Normal Stress Balance Research Articles

Rayleigh–Taylor instability in a Casson fluid layer with heat and mass transfer

This study investigates the stability of the interface between two fluids, a Casson fluid on top and a viscous fluid below, with heat and mass transfer occurring between them. We used the potential flow theory, which simplifies the problem by ignoring tangential stresses and focusing on normal stress balance. We did not consider no-slip conditions at the boundaries, assuming zero normal velocity at these rigid boundaries instead. A second-order polynomial equation was developed to calculate the growth rate of instabilities and solved numerically using the Newton–Raphson method. The results showed that heat and mass transfer improved the stability of the interface, even though the Atwood number, a measure of instability, remained high.

A balanced outflow boundary condition for swirling flows

In open flow simulations, the dispersion characteristics of disturbances near synthetic boundaries can lead to unphysical boundary scattering interactions that contaminate the resolved flow upstream by propagating numerical artifacts back into the domain interior. This issue is exacerbated in flows influenced by real or apparent body forces, which can significantly disrupt the normal stress balance along outflow boundaries and generate spurious pressure disturbances. To address this problem, this paper develops a zero-parameter, physics-based outflow boundary condition (BC) designed to minimize pressure scattering from body forces and pseudo-forces and enhance transparency of the artificial boundary. This “balanced outflow BC” is then compared against other common BCs from the literature using example axisymmetric and three-dimensional open swirling flow computations. Due to centrifugal and Coriolis forces, swirling flows are known to be particularly challenging to simulate in open geometries, as these apparent forces induce non-trivial hydrostatic stress distributions along artificial boundaries that cause scattering issues. In this context, the balanced outflow BC is shown to correspond to a geostrophic hydrostatic stress correction that balances the induced pressure gradients. Unlike the alternatives, the balanced outflow BC yields accurate results in truncated domains for both linear and nonlinear computations without requiring assumptions about wave characteristics along the boundary.

Viscous correction to the potential flow analysis of Rayleigh–Taylor instability in a Rivlin–Ericksen viscoelastic fluid layer with heat and mass transfer

AbstractThe current investigation focuses on examining viscous corrections for viscous potential flow (VCVPF) analysis concerning the Rayleigh–Taylor instability occurring at the interface of a Rivlin–Ericksen (R–E) viscoelastic fluid and a viscous fluid during the transfer of heat and mass between phases. The R–E model is a fundamental framework in the study of viscoelastic fluids, providing insights into their complex rheological behavior. It characterizes the material's response to both deformation and flow, offering valuable predictions for various industrial and biological applications. Within the framework of viscous potential flow (VPF) theory, viscosity is exclusively accounted for in the normal stress balance equation, disregarding the influence of shearing stress entirely. This study introduces a viscous pressure term into the normal stress balance equation alongside the irrotational pressure, presuming that this addition will improve the discontinuity of tangential stresses at the fluid interface. Through derivation of a dispersion relationship and subsequent theoretical and numerical stability analyses, the stability of the interface is investigated across various physical parameters. Multiple plots are generated using the dispersion relation, and a comparative analysis between VPF and VCVPF is conducted to establish improved stability criteria. The investigation highlights that the combined impact of heat/mass transport and shearing stress serves to delay the instability of the interface.

STABILITY ANALYSIS OF RIVLIN-ERICKSEN FLUID FILM WITH HEAT TRANSFER THROUGH AN ANNULAR POROUS MEDIUM

A theoretical study was conducted to analyze the stability of the interface between viscous and viscoelastic fluids, considering heat and mass transfer between the phases. The study used Rivlin-Ericksen's model for viscoelastic liquid and an irrotational flow theory to analyze the problem. A rigid cylinder surrounded the annular region of the viscoelastic liquid and viscous fluid. The viscous fluid was located in the inner region, while the viscoelastic liquid was in the outer region. The analysis did not take into account tangential stress, considered viscosity through normal stress balance, and did not impose no-slip conditions at the two cylindrical rigid boundaries. The results indicate that the interface stability is enhanced under porous conditions due to the heat/mass transfer. A medium with higher permeability stabilizes the interface, while a medium with higher porosity negatively affects the viscoelastic interface. The study compared the stabilizing characteristics of the Rivlin-Ericksen fluid interface under a nonporous and porous medium and found that the interface is more stable under porous media.

The evolution of a three-dimensional microbubble at a corner in a Maxwell fluid

Bubbles often appear in non-Newtonian liquids from nature, engineering to biomedical applications, but their study has been under research compared to their Newtonian counterpart. Here, we extend the axisymmetric modeling of Lind and Phillips to three-dimensional modeling. The approach is based on the boundary integral method coupled with the Maxwell constitutive equation. The flow is assumed to have moderate to high Reynolds numbers and, thus, is irrotational in the bulk domain. The viscoelastic effects are incorporated approximately in the normal stress balance at the bubble surface. The numerical model has excellent agreement with the corresponding Rayleigh–Plesset equation for spherical bubbles in a non-Newtonian liquid. Computations are carried out for a bubble near a corner at various angles. The numerical results agree very well with the experiments for bubbles in a Newtonian fluid in a corner. As the Deborah number increases, the amplitude and period of the bubble oscillation increase, the bubble migration to the corner enhances, and the bubble jet is broader, flatter, and inclined more to the further boundary. This implies an improvement to surface cleaning of all surrounding boundaries for ultrasonic cavitation cleaning and results in greater administration of noninvasive therapy and drug delivery.

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.

The separation angle of the free surface of a viscous fluid at a straight edge

We rework some of the die-swell singularity analysis for Stokes flow, originally by Ramalingam (Ramalingam, 1994 Fiber spinning and rheology of liquid-crystalline polymers, PhD thesis, Massachusetts Institute of Technology) in appendix A of his PhD thesis, in an attempt to demonstrate that for capillary numbers in the range $(0,\infty )$ the curvature may enter into the normal stress balance on the free surface and lead to separation angles exceeding $180^{\circ }$ and infinite curvature at the separation point. The singular coefficients in the asymptotic solution and the free surface shape in a neighbourhood of the separation point cannot be determined by a local analysis of the Michael type (Michael, Mathematika, vol. 5, 1958, pp. 82–84) but must be found from matching with the solution valid away from the die edge. The numerical method that we use in the truncated die-swell domain is a boundary element method incorporating the singular solution near the separation point. Although there is some variation in the extrudate swell ratios at different capillary numbers reported in the numerical literature, our results for capillary numbers $Ca$ from $1$ to $1000$ are within the range of values published in earlier papers. The computed separation angles at different values of $Ca$ agree well with the range of separation angles to be found in experimental and numerical papers. The separation angle appears to converge to a value different from $180^{\circ }$ as $Ca$ increases, leading us to conclude that the case of zero surface tension ( $Ca=\infty$ , with corresponding separation angle of $180^{\circ }$ ), is a singular limit.

Predicting transition from selective withdrawal to entrainment in two-fluid stratified systems.

Selective withdrawal is a desired phenomenon in transferring oil from large caverns in the U.S. Strategic Petroleum Reserve (SPR), because entrainment of oil at the time during withdrawal poses a risk of contaminating the environment. Motivated to understand selective withdrawal in an SPR-like orientation, we performed experiments in order to investigate the critical submergence depth as a function of critical flow rate. For the experiments, a tube was positioned through a liquid-liquid interface that draws the lower liquid upward, avoiding entrainment of the upper fluid. Analysis of the normal stress balance across the interface produced a Weber number, utilizing dynamic pressure scaling, that predicted the transition to entrainment. Additionally, an inviscid flow analysis was performed assuming an ellipsoidal control volume surface that produced a linear relationship between the Weber number and the scaled critical submergence depth. This analytical model was validated using the experimental data, resulting in a robust model for predicting transition from selective withdrawal to entrainment.

Instability of Rivlin-Ericksen fluid film with heat and mass transfer

The stability of the viscous- viscoelastic fluid interface is theoretically examined when the heat and mass transfer is allowed from one phase to another phase. The viscoelastic liquid and viscous fluid lie in an annular region and are enclosed by two rigid cylinders. The viscoelastic liquid is taken in the outer region while viscous fluid lies in the inner region. The analysis was performed by using the potential flow theory for viscoelastic liquid satisfying the Rivlin-Ericksen model. In potential flow theory, the tangential stresses are not considered, the viscosity enters through normal stress balance and the no-slip condition at the two cylindrical rigid boundaries is not enforced. A second-order polynomial in terms of growth rate is achieved and analyzed numerically using the well-known Newton-Raphson method. The results show that the impact of the transfer of heat/mass is found to boost the stability of the interface.

Stability and tip streaming of a surfactant-loaded drop in an extensional flow. Influence of surface viscosity

We study numerically the nonlinear stationary states of a droplet covered with an insoluble surfactant in a uniaxial extensional flow. We calculate both the eigenfunctions to reveal the instability mechanism and the time-dependent states resulting from it, which provides a coherent picture of the phenomenon. The transition is of the saddle-node type, both with and without surfactant. The flow becomes unstable under stationary linear perturbations. Surfactant considerably reduces the interval of stable capillary numbers. Inertia increases the droplet deformation and decreases the critical capillary number. In the presence of the surfactant monolayer, neither the droplet deformation nor the stability is significantly affected by the droplet viscosity. The transient state resulting from instability is fundamentally different for drops with and without surfactant. Tip streaming occurs only in the presence of surfactants. The critical eigenmode leading to tip streaming is qualitatively the same as that yielding the central pinching mode for a clean interface, which indicates that the small local scale characterizing tip streaming is set during the nonlinear droplet deformation. The viscous surface stress does not significantly affect the droplet deformation and the critical capillary number. However, the damping rate of the dominant mode considerably decreases for viscous surfactants. Interestingly, shear viscous surface stress considerably alters the tip streaming arising in the supercritical regime, even for very small surface viscosities. The viscous surface stresses alter the balance of normal interfacial stresses and affect the surfactant transport over the stretched interface.

Effect of time-varying magnetic field on metal droplet profiles

A nonmagnetic metal droplet with dynamic interfaces under a time-varying magnetic field was considered. Considering that the thermal effect of the magnetic field and the droplet evaporation were disregarded, we obtained the magnetically augmented Young–Laplace equation theoretically according to the normal stress balance principles. The droplet contact angle was further determined according to the droplet volume conservation. According to the curvature and contact angle, the droplet profile was mathematically determined. Results showed that a downward magnetic force caused droplet contraction and an upward force caused expansion. The increase in surface tension hindered the droplet spreading without affecting the change trend of droplet shape parameters. The increase in angular frequency (ω) decreased the variation amplitude of the droplet curvature and contact angle, and the increase in damping factor (τ) slowed down the decay of the droplet curvature and contact angle.

Two-dimensional flows in finite-width channels partially filled with porous medium

A stationary two-dimensional liquid flow in a channel partially filled with a porous non-deformable homogeneous medium is investigated numerically based on the proposed mathematical model. The homogeneous liquid is bounded by a solid or free rigid boundary, while the channel bottom is solid. The model is constructed based on the Berman transformation. The free flow is described by the Navier-Stokes equations and the filtration flow is described by the Darcy-Brinkman model. The interfacial boundary conditions include the velocity continuity and the balance of tangential and normal viscous stresses; the tangential stresses may have a discontinuity at the boundary. The numerical solution is found by the finite-difference relaxation method. It has been shown that the flow of the liquid into porous medium takes place for a wide parameter range. However in most cases, the transversal velocity is about 10-7of the maximum longitudinal velocity. The dependence of the transversal velocity on the permeability of porous medium and its thickness is examined. The velocity profiles are plotted for different sets of parameters. The effect of the relative thickness of porous medium on the transversal velocity maximum and its position is also considered. It has been found that the effect of the transversal flow becomes significant if the width of the free portion of the channel is less than 10% of the width of the entire system. In spite of the smallness of the transversal velocity, the total flow rate through the interface is about 0.1% of the total flow rate through the channel provided that the channel length is 105÷106of the channel width.

Influence of thermal effects on stability of nanoscale films and filaments on thermally conductive substrates

We consider fluid films and filaments of nanoscale thickness on thermally conductive substrates exposed to external heating and discuss the influence of the variation of material parameters with temperature on film stability. Particular focus is on metal films exposed to laser irradiation. Due to the short length scales involved, the absorption of heat in the metal is directly coupled to the film evolution, since the absorption length and the film thickness are comparable. Such a setup requires self-consistent consideration of fluid mechanical and thermal effects. We approach the problem via volume-of-fluid-based simulations that include destabilizing liquid metal–solid substrate interaction potentials. These simulations couple fluid dynamics directly with the spatio-temporal evolution of the temperature field both in the fluid and in the substrate. We focus on the influence of the temperature variation of material parameters, in particular of surface tension and viscosity. Regarding variation of surface tension with temperature, the main finding is that while the Marangoni effect may not play a significant role in the considered setting, the temporal variation of surface tension (modifying normal stress balance) is significant and could lead to complex evolution including oscillatory evolution of the liquid metal-air interface. Temperature variation of film viscosity is also found to be relevant. Therefore, the variations of surface tensions and viscosity could both influence the emerging wavelengths in experiments. By contrast, the filament geometry is found to be much less sensitive to a variation of material parameters with temperature.

The Effect of a Magnetic Field on the Profile of Sessile Magnetic Nanofluid Droplets

Surface wettability plays an important role in droplet formation, removal, and resistance to fouling. The sessile droplet profile is one of the most important parameters characterizing surface wettability. In this study, an investigation was carried out into the effects of a magnetic field on the profile of sessile droplets of magnetic nanofluid. We started with a revision of the classic Young–Laplace equation according to normal stress balance principles by considering a magnetic nanofluid droplet in an applied magnetic field gradient. The secant method was then used to solve the non-linear differential equation and predict the sessile droplet profile. The results showed that a downward magnetic force caused an expansion of the profile and an upwards magnetic force caused contraction. In other words, droplets on an hydrophilic or hydrophobic surface could be tuned by magnetic field gradients of different strength and direction. This magnetowetting effect could be further adjusted by selection of different fluid, material, and particle concentration.

A simple model for viscoplastic thin film formation for coating flows

A simple model of viscoplastic thin film formation flow, which is designed to describe a prototype downstream slot-coating problem, is proposed. Initially, a coating material, typically a dense suspension, is completely yielded due to an imposed shear stress caused by the stationary coating die lip and the moving substrate. After the die, the material experiences a sudden absence of shear stress at the gas/liquid interface on the top, while the bottom maintains a still shear-dominant flow. The asymptotic analysis of this film formation flow reveals that the zeroth order terms alone cannot properly explain stress evolution, hence predicting the shape and location of the yield plane inside this flow. When higher-order terms are included, an augmented pressure jump across the interface is observed in the normal stress balance due to yield stress. The results are compared with solutions of two-dimensional mass and momentum balance equations with the same constitutive equation using the finite volume method. Implications for desirable operating conditions for slot coating are discussed.

The impact and bounce of air bubbles at a flat fluid interface

The rise and impact of bubbles at an initially flat but deformable liquid-air interface in ultraclean liquid systems are modelled by taking into account the buoyancy force, hydrodynamic drag, inertial added mass effect and drainage of the thin film between the bubble and the interface. The bubble-surface interaction is analyzed using lubrication theory that allows for both bubble and surface deformation under a balance of normal stresses and surface tension as well as the long-range nature of deformation along the interface. The quantitative result for collision and bounce is sensitive to the impact velocity of the rising bubble. This velocity is controlled by the combined effects of interfacial tension via the Young-Laplace equation and hydrodynamic stress on the surface, which determine the deformation of the bubble. The drag force that arises from the hydrodynamic stress in turn depends on the hydrodynamic boundary conditions on the bubble surface and its shape. These interrelated factors are accounted for in a consistent manner. The model can predict the rise velocity and shape of millimeter-size bubbles in ultra-clean water, in two silicone oils of different densities and viscosities and in ethanol without any adjustable parameters. The collision and bounce of such bubbles with a flat water/air, silicone oil/air and ethanol/air interface can then be predicted with excellent agreement when compared to experimental observations.

Pressure Corrections for the Potential Flow Analysis of Electrohydrodynamic Kelvin-Helmholtz Instability

The present paper deals with the study of the pressure corrections to the viscous potential flow analysis of Kelvin-Helmholtz instability with tangential electric field at the interface of two viscous fluids. Viscosity enters through normal stress balance in the viscous potential flow theory and tangential stresses for two fluids are not continuous at the interface. Here we have considered viscous pressure in the normal stress balance along with the irrotational pressure and it is assumed that the addition of this viscous pressure will resolve the discontinuity between the tangential stresses and the tangential velocities at the interface of two fluids. The viscous pressure is derived by mechanical energy balance equation and this pressure correction applied to compute the growth rate of electrohydrodynamic Kelvin-Helmholtz instability. A dispersion relation is obtained and stability criterion is given in the terms of critical value of relative velocity. It has been observed that the inclusion of irrotational shearing stresses have stabilizing effect on the stability of the system.

Capillary meniscus dynamometry – Method for determining the surface tension of drops and bubbles with isotropic and anisotropic surface stress distributions

The stresses acting in interfacial adsorption layers with surface shear elasticity are, in general, anisotropic and non-uniform. If a pendant drop or buoyant bubble is covered with such elastic layer, the components of surface tension acting along the “meridians” and “parallels”, σs and σφ, can be different and, then, the conventional drop shape analysis (DSA) is inapplicable. Here, a method for determining σs and σφ is developed for axisymmetric menisci. This method, called ‘capillary meniscus dynamometry’ (CMD), is based on processing data for the digitized drop/bubble profile and capillary pressure. The principle of the CMD procedure for data processing is essentially different from that of DSA. Applying the tangential and normal surface stress balance equations, σs and σφ are determined in each interfacial point without using any rheological model. The computational procedure is fast and could be used in real time, during a given process. The method is applied to determine σs and σφ for bubbles and drops formed on the tip of a capillary immersed in solutions of the protein HFBII hydrophobin. Upon a surface compression, meridional wrinkles appear on the bubble surface below the bubble “equator”, where the azimuthal tension σφ takes negative values. The CMD method allows one to determine the local tensions acting in anisotropic interfacial layers (films, membranes), like those formed from proteins, polymers, asphaltenes and phospholipids. The CMD is applicable also to fluid interfaces (e.g. surfactant solutions), for which it gives the same surface tension as the conventional methods.

Electro-viscous instability of a swirling compound jet

The aim of this work is to study the effects of a combination of axial and swirl velocity on the instability of an annular liquid jet. A linear instability analysis yields an analytical dispersion relation, which governs the temporal growth of perturbations to an electrified streaming annular viscous liquid jet coated by two cylindrical shells of streaming swirling inviscid gases. The three-layer system is subjected to an electric field normal to the interfaces. In our study, the electric body force vanishes, and the electric field and fluid dynamics are coupled only at the interfaces through the tangential and normal interfacial stress balance equations. The governing electrohydrodynamic equations, along with the hydrodynamic and electrical boundary conditions, are solved, and then the corresponding dispersion relation is obtained. The instability analysis of a inviscid jet is discussed as a special case. Numerical results on the effect of the various problem parameters on the instability and breakup of a liquid jet are presented. It is observed, for instance, that the swirling ratio of the outer gas layer always enhances the stability of flow in the case of axisymmetric disturbances, whereas it has a dual effect (destabilizing effect) corresponding to the case of asymmetric disturbances. It has been found that the higher electrical potential, the larger the maximum growth rate and critical wave number of the disturbances and, consequently, the initial applied electric potential tends the instability and breakup of the liquid jet.

Kelvin-Helmholtz instability with mass transfer through porous media: Effect of irrotational viscous pressure

This paper studies the effect of the irrotational viscous pressure on Kelvin-Helmholtz instability of the plane interface of two viscous and incompressible fluids in a fully saturated porous media with mass and heat transfers across the interface. In the earlier work, the instability of the plane interface of two viscous and streaming miscible fluids through porous media was studied by assuming that the motion and the pressure are irrotational and the viscosity enters the normal stress balance. This theory is called the viscous potential flow theory. Here, we use another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by considering viscous contributions of the irrotational pressure. The Darcy-Brinkman model is used in the investigation and the stability criterion is formulated in terms of a critical value of the relative velocity. It is observed that the heat and mass transfer has a destabilizing effect on the stability of the system while the irrotational shearing stresses stabilize the system.