Newtonian drops

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

Other models are based upon the immiscibility of polymer blends described above, and they model the system as Newtonian drops of the dispersed polymer with concentration (pi in a Newtonian medium of the second polymer with concentration (p2 = — (pi. There exists some concentration, cpu = cp2i — 1, at which phase inversion takes place that is, at snfficiently high concentration, the droplet phase suddenly becomes continuous, and the second phase forms droplets. The phase inversion concentration has been shown to correlate with the viscosity ratio, A. = r i/r 2, and the intrinsic viscosity for at least a dozen polymer alloys and blends ... 

The simplest case to consider is steady flow of a dilute suspension of Newtonian drops or bubbles in a Newtonian medium. If the capillary number y a / F is small, so that the drops or bubbles do not deform under flow, then at steady state the viscosity of the suspension is given by Taylor s (1932) extension of the Einstein formula for solid spheres ... 

Keywords Impact, Wetting, Filtration, Non-Newtonian Drop, Power-Law Model... 

The order of contents in this chapter is as follows In Section 2 the governing equations and numerical methodology are given. Validation of the obtained results for Newtonian fluids in comparison with experimental observations presented by Lorenceau et al. [1] is provided in Section 3.1. A general description of the observations for non-Newtonian droplets captured by horizontal fiber is presented in Section 3.2, and the behavior of non-Newtonian drops affecting thin fibers is explained. Finally, a summary of the results and conclusions is provided. 

Since, for = 0 to the quantity in the square bracket ranges from 1.00 to 1.18, the drop deformability D = 0.55k. Thus, a small deformation of Newtonian drops in Newtonian matrix varies linearly with the capillarity number. This proportionality was indeed demonstrated in Couette-type rheometer for a series of com symp/silicon oil emulsions [Elemans, 1989]. 

For Newtonian drops suspended in viscoelastic fluid, Flumerfelt [1972] reported the existence of a minimum drop size below which breakup cannot be achieved. The author pointed out that the elasticity of the medium tends to increase this minimum value for breakup, that is, to stabilize the droplets. 

Higher for viscoelastic system than for Newtonian drops. 

For Newtonian drops in viscoelastic fluid, elasticity of the medium stabilizes the drops. 

In viscoelastic systems is always higher than in Newtonian — drop elasticity hinders the drop breakup, whatever the X value. 

The deformability of the viscoelastic drops in Newtonian matrix was studied in the convergent slit flow. Both, the experimental observations and the boundary element method computations were carried out. It was reported that deformation of the Boger fluid drop, was quite low —about 1/3 of that recorded for the deformability of a strongly shear-thinning, viscoelastic solution. The latter drops showed deformability similar to these observed for Newtonian drops of similar viscosities. 

A set of empirical equations was obtained by Wu to describe the dispersed phase average particle size obtained after dispersive mixing in an extruder (13). The equations were based on the case of a Newtonian drop suspended in a Newtonian matrix, that is, Taylor s theory (16,17) with an extension to the case of a viscoelastic drop in a viscoelastic matrix. The empirical data employed were for blends containing 15 wt% dispersed phase and 85 wt% matrix phase. The particle size was found to be critically dependent on the ratio of the dispersed phase to the matrix phase melt... 

In contrast and for comparison, the theoretical equation from Taylor s theory (16, 17) for a Newtonian drop suspended in a Newtonian matrix with the concentration of the dispersed phase particle assumed to be vanishingly small is... 

Keywords Bag breakup Breakup mode Breakup time Catastrophic breakup Fragments Fragment size distribution Initiation time Multimode breakup Newtonian drops Non-Newtonian drops Ohnesorge number (Oh) Secondary atomization Secondary breakup Sheet-thinning breakup Total breakup time Vibrational breakup Weber number (We)... 

Figure 6.1 shows typical breakup of Newtonian drops. Time increases from left to right and aerodynamic forces increase from top to bottom. The initial ambient velocity relative to the drop, Uo, acts in the direction shown. 

It also remains unclear as to which dimensionless groups play dominant roles in the purely viscous and viscoelastic cases of non-Newtonian drop breakup. As such, dependence of transition We on other nondimensional parameters is unavailable. 

The initial and total breakup times also need to be determined, as well as fragment size distribution information. Unanswered questions include Does Simmons scaling rule (MMD/D32 1.2) hold for non-Newtonian drop secondary breakup and Will non-Newtonian drop secondary breakup produce a root-normal fragment size distribution ... 

The few studies focused on secondary breakup of non-Newtonian drops are reviewed below. 

The first stage of breakup is deformation of the drop into a shape that resembles an oblate spheroid (see Fig. 6.2). Given the similarity to Newtonian drops, it is reasonable to assume that the same physical mechanisms apply, namely, unequal static pressure distribution over the drop surface. 

The sheet-thinning mechanism observed for non-Newtonian drops resembles that observed for Newtonian liquids in some aspects. For instance, it is found at... 

Contrary to Newtonian drop behavior, [27-28] observed that once ligaments peeled off the drop surface they were joined together by a thin sheet. The sheet expanded and the hgaments elongated and extended downstream undergoing additional splitting. 

Non-Newtonian liquids also exhibit a transitional multimode regime between bag and sheet-thinning breakup. It is important to note that only bag/plume-type breakup has been observed. Other breakup structures discussed in the Newtonian section have not been observed for non-Newtonian drops. It is unclear if this is due to a lack of available data or some rheological difference. 

In the multimode case, non-Newtonian drops form a much more pronounced stamen that has a much longer lifetime. This stamen eventually forms many large fragments when it finally breaks up. In the sheet-thinning case, non-Newtonian breakup proceeds through two steps - the thinning of the sheet followed by the drop core forming a bag that experiences multimode breakup. 

Higher Kcra for viscoelastic system than for Newtonian drops Gauthier et al. 1971... 

Starting with cell model of creeping flow, Choi and Schowalter [113] derived a constitutive equation for an emulsion of deformable Newtonian drops in a Newtonian matrix. The authors characterized the interphase with an ill-defined interfacial tension coefficient, Vu, affecting the capillarity number, k = (Judfvu. The analysis indicated that depending on magnitude of /cy the emulsion may be elastic, characterized by two relaxation times. For the steady-state shearing, the authors expressed the relative viscosity of emulsions and the first normal stress difference as ... 

Although Taylor s leaky dielectric theory provides a good qualitative description for the deformation of a Newtonian drop in an electric field, the validity of its analytical results is strictly limited to the drop experiencing small deformation in an infinitely extended domain. Extensive experiments showed a serions difference in this theoretical prediction [34]. 

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