Thermocapillary motion of droplets in complex fluid flows

Understanding how the presence of thermal gradients affects the motion of bubbles and drops is a subject of great relevance both from a theoretical and a practical standpoint, particularly when gravitational effects are minimal or completely uninfluential. In the past half century, considerable progress has been made onthe investigation of the so-called thermocapillary phenomenon in an attempt to clarify the mechanisms at work in multiphase systems with liquid-liquid or liquid-gas interfaces.;Given the complexity of the problem, most of these investigations have been carried out under simplified conditions, assuming unbounded flows or considering relatively simple geometries in which the presence of solid boundaries was not explicitly taken into account. Additionally, even though non-Newtonian fluids are ubiquitous in engineering and science, the majority of these works have been carried out assuming Newtonian phases.;The aim of the present thesis is to study the thermocapillary migration of a droplet in systems exhibiting an added level of complexity, specifically in terms of wall effects, domain shape and rheological properties of the fluids. To accomplish these objectives, we rely on a concerted approach based on well-established numerical strategies and, where possible, we derive analytical solutions.;A thermocapillary solver based on a hybrid Level Set-Volume of Fluid method availablein OpenFOAM has been implemented and validated against previous analytical results, numerical solutions and experimental observations obtained in reduced gravity conditions (Sect. 3.5). In the first part of the study, we investigate the problem of a droplet interacting with the boundaries of a parallelepipedic domain.;The case study has been assessed by releasing the droplet in proximity to the lateral walls of the domain considering both adiabatic and purely conductive boundary conditions. The results showed that the droplet can experience a secondary motion perpendicular to the main direction of motion. In particular, it was observed that the droplet can either move away or towards the walls depending on the thermal boundary conditions at the wall (i.e., whether the wall is adiabatic or purely conductive) and on the extent of convective phenomena.;The investigation was then extended by adopting more complex geometries (converging and diverging channels), which were found to produce distortion of the thermal field distribution with direct consequences on the migration process (Sect. 4.2.1 and 4.2.2). In the second part of the thesis, non-Newtonian effects have been expressly considered. Specifically, the role played by the fluid's elasticity (while neglecting convective transport of energy and momentum) has been accounted for by modelling the continuous phase on the basis of constant-viscosity viscoelastic models, namely the Oldroyd-B model and FENE-CR model.;The numerical simulations were carried out for a specific value of the Capillary number and assuming thesame material properties for both phases. We investigated the effects of the various model parameters (i.e., polymer concentration and extensibility parameter) and Deborah number on the droplet motion. The results showed that the droplet speed, evaluated as a function of the Deborah number, initially decreases following a quadratic trend.;For larger Deborah number, the trend reverts its concavity and eventually reaches a plateau. In terms of shape, the results have shown that under the prescribed conditions the droplet deforms in a prolate manner and, for sufficiently large values of the Deborah number (having fixed the Capillary number), the viscoelastic stresses localised at the rear stagnation point are responsible for the formation of a pointed tail.;The viscoelastic problem was also tackled by means of perturbation techniques under the assumption of absence of confinement and weak viscoelastic effects, which allowed the derivation of corrective formulae for the droplet migration velocity and expressions describing the shape of the deformed drop. The results of the analytical solutions were found to be in fairly good agreement with the outcomes of the computations, both interms of drop shape and migration speed.Understanding how the presence of thermal gradients affects the motion of bubbles and drops is a subject of great relevance both from a theoretical and a practical standpoint, particularly when gravitational effects are minimal or completely uninfluential. In the past half century, considerable progress has been made onthe investigation of the so-called thermocapillary phenomenon in an attempt to clarify the mechanisms at work in multiphase systems with liquid-liquid or liquid-gas interfaces.;Given the complexity of the problem, most of these investigations have been carried out under simplified conditions, assuming unbounded flows or considering relatively simple geometries in which the presence of solid boundaries was not explicitly taken into account. Additionally, even though non-Newtonian fluids are ubiquitous in engineering and science, the majority of these works have been carried out assuming Newtonian phases.;The aim of the present thesis is to study the thermocapillary migration of a droplet in systems exhibiting an added level of complexity, specifically in terms of wall effects, domain shape and rheological properties of the fluids. To accomplish these objectives, we rely on a concerted approach based on well-established numerical strategies and, where possible, we derive analytical solutions.;A thermocapillary solver based on a hybrid Level Set-Volume of Fluid method availablein OpenFOAM has been implemented and validated against previous analytical results, numerical solutions and experimental observations obtained in reduced gravity conditions (Sect. 3.5). In the first part of the study, we investigate the problem of a droplet interacting with the boundaries of a parallelepipedic domain.;The case study has been assessed by releasing the droplet in proximity to the lateral walls of the domain considering both adiabatic and purely conductive boundary conditions. The results showed that the droplet can experience a secondary motion perpendicular to the main direction of motion. In particular, it was observed that the droplet can either move away or towards the walls depending on the thermal boundary conditions at the wall (i.e., whether the wall is adiabatic or purely conductive) and on the extent of convective phenomena.;The investigation was then extended by adopting more complex geometries (converging and diverging channels), which were found to produce distortion of the thermal field distribution with direct consequences on the migration process (Sect. 4.2.1 and 4.2.2). In the second part of the thesis, non-Newtonian effects have been expressly considered. Specifically, the role played by the fluid's elasticity (while neglecting convective transport of energy and momentum) has been accounted for by modelling the continuous phase on the basis of constant-viscosity viscoelastic models, namely the Oldroyd-B model and FENE-CR model.;The numerical simulations were carried out for a specific value of the Capillary number and assuming thesame material properties for both phases. We investigated the effects of the various model parameters (i.e., polymer concentration and extensibility parameter) and Deborah number on the droplet motion. The results showed that the droplet speed, evaluated as a function of the Deborah number, initially decreases following a quadratic trend.;For larger Deborah number, the trend reverts its concavity and eventually reaches a plateau. In terms of shape, the results have shown that under the prescribed conditions the droplet deforms in a prolate manner and, for sufficiently large values of the Deborah number (having fixed the Capillary number), the viscoelastic stresses localised at the rear stagnation point are responsible for the formation of a pointed tail.;The viscoelastic problem was also tackled by means of perturbation techniques under the assumption of absence of confinement and weak viscoelastic effects, which allowed the derivation of corrective formulae for the droplet migration velocity and expressions describing the shape of the deformed drop. The results of the analytical solutions were found to be in fairly good agreement with the outcomes of the computations, both interms of drop shape and migration speed

Particle accumulation structures in non-cylindrical liquid bridges under microgravity conditions

The emergence of particle accumulation structures (PAS) in noncylindrical liquid bridges (LBs) is studied numerically for a high Prandtl number liquid considering microgravity conditions. Simulations are conducted in the framework of a finite-volume (Eulerian) approach with nonisodense particles tracked using a Lagrangian, one-way coupling scheme. First, the threshold of the Marangoni-flow instability is determined as a function of the aspect ratio and the volume of liquid held between the supporting disks, thereafter, PAS formation is investigated for supercritical conditions. The overall approach is specifically conceived to provide details about the morphological evolution of these structures as the main control parameters are varied. For this reason a set of dedicated notions and definitions (such as the linear extension of the PAS, its inner core radius, and the area of the "petals"or "blades") are introduced to allow a precise quantification of a series of purely geometrical effects. Though the analysis is deliberately limited to illustrating the macroscopic patterning behavior and its relationship with the overarching factors, a model is proposed to interpret the increased ability of slender (concave) LBs to support the formation of PAS over extended ranges of values of the particle Stokes number. This model yet relies on essentially geometrical arguments, that is, the triadic relationship among the curvature of the free surface, the topology of fluid streamlines, and particle mass effects

Thermal Marangoni migration of droplets in an Oldroyd-B fluid under creeping flow conditions

In this work we investigate the impact of elasticity on the thermocapillary motion of droplets for the case of vanishing Reynolds and Marangoni numbers, i.e. when inertial terms in the momentum equation and convective-transport contribution in the energy equation are negligible. The study has been carried out employing a coupled Level-Set-Volume of Fluid approach and an adaptive mesh refinement strategy implemented in the framework of the CFD toolbox OpenFOAM

Walls and domain shape effects on the thermal marangoni migration of three-dimensional droplets

The thermocapillary motion of liquid droplets in fluid media depends on a variety of influential factors, including the not yet fully understood role played by the presence of the walls and other geometrical constraints. In order to address this specific question, in the present work we rely on a rigorous mathematical and numerical framework (including an adaptive mesh strategy), which are key to perform physically consistent and computationally reliable simulations of such a problem given the different space scales it involves. Our final aim is the proper discernment of the triadic relationship established among viscous phenomena, thermal effects and other specific behaviour due to the proximity of the droplet to a solid boundary. Different geometric configurations are considered (e.g., straight, converging and diverging channels, droplets located near a single or adjacent walls) and distinct regimes are examined (including both (Ma, Re)->0 and finite Ma flows). The results show that for straight channels the droplet generally undergoes a decrease in the migration velocity due to its proximity to the wall. Such a departure becomes larger as the Marangoni number is increased. In addition, a velocity component directed perpendicularly to the wall emerges. This effect tends to “pull” the droplet away from the solid boundary if adiabatic conditions are considered, whereas for thermally conducting sidewalls and relatively large values of the Marangoni number, the distortion of the temperature field in the region between the droplet and the wall results in a net force with a component directed towards the surface. For non-straight channels, the dynamics depend essentially on the balance between two counteracting factors, namely, the effective distribution of temperature established in the channel (for which we provide analytic solutions in the limit as Re->0) and the “blockage effect” due to the non-parallel configuration of the walls. The relative importance of these mechanisms is found to change according to the specific regime considered (creeping flow or Re=O(1))

Particle accumulation structures in a 5 cSt silicone oil liquid bridge : new data for the preparation of the JEREMI experiment

Systems of solid particles in suspension driven by a time-periodic flow tend to create structures in the carrier fluid that are reminiscent of highly regular geometrical items. Within such a line of inquiry, the present study provides numerical results in support of the space experiments JEREMI (Japanese and European Research Experiment on Marangoni flow Instabilities) planned for execution onboard the International Space Station. The problem is tackled by solving the unsteady non-linear governing equations for the same conditions that will be established in space (microgravity, 5 cSt silicone oil and different aspect ratios of the liquid bridge). The results reveal that for a fixed supporting disk radius, the dynamics are deeply influenced by the height of the liquid column. In addition to its expected link with the critical threshold for the onset of instability (which makes Marangoni flow time-periodic), this geometrical parameter can have a significant impact on the emerging waveform and therefore the topology of particle structures. While for shallow liquid bridges, pulsating flows are the preferred mode of convection, for tall floating columns the dominant outcome is represented by rotating fluid-dynamic disturbance. In the former situation, particles self-organize in circular sectors bounded internally by regions of particle depletion, whereas in the latter case, particles are forced to accumulate in a spiral-like structure. The properties of some of these particle attractors have rarely been observed in earlier studies concerned with fluids characterized by smaller values of the Prandtl number

Thermocapillary motion of a Newtonian drop in a dilute viscoelastic fluid

In this work we investigate the role played by viscoelasticity on the thermocapillary motion of a deformable Newtonian droplet embedded in an immiscible, otherwise quiescent non-Newtonian fluid. We consider a regime in which inertia and convective transport of energy are both negligible (represented by the limit condition of vanishingly small Reynolds and Marangoni numbers) and free from gravitational effects. A constant temperature gradient is maintained by keeping two opposite sides of the computational domain at different temperatures. Consequently the droplet experiences a motion driven by the mismatch of interfacial stresses induced by the non-uniform temperature distribution on its boundary. The departures from the Newtonian behaviour are quantified via the “thermal” Deborah number, De T and are accounted for by adopting either the Oldroyd-B model, for relatively small De T, or the FENE-CR constitutive law for a larger range of De T. In addition, the effects of model parameters, such as the concentration parameter c=1−β (where β is the viscoelastic viscosity ratio), or the extensibility parameter, L 2, have been studied numerically using a hybrid volume of fluid-level set method. The numerical results show that the steady-state droplet velocity behaves as a monotonically decreasing function of De T, whilst its shape deforms prolately. For increasing values of De T, the viscoelastic stresses show the tendency to be concentrated near the rear stagnation point, contributing to an increase in its local interface curvature

Development of a multiphase solver for numerical simulations of thermally driven marangoni flows

Thermocapillary flows, also known as thermal Marangoni flows, have extensive applications in a variety of different fields. Applications can be found in metal welding, growth of crystals and processing of alloys (both organic and metallic). They also have significant implications in the study of multilayer non-isothermal configurations, as well as microdroplet migration and coalescence. In this work we discuss the implementation of a non-isothermal multiphase solver based on Volume of Fluid (VOF) implemented within the open source toolbox OpenFOAM® for the numerical simulation of such flows. Interfacial tension gradients may appear consequently of a non-uniform temperature distribution along a free liquid-liquid or liquid-gas interface. The imbalance of tensile stresses, which derives from such circumstances, generates a fluid motion even in absence of any other force or external pressure gradients. The interfacial stresses are modelled via an additional body force term added to the momentum equation using a “Continuum Surface Force” (CSF) model (Brackbill et al. 1992). An energy transport equation is also solved in order to determine the temperature field evolution, which in turn influences the flow field through the Marangoni forces herein considered. To date, our solver has been tested in 2D configurations of thermally driven stratified flows. We compared our numerical simulations using a well-established code (Lappa, 2005) and good agreement was found in terms of velocity and temperature fields. Next, we aim to extend our simulations to more complex flow configurations and to consider the effect of the Marangoni stresses in non-Newtonian viscoelastic flows under non-isothermal conditions

Deformation of a ferrofluid droplet in a simple shear flow under the effect of a constant magnetic field

Abstract: In the present work, we investigate the dynamics of a droplet of ferrofluid placed in a shear flow field subjected to the additional action produced by the application of a magnetic field in a direction perpendicular to the flow. The problem is solved in the framework of a moving-boundary method based on the solution of the Navier-Stokes equations complemented with the additional equations required for the determination of the magnetic force. The results reveal interesting changes in the trends displayed by the droplet deformation and inclination angle as a function of the capillary number when the intensity of the magnetic field is varied while maintaining flow conditions corresponding to the Stokes regime. The mechanism of droplet relaxation from equilibrium when the magnetic force is suddenly removed is also investigated. According to our numerical experiments the deformation evolves in time following a harmonic decaying process, which, in the limit of small capillary number, i.e. for very small deformations, can be fairly well represented by the temporal evolution of a simple damped harmonic oscillator

Flow focusing with miscible fluids in microfluidic devices

In this work, a series of experiments and numerical simulations performed using a Volume-of-Fluid approach were carried out to investigate the flow of miscible viscous fluid systems through microfluidic flow focusing devices with one central inlet stream (with 'Fluid 1') and two lateral inlet streams (with 'Fluid 2'). The combined effect of the fluid viscosity ratio and the inlet velocity ratio on the characteristics of the central focused outlet stream was assessed in microfluidic channels with different aspect ratios. An analytical expression for the two-dimensional (2D) case, relating the width of the central focused stream in the outlet channel with the velocity ratio and the viscosity ratio, was also derived from first principles. The analytical results are in excellent agreement with the two-dimensional numerical results, and the expression is also able to represent well the experimental findings for the configuration with an aspect ratio of 0.84. The width of the central focused outlet stream at the centre plane is seen to decrease with both the velocity ratio and the viscosity ratio. The results of the three-dimensional numerical simulations and experimental measurements are in good agreement, producing further insight into the curved interface known to exist when high viscosity contrasts are present in parallel flow systems. It was observed that the interface curvature across the depth of the channel cross section is strongly dependent on the ratio of inlet viscosities and microchannel aspect ratio, highlighting the three-dimensional (3D) nature of the flow, in which confinement plays a significant role

Numerical simulations of the thermocapillary migration of a deformable Newtonian droplet in an Oldroyd-B matrix fluid in stokes flow conditions

In this work we investigate the role of elasticity on the thermal Marangoni migration of a Newtonian droplet surrounded by a viscoelastic fluid matrix in a three‐ dimensional geometry for the case of small Reynolds and Marangoni numbers. The study has been conducted in the framework of a coupled Level‐Set‐Volume of Fluid method implemented using the CFD toolbox OpenFOAM. The resulting approach was validated in a variety of flow conditions by comparing our results with analytical correlations and relevant experimental data available in literature. In the present numerical experiments, we consider a neutrally buoyant system of a Newtonian droplet placed in a container with square cross‐section filled with an Oldroyd‐B fluid (a viscoelastic fluid of constant shear viscosity). We apply a thermal gradient by keeping two sides of the box at a different constant temperature so that the temperature gradients at the liquid‐liquid interface generate an imbalance in the interfacial stresses. Such imbalance in turn is responsible of the motion of the fluid from the higher temperature region to the lower temperature region. This mechanism results in the drop moving in the opposite direction due to the thrust generated by the counter motion of the surrounding phase. In order to quantify the viscoelastic effects, we introduce a new dimensionless parameter measuring the relative importance of thermocapillary and elastic stresses. According to the numerical results, the droplet migration speed and shape are significantly different from those observed for the Newtonian‐Newtonian system. This departure of the observed dynamics from Newtonian behaviour can be ascribed to the complex interplay between different effects, including droplet morphological evolution and related distribution of surface‐tension‐driven and elastic stresses at the interface