Surface Dilatational Viscosity Research Articles

Transient instability characteristics of fluid film bearings induced by bubble inclusion

Bubble inclusion in the tribo-pair leads to two-phase fluid lubrication. Upon the initial introduction of air bubbles to the tribo-pair, it can lead to instability in the operational state. A numerical model is formulated by coupling the fluid Reynolds equation, bubble dynamics equation, and rotor dynamics equation. Various parameters, such as hydrodynamic pressure, fluid carrying capacity, rotor trajectory, and equilibrium position, are employed to characterize the impact of operational and fluid interface parameters on the bubble entrainment process. The findings reveal that the hydrodynamic pressure plays a crucial role in establishing the correlation between velocity and fluctuations in kinetic parameters. Surface tension predominantly influences bubbles within the dispersion region, while surface dilatational viscosity affects the entire domain. Lower surface dilatational viscosity or neglecting surface tension can trigger larger fluctuations in the rotor trajectory. Changes in liquid-phase viscosity result in fluctuations in bubble behavior and dynamics parameters, influenced by the equilibrium position and the effect of hydrodynamic pressure. Higher initial gas-phase volume fractions lead to a more pronounced reduction in fluid-carrying capacity and increased system instability.

Numerical investigation of the effect of bubble properties on the linear resonance frequency shift due to inter-bubble interactions in ultrasonically excited lipid coated microbubbles

Ultrasonically excited microbubbles (MBs) have numerous applications in various fields, such as drug delivery, and imaging. Ultrasonically excited MBs are known to be nonlinear oscillators that generate secondary acoustic emissions in the media when excited by a primary ultrasound wave. The propagation of acoustic waves in the liquid is limited to the speed of sound, resulting in each MB receiving the primary and secondary waves at different times depending on their distance from the ultrasound source and the distance between MBs. These delays are referred to as primary and secondary delays, respectively. A previous study demonstrated that the inclusion of secondary delays in a model describing the interactions between MBs exposed to ultrasound results in an increase in the linear resonance frequency of MBs as they approach each other. This work investigates the impact of various MB properties on the change in linear resonance frequency resulting from changes in inter-bubble distances. The effects of shell properties, including the initial surface tension, surface dilatational viscosity of the shell monolayer, elastic compression modulus of the shell, and the initial radius of the MBs, are examined. MB size is a significant factor influencing the rate of linear resonance frequency increase with increasing concentration. Moreover, it is found that the shell properties of MBs play a negligible role in the rate of change in linear resonance frequency of MBs as the inter-bubble distances change.The findings of this study have implications for various applications of MBs in the biomedical field. By understanding the impact of inter-bubble distances and shell properties on the linear resonance frequency of MBs, the utilization of MBs in applications reliant on their resonant behavior can be optimized.

Research on two-phase thermo-hydrodynamic lubrication performance considering bubble dynamics

Gas-liquid two-phase fluid in the lubrication gap exist in many important engineering tribo-pairs. Based on the Reynolds equation, the bubble dynamics equation, and the full energy equation for two-phase fluids, the thermo-hydrodynamic lubrication problem in the lubrication gap is investigated. The evolution of the fluid phase composition with operating time and its impact on the thermodynamic properties are studied using the developed model. The mechanisms by which sliding velocity, clearance, surface dilatational viscosity, surface tension, and initial gas volume fraction affect two-phase fluid lubrication behavior are evaluated. The results show that hydrodynamic pressure-driven bubbles eventually condense in the dispersion area and have a major impact on the temperature distribution. Velocity and clearance affect the gas-phase expansion region and initial formation location in the steady state, respectively. The pressure amplitude, bubble expansion, and horizontal and vertical gas phase expansion in the dispersion zone were affected differently by surface dilatational viscosity and surface tension. It becomes difficult to preserve the initial gas-phase composition inside the pressure convergence zone when bubble migration is taken into account, which may be crucial for establishing gas-liquid two-phase lubrication.

Thermocapillary migration of a deformed droplet in the combined vertical temperature gradient and thermal radiation

Thermocapillary migration of a deformed droplet in the combined vertical temperature gradient and thermal radiations with uniform and non-uniform fluxes is first analyzed. The creeping flow solutions show that the deformed droplet has a slender or a cardioid shape, which depends on the form of the radiation flux. The deviation from a sphere depends not only on the viscosity and the conductivity ratios of two-phase fluids but also on capillary and thermal radiation numbers. Moreover, in the roles of interfacial rheology on thermocapillary migration of a deformed droplet, only the surface dilatational viscosity and the surface internal energy can reduce the steady migration velocity, but the surface shear viscosity has not any effects on the steady migration velocity. The surface shear and dilatational viscosities affect the deformation of the droplet by increasing the viscosity ratio of two-phase fluids. The surface internal energy directly reduces the deformation of the droplet. However, the deformed droplet still keeps its original shape without the influence of interfacial rheology. Furthermore, it is found that, based on the net force balance condition of the droplet, the normal stress balance at the interface can be used to determine the steady migration velocity, which is not affected by the surface deformation in the creeping flow. From the expressions of the normal/the tangential stress balance, it can be proved that the surface shear viscosity does not affect the steady migration velocity. The results could not only provide a valuable understanding of thermocapillary migration of a deformed droplet with/without the interfacial rheology in a vertical temperature gradient controlled by thermal radiation but also inspire its potential practical applications in microgravity and microfluidic fields.

Numerical investigation of the effect of surface viscosity on droplet breakup and relaxation under axisymmetric extensional flow

In this study, we perform boundary-integral simulations to investigate the role of interfacial viscosity in the deformation and breakup of a single droplet suspended in an axisymmetric extensional flow under the Stokes flow regime. We model the insoluble surfactant monolayer using the Boussinesq–Scriven constitutive relationship for a Newtonian interface. We compare the deformation and breakup results from our boundary-element simulations with results from small deformation perturbation theories. We observe that the surface shear/dilational viscosity increases/decreases the critical capillary number beyond which the droplet becomes unstable and breaks apart by reducing/increasing the droplet deformation at a given capillary number compared with a clean droplet. We present the relative importance of surface shear and dilational viscosity on droplet stability based on their measured values reported in experimental studies on surfactants, lipid bilayers and proteins. In the second half of the paper, we incorporate the effect of surfactant transport by solving the time-dependent convection–diffusion equation and consider a nonlinear equation of state (Langmuir adsorption isotherm) to correlate the interfacial tension with the changes in surfactant concentration. We explore the coupled influence of pressure-dependent surface viscosity and Marangoni stresses on droplet deformation and breakup. In the case of a droplet with pressure-dependent surface shear viscosity, we find that a droplet with pressure-thinning/thickening surfactant is less/more deformed than a droplet with pressure-independent surfactant. We conclude by discussing the combined impact of surface viscosity and surfactant transport on the relaxation of an initially extended droplet in a quiescent external fluid.

A model for bubble dynamics in a protein solution

This paper presents a model for bubble dynamics in a protein solution, including varying surface tension and dynamic adsorption/desorption of protein onto the bubble surface. We model the protein-coated bubble surface as a linear viscoelastic interface. Two cases are studied to understand the importance of the surface excess stress: (1) bubble dissolution in a protein solution due to gas diffusion; and (2) bubble response to an imposed fluctuating pressure field. In the first case study, the surface excess stress stabilizes the bubble against dissolution. Initially, the surface excess stress is negligible, and the dissolution rate is governed by the Weber number, which compares the gas inertial force and the surface tension force. As the bubble shrinks, the surface excess stress grows and eventually balances the surface tension. After that, the dissolution rate is governed by the protein desorption rate and the elasto-capillary number, which compares the surface tension and the surface dilatational elasticity. Our model predictions for the dissolution process agree with experiments before the bubble buckles. In the second case study, the surface dilatational viscosity and dilatational elasticity add resistance and stiffness to the system, respectively. Including the surface excess stress increases (or reduces) the amplitude of the bubble radius if the frequency of the imposed fluctuating pressure is greater (or less) than a critical value. These results highlight the importance of the surface rheology on the protein-coated bubble dynamics, which has applications in drug delivery and ultrasound contrast agents.

The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium

Abstract In the present article, the analytical solution for creeping motion of a drop/bubble characterized by insoluble surfactant is examined at the instant it passes the center of a spherical container filled with Newtonian fluid at low Reynolds number. The presence of surfactant characterizes the interfacial region by an axisymmetric interfacial tension gradient and coefficient of surface dilatational viscosity. Under the assumption of the small capillary number, the deformation of spherical phase interface is not taken into account. The computations not only yield information on drag force and wall correction factor, but also on interfacial velocity and flow field for different values of surface tension gradient and surface dilatational viscosity. In the limiting cases, the analytical solutions describing the drag force and wall correction factor for a drop in a bounded medium reduces to expressions previously stated by other authors in literature. The results reveal the strong influence of the surface dilatational viscosity and surface tension gradient on the motion of drop/bubble. Increasing the surface tension gradient and surface dilatational viscosity, results in linear variation of drag force. When the surface tension gradient increases, the drag force for unbounded medium increases more as compared to the bounded medium hence wall correction factor decreases with increase in surface tension gradient whereas it increases with increase in surface dilatational viscosity.

A natural source of saponin: Comprehensive study on interfacial properties of Chubak (Acanthophyllum Glandulosum) root extract and related saponins

Chubak (Acanthophyllum Caryophyllaceae) is a shrubby plant with saponin-rich roots. This feature makes this plant to a new source of natural surfactant. This article presents some basic information about the surface activity of Chubak root extract (CRE) and the corresponding purified saponin. The Calendasaponin B and Calendasaponin D were characterized in the purified saponin using Liquid Chromatography-Mass Spectrometry as the two main saponins. The dynamic surface tension and dilational surface visco-elasticity of CRE and the purified saponin were studied using a profile analysis tensiometer. The equilibrium surface tensions of CRE and the purified saponin in the concentration range up to the CMC show that the used purification method is not efficient to separate the complete saponin content of CRE. The surface activity of CRE started at 0.005 g/l, while the minimum saponin concentration for a first visible surface tension decrease was 0.0012 g/l. The uniform trend in surface dilational elasticity and viscosity of CRE and the purified saponin is the evidence for the similarity of dominant surface-active compounds in both samples. Comparing the short time surface tension of oscillating and steady drops showed the impact of adsorption layer oscillations on the surface tension variations.

Impact of surface viscosity on the stability of a droplet translating through a stagnant fluid

This study examines the impact of interfacial viscosity on the stability of an initially deformed droplet translating through an unbounded quiescent fluid. The boundary-integral formulation is employed to investigate the time evolution of a droplet in the Stokes flow limit. The droplet interface is modelled using the Boussinesq–Scriven constitutive relationship having surface shear viscosity $\eta _\mu$ and surface dilatational viscosity $\eta _\kappa$ . We observe that, below a critical value of the capillary number, $Ca_C$ , the initially perturbed droplet reverts to its spherical shape. Above $Ca_C$ , the translating droplet deforms continuously, growing a tail at the rear end for initial prolate perturbations and a cavity for initial oblate perturbations. We find that surface shear viscosity inhibits the tail/cavity growth at the droplet's rear end and increases the $Ca_C$ compared with a clean droplet. In contrast, surface dilatational viscosity increases tail/cavity growth and lowers $Ca_C$ compared with a clean droplet. Surprisingly, both shear and dilatational surface viscosity appear to delay the time at which pinch off occurs, and hence satellite droplets form. Lastly, we explore the combined influence of surface viscosity and surfactant transport on droplet stability by assuming a linear dependence of surface tension on surfactant concentration and exponential dependence of interfacial viscosities on the surface pressure. We find that pressure-thinning/thickening effects significantly affect the droplet dynamics for surface shear viscosity but play a small role for surface dilatational viscosity. We lastly provide phase diagrams for the critical capillary number for different values of the droplet's viscosity ratio and initial Taylor deformation parameter.

Modelling Lipid-Coated Microbubbles in Focused Ultrasound Applications at Subresonance Frequencies

We present a computational study of the behaviour of a lipid-coated SonoVue microbubble with initial radius 1 µm ≤ R0 ≤ 2 µm, excited at frequencies (200–1500 kHz) significantly below the linear resonance frequency and pressure amplitudes of up to 1500 kPa—an excitation regime used in many applications of focused ultrasound. The bubble dynamics are simulated using the Rayleigh–Plesset equation and the Gilmore equation, in conjunction with the Marmottant model for the lipid monolayer coating. Also, a new continuously differentiable variant of the Marmottant model is introduced. Below the onset of inertial cavitation, a linear regime is identified in which the maximum pressure at the bubble wall is linearly proportional to the excitation pressure amplitude and the mechanical index. This linear regime is bounded by the Blake pressure, and, in line with recent in vitro experiments, the onset of inertial cavitation is found to occur at an excitation pressure amplitude of approximately 130–190 kPa, depending on the initial bubble size. In the nonlinear regime the maximum pressure at the bubble wall is found to be readily predicted by the maximum bubble radius, and both the Rayleigh–Plesset and Gilmore equations are shown to predict the onset of sub- and ultraharmonic frequencies of the acoustic emissions compared with in vitro experiments. Neither the surface dilational viscosity of the lipid monolayer nor the compressibility of the liquid has a discernible influence on the quantities studied, but accounting for the lipid coating is critical for accurate prediction of the bubble behaviour. The Gilmore equation is shown to be valid for the bubbles and excitation regime considered, and the Rayleigh–Plesset equation also provides accurate qualitative predictions, even though it is outside its range of validity for many of the cases considered.

Drag on a spherical particle at the air–liquid interface: Interplay between compressibility, Marangoni flow, and surface viscosities

We investigate the flow of viscous interfaces carrying an insoluble surface active material, using numerical methods to shed light on the complex interplay between Marangoni stresses, compressibility, and surface shear and dilatational viscosities. We find quantitative relations between the drag on a particle and interfacial properties as they are required in microrheology, i.e., going beyond the asymptotic limits. To this end, we move a spherical particle probe at constant tangential velocity, symmetrically immersed at either the incompressible or compressible interface, in the presence and absence of surfactants, for a wide range of system parameters. A full three-dimensional finite element calculation is used to reveal the intimate coupling between the bulk and interfacial flows and the subtle effects of the different physical effects on the mixed-type velocity field that affects the drag coefficient, both in the bulk and at the interface. For an inviscid interface, the directed motion of the particle leads to a gradient in the concentration of the surface active species, which in turn drives a Marangoni flow in the opposite direction, giving rise to a force exerted on the particle. We show that the drag coefficient at incompressible interfaces is independent of the origin of the incompressibility (dilatational viscosity, Marangoni effects or a combination of both) and that its higher value can not only be related to the Marangoni effects, as suggested earlier. In confined flows, we show how the interface shear viscosity suppresses the vortex at the interface, generates a uniform flow, and consequently increases the interface compressibility and the Marangoni force on the particle. We mention available experimental data and provide analytical approximations for the drag coefficient that can be used to extract surface viscosities.

Universal Thinning of Liquid Filaments under Dominant Surface Forces.

Theory and numerical simulations of the thinning of liquid threads at high superficial concentration of surfactants suggest the existence of an asymptotic regime where surface tension balances surface viscous stresses, leading to an exponential thinning with an e-fold time F(Θ)(3μ_{s}+κ_{s})/σ, where μ_{s} and κ_{s} are the surface shear and dilatational viscosity coefficients, σ is the interfacial tension, Θ=κ_{s}/μ_{s}, and F(Θ) is a universal function. The potential use of this phenomenon to measure the surface viscosity coefficients is discussed.

Deformation and burst of a liquid droplet with viscous surface moduli in a linear flow field

We investigate conditions for the breakup of a droplet with viscous surface moduli, under the assumption of weak flow and negligible Marangoni forces. The viscous interface is treated as a homogenous fluid obeying the Boussinesq--Scriven constitutive law. We observed that the presence of surface shear viscosity stabilizes the droplet, while that of surface dilational viscosity destabilizes it. The destabilizing effect of surface dilational viscosity appears similar to surfactant convection effects, while the stabilizing impact of surface shear viscosity appears similar to surfactant dilution.

Slow viscous flow past cylindrical particles with thin liquid layer: Cell model

In order to investigate the hydrodynamic interaction between an interface and a cylindrical particle, it is essential to compute the drag and other physical properties exerted on the cylinder in the vicinity of the interface. The present study examines the analytical solution of slow viscous flow past a swarm of cylindrical particles, where each particle consists of a solid core covered by a liquid shell coated with thin monomolecular layer of surfactant using cell model technique. We have assumed that each particle is enclosed by a hypothetical cell. The boundary conditions of Happel, Kuwabara, Kvashnin and Mehta–Morse models are considered on hypothetical cell. The effects of Surfactants are accommodated by considering the Scriven boundary conditions at the shell involving Surface Shear viscosity and Surface Dilatational viscosity. The variation of drag force and hydrodynamic permeability with various parameters is graphically presented. The result reveals in all cases the strong influence of surface viscosity on the motion of cylindrical particle, following cell model technique. It is seen that the presence of surfactant layer increases drag force on the contrary decreases hydrodynamic permeability. In the limiting cases, the analytical solutions describing the drag force reduces to known results of Happel, Kuwabara, Kvashin and Mehta-Morse.

Influence of complex interfacial rheology on the thermocapillary migration of a surfactant-laden droplet in Poiseuille flow

The effect of surface viscosity on the motion of a surfactant-laden droplet in the presence of a non-isothermal Poiseuille flow is studied, both analytically and numerically. The presence of bulk-insoluble surfactants along the droplet surface results in interfacial shear and dilatational viscosities. This, in turn, is responsible for the generation of surface-excess viscous stresses that obey the Boussinesq-Scriven constitutive law for constant values of surface shear and dilatational viscosities. The present study is primarily focused on finding out how this confluence can be used to modulate droplet dynamics in the presence of Marangoni stress induced by nonuniform distribution of surfactants and temperature along the droplet surface, by exploiting an intricate interplay of the respective forcing parameters influencing the interfacial stresses. Under the assumption of negligible fluid inertia and thermal convection, the steady-state migration velocity of a non-deformable spherical droplet, placed at the centerline of an imposed unbounded Poiseuille flow, is obtained for the limiting case when the surfactant transport along the interface is dominated by surface diffusion. Our analysis proves that the droplet migration velocity is unaffected by the shear viscosity whereas the dilatational viscosity has a significant effect on the same. The surface viscous effects always retard the migration of a surfactant-laden droplet when the temperature in the far-field increases in the direction of the imposed flow although the droplet always migrates towards the hotter region. On the contrary, if a large temperature gradient is applied in a direction opposite to that of the imposed flow, the direction of droplet migration gets reversed. However, for a sufficiently high value of dilatational surface viscosity, the direction of droplet migration reverses. For the limiting case in which the surfactant transport along the droplet surface is dominated by surface convection, on the other hand, surface viscosities do not have any effect on the motion of the droplet. These results are likely to have far-reaching consequences in designing an optimal migration path in droplet-based microfluidic technology.

Erratum to: Modeling of complex interfaces for pendant drop experiments

The original version of this article unfortunately contained mistakes. Theo A. Tervoort was not listed among the authors. The correct information is given above. In Balemans et al. (2016), an axisymmetric finite element model is presented to study the behaviour of complex interfaces in pendant drop experiments. While the bulk fluid of the pendant drop is modeled as a Newtonian fluid, the interface behaviour is described using a quasi-linear Kelvin-Voigt constitutive model. To study the influence of the five model parameters of this constitutive model, a parameter study is performed. Furthermore, small amplitude oscillatory measurements are simulated to demonstrate the applicability of the numerical model. The quasi-linear two-dimensional Kelvin-Voigt viscoelastic constitutive equation as proposed by Verwijlen et al. (2014) and used in Balemans et al. (2016) is not material frame indifferent, because the Cauchy-Green deformation tensor Cs is directly coupled to the surface stress tensor ς. Instead the Finger tensor Bs should be used to construct a material frame indifferent constitutive equation. The correct formulation of the quasi-linear twodimensional Kelvin-Voigt constitutive equation becomes (Formula presented.) where J = det(Bs). The constitutive equation has both viscous and elastic properties, and all simulations for Balemans et al. (2016) have been repeated after implementation of Eq. 1 into the numerical model. The results show minor differences for small deformations and small values of the surface elasticity parameters. However, the results presented in Figure 18 (only K = 100 μN/mm) and Figure 21 are significantly different. These deviating results are summarized in Fig. 1, where the apex length in time for G = 1, 10, 50, 100 μN/mm is shown at a surface tension γ = 50 μN/mm, surface dilatational viscosity κ = 1 μNs/mm, surface shear viscosity μ = 1 μNs/mm, and surface dilatational elasticityK = 100 μN/mmas presented before using the Cauchy-Green tensor Cs (dotted lines) and with the use of the Finger tensor Bs (solid lines). Using the new results, Figures 18 and 21 of the article by Balemans et al. (2016) should be replaced with Figs. 2 and 3, respectively. The conclusions of the article do not change. (Figure presented.).

Two-way coupled Reynolds, Rayleigh-Plesset-Scriven and energy equations for fully transient cavitation and heat transfer modeling

The Rayleigh-Plesset-Scriven (RPS) equation representing the source of void production is coupled with the Reynolds (RE) and energy equations in a fully transient solution with feedback between these equations. Temperature effects are accounted for by using an energy equation integrated across the bearing film. Dynamic enlargement of the surface of each of the discrete bubbles forming the pseudo-cavitation zone is accounted for by introducing the surface dilatational viscosity term in the RPS equation; this represents a continuation of previous work by Snyder et al [1] that established parametrically the significance of the surface dilatational viscosity (ks=3.75*10-3 N.s/m) both in the development of the cavitation bubble and the formation and sustenance of the subcavitation tensile forces. The study presents the interlaced effects of residual fluid internal energy, eccentricity, angular velocity and heat transfer coefficient on the pressure and pseudo-cavitation development.

Influence of the Surface Viscosity on the Breakup of a Surfactant-Laden Drop.

We examine both theoretically and experimentally the breakup of a pendant drop loaded with an insoluble surfactant. The experiments show that a significant amount of surfactant is trapped in the resulting satellite droplet. This result contradicts previous theoretical predictions, where the effects of surface tension variation were limited to solutocapillarity and Marangoni stresses. We solve numerically the hydrodynamic equations, including not only those effects but also those of surface shear and dilatational viscosities. We show that surface viscosities play a critical role to explain the accumulation of surfactant in the satellite droplet.

Dilatational viscosity of dilute particle-laden fluid interface at different contact angles.

We consider a solid spherical particle adsorbed at a flat interface between two immiscible fluids and having arbitrary contact angle at the triple contact line. We derive analytically the flow field corresponding to dilatational surface flow in the case of a large ratio of dynamic shear viscosities of two fluids. Considering a dilute assembly of such particles we calculate numerically the dependence on the contact angle of the effective surface dilatational viscosity particle-laden fluid interface. The effective surface dilatational viscosity is proportional to the size and surface concentration of particles and monotonically increases with the increase in protrusion of particles into the fluid with larger shear viscosity.

Interpreting attenuation at different excitation amplitudes to estimate strain-dependent interfacial rheological properties of monodisperse contrast microbubbles

Broadband attenuation of ultrasound measured at different excitation pressures being different raises a serious theoretical concern, because the underlying assumption of linear and independent propagation of different frequency components nominally requires attenuation to be independent of excitation. This issue is investigated by examining ultrasound attenuation through a monodisperse lipid coated microbubble suspension measured at four different acoustic excitation amplitudes. We use the attenuation data to determine interfacial rheological properties (surface tension, surface dilatational elasticity and surface dilatational viscosity) of the encapsulation according to three different models. Although different models result in similar rheological properties, attenuation measured at different excitation levels (4–110 kPa) leads to different values for them; the dilatation elasticity (0.56 N/m to 0.18 N/m) and viscosity (2.4×10−8 Ns/m to 1.52×10−8 Ns/m) both decrease with increasing pressure. Numerically simulating the scattered response, nonlinear energy transfer between frequencies are shown to be negligible, thereby demonstrating the linearity in propagation and validating the attenuation analysis. There is a second concern to the characterization arising from shell properties being dependent on excitation amplitude which is not a proper constitutive variable. It is resolved by arriving at a strain-dependent rheology for the encapsulation.