Numerical investigation of droplet deformation in contraction nozzles containing shear-thinning fluids

2.1. Constitutive relation

The non-Newtonian continuous phases follow the Bird-Carreau model:

where ${\eta }_0$ and ${\eta }_{\infty }$ are the zero- and infinite-shear viscosities, $\tau$ is a characteristic time, and $n$ is the flow-behavior index. The two non-Newtonian cases share ${\eta }_0$, $\tau$, and $n$ but have different ${\eta }_{\infty }$, producing different reductions of apparent viscosity.

2.2. Geometry, fluid properties and boundary conditions

The computational geometry represents the converging part of a spraying nozzle. The upstream radius is $R_u=$ 3 mm and the downstream radius is $R_d=$ 0.5 mm, giving $R_u/R_d=$ 6. The downstream channel is extended to 20 mm so that a strongly deformed droplet can evolve away from the outlet while breakup is assessed.

For both non-Newtonian fluids, $\tau =$ 0.0049 s and $n=$ 0.01323, while the interfacial tension is $\sigma =$ 5.75×10^-3 N/m. A fully developed inlet profile is prescribed with average velocity $U=$ 0.0295 m/s; no-slip and impermeability conditions are imposed at the wall, the centerline is a symmetry boundary, and the outlet uses a velocity zero-gradient condition with a fixed reference pressure.

Fig. 1Axisymmetric contraction geometry used in the simulations

2.3. Discretization, adaptive refinement and dimensionless parameters

The equations are discretized by the finite-volume formulation implemented in OpenFOAM. A modified interDyMFoam solver permits repeated refinement and unrefinement as the droplet travels through the nozzle. Refinement is concentrated around cells crossed by the interface, especially near breakup where thin necks and highly curved regions develop.

Mesh sensitivity was assessed using successively refined meshes by comparing velocity, pressure, interface evolution, and breakup response. The retained base mesh contains approximately 15 000 control volumes. Local cell sizes in the contraction and downstream region satisfy 0.02 mm $\le h\le$0.09 mm, with additional dynamic refinement restricted to the interface neighborhood.

The principal dimensionless groups are:

where $D=2R_d$. For shear-thinning fluids, ${\eta }_c$ is the local Bird-Carreau apparent viscosity. The inlet condition gives $Re=$ 0.764 and $\lambda =$ 0.561 for the Newtonian case; the corresponding ranges are $Re=$ 0.764-1.08 and $\lambda =$ 0.561-0.793 for continuous phase II, and $Re=$ 0.764-8.64 and $\lambda =$ 0.561-6.34 for continuous phase III.

3.1. Single-phase loading field

Before introducing a droplet, a steady single-phase solution is computed for each continuous phase. This separates the rheological effect from the disturbance produced by the droplet and provides the deformation-rate and stress histories subsequently experienced along the selected trajectories.

The shear rate increases sharply as the fluid accelerates through the contraction and remains important downstream. The elongational component has a more localized history: pronounced peaks occur as a streamline enters the contraction, after which the elongation rate falls toward zero. Thus, the droplet experiences strong elongation mainly during entry into the contraction, whereas shear supplies a longer-lasting downstream load. Rheology redistributes the deformation rate; the strongly shear-thinning fluid can exhibit a steep near-wall increase in shear rate even though its apparent viscosity is substantially reduced.

Fig. 2Representative loading histories along selected streamlines

a) Shear rate

b) Elongation rate

Fig. 3Fully developed downstream profiles for the three continuous phases

a) Shear rate

b) Shear stress

The downstream profiles emphasize the distinction. Shear rate rises from the axis toward the wall for all three liquids and is steepest for the strongest shear-thinning case. The stress ordering is reversed: the Newtonian liquid carries the largest shear stress and the strongly shear-thinning liquid the smallest. Hence a near-wall droplet can encounter a high shear rate without experiencing the highest viscous stress.

3.2. Droplet tracking and breakup mechanisms

Droplets are released on selected streamlines in the fully developed upstream section. Their initial transverse positions are represented by $Y=y/R_u$. For each streamline and continuous phase, the initial radius is varied in successive simulations until the transition between stable deformation and breakup is bracketed. The reported critical radius is estimated from the largest tested radius that remains intact and the neighboring radius that fragments.

The dynamic mesh remains active throughout this search. As the droplet elongates, the finest cells follow the interface while the remainder of the domain stays comparatively coarse. This is particularly important close to breakup, when thin necks and strongly curved regions appear.

Two distinct breakup sequences are obtained. In filament fracture, the droplet stretches into a long filament, several necks form, and continued thinning divides the filament into daughter droplets. This occurs at the lower end of the breakup capillary-number range considered. In the second mode, the central portion remains comparatively thick while a thin trailing region releases small satellite droplets. Within the axisymmetric model this behavior is referred to as tip streaming. Both modes are found for Newtonian and shear-thinning continuous phases, with tip streaming appearing as the local capillary number increases.

Fig. 4Adaptive mesh refinement concentrated around the moving droplet interface

Fig. 5Two simulated droplet-breakup sequences

a) Filament fracture

b) Tip streaming

3.3. Critical droplet radius

The critical-radius calculations are repeated for each selected streamline and continuous phase. The resulting threshold is a numerically resolved stable/breaking bracket rather than an exact analytical critical value.

Fig. 6Normalized critical droplet radius for Newtonian and non-Newtonian continuous phases

a) Versus streamline position $\mathrm{Y}$

b) Versus downstream shear rate

The strongest variation occurs between the inner and intermediate streamlines. As $Y$ increases, the normalized critical radius falls sharply; farther toward the wall, the threshold approaches a small plateau. Plotting the same data against downstream shear rate gives a similar overall decrease, but the three fluids remain separated. This confirms that shear rate alone does not incorporate the rheology-induced reduction in apparent viscosity.

3.4. Critical capillary number and stress-based representation

For each critical-radius point, a critical capillary number is formed using the downstream apparent viscosity and shear rate. It decreases markedly from the inner streamlines and changes much less as the wall is approached. The largest differences among rheologies occur where both the apparent viscosity and critical droplet size are changing rapidly.

Fig. 7Critical capillary number for the investigated trajectories

a)$C{a}_{crit}$ versus $Y$

b)$C{a}_{crit}$ versus shear stress

Because shear rate does not account for viscosity reduction, the breakup data are also represented against downstream shear stress. The critical radius decreases as stress increases, and the critical capillary numbers follow a similar progression for all three liquids. Compared with the shear-rate representation, the stress-based plots make the common behavior of the Newtonian and shear-thinning cases more apparent.

Fig. 8Breakup thresholds represented against downstream shear stress

a) Critical radius

b) Critical capillary number

A final comparison uses a scaled downstream shear-stress coordinate. The scaling is applied to the stress coordinate rather than to the observed breakup outcome, so the degree of collapse directly indicates whether this normalization is useful for comparing different continuous-phase rheologies.

After rescaling, the data lie close to common curves despite the substantial difference between the Newtonian viscosity and the high-shear viscosities of the Bird–Carreau liquids. Taken together, the results show that the breakup threshold is controlled by more than the local shear rate. Shear thinning changes apparent viscosity, viscous stress, and viscosity ratio simultaneously. This explains why the fluids remain separated in a shear-rate plot but approach a common trend when a scaled downstream-stress coordinate is used. The stress-based representation should not be interpreted as a universal breakup law; the present calculations concern one axisymmetric contraction, one dispersed-phase viscosity, fixed interfacial tension, and a limited set of Bird-Carreau parameters.

Fig. 9Breakup characteristics represented by scaled downstream shear stress

a) Scaled critical radius

b) Scaled critical capillary number