Pressure corrections for viscoelastic potential flow analysis of capillary instability

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1 ve-july29-4.tex 1 Pressure corrections for viscoelastic potential flow analysis of capillary instability J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering and Mechanics, University of Minnesota, 110 Union St. SE, Minneapolis, MN 55455, USA Department of Digital Engineering, Numazu College of Technology, 3600 Ooka, Numazu, Shizuoka, , Japan (Received, July 2004) Abstract Linear stability analysis of the capillary instability of a viscoelastic liquid thread was carried out by Funada and Joseph (2003), who assumed irrotational flow but retained the effect of viscoelasticity (viscoelastic potential flow, VPF). They compared their results with the unapproximated normal mode solution of the linearized fully viscoelastic flow (FVF). The comparison showed that the growth rates from VPF are several times larger than from FVF when the Reynolds number is not large (of the order of 1 or less). Joseph and Wang (2004) presented a method for computing a viscous correction of the irrotational pressure to resolve the discrepancy between the non-zero irrotational shear stress and the zero shear stress boundary condition at a free surface. Here we extend the pressure correction formulation to potential flows of viscoelastic fluids in flows governed by linearized equations, and apply this pressure correction to capillary instability of viscoelastic fluids. The growth rates from this viscoelastic correction of VPF (VCVPF) are in excellent agreement with those from the unapproximated FVF solution. Keywords: Capillary instability, Viscoelastic potential flow, Pressure correction, Dissipation method Author to whom correspondence should be addressed. Telephone: (612) ; fax (612) ; electronic mail: joseph@aem.umn.edu

2 ve-july29-4.tex 2 1 Introduction Capillary instability of a liquid cylinder of mean radius R leading to capillary collapse can be described as a neckdown due to surface tension γ in which fluid is ejected from the throat of the neck, leading to a smaller neck and greater neckdown capillary force as seen in the diagram in Fig. 1. Capillary instability of Newtonian fluids was studied by Rayleigh [1] following earlier work by Plateau [2] who showed that a long cylinder of liquid is unstable to disturbances with wave lengths greater than 2πR. The analysis of Rayleigh is based on potential flow of an inviscid liquid. Tomotika [3] studied the capillary instability and gave an exact normal mode solution of the linearized fully viscous Navier-Stokes equations. The linear analysis of capillary instability of viscoelastic fluids has been done by Middleman [4], Goldin et al. [5], Goren and Gottlieb [6]. They showed that the growth rates are larger for the viscoelastic fluids than for the equivalent Newtonian fluids. Funada and Joseph [7], [8] presented potential flow analyses of capillary instability of viscous and viscoelastic fluids. In their studies, the flow is assumed to be irrotational but the viscous and viscoelastic effects are retained. The viscous and viscoelastic stresses enter into the analyses through the normal stress balance at the interface. Funada and Joseph compared their results based on potential flow to the unapproximated normal mode results (Tomotika [3]). They showed that the results with viscous and viscoelastic effects retained are in better agreement with the unapproximated results than those assuming inviscid fluids. r = R+η R u Capillary Force γ/r Figure 1: Capillary instability. The force γ/r drives fluid away from the throat, leading to collapse.

3 ve-july29-4.tex 3 The capillary instability can be viewed as a free surface problem when either the interior or the exterior fluid is a gas of negligible density and viscosity. One difficulty in the potential flow analyses of free surface problems is that the non-zero irrotational shear stress violates the zero shear stress condition at the free surface. Joseph and Wang [9] derived a viscous correction for the irrotational pressure, which could resolve the unphysical discontinuity of the shear stress. Wang, Joseph and Funada [10] applied this pressure correction to the potential flow analysis of capillary instability of Newtonian fluids. They showed that the results computed with the pressure correction are almost indistinguishable from the exact results. Here we extend the pressure correction formulation to potential flows of viscoelastic fluids in flows governed by linearized equations, and apply this pressure correction to capillary instability of viscoelastic liquid filaments of Jeffreys type. The results are in remarkably good agreement with those obtained from the unapproximated normal mode analysis for viscoelastic fluids. The linear stability analysis given here and elsewhere indicates that the liquid jets are less stable with increasing elasticity, which contradicts the observation in experiments. A possible explanation of this contradiction is related to the linear stability analysis of a stressed filament at rest (Entov [11]). One difficulty is that a stressed filament at rest is not a permanent solution. 2 Linear stability equations and fully viscoelastic flow (FVF) analysis In an undisturbed rest state, the long cylinder of a viscoelastic liquid is surrounded by a gas of negligible density and viscosity. We use cylindrical coordinates (r, θ, z) and consider small axisymmetric disturbances. The linearized governing equations of the interior liquid are u =0, (1) ρ u t = p + τ (2)

4 ve-july29-4.tex 4 where u = ue r + we z is the velocity, ρ is the density, p is the pressure, and τ is the extra stress. The extra stress may be modeled by Jeffreys model ( ) τ τ + λ 1 t =2µ D D + λ 2, (3) t where D is the rate of strain tensor, µ is the viscosity, λ 1 and λ 2 are relaxation times. Suppose that we have normal mode solutions with the growth rate σ: τ = exp(σt) τ and D = exp(σt) D, (4) (3) leads to The momentum equation (2) becomes ρ u t The shear and normal stress boundary conditions are τ = 1+λ 2σ 1+λ 1 σ 2µ D τ = 1+λ 2σ 2µD. (5) 1+λ 1 σ ( ) 1+λ2 σ = p + 1+λ 1 σ 2µD = p + 1+λ 2σ 1+λ 1 σ µ 2 u. (6) ( 1+λ 2 σ u 1+λ 1 σ µ z + w ) =0; (7) r p + 1+λ ( 2σ 2 1+λ 1 σ 2µ u r = γ η z 2 + η ) R 2, (8) where η is the varicose displacement. The governing equations (1) and (6) and boundary conditions (8) and (7) are the same as those for a Newtonian fluid except that 1+λ 2σ µ replaces µ. 1+λ 1 σ Following scales are used to construct dimensionless governing equations: the cylinder diameter D for length, U = γ/(ρd) for velocity, T = D/U for time and p 0 = ρu 2 for pressure. The dimensionless momentum equation is (we use the same symbols for dimensionless variables) u t = p + ˆµ J 2 u, (9)

5 ve-july29-4.tex 5 where ˆµ = 1+ˆλ 2 σ 1+ˆλ 1 σ (10) with U γ ˆλ 1 = λ 1 D = λ 1 ρd 3 and ˆλ U γ 2 = λ 2 D = λ 2 ρd 3, (11) and J = ργd/µ 2 (12) is the Reynolds number and J is the Ohnesorge number. The dimensionless boundary conditions are p +2 ˆµ J u r = 2 η z 2 + η R 2 ; (13) ( ˆµ u J z + w ) =0. (14) r A solution of (9) which satisfies both the boundary conditions (13) and (14) takes the following form: ψ =[A 1 ri 1 (kr)+a 2 ri 1 (k v r)] exp(σt + ikz), (15) η = H exp(σt + ikz), (16) where k is the wavenumber and I 1 denotes the first kind modified Bessel function of the first order. Substitution of (15) and (16) into (13) and (14) leads to the solvability condition, which is given as the dispersion relation of σ: where 2k 2 I 1 (kr) ( k 2 + kv 2 ) I1 (k v R) F 1 F 2 =0 (17) ( ) ( ) F 1 = σi (kr)+2ˆµk2 di1 (kr) 1 k 0 J d(kr) R 2 k2 σ I 1(kR), (18)

6 ve-july29-4.tex 6 with k v = k 2 + J ˆµ σ. F =2ˆµkk ( ) ( ) v di1 (k 2 v R) 1 k J d(k v R) R 2 k2 σ I 1(k v R), (19) 3 Viscoelastic Potential flow (VPF) It is easy to show that the momentum equation (9) admits potential flow solutions. Take curl of equation (9) and use u = φ, we obtain φ t = ( p)+ ˆµ J 2 φ. (20) Both sides of (20) are zero, therefore potential flow solutions are compatible in this problem. The pressure integral can also be easily obtained from (9), ( ) φ = p p + ˆµ 2 φ p p = φ t J t, (21) where p p denotes the pressure from the potential flow solution and it is equal to the pressure from the inviscid potential flow. The potential flow solution is given by φ = AiI 0 (kr) exp(σt + ikz), (22) η = H exp(σt + ikz). (23) Substitution of the potential flow solution into the normal stress balance (13) leads to the dispersion relation I 0 (kr) ( ) ( ) [ I 1 (kr) σ2 1+ˆλ 1 σ + 1+ˆλ 2 σ σ 2k2 I0 (kr) J I 1 (kr) 1 ] ( ) 1 ( k kr R 2 k2 1+ˆλ 1 σ) =0, (24) which is a cubic equation of σ and has explicit solutions.

7 ve-july29-4.tex 7 When J, equation (24) reduces to ( ) I 0 (kr) 1 I 1 (kr) σ2 = k R 2 k2, (25) which is the dispersion relation for inviscid potential flow (IPF) solution. The IPF solution does not allow viscous or viscoelastic effects. 4 Dissipation and pressure correction formulation Joseph and Wang [9] derived a viscous pressure correction for the potential flow solutions of Newtonian fluids by considering the dissipation of energy. Here we extend the analysis to a viscoelastic fluid of Jeffereys type in flows governed by linearized equations. We start from the momentum equation ρ du dt = T u ρdu dt =( T) u, (26) where T is the total stress. It follows that ρ d dt ( ) 1 2 u u = (T u) u : T = (T u) (D + Ω) :( p1 +2ˆµµD) = (T u) D :( p1 +2ˆµµD) = (T u) 2ˆµµD : D. It follows that d ( ρ ) dt V 2 u u dv = n (T u) da 2ˆµµ D : DdV, (27) A V where V is the volume occupied by the viscoelastic fluid, A is the boundary of V, and n is the outward normal of V on A. If we use viscoelastic potential flow, (27) becomes d ( ρ ) dt V 2 u u dv = [( p p + τ rr )u + τ rz w] da 2ˆµµ D : D dv. (28) A V

8 ve-july29-4.tex 8 If we add a pressure correction p c and set the shear stress to be zero at the interface, (27) becomes Comparing (28) and (29), we obtain d ( ρ ) dt V 2 u u dv = [( p p p c + τ rr )u] da 2ˆµµ D : D dv. (29) A V A τ rz w da = A ( p c )u da, (30) which is the same as the pressure correction formulation as in the potential flow of a viscous Newtonian fluid (Joseph and Wang [9]). However, the calculation of τ rz in viscoelastic fluids is different than in Newtonian fluids. 5 Pressure correction for capillary instability Now we consider the pressure correction of the potential flow analysis of capillary instability. Joseph and Wang [9] showed that in linearized problems, the governing equation for the pressure correction is 2 p c =0. (31) It is easy to show that (31) holds for the viscoelastic fluid under consideration here. Solving (31), we obtain p c = C j ii 0 (jr) exp(σt + ijz), (32) j=0 where C j are constants. With the pressure correction, the normal stress balance becomes p p p c +2ˆµ 1 J u r = 2 η z 2 + η R 2, (33) which gives rise to [ {A 1 σi 0 (kr)+c k I 0 (kr)+ 2ˆµk2 A 1 I 0 (kr) I 1(kR) J kr + j k ( k 1 C j I 0 (jr) exp(σt + ijz) =A 1 σ I 1(kR) R 2 k2 ]} exp(σt + ikz) ) exp(σt + ikz). (34)

9 ve-july29-4.tex 9 By orthogonality of Fourier series, C j =0if j k. The coefficient C k can be determined using (30). The left hand side of (30) is A τ rz w da = ˆµ J 4πlA 2 1 k3 I 0 (kr)i 1 (kr) exp(2σt), (35) where l is the length of one wave period and * denotes conjugate variables. On the other hand, It follows that C k =2 ˆµ J A 1 k 2 and A ( p c )u da =2πlC k A 1 ki 0 (kr)i 1 (kr) exp(2σt). (36) p c = ia 1 k 2 2ˆµ J I 0 (kr) exp(σt + ikz). (37) Inserting C k into (34), we obtain [ σi 0 (kr)+ 2ˆµk2 I 0 (kr)+ 2ˆµk2 I 0 (kr) I ] 1(kR) = k ( ) 1 J J kr σ I 1(kR) R 2 k2, which can be written as I 0 (kr) ( ) ( ) [ I 1 (kr) σ2 1+ˆλ 1 σ + 1+ˆλ 2 σ σ 2k2 2I0 (kr) J I 1 (kr) 1 ] ( ) 1 ( k kr R 2 k2 1+ˆλ 1 σ) =0. (38) Equation (38) is the dispersion relation from the viscoelastic correction of VPF (VCVPF). 6 Comparison of the growth rate We compare the dispersion relation (38) from VCVPF with (24) from VPF, (25) from IPF and (17) from FVF. Equations (17), (24), (25) and (38) are solved by numerical methods for the growth rate σ and the values of σ are compared. First we examine two practical cases: 2% PAA in air and 2% PO in air (following Funada and Joseph [8]). We choose the diameter of the fluid cylinder to be 1 cm. The σ vs. k plots for 2% PAA and 2% PO are shown in Figs. 2 and 3, respectively. These figures show that the results from VCVPF are almost indistinguishable from FVF, whereas IPF and VPF overestimates σ significantly.

10 ve-july29-4.tex IPF 0.06 VPF 0.05 σ FVF VCVPF k Figure 2: The growth rate σ vs. k from IPF, VPF, VCVPF and FVF. The growth rates for FVF and VCVPF are almost the same (see Table 1). The fluid is 2% PAA, ρ =0.99 g cm 1, µ =96 P, γ =45.0 dyn cm 1, λ 1 =0.039 s, λ 2 =0s, J = , ˆλ 1 =0.263.

11 ve-july29-4.tex IPF VPF σ FVF VCVPF k Figure 3: The growth rate σ vs. k from IPF, VPF, VCVPF and FVF. The results of FVF and VCVPF are almost the same (see Table 1). The fluid is 2% PO, ρ =0.99 gcm 1, µ = 350 P, γ =63.0 dyn cm 1, λ 1 =0.21 s, λ 2 =0s, J = , ˆλ 1 =1.676.

12 ve-july29-4.tex 12 Capillary instability is controlled by three dimensionless numbers: J, ˆλ 1, and ˆλ 2. We vary these parameters and present the computed growth rate in Figs The Reynolds number J ranges from 10 4 to 10 4, ˆλ 1 ranges from 0.1 to 1000, and ˆλ 2 ranges from 0 to 100. In all the cases, the growth rates from VCVPF are in excellent agreement with FVF, indicating that our pressure correction is valid for a wide range of controlling parameters IPF VPF σ FVF VCVPF k Figure 4: The growth rate σ vs. k; J =10 4, ˆλ 1 =0.1, ˆλ 2 =0. Figures 4 and 5 show that the growth rates increase with ˆλ 1 when J and ˆλ 2 are fixed. Comparing Figs. 5 and 6, it can be seen that the effect of ˆλ 2 is opposite to that of ˆλ 1 ; the growth rates decreases with ˆλ 2. When ˆλ 1 = ˆλ 2, the fluid becomes Newtonian. When the Reynolds number is as high as 10 4 (Fig. 7), IPF and VPF slightly over-estimate the maximum growth rate whereas the VCVPF results are almost the same as the FVF solution. Acknowledgement This work was supported in part by the NSF under grants from Chemical Transport Systems.

13 ve-july29-4.tex VPF FVF IPF σ VCVPF k Figure 5: The growth rate σ vs. k; J =10 4, ˆλ 1 = 1000, ˆλ 2 = IPF VPF 0.05 σ FVF VCVPF k Figure 6: The growth rate σ vs. k; J =10 4, ˆλ 1 = 1000, ˆλ 2 = 100.

14 ve-july29-4.tex σ k 1 (a) 0.98 σ IPF FVF VPF 0.92 VCVPF k (b) Figure 7: (a) The growth rate σ vs. k; J =10 4, ˆλ 1 =0.1, ˆλ 2 =0. When the Reynolds number J is large, viscoelastic effects are relatively small, and the four curves are close; but differences among them can be seen near the peak growth rate. (b) The amplified plot for the region near the peak growth rate. VCVPF is the best approximation to FVF.

15 ve-july29-4.tex 15 Table 1: Maximum growth rate σ m and the associated wavenumber k m for VPF, VCVPF and FVF in Figs For IPF solution, k m =1.394 and σ m = in all the 6 cases. VPF VCVPF FVF Fig. k m σ m k m σ m k m σ m e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-01 References [1] L. Rayleigh, On the capillary phenomena of jets. Proc. Roy. Soc. London A 29 (1879), 71. [2] Plateau, Statique experimentale et theorique des liquide soumis aux seules forces moleculaire, vol. ii (1873), 231. [3] S. Tomotika, On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. Roy. Soc. London A 150 (1935), 322. [4] S. Middleman, Stability of a viscoelastic jet. Chem. Eng. Sci., 20 (1965), [5] M. Goldin, J. Yerushalmi, R. Pfeffer and R. Shinnar, Breakup of a laminar capillary jet of a viscoelastic fluid. J. Fluid Mech. 38 (1969), 689. [6] S. L. Goren, and M. Gottlieb, Surface-tension-driven breakup of viscoelstic liquid threads. J. Fluid Mech. 120 (1982), 245.

16 ve-july29-4.tex 16 [7] T. Funada, and D. D. Joseph, Viscous potential flow analysis of capillary instability. Inter. J. Multiphase Flow, 28 (2002), [8] T. Funada, and D. D. Joseph, Viscoelastic potential flow analysis of capillary instability. J. Non-Newtonian Fluid Mech. 111 (2003), 87. [9] D. D. Joseph, and J. Wang, The dissipation approximation and viscous potential flow. J. Fluid Mech. 505 (2004), 365. [10] J. Wang, D. D. Joseph, and T. Funada, Pressure correction for potential flow analysis of capillary instability of viscous fluids. Under review (2004). [11] V. M. Entov, J. Eng. Phys. 34 (1978), 159.

Purely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder

ve-march17.tex 1 Purely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering