Newtonian Drop Research Articles

On the influence of heat-induced evaporation over interfacial atomization driven by surface acoustic waves

In this work, the influence of evaporative flux over the atomization driven by surface acoustic waves (SAWs) of a Newtonian water drop is studied. The drop is placed on a heated substrate at constant temperature, higher than the saturation temperature at a given vapor pressure. In this manner, an interfacial temperature distribution arises along the drop free surface in terms evaporative mass flux and vapor recoil, which repercussion over aerosol size is studied by determining the asymptotic evolution equation governing the acoustically driven free surface. At such scenario, the connection between surface tension and temperature is also considered; thus, thermocapillary flow is incorporated into our drop model, described in terms of fundamental parameters, like the evaporation number, Marangoni number, and acoustic capillary number. Numerical solution of the evolution equation led us to obtain a simplified representation of the drop interfacial deformation mechanism, capable of predicting atomization and portraying the influence of evaporation over atomization. Subsequent analysis shows that the incorporation of evaporation at SAW atomization traduces in normal stresses counteracting the acoustic and thermocapillary effect, leading to the development of smaller drop aspect ratios with respect to the no-evaporative case. Being aware that the aerosol size is deeply related to the aspect ratio, we propose an analytical expression to estimate aerosol diameter under evaporative conditions. The results show that aspect ratio reduction leads to a decrement on aerosol size, up to two orders of magnitude, with respect to the no-evaporative case. Our study is a first approach providing insight about the importance of evaporation on aerosol regulation at SAW atomization.

From weakly to strongly nonlinear viscous drop shape oscillations: An analytical and numerical study

Nonlinear axisymmetric shape oscillations of a Newtonian drop in a vacuum are investigated using two different theoretical methods, for fundamental interest and for the significance of the oscillations in transport processes across the drop surface. The extended discontinuous Galerkin method is contrasted to the weakly nonlinear theory. While the former allows large drop surface deformation amplitudes to be analyzed with high precision and drop volume errors below 0.11% even at the largest deformations, the latter provides analytical insight into the origin of quasiperiodic time behavior of the oscillations and reveals the oscillation modes coupled in the nonlinear motion. Results from both methods for moderate initial deformation amplitudes at modes of initial drop deformation m=2, 3, and 4 are in excellent agreement, showing the time asymmetry of the oscillation and the decrease of the oscillation frequency with increasing deformation amplitude. The Fourier power spectra for the first oscillation period exhibit decreased dominant frequencies as compared to the linear results as well as the mode coupling as nonlinear effects. The numerical method is used to compute the oscillatory and damping behavior of viscous drops, as well as the interconversion of kinetic and surface energies during the oscillations at strong initial deformations. Published by the American Physical Society 2024

Spreading of a viscoelastic drop on a solid substrate

We study the spreading of Newtonian viscous (aqueous glycerin solution) and viscoelastic (aqueous polymer solution) drops on solid substrates with different wettabilities. For drops of the same zero-shear viscosity, we find in the early stages of spreading that viscoelastic drops (i) spread faster and (ii) their contact radius shows a different power law vs time than Newtonian drops. We argue that the effect of viscoelasticity is only observable for experimental time scales of the order of or larger than the internal relaxation time of the viscoelastic polymer solution. We attribute this behaviour to the shear thinning of the viscoelastic polymer solution. When approaching the contact line, the shear rate increases and the steady-state viscosity of the viscoelastic drop is lower than that of the Newtonian drop. We support our experimental findings with a simple (first-order) perturbation model that qualitatively agrees with our findings.

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.

Plastocapillarity: Partial and full Newtonian drop embedding into immiscible yield stress substrates

HypothesisRecent advances have been made in elastocapillarity; reversible 3D deformations of solid substrates with low elastic moduli from the surface tension of deposited drops. This study explores permanent deformations caused by liquid drops on immiscible yield stress substrates. We hypothesize that the substrate's rheological properties play a major role in determining the shape and stability of the drop-substrate interface, and govern partial or full embedding into the substrate. ExperimentsSubstrate yield stress magnitudes are modified through altering the mixture ratios of petroleum jelly to paraffin oil. Water drops are deposited on substrates and deformation profiles of the deformed interface are quantified. FindingsAbove a critical Bingham-Capillary number, which characterizes the ratio of yield stress magnitude to surface tension, deposited water drops deform the substrate surface permanently, but minimally. Below this value, drops become increasingly embedded as the substrate yield stress magnitude decreases, with larger indentation depths and increased circumferential ridge heights. With sufficiently low yield stress magnitudes, where surface tension forces dominate over yield stress forces, the plastically deformed ridges fully encapsulate the liquid drop surface, resulting in full drop embedding within the substrate. These results advance knowledge of interfacial wetting on soft yield stress substrates and has implications for binary fluids, functional materials, and new drug delivery systems.

On the Hinch–Kim dualism between singularity and Faxén operators in the hydromechanics of arbitrary bodies in Stokes flows

We generalize the multipole expansion and the structure of the Faxén operator in Stokes flows obtained for bodies with no-slip to generic boundary conditions, addressing the assumptions under which this generalization is conceivable. We show that a disturbance field generated by a body immersed in an ambient flow can be expressed, independently on the boundary conditions, as a multipole expansion, the coefficients of which are the moments of the volume forces. We find that the dualism between the operator giving the disturbance field of an nth order ambient flow and the nth order Faxén operator, referred to as the Hinch–Kim dualism, holds only if the boundary conditions satisfy a property that we call Boundary-Condition reciprocity (BC-reciprocity). If this property is fulfilled, the Faxén operators can be expressed in terms of the (m, n)th order geometrical moments of the volume forces (defined in the article). In addition, it is shown that in these cases, the hydromechanics of the fluid-body system is completely determined by the entire set of the Faxén operators. Finally, classical boundary conditions of hydrodynamic applications are investigated in light of this property: boundary conditions for rigid bodies, Newtonian drops at the mechanical equilibrium, porous bodies modeled by the Brinkman equations are BC-reciprocal, while deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian drops, and porous bodies modeled by the Darcy equations do not have this property. For Navier-slip boundary conditions on a rigid body, we find the analytical expression for low order Faxén operators. By using these operators, the closed form expressions for the flow past a sphere with arbitrary slip length immersed in shear and quadratic flows are obtained.

The role of viscoplastic drop shape in impact

The impact of fluid drops on solid substrates is a cardinal fluid dynamics phenomenon intrinsically related to many fields. Although these impacting objects are very often non-spherical and non-Newtonian, previous studies have mainly focused on spherical Newtonian drops. As a result, both shape and rheological effects on the drop-spreading dynamics remain largely unexplored. In the present work we use a mixed approach combining experiments with multiphase three-dimensional numerical simulations to extend the work reported by Luu & Forterre (J. Fluid Mech., vol. 632, 2009, pp. 301–327) by highlighting the fundamental role of shape in the normal impact of viscoplastic drops. Such complex fluids are highly common in various industrial domains and ideally behave either like a rigid body or a shear-rate-dependent liquid, according to the stress solicitation. Spherical, prolate, cylindrical and prismatic drops are considered. The results show that, under negligible capillary effects, the impacting kinetic energy of the drop is dissipated through viscoplastic effects during the spreading process, giving rise to three flow regimes: (i) inertio-viscous, (ii) inertio-plastic, and (iii) mixed inertio-visco-plastic. These regimes are deeply affected by the drop initial aspect ratio, which in turn reveals the possibility of using drop shape to control spreading. The physical mechanisms driving the considered phenomenon are underlined by energy budget analyses and scaling laws. The results are summarised in a two-dimensional diagram linking the drop maximum spreading, minimum height and final shape with different spreading regimes through a single dimensionless parameter, here called the impact number.

Heat and mass transfer in peristaltic flow of MHD non-miscible micropolar and Newtonian fluid through a porous saturated asymmetric channel

The current flow problem primarily focuses on the production of entropy as a result of radiative heat and mass transfer during peristaltic transport of non-miscible Newtonian and micropolar fluid through a porous saturated channel. The movement of immiscible fluid has been simulated using a two-phase flow model with Newtonian fluid at the channel's periphery and micropolar fluid in the core area. Here, under the long-wavelength approximation and low Reynolds number assumption, the linearized governing fluid flow equations were solved by classical technique and obtained closed-form solutions for pressure rise, volume flow rate, velocity, temperature distribution, and concentration. In the present peristaltic flow problem, authors evaluated the entropy analysis in non-miscible type of Newtonian and micropolar fluid through the porous saturated channel under the influence of heat radiation, mass transfer, and an oriented magnetic field. The production of entropy and Bejan number due to concentration distribution in peristaltic motion of non-miscible type of Newtonian and micropolar fluid in a porous medium is the novelty of the work. From this study, we obtained that on raising the Soret and Schmidt number, the concentration of the immiscible micropolar and Newtonian fluid drops, and entropy generation number rises. Also, the temperature and flow distribution of the immiscible micropolar and Newtonian fluid get decreased on raising the permeability parameter and Hartmann number. The authors come to the significant conclusion that, during the peristaltic motion of immiscible fluid in a porous saturated asymmetric channel, naturally occurring porous material achieved higher values of flow properties like axial pressure gradient, temperature profile, concentration, and entropy generation number than man-made porous material. The newly obtained findings of the present peristaltic flow problem are validated with the past published research work. The findings of this study may be used in biomedical engineering, applications, including the thermal therapy procedure.

Drop rise and interfacial coalescence initiation in Bingham materials

Drop coalescence in viscoplastic materials is present in various industrial applications and the environment. The plasticity of the surrounding material affects drop rise, collision, and film drainage dynamics, and knowledge about these is essential for designing and operating industrial mixer and separator units. Despite its importance, the coalescence of drops in yield stress materials is not entirely understood. In this work, we investigate the effects of the surrounding material plasticity on the rise and interfacial coalescence initiation of Newtonian drops using direct numerical simulations. Plastic effects contribute to the formation of smaller and spherical films, which facilitate the film drainage process on the one hand, but also to an increase in the resistance of the film to flow, which makes the drainage process more difficult on the other hand. The fluids’ interfaces are less deformable for high surface tension regimes, and the flow resistance effects become more significant than the film shape change effects. As a result, the drainage time tends to increase with the level of plasticity. For low surface tension regimes, the fluid interface is more deformable, and film drainage time tends to decrease with an increase in the level of plasticity.

Deformation of a Hele–Shaw drop undergoing quadratic flow

A Hele–Shaw quadratic flow in the form of a six-port microfluidic device is employed to study the deformation of a single channel-spanning or Hele–Shaw Newtonian drop suspended in a Newtonian medium. An initially circular drop in a quadratic flow deforms into a regular triarcle, i.e., a rounded-corner triangle. Theoretically, the deformation is calculated in the limit of small capillary number, which is defined as Ca=Cμa4/(4γb2), where C is the quadratic flow rate, μ is the suspending fluid viscosity, a is the drop radius, γ is the interfacial tension, and b is the channel depth. The theory matches reasonably well with the experiment for small capillary numbers. This is the new way of deforming drops on a Hele–Shaw quadratic platform experimentally and may lead to measurements such as complex interfacial properties and breakup.

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.

Numerical simulation of the coalescence-induced polymeric droplet jumping on superhydrophobic surfaces

Self-propelled jumping of two polymeric droplets on superhydrophobic surfaces is investigated by three-dimensional direct numerical simulations. Two identical droplets of a viscoelastic fluid slide, meet and coalesce on a surface with contact angle 180 degrees. The droplets are modelled by the Giesekus constitutive equation, introducing both viscoelasticity and a shear-thinning effects. The Cahn-Hilliard Phase-Field method is used to capture the droplet interface. The simulations capture the spontaneous coalescence and jumping of the droplets. The effect of elasticity and shear-thinning on the coalescence and jumping is investigated at capillary-inertial and viscous regimes. The results reveal that the elasticity of the droplet changes the known capillary-inertial velocity scaling of the Newtonian drops at large Ohnesorge numbers; the resulting viscoelastic droplet jumps from the surface at larger Ohnesorge numbers than a Newtonian drop, when elasticity gives rise to visible shape oscillations of the merged droplet. The numerical results show that polymer chains are stretched during the coalescence and prior to the departure of two drops, and the resulting elastic stresses at the interface induce the jumping of the liquid out of the surface. This study shows that viscoelasticity, typical of many biological and industrial applications, affects the droplet behaviour on superhydrophobic and self-cleaning surfaces.

Numerical simulation of viscoelastic effects in peristaltic transport of drops

• Under creeping-flow conditions, elasticity of a drop slightly hinders its peristaltic transport. • Elasticity of the liquid surrounding the drop has a significant but non-monotonic effect on the drop's peristaltic transport. • Viscosity of the bulk liquid can have an accelerating or decelerating effect on the drop's velocity. • Peristalsis can be regarded as a viable option for the transport of elastic drops in microfluidic systems. Peristalsis is known to be a viable option for the transport of Newtonian drops suspended in another Newtonian liquid. In certain applications, the drop or the bulk liquid is known to be, more or less, viscoelastic. In the present work the effect of a liquid's elasticity is investigated on the peristaltic transport of a single, circular drop. Having assumed that the liquid obeys the Oldroyd-B model, numerical results have been obtained at Deborah numbers up to 15 for two different cases: (1) the drop is viscoelastic, and (2) the liquid surrounding the drop is viscoelastic. Based on the results obtained in this work, it is predicted that under creeping-flow conditions, a drop's elasticity has a negligible effect on its transport velocity provided it is placed on the centerline. But for off-center viscoelastic drops the effect of elasticity is significant and delay their migration to the centerline. For the case in which only the liquid surrounding the drop is viscoelastic, elastic stresses are predicted to have a significant but non-monotonic effect on the drop velocity. In both cases, the response of a drop is predicted to strongly depend on its size and also on the cross-viscosity ratio.

Weakly nonlinear shape oscillations of a Newtonian drop

Nonlinear axisymmetric shape oscillations of a Newtonian drop in a vacuum are investigated theoretically, for fundamental interest and for their relevance in transport processes across the drop surface. The weakly nonlinear analysis is carried out for, but not limited to, the modes of initial drop deformation up to m = 4. The drop Ohnesorge number covers the range between 0.01 and 1. The weakly nonlinear approach, which is carried to third order, accounts for the coupling of different oscillation modes. With increasing surface deformation, the oscillations develop an asymmetry of the times during one period the drop spends in different states of deformation, a frequency decrease below the linear value, and quasi-periodicity of the motion. In contrast to the inviscid case [D. Zrnić and G. Brenn, “Weakly nonlinear shape oscillations of inviscid drops,” J. Fluid Mech. 923, A9 (2021)], the present analysis reveals the frequency decrease and the quasi-periodicity already in the second-order approximation. The results are positively validated against relevant literature. The theory quantifies the effects of viscosity, measured by the drop Ohnesorge number, dampening the nonlinear behavior and enhancing the coupling of different oscillation modes [E. Becker et al., “Nonlinear dynamics of viscous droplets,” J. Fluid Mech. 258, 191 (1994)]. The present theory reveals the quasi-periodicity of nonlinear viscous drop shape oscillations at strong deformation. The resultant drop motion, starting from a higher-order mode of initial deformation, for which the drop exhibits aperiodic linear behavior, may turn into damped oscillatory with ongoing time due to the coupling to lower-order modes.

Coalescence of Microscopic Polymeric Drops: Effect of Drop Impact Velocities.

In this paper, we employ the direct numerical simulation (DNS) method for probing three-dimensional, axisymmetric coalescence of microscale, power-law-obeying, and shear-thinning polymeric liquid drops of identical sizes impacting a solid, solvophilic substrate with a finite velocity. Unlike the cases of drop coalescence of Newtonian liquid drops, coalescence of non-Newtonian polymeric drops has received very little attention. Our study bridges this gap by providing (1) the time-dependent, three-dimensional (3D) velocity field and 3D velocity vectors inside two coalescing polymeric drops in the presence of a solid substrate and (2) the effect of the drop impact velocity (on the solid substrate), quantified by the Weber number (We), on the coalescence dynamics. Our simulations reveal that the drop coalescence is qualitatively similar for different We values, although the velocity magnitudes involved, the time required to attain different stages of coalescence, and the time needed to attain equilibrium vary drastically for finitely large We values. Finally, we provide detailed simulation-based, as well as physics-based, scaling laws describing the growth of the height and the width of the bridge (formed due to coalescence) dictating the 3D coalescence event. Our analyses reveal distinct scaling laws for the growth of bridge height and width for early and late stages of coalescence as a function of We. We also provide simulation-based coalescence results for the case of two unequal sized drops impacting on a substrate (nonaxisymmetric coalescence) as well as results for axisymmetric coalescence for drops of different rheology. We anticipate that our findings will be critical in better understanding events such as inkjet or aerosol jet polymer printing, dynamics of polymer blends, and many more.

The influence of the negative wake on the deformation and breakup of viscoelastic droplets

Experiments were performed using visual and PIV techniques in order to study the appearance of a negative wake as well as its influence upon the deformation and breakup of droplets rising in viscoelastic fluids. In this report, Newtonian and viscoelastic drops were injected through different viscoelastic fluids; the deformation of the droplets was then followed and analyzed. In the case of Newtonian drops traveling through a viscoelastic fluid, a tail appears which later breaks into satellite droplets; a negative wake is present on the sides of the tail. The viscoelastic drops also exhibit a tail which is more resistant to rupture and the negative wake appears after the tail; additionally, a bump appears at the tip of the tail which enhances its elongation and determines the onset of breakup.

Effect of rheology and interfacial tension on spreading of emulsion drops impacting a solid surface

This paper presents an experimental and theoretical study of Newtonian and non-Newtonian (Bingham plastic) emulsion drop impact on a solid non-heated surface. The utilization of different emulsifiers at a constant concentration of continuous and dispersed phases in emulsions allows the considerable variation of the surface tension at the liquid–liquid interface. Our data for the maximum spreading diameter of water, n-decane, and emulsion drops impacting on a surface are compared with that predicted from the existing models for single-phase liquid drops. All selected models underpredict the experimental data. As a result, the importance of considering the capillary effects at the internal interfaces of the emulsion drops and the careful examination of all rheological properties in the case of Bingham plastic fluids is confirmed experimentally and is taken into account theoretically. The models of Pasandideh-Fard et al. [“Capillary effects during droplet impact on a solid surface,” Phys. Fluids 8, 650 (1996)] and Ukiwe and Kwok [“On the maximum spreading diameter of impacting droplets on well-prepared solid surfaces,” Langmuir 21, 666–673 (2005)] are modified and adapted to the emulsion drop by means of including the additional surface energy term at the liquid–liquid interface of the emulsion drop in the energy conservation equation and the non-Newtonian Reynolds number. The predictions of the maximum spreading diameter give good agreement with the measured one. Several constraints and future lines of research that relate to a specific behavior of the compound liquid drops at the impact on a solid surface are highlighted.

Benchmark solutions for flows with rheologically complex interfaces

Complex fluid–fluid interfaces determine for an outsized part the macroscopic material properties of foams and emulsions that appear in applications like food, materials processing and consumer care products. As a step towards predicting these properties, a 2D axisymmetric and a 3D finite element model are developed for simulating the dynamics of one Newtonian drop by a Newtonian matrix fluid with a rheologically complex sharp interface in between. Interfaces with constant interfacial surface tension and with viscous, elastic and viscoelastic extra interfacial stresses are considered. The model has been validated by means of the tactic of manufactured solutions and by comparison with results from other studies using different discretisation methods. Higher-order convergence in space and time is obtained, demonstrating correct implementation of our numerical methods. Benchmark solutions for a drop with a Kelvin–Voigt interface under simple shear flow are provided. Compared to a viscous interface, the drop deformation becomes smaller and therefore the drop becomes less oriented within the direction of flow if interfacial elasticity is added. Fluid–fluid interfaces play a key role within the processing and functioning of multicomponent materials. samples of such materials are emulsions, where fluid drops of 1 phase are dispersed into another fluid phase immiscibly. Dispersion are often useful for designing material structures to e.g. gain synergy of properties. Emulsions are partly for this reason utilized in a spread of applications like in food, paint, cosmetics, materials processing (e.g. polymer blends) and within the industry , but they also appear in nature and biology. The interfacial area to volume ratio in emulsions is comparatively large, since albeit the drops in e.g. polymer blends are small in size (about tens of); they're present in large quantities. Therefore, the interfacial properties determine for an outsized part the general macroscopic material behaviour. When applying flow to an emulsion, the structure is continuously developing through the consecutive and simultaneous deformation, break-up and coalescence of drops interacting with one another and therefore the surrounding fluid during a complex manner. Curvature changes of the interface are opposed or helped by interfacial surface tension between the phases. Surfactants that are often added as stabilisers to the mixture and adsorb at the interface, lower the interfacial surface tension to hamper or hinder coalescence Hence, surfactants are required to get stable emulsions, but they also further complicate the behaviour of the interface. Surfactant concentration gradients within the interface cause spatially varying Marangoni stresses. Surfactant transport within the interface or between the interface and therefore the bulk introduces non-trivial time scales. Besides, interactions within the interface thanks to a big microstructure cause extra and deviatoric interfacial stresses and possibly rheologically complex behaviour of the interface, These complex interfaces can behave intrinsically viscoelastic.

Viscoelasticity and elastocapillarity effects in the impact of drops on a repellent surface.

We investigate freely expanding viscoelastic sheets. The sheets are produced by the impact of drops on a quartz plate covered with a thin layer of liquid nitrogen that suppresses shear viscous dissipation as a result of the cold Leidenfrost effect. The time evolution of the sheet is simultaneously recorded from top and side views using high-speed cameras. The investigated viscoelastic fluids are Maxwell fluids, which are characterized by low elastic moduli, and relaxation times that vary over almost two orders of magnitude, thus giving access to a large spectrum of viscoelastic and elastocapillary effects. For the purposes of comparison, Newtonian fluids, with viscosity varying over three orders of magnitude, are also investigated. In this study, dmax, the maximal expansion of the sheets, and tmax the time to reach this maximal expansion from the time at impact, are measured as a function of the impact velocity. By using a generalized damped harmonic oscillator model, we rationalize the role of capillarity, bulk elasticity and viscous dissipation in the expansion dynamics of all investigated samples. In the model, the spring constant is a combination of the surface tension and the bulk dynamic elastic modulus. The time-varying damping coefficient is associated to biaxial extensional viscous dissipation and is proportional to the dynamic loss modulus. For all samples, we find that the model reproduces accurately the experimental data for dmax and tmax.

Decoupled rolling, sliding and sticking of a viscoplastic drop on a superhydrophobic surface

Abstract