v0 = 0. We nondimensionalize the equations and physical variables using the characteristic time and length scales defined by

tll =

y

O's

,

r0 = ~b0,

(26)

from which the dimensionless physical variables and parameters are: v = V

cko to

~

C

c-

ito '

)~ = Ato,

k = K / (oo,

(27) r = R@),

1/ W

°'st02

1,

Calcp

otq~o

where upper case letters denote dimensionless variables, coordinates, and parameters. W is the Weber n u m b e r and Calcp is an important parameter we refer to as the L C P capillary n u m b e r as it measures LCP orientational energy relative to surface tension. This competition between interracial tension and the orientational strength of the LCP turns out to be a critical parameter in the degree to which microstructure affects the Rayleigh instability. Since we have set the axial velocity to zero, the natural choice for characteristic time scale is that given above, from which it follows that W = 1. Note that Re(ic) = t o l R e ( C ) , so the stability classification above reverts to the sign of Re[C(K)]. We address the stability of the highly aligned, prolate family in this section, and then the isotropic family at low LCP concentrations N is treated in Section 5. We note that partial results for the highly aligned prolate phase have been announced in [8]; the analysis and complete picture are presented in the following. 4.1. Dispersion equation f o r the highly aligned anisotropic phase

When so is a root of U ( s o ) / s o = 0, rrz(r) = 0; this leads to the following ode for the dimensionless radial velocity component, Vr:

(28)

R2Vr" (R) -4- R V r'( R ) - ( ( v K ) 2 R 2 -4- 1)Vr(R) -= O, where v2

C t.:(l)t.z(2 ) -~ ~-T~O ~0 '

GI =

CG(1)t,-7(2) 0 ~0

~(1) = C + a N ~0

--~-s0,

2

2

(I)

(2)

(TOt(so) G(O2) = C + - - A , 4~g7(I)

+ CalcpK [ 2 A s o G 0 G O - ( U ' ( s o ) ( ( 2 s 0 - ~so~ - ~ J ~ o

-+- 4aN 9A (1 "" -- S0)2S0) -- -4N ' 9 - 'tl -- sO)2soG(02))]"

(29)

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This is a parametric Bessel equation. The solution, obeying the centerline condition Vr(0) = 0, is given by Vr(R) = dl II (v K R),

(30)

where Ii (x) is the generalized Bessel function of the first kind and dt is an arbitrary constant. Substituting (30) into the kinematic and kinetic boundary conditions, we obtain the boundary condition for Vr(R) atR= 1:

C[GIVr(1)+G3Vr(I)] = G 0(1)G O (2) K 2( I - K ' ) V r ( 1 ) ,

(31)

where (1) (2) 2 2 (1) (2) I "~ G3 = C G 0 G o + CalcpK [2AsoG 0 G O - (U (so)((2s 6 - 4so .

4~N 9A (1 -- S0)2S0) -~- ~ - ( l

--

2~ll)

-

-

so)2soG~?))].

J

~

0

(32)

This boundary condition, in fact, yields the linearized dispersion equation for this boundary value problem after we insert the explicit radial eigenfunction from (30):

C ( G I I o ( v K ) - G,G(2) o

~

,]

( 1 - K 2) II(VK)v___K_,

(33)

where

G2 = ~CalcpK2Nso(1 - so).

(34)

Remarks. This transcendental dispersion equation has a simple factor, providing one explicit constant wavespeed, clearly tied to the orientation dynamics: -

aN C = -s0-A

-

-

(35)

This result follows from the observation that G~1) = 0 exactly satisfies the dispersion equation. Note that this condition yields v = 0, and the Bessel equation reduces to a Cauchy-Euler equation with bounded solution Vr(R) = d2 R, with d2 constant. The reader can follow through the details of this degenerate case. Referring back to the orientational dynamics captured in Fig. 1, we identify this growthratefunction with the biaxial order parameter instability of the anisotropic equilibrium branches. Indeed this growthrate is completely unaltered by the flow coupling in the linear approximation, which should have significant implications for the unstable phases because all wave numbers are unstable, which will then couple to the hydrodynamic modes in the nonlinear amplitude regime. For the highly aligned prolate phase of interest here, this orientation-dominated dispersion relation indicates stability. If Calcp = 0, this dispersion equation reproduces Rayleigh's exact, inviscid result (1) by collapsing the parameter v 2 to 1, so that the radial velocity eigenfunction is uniquely defined. Thus, the dispersion equation (30) collapses from a transcendental equation involving v(C) and C to a simple algebraic equation for C 2 ! This transcendental dispersion equation for C(K) a priori has a countable number of root branches, C lj) (K). Each root gives a potential growthrate, but also constitutes an eigenvalue/eigenfunction pair of the parametric Sturm-Liouville boundary value problem, consisting of Eq. (28) and boundary conditions Vr(0) = 0 and (31).

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171

For any given positive-real-part dispersion relation, C tj) (K), we are interested in the wave number instability band, or the set of wave numbers K for which C (j) has positive real part. These instability bands are most efficiently found by solving for the so-called cutoffwave numbers, Kc, defined by Re[C Ij) (Kc)] = 0. In practice, a simple algorithm to locate branches C (j) is to impose Re(C) = 0 in (33) and then solve for the cutoff wave numbers associated with the posited neutrally stable wavespeeds. (Another possible root finding strategy is to fix a value of K, say 0, and then search for all positive real part roots for each fixed K.) Note however from the two decoupled stability results of the previous section, that absent of coupling both the Rayleigh instability and the orientational instabilities all occur with Ira(C) = 0. Thus when we impose the stronger condition, C = 0, we immediately identify three branches of dispersion functions; not surprisingly, these three branches are anchored in the pure Rayleigh capillary instability, the pure uniaxial instability of the anisotropic nematic phase, and the pure director instability of the anisotropic phase. - There is a complementary, analytical method that we employ to benchmark the numerical algorithm on the full dispersion equation. This analysis can also be used as a "predictor" step in identifying precise estimates of at least the three sources of instability that we know exist independent of the coupling. That is, we know from the exact Rayleigh analysis and from our own orientation stability analysis that the capillary instability and the orientation instabilities all reside in the longwave regime. Therefore, we can either perform a longwave asymptotic expansion of the exact dispersion equation (33) or appeal to our slender longwave asymptotic models [10] and study the linearization of cylindrical LCP jets within that longwave approximation. Both calculations provide identical, accurate approximations for K ~ 0 of the three distinguished dispersion functions associated with anisotropic phases. Next we pursue the first strategy, expanding the exact linearized dispersion equation (33) for K ~v 0. - Recall that our focus in this paper is on the LCP phases which do not suffer orientational instabilities. Thus once we find the Rayleigh-dominated dispersion function, and confirm this is the only positive real part linearized growthrate, we only have to continue this root of the dispersion equation for all K (which is done numerically) and address the microstructural impact on the growthrates and wave number instability band. The remaining issues of orientation-dominated instabilities, and very intriguing phenomena born in the flow/orientation coupling, are not relevant to these LCP phases and are therefore addressed elsewhere [9].

4.2. Long wave approximation, 0 < K << 1, of the dispersion equation for anisotropic phases Here, we seek longwave (K ~ 0) asymptotic solutions of the dispersion equation for the wavespeeds C ~.i) in the form c ), =

cy',,-,.

(36)

i=0

Substitution into the dispersion equation (33) yields three possible unstable growthrates:

c{IlI =0,

C¢11 =

C~)2) = ---Uk cr , (so),

,

C~I ) - _

C(12, = 0,

~Calcp

2Aso+--(1-so)(Zso+l)a

C~2) -- ACalcp a (1 - s0)(2s0 + 1),

' C,3, = - aAN s0.

(37)

These formulas yield precise information for K ~ 0 for three dispersion functions associated with potential linearized instabilities: the first coefficient gives Re(C~J))(0), which is the growthrate of the K = 0 perturbed mode; the second coefficient gives Re(C(IJ))(0), which is the slope of the growthrate curve at K = 0; the next

M.G. Forest, Q. Wang/PhysicaD 123 (1998) 161-182

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coefficient gives concavity of the growthrate curve at K = 0. From these three wavespeed expansions, and the exact stability results of the decoupled hydrodynamics and orientation from Section 4.1, we conclude: - The first wavespeed C (1) is the flow/orientation deformation of the Rayleigh capillary instability. Since Co~1) = 0, the K = 0 mode is neutrally stable. - The positive slope of 1 / ~ implies instability for K positive and near zero, for all LCP phases. Moreover, this growthrate agrees precisely with the longwave expansion of the Rayleigh inviscid growthrate, Pinv( K ) in (1), where we recall W = ! with our scaling; the exact expansion for K ~ 0 of (1) can be found in [1]. Therefore this branch C(1)(K) of the dispersion equation matches the Rayleigh inviscid formula through two orders (in value and slope) for K ~ 0. - Furthermore, the order K 2 term, L~(1) 2 , explicitly characterizes the orientation-induced deformation of the Rayleigh instability in the longwave regime near K = 0. We note there is no hydrodynamic contribution at order K 2 for inviscid fluids as noted in [1]. Thus the microstructure alone controls the concavity of the growthrate curve in the longwave regime for K ~ 0. - The key information then is the sign of this term, C~ L), which is strictly negative for any LCP equilibrium phase since - ½ < so < 1! The next order inviscid hydrodynamic correction is at order K 3, which is also negative, further bending this capillary-dominated growthrate curve downward below the linear approximation

~v/~- I/2 K. - The other two dispersion functions, C 12)'(3), are clearly orientation-dominated. Referring back to Section 3.2, and restricting to the torsion-free class of Q, we determine: The growthrate C (3) is associated with the biaxial instability specific to the oblate anisotropic phase, so < 0, N > 3, for which U~(so) > 0. Remarkably, this instability is independent of K. Including the modified Rayleigh instability, the oblate phase is at least two-dimensionally unstable. The growthrate C (2) is associated with the uniaxial instability specific to the less aligned prolate phase, 0 < so < 1 4, 38 < N < 3, for which U~(so) < 0. There is a K-dependent, hydrodynamic correction to the pure orientation rate Co~2); the details are not pursued here. Including the capillary-dominated rate, this moderately aligned prolate phase is (at least) two-dimensionally unstable. Since we are restricting attention to the more highly aligned prolate phase, we note that this phase is stable at least for K "-~ 0 for these orientation-dominated wavespeeds, and thus is only one-dimensionally unstable for small K with capillary-dominated growthrate characterized above. To generalize these estimates to arbitrary wave numbers, we must resort to numerical computations. First, we examine stability in the hyper-anisotropic drag limit where a = 0. In our previous slender jet studies [ 10], we discovered this limiting case provides sharp lower bounds on the reduced Rayleigh-dominated growthrate: the lower bound is zero for Calcp above a critical value and finite, nonzero for capillary numbers below the critical value. Here we establish rigorous 3D lower bounds on the Rayleigh-dominated growthrate curve where again there is a critical capillary number condition that separates the ability for microstructure to reduce the capillary instability completely or by a measured nonzero fraction.

4.3. Hyper-anisotropic macromolecular drag limit (~ = O) When the motion of the LCP molecules is subject to a fully anisotropic hydrodynamic drag, i.e., drag along the molecular dumbbell axis is zero relative to drag orthogonal to the dumbbell axis, then the dispersion equation for the anisotropic phases reduces to

Ii ( v g ) C HIIo(vK) - He - ~

K2(I -

K2 ) I1 (vK) ] oK J

0,

(38)

M.G. Forest, Q. Wang/Physica D 123 (1998) 161-182

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where H, -----{C 2 + CalcpK2[2CAso - U:(so)(2s 2 - 2s o - 4) + ~N(1 - s0)2s0]}, (39) H2 ---- G2 -~ 3CalcpKZN(1 - so)so.

The trivial solution C = 0 is the ~r = 0 limit of the constant factor noted above; now we address the non-trivial factor. A longwave asymptotic approximation (36) yields a single wavespeed with possible nonpositive real part, again for the highly aligned prolate phase so > a'I. Co = 0,

Cl = v / ~ ,

(40)

where A = Calcph(so)U'(so)

I 2'

h(so) = (1 - s0)(2s0 + 1).

(41)

The expression A depends on the balance between the LCP orientational energy and the interfacial tension. The sign of this simple diagnostic, from (40), determines whether the growthrate is real or purely imaginary to linear order in K; as we soon discover, this asymptotically derived condition is conclusive for all K! (Anecdotal remark: we discovered this simple condition in this limit in the context of slender longwave models [10], and were quite surprised to discover its generality.) We proceed now to interpret the nature of this condition on the sign of A. Recall that h(so) > 0 for N finite; moreover, for this highly oriented phase, U'(so) > 0. Therefore, A > 0 if and only if the LCP capillary number Calcp is sufficiently large: x~ > 0

<---->

Calcp > [2h(so)U'(so)] - I > 0.

(42)

From the definitions of so (which depends only on N), h (a simple quadratic), and U (the bulk tree energy density which also depends only on N), the positivity of A translates to: the LCP capillary number must lie above a critical function of LCP concentration N; see Fig. 3(a). We refer to this function of N as the critical LCP capillary number, C _crit. Ulcp • Calccril p (N) = [2h(so)U'(so)] -1 .

(43)

Recall this phase is only defined for N > 8 The associated critical capillary number is positive, unbounded as N approaches 8, then decreases monotonically toward a finite nonzero bound, ~-, as N approaches infinity. This implies that for any fixed capillary number greater than -~, there is a critical concentration Nc above which ,4 > 0. On the other hand, for all capillary numbers Ca|cp < 1, the growthrate slope at K = 0 is strictly positive, so the Rayleigh instability cannot be diminished below this calculable longwave growthrate. Therefore in this low capillary number range, orientation can alter (indeed below we confirm always weakens) but not completely suppress the instability; Fig. 2(b) provides an illustration of this lower bound on the reduction due to orientation. The consequence o f the sign o f A is dramatic: -

If A > 0, i.e. Calcp lies above Fig. 3(a) curve, then for K ~ 0 the Rayleigh-dominated capillary growthrate vanishes at least through order K! The growthrate function is purely imaginary through order K (CI is purely imaginary). The implication, confirmed below by numerical evaluation for all K, is that orientation can completely suppress the Rayleigh instability in this hyper-anisotropic drag limit if the LCP capillary number is above this critical, concentration-dependent, value.

M.G. Forest, Q. Wang/Physica D 123 (1998) 161-182

174

~lcp

0.4

0.4-

0.3

0.3

0.2

lllep 0.2

,/ ,"

0.1

O

" 022

\

0£4

026

\

0.1

\ \\ f012

" 028 "

014. . . .

K

k i/ ~'i

06

08

K

(b) Fig. 2. Growthrates for the highly aligned prolate steady state (so = 1 (1 + 3 d l - 8 / 3 N ) ) in the hyper-anisotropic regime (a = 0). (a) 8 to ~O 89 in increments of 0.1. The solid curve is the growthrate The parameter values are Calc p = 1, A =- 1, with N ranging from N = .~ of the inviscid fluid and dotted curves are the LCP growthrates. This capillary number is greater than ~, so for N sufficiently low the Rayleigh instability survives but for N sufficiently large it is suppressed. At N = 8, the concentration which first supports this prolate equilibrium, the LCP jet has slightly lower growthrate. When N increases, both the cutoff wave numbers and the growthrate curves decrease to zero. This behavior occurs since the diagnostic A is negative until N reaches Nc = 3, the critical concentration for this value of Calcp. (b) The capillary number is lowered from (a) to the value Calcp = 0.1, which is below the lower bound of 1, so A < 0 for all concentrations N. As a result, there is a positive lower bound to the growthrate reduction, which the curves approach as N is increased.

10

i ! }

6 ¸

I

5;

l

C cvit alep

CalCcript 4!

!

4

i I

i

/

\

' ~

'315 . . . .

,i

'

415'

N

"ols'"

i'"als'

I'

215'

N

(b) Fig. 3. The critical LCP capillary number ' Ca~I rit as a function of N"" C a Icrit = (2h(so)Ur(so)) - I " (a) The critical capillary number vs. N p ' cp for the highly aligned prolate phase, which exists for N > _ 8 . Note that l i m N ~ oc Calccrit p (N) = / (b) The critical capillary number vs. N for the stable isotropic phase, C a crit = 3[3 - N] -1 , 0 < N < 3. lcp

M.G. Forest, Q. Wang/Physica D 123 (1998) 161-182

175

If A < 0, i.e., Caicp lies below this function of N, then the capillary-dominated growthrate is positive for K ~ 0, with K = 0 neutrally stable; the growthrate curve has positive slope at K = 0 given by IVq-~. Thus if N > 8 is fixed, this establishes that the Rayleigh instability of the highly aligned prolate phase persists for sufficiently low capillary numbers• But, note that this inequality is equivalent to the condition that the slope of the growthrate curve, C I , must be less than ½-the inviscid growthrate slope. In other words, in the longwave regime K ~ O, for sufficient/), low LCP capillary numbers, the capillary-dominated growthrates are positive but lie strictly below the Rayleigh inviscid growthrates. The full extent of the LCP capillary growthrate curve versus the inviscid curve is addressed below. - The next natural information to seek is whether the inviscid unstable wave number cutoff, K cutoff inv = 1 is altered by orientation. We have already seen that in this hyper-anisotropic drag limit, for sufficiently large Calcp the cutoff can be pushed all the way to K = 0! In this same a = 0 special limit, we now quantify how the instability cutoff varies in the low capillary number regime, where we know thus far only that the Rayleigh instability is weakened for K "-~ 0. By imposing C = 0 in the non-trivial factor of (39), in addition to K = 0 we deduce a dimensionless wave number cutoff less than one: -

Kcutofr = ~

< I.

(44)

For this prolate phase, --2A = 1 -- 2Calcp h (so)U' (s0) which is clearly less than 1. Moreover, 2h U' is monotone increasing with N with an upper bound of 6 from Fig. 3(a), which implies Kcutoff is monotone decreasing as N I increases with a lower bound that is positive for all Calcp < g. - Indeed, from Fig. 3(a) it follows that for any fixed Calcp < ~, I A must be negative for all N, the capillary instability persists and the formula above gives the instability band which lies inside the inviscid band 0 < K < 1. On the other hand, for any fixed Calcp > / for low LCP concentrations N the instability persists but for high N the instability is completely suppressed. N low enough yields conditions below the critical curve, the prolate phase is unstable, and Kcutoff < 1. As N increases however, holding Calcp fixed, there is a critical N above which A > 0, and therefore the LCP jet with this phase is stable to capillary instability. - Fig. 2(a) shows a numerically generated family of LCP capillary growthrate curves with A < 0, for Calcp fixed above g,I for several values of N. Since the capillary number is greater than g,I there is a critical concentration Nc (precisely Nc = 3 for this graph) at which one hits the critical curve in Fig. 3(a), i.e., at which A = 0. The growthrate curves confirm all estimates provided above: as N increases from _~ s (where the LCP growthrates are very close to the inviscid Rayleigh curve), the slope at K = 0 drops toward 0, the cutoff wave number drops toward 0, and the strength of the instability uniformly drops. Note also that the most unstable wave number, corresponding to the peak of the growthrate curve, also decreases as N increases. (We defer graphs of this particular information about the dominant mode of instability until we address the more physically plausible, nonzero values of or.) • I - Fig. 2(b) shows the growthrate curves for various values of N and specified Calcp < g, for which A < 0 for all N. In this regime, one cannot vary N and rise above the critical capillary number curve in Fig. 3(a). The closest one can get to this neutral stability curve, with Calcp fixed, is by taking N arbitrarily large; the entire growthrate curve drops as N increases, so the absolute lowest possible growthrate curve for this low capillary number is 2o is already essentially at the saturation point of achieved by taking N to infinity. In Fig. 2(b), the value N = T the growthrate curve. I This observation yields a sharp, positive lower bound on the LCP capillary growthrate curve for any Calcp < ~. Simply calculate the growthrates corresponding to N large, which yields a curve similar to the lower curve in Fig. 2(b), and the actual growthrate curve for N fixed must lie between this lower bound and the Rayleigh upper bound.

M.G. Forest, Q. Wang/PhysicaD 123 (1998) 161-182

176

4.4. Numerical resolution of the Rayleigh-dominated growthrate curves for all K (0 < ~r < 1) As characterized by the longwave asymptotic analysis, there is a persistent capillary-dominated growthrate for small IKI whenever tr # 0. Intermediate and shortwave information, such as whether there is a finite instability cutoff, /(cutoff, or even multiple instability bands, has to be determined from a numerical implementation of the exact dispersion equation (33). Figs. 4 - 6 convey the essential behavior of the growthrates for an LCP cylindrical jet in the highly aligned prolate phase. We draw conclusions from these numerically generated solutions of the exact dispersion equation. - The capillary-dominated LCP growthrates are positive for all 0 < K < l, become zero at K = l, and remain negative for all K > 1. The geometric condition for capillary instability is preserved whenever the anisotropic drag is nonzero: the LCP cylindrical jet is unstable to all streamwise perturbations of wavelength greater than the jet circumference. - The magnitude o f the LCP growthrate curve is unconditionally lower than the Rayleigh inviscid rate. The effect o f microstructure is to weaken the capillary instability uniformly for all 0 < K < I. - The LCP capillary growthrate curve further drops, uniformly for all 0 < K < 1, as the anisotropic drag parameter is lowered toward zero. The more anisotropic the drag, the lower the growthrate. Graphic illustrations are given in Figs. 4 and 5. - The absolute magnitude of reduction due to drag anisotropy depends critically on the LCP capillary number, with a simple condition identified in the previous section in the tr = 0 limit (Figs. 3(a)). Fig. 4 illustrates: if the LCP capillary number lies above the concentration-dependent critical capillary number, then the LCP capillary instability can be lowered arbitrarily close to zero in the highly anisotropic drag limit. Fig. 5 illustrates: crit then there is a sharp, positive lower bound on the reduction of the Rayleigh inviscid rates due if Calc p < Lalcp, to orientation. 0.4

1

03

/ /

l]lcp

0.2

0.8

\

/

\ \?%.-..

\

0.6 1-'1cp

/

b'in v

0.4

0.2

\

/

\ o

....

6::~

. . . .

".~:~, " - '.:,k

.......

T

o~,~ :~ :--:o-Y6~--'~%?~ .......

K

K

(a)

(b)

Fig. 4. Growthrate curves for the highly aligned prolate equilibrium so = ¼[l + 3V/1 - 8/3N] in the unstable wave number band 0 < K < 1, in a regime where LCP orientational effects are dominant over surface tension (d > 0). The parameter values are A = 1, N = 4, o- = 0.01, 0.25, 0.5, 0.75, 1, C a l c p = 1. Arrows indicate the direction of decreasing a. (a) The solid curve is the growth rate of an inviscid fluid while the dotted curves are growthrates of LCP filaments for varying tr. (b) The same curves scaled relative to the inviscid Rayleigh growthrate curve.

M.G. Forest, Q. Wang/Physica D 123 (1998) 161-182

177

0.4........ -......:.--..'-~...... ~..,,~,.....

0.3-

~"-... "",,":,'~:,, \ .......\ %

I

I

l

Illcp

\

O'6t

0.2

"\

\~:~

~'x \!1

vinv

\

°'I

0.1

0

.

I

2

~t

I:

~

0

1

K

K

(a)

(b)

Fig. 5. Growthrate curves for the highly aligned prolate equilibrium so = ¼[1 + 3~/1 - 8/3N] in the wave number band 0 < K < l in a parameter regime, where surface tension is dominant over LCP orientational effects (A < 0). The parameter values are A = 1, N = 4, a = 0.05, 0.25, 0.5.0.75, 1, Calco = 0.1. Arrows indicate the direction of decreasing a. (a) The solid curve is the Rayleigh growthrate of an inviscid fluid while the dotted curves are growthrates of LCP filaments. The lowest curve, approached as a goes to zero, is the exact growthrate curve for the hyper-anisotropic limit cr = 0. (b) The same curves in (a) scaled relative to the inviscid Rayleigh growthrate.

0

.

7

~

0.4

0.6 0.3

0.5

/

0.4 Kraax

o.

I

l

c

,,

p

=0.1 -_

0,3 o,2F

~

/Jlr~;zO-2

"" I

.........

/

0.2

CO.lc p

=

CatcZa

=

1

Calc p

=

2

-"

1/./

--

Catcp= 1

~1/

.......

c~,~

=

0.1

2

/

' '012

'

0!4

' 'O16'

'

'018'

i

O" (b) Fig. 6. (a) The wave number Kmax of the most unstable linearized mode as a function of a for varying corresponding maximum unstable growthrate vmax for (a). Lcp - To r e c o n c i l e the rigid c u t o f f w a v e n u m b e r Kcutoff =

Calc p =

2, 1.0.2, 0.1. (b) The

1 for all n o n z e r o a v e r s u s the l o w e r c u t o f f for a =

1,

F i g s . 4 ( a ) a n d 5(a) s h o w the b e h a v i o r near K = 1 in the h i g h a n d l o w c a p i l l a r y n u m b e r c a s e s , r e s p e c t i v e l y . A b o v e the critical c a p i l l a r y n u m b e r (Fig. 4 ( a ) ) , the s l o p e o f the g r o w t h r a t e c u r v e at K = 1 s m o o t h l y drops t o w a r d zero w i t h u n i f o r m c o n c a v i t y in the entire c u r v e all the w a y to a = 0. B e l o w the critical c a p i l l a r y n u m b e r (Fig. 5(a)),

M.G. Forest, Q. Wang/PhysicaD 123 (1998) 161-182

178

as cr drops toward zero, the curves have to develop an inflection point to accommodate the convergence to the previously determined lower bound with 0 < Kcutoff < 1. - The relative rates, Figs. 4(b) and 5(b), confirm a clear wave number selection principle. As analytically determined above, the LCP growthrate curve is tangent to the Rayleigh inviscid curve at K = 0; this implies, and Figs. 4(b) and 5(b) confirm, that the instability is very weakly affected in the longwave limit. (The only exception to this result occurs in the unphysical limit a = 0.) Indeed, this is essentially why the instability persists (except in the limit ~r = 0). However, the magnitude of the reduction due to orientation grows as K increases, sharply in the high capillary number regimes (Fig. 4(b)), but more gradually in the low capillary number regimes (Fig. 5(b)). The upshot is that the orientation-induced deformation of the capillary instability enters at order K 2 for K -,~ 0, driving the inviscid curve's concavity downward, but retaining contact through order K. As K increases, the relative effect of the microstructure increases, with the ratio of LCP to inviscid rates decreasing toward zero as K approaches the instability cutoff K = 1; as a result the LCP growthrates lie well below the Rayleigh curve for K < 1, then intersect again at K = 1, but transversely with a lower-magnitude slope of all LCP curves. - The orientation effects on the dominant wavelength of the instability and the associated maximum growthrate are given in Fig. 6. In Fig. 6(a), the top two curves correspond to Caicp below the critical capillary number, while the bottom two curves have high capillary number. For a fixed nonzero drag parameter, cr, one sees how the most unstable mode shifts to longer wavelength (K smaller) as Calcp increases, while the strength of the instability drops. (Recall the LCP capillary number is a ratio of polymer-induced tension to interfacial tension; it may be increased by lower surface tension or by higher polymer density or temperature.) As detailed earlier, we further observe that the most unstable wave number and associated highest growthrate both converge in the hyper-anisotropic drag limit (~ = 0): to finite, positive values for Calcp below the critical capillary number; and to zero for Calcp above critical.

5. S t a b i l i t y for L C P j e t flows in the i s o t r o p i c p h a s e (so = 0) at l o w c o n c e n t r a t i o n s (0 < N < 3)

In the isotropic phase (so = 0), the perturbation of the shear stress is nonzero, rr~(r) # 0. The dispersion equation derived for anisotropic phases, presented above, is no longer valid. Remarkably, the linearized equations are identical in form to the linearized Navier-Stokes free surface equations, with an effective viscosity played by the following orientation-dependent expression:

et

1 - N/3

~" = 3 C + (~/A)(1 - N/3)"

(45)

Following Chandrasekhar [3], we can simply invoke the viscous Rayleigh results; the new dispersion equation for this steady state is given by

If(K) [

2K2(2K 2 +RemC)/o--~

1

2KY I I ( K ) I((Y) ] K 2 + y2 I1 (Y) I~(K) J + RemC(2K2 + RemC)

= Re2mK( 1 _ K2 ) 11 (K)

(46)

Io(K)' where 3 C + (a/A)(1 - N/3) Rem = Calcp 1 - N/3 '

y2 = K 2 + RemC.

(47)

M.G. Forest, Q. Wang/Physica D 123 (1998) 161-182

179

Note that 1 c=o -- CatcpA Rein 3o- '

if N :fi 3,

1 c 0 =0' Re,n

if N = 3 .

(48)

Analogous to the dispersion equation in the anisotropic phases, and consistent with the pure orientational, uniaxial instability of the isotropic phase, this dispersion equation admits a constant solution given by

C--

A

,49

This growthrate is negative (a stable decay rate) when N < 3 and positive for N > 3. We are only interested for this paper in the orientationally stable phases, so we restrict to N < 3 for this isotropic phase. Note that the dispersion equation reduces to the inviscid Rayleigh formula when N = 3. In addition, there exists a unique positive growthrate for this isotropic phase, that agrees precisely with the Rayleigh inviscid formula through order K for K "-, 0. The numerical resolution of the transcendental dispersion relation (46) shows instability in the wave number band 0 < K < 1 for all 0 < N < 3, with growthrates bounded above by the Rayleigh growthrate for inviscid fluids. As before, we first analyze the special a = 0 limit, which will provide lower bounds as well as reveal critical parameters. 5.1. Hyper-anisotropic regime (t~ ~ O) The stability properties for the isotropic phase with N < 3 are qualitatively the same as those of the highly aligned prolate phase. - The cutoff wave number is given by Kcutoff --= max{0, K0},

-

--

-

(50)

where K0 is the solution of K = x / ( - 2 A C a l c p l o ( K ) K / l l (K). When A > 0, the cutoff wave number is zero, and the LCP jet in the isotropic phase does not suffer a capillary instability. If A < 0 (since h(0)U'(0) = 1 - I3N > 0 for N < 3), 0 < Kcutoff < 1. The critical capillary number for the isotropic phase is ~ a critlcp• 3,[3 -- N ] - I. For capillary numbers below this curve, Fig. 3(b), the capillary instability can only be reduced to some limiting positive growthrate. Above this capillary number, the Rayleigh instability can be entirely suppressed in this a = 0 limit. This critical curve, as opposed to the prolate phase, has a minimum value of ½ as N goes to 0, then increases monotonically to infinity as N approaches 3. Similar results as before follow. For all Calcp < ~, 1 the isotropic phase at low concentrations (0 < N < 3) has a reduced capillary instability relative to the inviscid Rayleigh result. There is a sharp, positive lower bound on the growthrate curve, given precisely by the N = 0 limit of the LCP growthrate. To save space, we do not give supporting figures; they are similar to Fig. 2(b). 1 For all Calcp > ~, the isotropic phase has a lower but positive capillary growthrate if N lies to the right of the critical curve in Fig. 3(b), but does not suffer capillary instability for sufficiently low concentrations to the left of the critical curve.

M.G. Forest, Q. Wang/PhysicaD 123 (1998) 161-182

180

0.4 0"41

/j.f............ . \ /

0.3 / l~lcp

/""- """\

0.3 .........

~

\

0.2.

,\

// IJlcp

0.2

/

/.~:"..-;-.." -%\ \

////~

,\

\

f,

....... ",,

\h.,

"'.,,

.."

\;

i

0.1

O12'~

'OJ4. . . . OJ6'

O18 1

0

'

0.2

0.4 '

K

K

(a)

(b)

0~ '"~":0.8..... " ~

Fig. 7. Growthrates for the isotropic equilibrium so = 0. The parameter values are Calcp = 1, a = 1, A = 1. The solid curve is the growthrate of the inviscid fluid and dotted curves are the LCP growthrates. (a) a = 1. N = 1.2; (b) N = 2, a = 0.01, 0.25, 0.5, 0.75, 1.

5.2. Anisotropic drag regime (0 < cr _< 1) For the isotropic steady state with 0 < N < 3, numerical evaluation of the linearized dispersion equation confirms that there is a unique positive growthrate in the wave number band (0, 1), whose magnitude is smaller than the inviscid Rayleigh growthrate. We refer to Fig. 7 for the following conclusions: - Fig. 7(a) enforces isotropic drag, a = l, which corresponds to the maximum possible LCP capillary growthrate. A n y other value of a will lower these two curves further below the Rayleigh exact curve (top curve). These two curves correspond to concentrations N = 1, 2, and appear to predict nearly equal growthrates; the smaller N -- l value yields the more reduced instability. This is consistent with the notion that as the LCP concentration gets closer to the density (N = 3) that supports anisotropic states, the isotropic state becomes less preferred. The difference between these two concentrations, N = 1 vs. N = 2, is not very significant with isotropic drag, but notice that the capillary number Calcp = 1 imposed here is above ~alc p g .crit : _ 3 [3 -- N ] - J for N = 1, but below critical for the higher concentration N = 2. Our analysis therefore implies that as the drag anisotropy is enhanced, with tr dropping toward 0, the N = 2 concentration encounters a sharp, positive lower bound curve (given by the cr = 0 growthrate) that the growthrate curves cannot pass below. This analytical prediction is confirmed in Fig. 7(b). For the lower, supercritical concentration N = 1, there is no lower bound to the growthrates as a goes to zero; we omit this curve to save space, and note it looks similar to Fig. 4(a) for the prolate phase. As N increases, the growthrate curves approach the inviscid limit and converge to the Rayleigh curve at N=3.

6.

Conclusion

By analyzing macroscopic flow/orientation free surface equations for axisymmetric free surface flows, we have generalized the classical Rayleigh instability for inviscid fluids to incorporate various orientational effects. These

M.G. Forest, Q. Wang/PhysicaD 123 (1998) 161-182

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equations admit steady state flows consisting of a cylindrical free surface, constant axial velocity, and an equilibrium uniaxial LCP phase. Of the three orientation phases, only the highly aligned prolate phase at high LCP concentrations (N > ~), and the isotropic phase at low LCP concentrations (0 < N < 3) are stable to orientational instabilities. We therefore focus on these phases as potential candidates for experimental confirmation of the primary result of this paper: The influence of microstructure, as modeled by these Doi-type 3D flow~orientation equations, unconditionally acts to tame the Rayleigh instability of these two orientation phases. - The strength of the unstable capillary-dominated growthrate is always reduced by the coupling of orientation. If one further explores special regions of LCP parameter space, precise conditions are given under which the capillary-dominated instability of LCP cylindrical filaments can be suppressed arbitrarily close to zero. If these special conditions are not met, then the LCP capillary instability can be bracketed between sharp upper and lower bounds; the upper bound is the Rayleigh inviscid growthrate curve, while the lower bound is provided by a hyper-anisotropic drag limit of the theory. - The wave number of the most unstable linearized mode is always lowered relative to the inviscid jet of the same geometry, flow and surface tension, indicating a longer dominant wavelength of instability. Under special conditions (large capillary number and highly anisotropic drag), the preferred wavelength of instability moves arbitrarily close to infinite wavelength; whereas, under complementary conditions (small capillary number) the wavelength of the most unstable linearized mode is longer but with a finite, precise upper bound. The implications of this analysis are consistent with observations of cylindrical filaments of fluids with complex microstructure, in both nature and technology. For example, when spiders spin silk to make their webs, the liquid they extrude is very similar to liquid crystalline polymer materials. The silk filaments solidify before suffering capillary-dominated instability. The suggestion here is that the LCP-like silk filaments are susceptible to a capillary instability that is of lower strength, longer timescale, and longer dominant wavelength than a simple isotropic fluid, and therefore solidification can occur well before the instability can develop. There are many papers in the literature over the past 20 years which attribute this weakened capillary instability to isotropic elastic material response; refer to [13] and references therein. A comparison with our own studies of polymeric, viscoelastic filaments show that the anisotropic microstructural interaction is more effective than elasticity in controlling capillary instability.

Acknowledgements

Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAR under grant number F49620-96-1-0131 and F49620-97-1-0001. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the US Government.

References

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