Newtonian Droplets Research Articles

Spreading and engulfment of a viscoelastic film onto a Newtonian droplet

Spreading and engulfment of a viscoelastic film onto a Newtonian droplet

Capillary wave-assisted collapse of non-Newtonian droplets

Understanding the peripheral capillary wave propagation during droplet impact is crucial for comprehending the physics of wetting onset and droplet fragmentation. Although Newtonian droplets have been extensively studied, we show how capillary waves deform non-Newtonian droplets in such a way that rheological features, such as the critical concentrations for the overlap (c*) and entangled polymer molecules (c**), may be directly obtained from the deformation history. Determining these critical concentrations is essential as they mark transitions in the rheological behavior of aqueous polymeric solutions, influencing viscosity, elasticity, and associated fluid dynamics. We have also compared capillary waves among Newtonian, shear-thinning, and Boger fluid droplets and found that although the fluid kinematics appear to be purely biaxial extensional flow, the infinite-shear properties of the droplets dominate the physics of capillary wave formation and propagation.

Dynamic behaviors of shear-thinning droplet impacting on a spherical particle surface

Non-Newtonian droplet impact is common in nature and industry. The viscosity of non-Newtonian liquids changes with shear stress, leading to different impact behaviors compared to Newtonian liquids. This study prepared shear-thinning solutions by adding HPMC polymer into deionized water and investigated its impact dynamics on hydrophobic spherical surface. The evolution patterns of droplet spreading factor, film thickness, and dynamic contact angle are investigated under different impact conditions. The results show that addition of polymers increases the energy dissipation during impact and suppresses droplet rebound from spherical surface compared to Newtonian droplet with similar viscosity. The droplet-impacting process is distinguished into three phases: (I) initial deformation, (II) inertia-dominated, and (III) viscosity-dominated phases. The temporal evolution of film thickness in Phases I and II is fitted by simple equations. In addition, an empirical formula that integrates Weber and Reynold numbers is proposed to predict the maximum spreading factor of HPMC droplets.

Comparison of Newtonian and glycerol-water solution-based SiO2 nanofluid droplets impacting on heated spherical surfaces

The dynamic interactions of liquid droplets with heated spherical particles, relevant across various scientific and engineering disciplines, display intriguing collision behaviors whose principles are yet to be fully deciphered. This investigation utilizes high-speed photography to explore the wetting and non-wetting responses of different Newtonian and glycerol-water solution-based SiO2 nanofluid droplets upon contact with heated particles. Findings indicate that nanofluid droplets disrupt the thermal field and modify the surface morphology of particles more extensively under wetting than non-wetting conditions. Wetting conditions lead to significant alterations in the liquid film thickness and notable velocity field disruptions for droplets of deionized water and glycerol solution. Conversely, in the non-wetting phase, droplets nearing full rebound cause less disturbance to the velocity field, a behavior consistent across all fluid types and particle temperatures. The shape changes nanofluid droplets undergo when rebounding and separating from the particle surface are similar to those seen in glycerol solution and water droplets. Droplets in the breakup-rebound phase experience shorter contact times than those merely rebounding. A phase diagram is presented, detailing wetting (receding-deposition and boiling-breakup) and non-wetting (rebound and breakup-rebound) actions. An increase in the Weber number necessitates a higher particle temperature for the transition from rebound to breakup-rebound, which corroborates with theoretical insights. Furthermore, droplets achieve their maximum spreading area during the boiling-breakup mode.

Numerical study on the dynamic behaviors of shear-thinning droplets impacting on a hydrophobic spherical surface

In this study, the dynamic behavior of shear-thinning droplets impacting on a hydrophobic spherical surface is numerically investigated using the volume of fluid method coupled with dynamic contact angle and a power-law model. The differences in dynamic behaviors between shear-thinning and Newtonian droplets are first studied. By analyzing the distribution of pressure and shear rate inside the droplet, it is found that the shear-thinning behavior of the droplets leads to an uneven distribution of apparent viscosity upon impact, which in turn prevents droplet rebound. The effects of various impacting conditions, such as apparent viscosity, impact velocity, surface tension and dynamic contact angle, on the spreading factor and liquid film thickness of shear-thinning droplets are investigated. According to the behaviors of droplets, the impacting process can be divided into three phases: (I) initial deformation, (II) inertia-dominated, and (III) viscosity-dominated phases. In order to reveal the physical mechanisms that prevent shear-thinning droplets from rebounding on the hydrophobic particle surface, the conversion of kinetic energy, viscous dissipation, surface energy, and potential energy during the three phases of the impact process is also analyzed in detail.

Three-dimensional internal flow evolution of an evaporating droplet and its role in particle deposition pattern

The internal flow within an evaporating sessile droplet is one of the driving mechanisms that lead to various particle deposition patterns seen in applications such as inkjet printing, surface patterning, and blood stain analysis. Despite decades of research, the causal link between droplet internal flow and particle deposition patterns has not been fully established. In this study, we employ a three-dimensional (3D) imaging technique based on digital inline holography to quantitatively assess the evolution of internal flow fields and particle migration in three distinct types of wetting droplets, namely water droplets, sucrose aqueous solution droplets, and sodium dodecylsulfate aqueous solution droplets, throughout their entire evaporation process. Our imaging reveals the three-stage evolution of the 3D internal flow regimes driven by changes in the relative importance of capillary flow, Marangoni flow, and droplet boundary movement during evaporation. Moreover, the migration of individual particles from their initial locations to deposition can be divided into five categories, with some particles depositing at the contact line and others inside the droplet. In particular, we observe the changing migration directions of particles due to competing Marangoni and capillary flows. We further develop an analytical model that predicts the droplet internal flow, deposition patterns, and determines the deposition mechanisms depending on their initial locations and evolving internal flow. The model, validated using different types of droplets from our experiment and the literature, can be further expanded to other Newtonian and non-Newtonian droplets, potentially serving as a real-time assessment tool for particle deposition in various applications.

A New Solution of Drag for Newtonian Fluid Droplets in a Power-Law Fluid

Understanding flow behaviors of multiple droplets in complex non-Newtonian fluids is crucial in many science and engineering applications. In this study, a new and improved analytical solution is developed based on the free surface cell model for the flow drag of swamp of Newtonian fluid drops through a power-law fluid. The developed solution is accurate and compares well to the numerical solutions. The improvement involves a new quantification of shear stress boundary condition at the interface and a more consistent approximation in linearizing the power-law fluid flow governing equation. The Newtonian fluid solutions can be reasonably used to linearize the flow governing equation. The approximation of the boundary conditions at the interface, however, has a major impact on the model prediction. The main improvement in the new solution is observed under the condition of comparable viscosities of the Newtonian drops and the outside power-law fluid when the results are sensitive to the interface boundary condition. Under the two extreme conditions of high viscosity ratio (approaching particles) and low ratio (approaching bubbles), the present and existing solutions converge.

Flow around a droplet suspended in a yield-stress fluid

We investigate the minimal yield-stress required in order to hold static an ellipsoidal Newtonian droplet inside a yield-stress liquid. This critical limit (Yc) is influenced by the droplet aspect ratio (χ), the interfacial tension (γ), and the viscosity ratio (M) between the droplet and the surrounding liquid, as well as the ratio of the yield-stress to the buoyancy stress (Y). The droplet will remain trapped by the liquid yield-stress for Y>Yc. Our study bridges the gap in the published results between those calculated for bubbles (M→0) and the solid rigid particles (M→∞), being of practical use for those estimating the design of stable yield-stress emulsions. In general, the critical yield number increases with the interfacial tension and the droplet aspect ratio and will decrease with the droplet viscosity. For spherical droplets, our results computed for yield numbers below Yc suggest that the spherical shaped droplet may propagate in steady motion.

Axisymmetric Stokes flow of a spherical droplet or slip particle in the presence of a circular disk

This article deals with the axisymmetric flow of a Newtonian fluid droplet moving along the axis of a thin circular disk under the conditions of a low Reynolds number in the quasi-steady limit. The system is immersed in a second Newtonian fluid. With the same geometry, the motion of a slip spherical particle is treated also. Basic solutions are constructed in cylindrical and spherical coordinates according to the appropriate flow region. A combined analytic-numerical solution procedure in conjunction with a dual integral equation is used. Numerical results show that the normalized drag force coefficient acting on the droplet/ slip particle is obtained with good convergence for various values of viscosity ratio, slip parameter, and clearance parameter. The findings demonstrate that the results of the drag coefficient are in good agreement with the limiting cases available in the literature.

Creeping thermocapillary motion of a Newtonian droplet suspended in a viscoelastic fluid

In this work we consider theoretically the problem of a Newtonian droplet moving in an otherwise quiescent infinite viscoelastic fluid under the influence of an externally applied temperature gradient. The outer fluid is modelled by the Oldroyd-B equation, and the problem is solved for small Weissenberg and Capillary numbers in terms of a double perturbation expansion. We assume microgravity conditions and neglect the convective transport of energy and momentum. We derive expressions for the droplet migration speed and its shape in terms of the properties of both fluids. In the absence of shape deformation, the droplet speed decreases monotonically for sufficiently viscous inner fluids, while for fluids with a smaller inner-to-outer viscosity ratio, the droplet speed first increases and then decreases as a function of the Weissenberg number. For small but finite values of the Capillary number, the droplet speed behaves monotonically as a function of the applied temperature gradient for a fixed ratio of the Capillary and Weissenberg numbers. We demonstrate that this behaviour is related to the polymeric stresses deforming the droplet in the direction of its migration, while the associated changes in its speed are Newtonian in nature, being related to a change in the droplet’s hydrodynamic resistance and its internal temperature distribution. When compared to the results of numerical simulations, our theory exhibits a good predictive power for sufficiently small values of the Capillary and Weissenberg numbers.

Predictions of the behavior of a single droplet and blends composed of Newtonian/viscoelastic minor phase and viscous major phase subjected to oscillatory shear flow

The oscillatory behavior of Newtonian and viscoelastic droplets in a Newtonian phase and blends composed of viscoelastic minor phase in a Newtonian major phase are theoretically investigated in this work. The non-Newtonian constrained volume model predictions are compared to experimental oscillatory shearing flow data of droplet and blends that are available in the literature. For a Newtonian droplet in a Newtonian phase, the model describes experimental droplet behavior well at high viscosity ratio and high strain amplitude. For viscoelastic droplet in a Newtonian phase, the model predicts less deformation for viscoelastic droplet than a comparable Newtonian droplet. For large amplitude oscillatory shear rheological data of blends composed of Boger fluid minor phase in a Newtonian major phase, the model shows improvement in prediction of the elastic modulus at high viscosity ratio, compared to the Newtonian model. The model also shows good agreement with large and minimum strain elastic moduli and large and minimum rate dynamic viscosities for small and large viscosity ratio viscoelastic polymer blends. For the blend of Boger fluid minor phase in a Newtonian major phase at viscosity ratio larger than one, we find that elasticity contributes to total stress from small to large strain amplitude values. For the blend of Boger fluid minor phase in a Newtonian major phase at viscosity ratio smaller than one, we find that elasticity is important only at large values of strain amplitude. Moreover, for the aforementioned blend at viscosity ratio larger than one, the predicted Lissajous Bowditch plots of excess stress do not reflect droplet shape/droplet orientation, and the opposite is true for the small viscosity ratio blend. Investigation of droplet long semi axis for the large viscosity ratio blend at LAOS conditions reveals oscillations at two to three times the imposed frequency.

Energy stable finite element strategy for simulating spreading, sliding and rolling flow dynamics of viscoelastic droplets

Moving mesh finite element method (FEM) for simulating dynamics of viscoelastic droplets on inclined surfaces is proposed. Viscoelasticity is incorporated into the governing system employing the Oldroyd–B constitutive model. The supporting (inclined) surface is allowed to have non–homogeneous properties incorporated into the mathematical model through the generalized Navier boundary conditions (GNBC). The droplet motion (sliding and/or rolling) is handled by employing arbitrary Lagrangian Eulerian (ALE) framework. Energy balance is derived from the governing system and it is a starting point in the derivation of the numerical scheme. The overall numerical strategy is designed in such a way that a counterpart of the (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system which, in turn, guarantees the stability of the scheme in the energy norm. The framework proposed is very general and encapsulates both two and three dimensional scenarios. Numerical studies in this work focus primarily on the three dimensional scenarios since such are significantly more challenging and seem to be much scarcer in the literature. The newly proposed numerical strategy is validated on several examples to confirm the theoretical predictions. The role of viscoelasticity in the overall droplet dynamics is briefly investigated and the behaviors of Newtonian and non–Newtonian droplets are compared.

An experimental study on the impact of Boger and Newtonian droplets on spherical surfaces

This study examined the impact of Boger droplets on spherical surfaces, considering the various properties of liquids and surfaces for the first time. The experimental setup involved spheres with three different diameters composed of hydrophilic and hydrophobic surfaces. The main innovative aspect of the present study lies in the examination of the effects of rheological properties including elasticity, elongational viscosity, and the first normal stress difference on the impact of Boger droplets on solid surfaces via stress analysis. For this purpose, the results of the impact of Boger droplets are compared to the impact of equivalent Newtonian droplets with the same viscosity and surface tension coefficient. The research also explored the influences of Weber number, diameter ratio, and surface wettability on the dynamic behaviors of viscoelastic droplets during the spreading and receding stages. The results suggested that increasing the Weber number increases the maximum spreading factor of the droplets. A smaller diameter ratio required more time to reach this factor. The surface wettability significantly affected the receding phase, while its effect on the spreading stage was minimal. Droplets exhibited more retraction on surfaces with higher contact angles. The Newtonian fluids showed greater spreading and retraction than viscoelastic fluids with the same viscosity, highlighting the influence of fluid elasticity. It is shown that the normal stress caused by extensional viscosity during droplet spreading is significantly higher than the normal stress difference resulting from shear deformation, showing different behaviors between these two types of stress concerning droplets.

3D manipulation and dynamics of soft materials in 3D flows

Flow-based manipulation of particles is an essential tool for studying soft materials, but prior work has nearly exclusively relied on using two-dimensional (2D) flows generated in planar microfluidic geometries. In this work, we demonstrate 3D trapping and manipulation of freely suspended particles, droplets, and giant unilamellar vesicles in 3D flow fields using automated flow control. Three-dimensional flow fields including uniaxial extension and biaxial extension are generated in 3D-printed fluidic devices combined with active feedback control for particle manipulation in 3D. Flow fields are characterized using particle tracking velocimetry complemented by finite-element simulations for all flow geometries. Single colloidal particles (3.4 μm diameter) are confined in low viscosity solvent (1.0 mPa s) near the stagnation points of uniaxial and biaxial extensional flow for long times (≥10 min) using active feedback control. Trap stiffness is experimentally determined by analyzing the power spectral density of particle position fluctuations. We further demonstrate precise manipulation of colloidal particles along user-defined trajectories in three dimensions using automated flow control. Newtonian liquid droplets and GUVs are trapped and deformed in precisely controlled uniaxial and biaxial extensional flows, which is a new demonstration for 3D flow fields. Overall, this work extends flow-based manipulation of particles and droplets to three dimensions, thereby enabling quantitative analysis of colloids and soft materials in complex nonequilibrium flows.

Impinging blood droplets on different wettable surfaces: Impact phenomena, contact line motion, post-impact oscillation and dried stains

Understanding the underlying hydrodynamics of impinging blood droplets and finding out the physical parameters determining the bloodstain characteristics are of great importance in blood related forensic investigations. In this work, the impact of non-Newtonian blood droplets on solid surfaces ranging from lyophilic to superlyophobic was systematically investigated and compared to that of Newtonian droplets with a similar dynamic shear viscosity. We show that impinging blood droplets behave as low-viscosity Newtonian droplets in the short-time spreading, which is dominated by capillary and inertial forces, but their non-Newtonian viscoelasticity would notably affect the droplet retraction and post-impact oscillation occurring in large timescales. Whereas the strong liquid–solid adhesion and the non-Newtonian elongational viscosity hinder droplet recoiling and thus alter the impact phenomena on lyophobic and superlyophobic surfaces, the shear and elongational viscosities are coupled to result in higher damping coefficients of oscillating blood droplets after deposition, in comparison to that of impinging Newtonian droplets. The size of the dried bloodstain was found to be different from both the maximum spreading radius of the droplet that can reach during impact and the final radius of the deposited droplet after oscillation, and their correlations are highly dependent on the impact velocity and surface wettability. Moreover, the morphologic characteristics of the bloodstains would also be changed by varying either the impact velocity or the surface wettability. We envision that these findings can not only find applications in the bloodstain pattern analysis, but also provide useful information for medical diagnosis based on blood droplet test.

Shock-induced aerobreakup of a polymeric droplet

Droplet atomization through aerobreakup is omnipresent in various natural and industrial processes. Atomization of Newtonian droplets is a well-studied area; however, non-Newtonian droplets have received less attention despite their being frequently encountered. By subjecting polymeric droplets of different concentrations to the induced airflow behind a moving shock wave, we explore the role of elasticity in modulating the aerobreakup of viscoelastic droplets. Three distinct modes of aerobreakup are identified for a wide range of Weber number $({\sim }10^2\unicode{x2013}10^4)$ and elasticity number $({\sim }10^{-4}\unicode{x2013}10^2)$ variation: these modes are vibrational, shear-induced entrainment and catastrophic breakup modes. Each mode is described as a three-stage process. Stage I is droplet deformation, stage II is the appearance and growth of hydrodynamic instabilities and stage III is the evolution of liquid mass morphology. It is observed that elasticity plays an insignificant role in the first two stages but a dominant role in the final stage. The results are described with the support of adequate mathematical analysis.

Flows inside polymer microfluidic droplets: Role of elasticity

The role of elasticity on the flow topology inside viscoelastic (Boger) droplets moving in a rectangular microchannel is examined experimentally by means of micro Particle Image Velocimetry (μPIV). Polyacrylamide (PAAM) water -glycerol solutions of different concentrations are employed to vary droplet elasticity. Varying the Wi number alters the flow topology inside the Boger microdroplets, progressively reducing the number of vortical structures observed until their complete disappearance. The flow structure resembles that of Newtonian inelastic droplets for low Wi and elasticities. However, when the Wi and elastic number increase above one-i.e. elastic effects become more important- a new flow structure is observed at the front of the droplets characterized by two recirculating regions either side of the droplet centreline. Spatiotemporal maps show that the flow in this new regime fluctuates periodically around the centreline indicating the onset of an elastic instability. This flow transition is attributed to the well documented coupled effects of polymer stretching and curved streamlines and resultant hoop stresses. The findings highlight the importance elasticity can have on the nature of microdroplet flows of complex fluids and the potential of tuning elasticity to engineer flow structures for given microfluidic applications.

Settling dynamics of circular particles in vibrating tanks filled with a yield-stress liquid

The effect of sinusoidal vibration is numerically investigated on the settling dynamics of a heavy particle of circular shape immersed in a viscoplastic fluid obeying the regularized Bingham–Papanastasiou (BP) model. Having modeled the solid particle as a highly viscous Newtonian droplet, we have relied on the finite element method for solving the equations of motion for the particle and the surrounding fluid. Our numerical results could closely recover the theoretical critical Bingham number of 0.0658 for circular particles settling in Bingham materials. Using a very small threshold velocity to decide whether a particle is stuck or unstuck, it is shown that stuck particles can be excited to fall in the BP liquid provided that, for any given frequency, the amplitude is larger than a minimum value. An increase in the size of the particle or its density is predicted to reduce the threshold amplitude. In general, vibration is found to have an accelerating effect on particle settling in the BP liquid although the effect is non-monotonic. The accelerative effect of vibration is attributed to the enlarged size of the yielded zone, while the non-monotonic behavior is attributed to the effect of the sidewalls. At high density ratios, a phase lag is predicted to arise between the particle and the vessel. A comparison between the obtained numerical results with published experimental data for spherical particles suggests that, in settling flows, circular particles can be used as a good paradigm for spherical particles.

Dynamics of a Newtonian droplet in the turbulent flow of a shear thinning fluid in a microchannel

The dynamics of a single Newtonian drop in a turbulent microscopic flow are strongly influenced by the shear thinning behavior of the continuous phase. The central aim of this study is to map and describe the various dynamic modes of a single Newtonian drop in terms of both the rheology of the continuous phase and the Reynolds number. In the presence of shear thinning the drops are elongated in the form of long filaments, and the stronger the shear thinning behavior of the matrix, the longer their characteristic break time is. Single drop dynamics is found to be dominated by local extension. Our findings could be useful for addressing the turbulent emulsification of non-Newtonian fluids.

Study on the Bouncing Behaviors of a Non-Newtonian Fluid Droplet Impacting on a Hydrophobic Surface

The control of a droplet bouncing on a substrate is of great importance not only in academic research but also in practical applications. In this work, we focus on a particular type of non-Newtonian fluid known as shear-thinning fluid. The rebound behaviors of shear-thinning fluid droplets impinging on a hydrophobic surface (equilibrium contact angle θeq ≈ 108°and contact angle hysteresis Δθ ≈ 20°) have been studied experimentally and numerically. The impact processes of Newtonian fluid droplets with various viscosities and non-Newtonian fluid droplets with dilute xanthan gum solutions were recorded by a high-speed imaging system under a range of Weber numbers (We) from 12 to 208. A numerical model of the droplet impact on the solid substrate was also constructed using a finite element scheme with the phase field method (PFM). The experimental results show that unlike the Newtonian fluid droplets where either partial rebound or deposition occurs, complete rebound behavior was observed for non-Newtonian fluid droplets under a certain range of We. Moreover, the minimum value of We required for complete rebound increases with xanthan concentration. The numerical simulations indicate that the shear-thinning property significantly affects the rebound behavior of the droplets. As the amount of xanthan increases, the high shear rate regions shift to the bottom of the droplet and the receding of the contact line accelerates. Once the high shear rate region appears only near the contact line, the droplet tends to fully rebound even on a hydrophobic surface. Through the impact maps of various droplets, we found that the maximum dimensionless height Hmax* of the droplet increases almost linearly with We as Hmax* ∼ αWe. In addition, a critical value Hmax,c* for the distinction between deposition and rebound for droplets on the hydrophobic surface has been theoretically derived. The prediction of the model shows good consistency with the experimental results.