Normal Viscous Stresses Research Articles

Rock failure mechanisms based on rheological dynamics

This paper investigates the mechanisms of rock failure related to axial splitting and shear failure due to hoop stresses in cylindrical specimens. The hoop stresses are caused by normal viscous stress. The rheological dynamics theory (RDT) is used, with the mechanical parameters being determined by P- and S-wave velocities. The angle of internal friction is determined by the ratio of Young's modulus and the dynamic modulus, while dynamic viscosity defines cohesion and normal viscous stress. The effect of frequency on cohesion is considered. The initial stress state is defined by the minimum cohesion at the elastic limit when axial splitting can occur. However, as radial cracks grow, the stress state becomes oblique and moves towards the shear plane. The maximum and nonlinear cohesions are defined by the rock parameters under compressive strength when the radial crack depth reaches a critical value. The efficacy and precision of RDT are validated through the presentation of ultrasonic measurements on sandstone and rock specimens sourced from the literature. The results presented in dimensionless diagrams can be utilized in microcrack zones in the absence of lateral pressure in rock masses that have undergone disintegration due to excavation.

Impact of swirling on capillary instability of Walter’s B viscoelastic fluid-viscous fluid interface with heat and mass transfer

The multifaceted applications of swirling interfaces featuring heat and mass transfer extend across diverse industries, encompassing energy, manufacturing, healthcare, and environmental protection. The present study is concerned with examining the swirling impact on the capillary instability occurring at the interface between a Walter’s B viscoelastic fluid and a viscous fluid, under the influence of heat and mass transfer. The setup consists of two rigid cylinders that form an annular region filled with viscous fluid on the inner side and Walter’s B viscoelastic fluid on the outer side. The outer cylinder rotates at a constant angular velocity, while the inner cylinder remains stationary. The study employs the Walter’s B viscoelastic model, which conforms to the potential flow theory for viscoelastic fluids. The proposed flow theory disregards tangential stresses and balances the difference in normal viscous stresses with the surface force at the free boundary. The perturbed equations yield a quadratic equation in growth rate, which is numerically analyzed. The results show that heat and mass transfer enhance the interface’s stability and swirling makes it more stable. Moreover, it is noteworthy that the Weber number exerts a stabilizing influence on the interface, while the density ratio also plays a role in promoting its stability. The centrifuge number exhibits a counteractive effect by impeding interface growth, thus contributing to its stabilization.

Rarefaction effect on the aerodynamics of bristled wings in miniature insects

All previous studies on the aerodynamics of bristled wings in miniature insects are based on continuum flows. However, the diameter of the bristle is very small, and the diameter-based Knudsen number (Kn) is approximately between 0.03 and 0.11, indicating that the flow around the bristle is in the slip-flow regime and rarefaction effect will be present. To investigate how the rarefaction will affect the aerodynamic force and flow field of the bristled wing, we calculated and analyzed the flow around a model bristled wing under two conditions: the continuum flow and the slip flow. The following is shown. Within the range of Kn considered in this study (0.01 ≤ Kn ≤ 0.1), the rarefaction has a very small effect on the aerodynamic force of the bristled wing: it decreases the aerodynamic force by less than 0.5% compared with that of the continuum flow. However, the rarefaction has a significant effect on the contributions of the viscous tangential and normal stress terms to the aerodynamic force: in the continuum flow, the force contribution of the viscous tangential stress is 50.7% and that of the viscous normal stress is zero, whereas in the slip flow, e.g., at Kn = 0.08, the contribution of the viscous tangential stress is only 37.7% and that of the viscous normal stress is 12.9% instead of zero; this is because the rarefaction-induced slip velocity in the slip flow changes the normal derivative of the velocity on the bristle surface compared with that of the continuum flow. Since the rarefaction has only a slight effect on the aerodynamic force, the results on the aerodynamic force of the bristled wing obtained based on continuum flows in previous studies are very good approximations to the correct results.

Instability of a thin film of chemotactic active suspension

In this paper, we analyse the instability of a thin-film suspension of micro-swimmers subjected to an attractant gradient. The imposed attractant gradient introduces a preferential swimming direction for the swimmers, resulting in an anisotropic orientation field. By modelling the swimmers as force dipoles, the study by Kasyap & Koch (Phys. Rev. Lett., vol. 108, issue 3, 2012, 038101, J. Fluid Mech., vol. 741, 2014, pp. 619–657) found an instability arising from the coupling between the active stress associated with an anisotropic orientation field and perturbations to the swimmer density field. In this work we begin by presenting a stability analysis that calculates the modification of this instability in the presence of an interface. We then detail the presence of a new mode of instability that arises solely from the deformation of the interface mediated by the active stress in the suspension. The resulting hydrodynamic instabilities in the system are observed to be sensitive to the direction of the attractant gradient relative to the interface. Furthermore, we show that the coupling between the two modes involving a jump in interfacial viscous stresses and normal stress differences within the suspension allows for an instability to manifest in a suspension of chemotactic pullers, previously thought to be unconditionally stable. We then use a long-wave theory to write down a set of nonlinear equations governing the film evolution. The numerical solution of the system using the long-wave theory aids in explaining the mechanism associated with the instability in addition to validating the predictions from the linear stability theory.

Numerical study on the deformation of a gas bubble in uniform flow

The deformation of a gas bubble in uniform flow is numerically investigated using a dynamic body force method. In our parametric range, bubble steady shape is the superposition of two types of deformation: O-type deformation (corresponding to oblate eccentricity) and D-type deformation (corresponding to fore-aft symmetry). The calculation results show that O-type deformation mainly depends on surface tension, so the aspect ratio is mainly correlated with We (ρldeUslip2/σ). D-type deformation is mainly depends on the competition between pressure and viscous normal stress on bubble surface. Therefore, bubble fore-aft symmetry has a more significant correlation with Re (ρldeUslip/μl). We find that the critical Re of fore-aft symmetry slightly increases as We increases, and the interval of critical Re becomes narrower. When We exceeds 7, bubble will loss its fore-aft symmetry. A new power-law correlation of aspect ratio related to We and a phase diagram of bubble symmetry mapped to Re–We are given to predict bubble shape.

Droplet deformation and breakup in shear-thinning viscoelastic fluid under simple shear flow

A three-dimensional lattice Boltzmann method, which couples the color-gradient model for two-phase fluid dynamics with a lattice diffusion-advection scheme for the elastic stress tensor, is developed to study the deformation and breakup of a Newtonian droplet in the Giesekus fluid matrix under simple shear flow. This method is first validated by the simulation of the single-phase Giesekus fluid in a steady shear flow and the droplet deformation in two different viscoelastic fluid systems. It is then used to investigate the effect of Deborah number De, mobility parameter α, and solvent viscosity ratio β on steady-state droplet deformation. We find for 0.025<α<0.5 that as De increases, the steady-state droplet deformation decreases until eventually approaching the one in the pure Newtonian case with the viscosity ratio of 1/β, which is attributed to the strong shear-thinning effect at high De. While for lower α, the droplet deformation exhibits a complex nonmonotonic variation with De. Under constant De, the droplet deformation decreases monotonically with α but increases with β. Force analysis shows that De modifies the droplet deformation by altering the normal viscous and elastic stresses at both poles and equators of the droplet, while α mainly alters the normal stresses at the poles. Finally, we explore the roles of De and α on the critical capillary number Cacr of the droplet breakup. By establishing both Ca–De and Ca–α phase diagrams, we find that the critical capillary number increases with De or α except that a plateau critical capillary number is observed in Ca–De phase diagram.

Viscous normal stresses and fingertip tripling in radial Hele-Shaw flows.

Viscous fingering in radial Hele-Shaw cells is markedly characterized by the occurrence of fingertip splitting, where growing fingered structures bifurcate at their tips, via a tip-doubling process. A much less studied pattern-forming phenomenon, which is also detected in experiments, is the development of fingertip tripling, where a finger divides into three. We investigate the problem theoretically, and employ a third-order perturbative mode-coupling scheme seeking to detect the onset of tip-tripling instabilities. Contrary to most existing theoretical studies of the viscous fingering instability, our theoretical description accounts for the effects of viscous normal stresses at the fluid-fluid interface. We show that accounting for such stresses allows one to capture the emergence of tip-tripling events at weakly nonlinear stages of the flow. Sensitivity of fingertip-tripling events to changes in the capillary number and in the viscosity contrast is also examined.

Water Film Drainage between a Very Viscous Oil Drop and a Mica Surface.

We investigate thin film drainage between a viscous oil drop and a mica surface, clearly illustrating the competing effects of Laplace pressure and viscous normal stress (τ_{v}) in the drop. τ_{v} dominates the initial stage of drainage, leading to dimple formation (h_{d}) at a smaller critical thickness with an increase in the drop viscosity (the dimple is the inversion of curvature of the drop in the film region). Surface forces and interfacial tension control the last stage of film drainage. A scaling analysis shows that h_{d} is a function of the drop size R and the capillary numbers of the film (Ca_{f}) and drop (Ca_{d}), which we estimate by h_{d}=0.5Rsqrt[Ca_{f}/(1+2Ca_{d})]. This equation clearly indicates that the drop viscosity needs to be considered when Ca_{d}>0.1. These results have implications for industrial systems where very viscous liquids are involved, for example, in 3D printing and heavy oil extraction process.

Impact of bulk viscosity on flow morphology of shock-accelerated cylindrical light bubble in diatomic and polyatomic gases

Shock-accelerated bubbles have long been an intriguing topic for understanding the fundamental physics of turbulence generation and mixing caused by the Richtmyer–Meshkov instability. In this study, the impact of bulk viscosity on the flow morphology of a shock-accelerated cylindrical light bubble in diatomic and polyatomic gases is investigated numerically. An explicit mixed-type modal discontinuous Galerkin scheme with uniform meshes is employed to solve a two-dimensional system of unsteady physical conservation laws derived rigorously from the Boltzmann–Curtiss kinetic equations. We also derive a new complete viscous compressible vorticity transport equation including the bulk viscosity. The numerical results show that, during the interaction between a planar shock wave and a cylindrical light bubble, the bulk viscosity associated with the viscous excess normal stress in diatomic and polyatomic gases plays an important role. The diatomic and polyatomic gases cause significant changes in flow morphology, resulting in complex wave patterns, vorticity generation, vortex formation, and bubble deformation. In contrast to monatomic gases, diatomic and polyatomic gases produce larger rolled-up vortex chains, various inward jet formations, and large mixing zones with strong, large-scale expansion. The effects of diatomic and polyatomic gases are explored in detail through phenomena such as the vorticity generation, degree of nonequilibrium, enstrophy, and dissipation rate. Furthermore, the evolution of the shock trajectories and interface features is investigated. Finally, the effects of bulk viscosity on the flow physics of shock-accelerated cylindrical light bubble are comprehensively analyzed.

On the use of peristaltic waves for the transport of soft particles: A numerical study

Peristaltic transport of inelastic circular droplets immersed in an immiscible viscous fluid is numerically studied in a planar two-dimensional channel using the finite-volume method. Numerical results could be obtained for a wide range of droplet’s material properties at large deformations. Based on the results obtained in this work, for a particle that is initially placed at the centerline, an increase in the droplet’s viscosity is predicted to increase its transport velocity, but the effect can saturate at viscosity ratios as small as two. The transport velocity is shown to linearly increase with the droplet’s density, but the effect turns out to be quite weak. An increase in the interfacial tension is found to lower the transport velocity although the effect appears to approach an asymptote. Depending on their size and the Weber number, droplets are predicted to move faster or slower than rigid particles. The transport velocity of droplets is found to increase with an increase in the wave speed or, equivalently, the Reynolds number. Off-center droplets are predicted to migrate toward the wall or toward the centerline. Droplets that migrate toward the centerline remain a short distance away from it under steady conditions. Distribution of surface forces is used to explain some of these results with viscous normal stress predicted to play a key role in controlling the dynamics of droplets in peristaltic flow.

On the total enthalpy behavior inside a shock wave

The total enthalpy behavior inside a shock wave in a dilute monatomic gas has been numerically studied for various values of Mach and Prandtl numbers with the continuum (the Navier–Stokes–Fourier equations) and kinetic (the Shakhov model and the direct simulation Monte Carlo method) approaches. A significant difference between the results by the continuum and kinetic approaches has been observed for the internal shock wave structure. In a wide range of the free-stream Mach numbers, the continuum approach predicts qualitatively similar behavior of total enthalpy distributions that can be of a concave, constant, or convex shape depending on the Prandtl number. The more sophisticated kinetic approach predicts a more complicated form of total enthalpy profiles: e.g., an inflection point for Mach numbers around two and Prandtl numbers close to unity. The evolution of the total enthalpy in the shock is determined by the balance of heat conduction and mechanical work of normal viscous stress—processes that are predicted inaccurately by using the Navier–Stokes–Fourier equations at high Mach numbers.

Low Reynolds number friction reduction with polymers and textures

We present an experimental investigation with surface textures and a Non-Newtonian polymer solution (polyisobutylene in oil) to determine synergistic effects on laminar hydrodynamic friction reduction. Gap controlled experiments were performed on a custom parallel disk tribo-rheometer to systematically vary the Reynolds number (0.1–16), Weissenberg number (0.1–30), and Deborah number (0.005–0.5) in bi-directional sliding motion with five different texture geometries: cylindrical holes with varying asymmetric angle β. Cavitation effects are not present, thus normal force is produced solely by the textures and the lubricant rheology. Contrary to Newtonian fluids without cavitation, the normal force is positive, independent of the direction of motion when shear normal forces dominate purely viscous effects. We observe that the normal thrust can be larger than a simple superposition of forces due to viscous hydrodynamics and shear normal stresses. This synergistic coupling between the asymmetric textures and the polymer solution (at high Wi) produces an optimal angle of texture asymmetry β for dramatically decreasing the effective coefficient of friction compared to the Newtonian oil without polymer.

Viscous stress tensor and stability of laminar contravortical flows

Introduction: coaxial layers in contravortical flows rotate in the opposite directions. This determines their complicated spatial structure. The relevance of the subject is in the uniquely effective mixing of the moving medium. This property has a great potential of application from microbiology and missile building for obtaining highly dispersed mixtures to heat engineering for increasing the intensity of heat transfer. However, contravortical flows have a high degree of hydrodynamic instability. This hinders effective development of these technologies. Contravortical flows are observed behind Francis hydroturbines, whose derated operation causes modes with a significant increase of hydraulic unit vibrations up to destruction of the units. The purpose of the study is to identify physical laws of the contravortical flow hydrodynamics, common for both laminar and turbulent fluid flow modes.
 Materials and methods: theoretical analysis of the viscous stress tensor and local stability zones of contravortical laminar flows.
 Results: the article provides a mathematical description of the tensor of viscous tangential (τij) and normal (σii) stresses as well as local stability zones of the flow according to Rayleigh (Ra) and Richardson (Ri) criteria. The graphs of the radial-axial distributions of the viscous stress components are given, local stability zones are shown and the point of “vortex breakdown” is indicated. The solutions are obtained in the form of Fourier – Bessel series. The hydrodynamic structure of the flow is analysed.
 Conclusions: it is established that the most significant viscous stresses are observed at the beginning of the interaction zone of contrarotating layers. It is established that the areas with the most unstable flow are localized in the flow vortex core. Three zones can be distinguished in the vortex core: a zone of weak instability with local Richardson numbers to Ri = –1, passing into a zone of flow destabilization with high negative values of Richardson numbers Ri = –10 to –100, in turn, transforming into a zone with rapidly increasing instability up to Ri = –1000. This is a zone of loss of flow stability, culminating in the “ortex breakdown”.

Hydrodynamic structure of laminar flows with oppositely-swirled coaxial layers

The article is devoted to the theoretical study of hydrodynamics of laminar flows with coaxial layers swirled in opposite directions and moving along the pipe. Such flows in a turbulent range have a wide practical application potential in technologies of dissipation of mechanical energy and mixing multiphase and heterogeneous media in microbiology, chemistry, ecology, heat engineering, power engineering, engine and rocket engineering. The article describes the tensor of viscous tangents (τii) and normal (σii) stresses. The questions of stability of flow according to the Rayleigh (Ra) and Richardson (Ri) criteria are considered. Calculation formulas and graphs of radial-axial distributions of viscous stress components, local stability zones are given, the point of “crisis and decay of the flow” or “vortex breakdown” is indicated. The solutions are obtained in the form of Fourier-Bessel series. The analysis of the hydrodynamic structure of the flow is made.

Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton’s theory to empirical models

The aim of this study is to derive accurate models for quantities characterizing the dynamics of droplets of non-vanishing viscosity in capillaries. In particular, we propose models for the uniform-film thickness separating the droplet from the tube walls, for the droplet front and rear curvatures and pressure jumps, and for the droplet velocity in a range of capillary numbers, Ca, from $$10^{-4}$$ to 1 and inner-to-outer viscosity ratios, $$\lambda$$ , from 0, i.e. a bubble, to high-viscosity droplets. Theoretical asymptotic results obtained in the limit of small capillary number are combined with accurate numerical simulations at larger Ca. With these models at hand, we can compute the pressure drop induced by the droplet. The film thickness at low capillary numbers ( $$Ca<10^{-3}$$ ) agrees well with Bretherton’s scaling for bubbles as long as $$\lambda <1$$ . For larger viscosity ratios, the film thickness increases monotonically, before saturating for $$\lambda>10^3$$ to a value $$2^{2/3}$$ times larger than the film thickness of a bubble. At larger capillary numbers, the film thickness follows the rational function proposed by Aussillous and Quere (Phys Fluids 12(10):2367–2371, 2000) for bubbles, with a fitting coefficient which is viscosity-ratio dependent. This coefficient modifies the value to which the film thickness saturates at large capillary numbers. The velocity of the droplet is found to be strongly dependent on the capillary number and viscosity ratio. We also show that the normal viscous stresses at the front and rear caps of the droplets cannot be neglected when calculating the pressure drop for $$Ca>10^{-3}$$ .

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.

Surfactant Effect on the Average Flow Generation Near Curved Interface

The present work is devoted to the average flow generation near curved interface with a surfactant adsorbed on the surface layer. The investigation was carried out for a liquid drop embedded in a viscous liquid with a different density. The liquid flows inside and outside the drop are generated by small amplitude and high frequency vibrations. Surfactant exchange between the drop surface and the surrounding liquid is limited by the process of adsorption-desorption. It was assumed that the surfactant is soluble in the surrounding liquid, but not soluble in the liquid drop. Surrounding liquid and the liquid in the drop are considered incompressible. Normal and shear viscous stresses balance at the interface is performed under the condition that the film thickness of the adsorbed surfactant is negligible. The problem is solved under assumption that the shape of the drop in the presence of adsorbed surfactant remains spherical symmetry. The effective boundary conditions for the tangential velocity jump and shear stress jump, describing the above generation have been obtained by matched asymptotic expansions method. The conditions under which the drop surface can be considered as a quasi-solid are determined. It is shown that in the case of the significant effect of surfactant on the surface tension, the dominant mechanism for the generation is the Schlichting mechanisms under vibrations.

Stability of horizontal viscous fluid layers in a vertical arbitrary time periodic electric field

The stability of a horizontal interface between two viscous fluids, one of which is conducting and the other is dielectric, acted upon by a vertical time-periodic electric field is considered theoretically. The two fluids are bounded by electrodes separated by a finite distance. For an applied ac electric field, the unstable interface deforms in a time periodic manner, owing to the time dependent Maxwell stress, and is characterized by the oscillation frequency which may or may not be the same as the frequency of the ac electric field. The stability curve, which relates the critical voltage, manifested through the Mason number—the ratio of normal electric stress and viscous stress, and the instability wavenumber at the onset of the instability, is obtained by means of the Floquet theory for a general arbitrary time periodic electric field. The limit of vanishing viscosities is shown to be in excellent agreement with the marginal stability curves predicted by means of a Mathieu equation. The influence of finite viscosity and electrode separation is discussed in relation to the ideal case of inviscid fluids. The methodology to obtain the marginal stability curves developed here is applicable to any arbitrary but time periodic signal, as demonstrated for the case of a signal with two different frequencies, and four different frequencies with a dc offset. The mode coupling in the interfacial normal stress leads to appearance of harmonic and subharmonic modes, characterized by the frequency of the oscillating interface at an integral or half-integral multiple of the applied frequency, respectively. This is in contrast to the application of a voltage with a single frequency which always leads to a harmonic mode oscillation of the interface. Whether a harmonic or subharmonic mode is the most unstable one depends on details of the excitation signal.

Acoustic microbubble dynamics with viscous effects

Microbubble dynamics subject to ultrasound are associated with important applications in biomedical ultrasonics, sonochemistry and cavitation cleaning. The viscous effects in this phenomenon is essential since the Reynolds number Re associated is about O(10). The flow field is characterized as being an irrotational flow in the bulk volume but with a thin vorticity layer at the bubble surface. This paper investigates the phenomenon using the boundary integral method based on the viscous potential flow theory. The viscous effects are incorporated into the model through including the normal viscous stress of the irrotational flow in the dynamic boundary condition at the bubble surface. The viscous correction pressure of Joseph & Wang (2004) is implemented to resolve the discrepancy between the non-zero shear stress of the irrotational flow at a free surface and the physical boundary condition of zero shear stress. The model agrees well with the Rayleigh–Plesset equation for a spherical bubble oscillating in a viscous liquid for several cycles of oscillation for Re=10. It correlates pretty closely with both the experimental data and the axisymmetric simulation based on the Navier-Stokes equations for transient bubble dynamics near a rigid boundary. We further analyze microbubble dynamics near a rigid boundary subject to ultrasound travelling perpendicular and parallel to the boundary, respectively, in parameter regions of clinical relevance. The viscous effects to acoustic microbubble dynamics are analyzed in terms of the jet velocity, bubble volume, centroid movement, Kelvin impulse and bubble energy.

A two-phase thermomechanical theory for granular suspensions

In this paper, a two-phase thermomechanical theory for granular suspensions is presented. Our approach is based on a mixture-theoretic formalism and is coupled with a nonlinear representation for the granular viscous stresses so as to capture the complex non-Newtonian behaviour of the suspensions of interest. This representation has a number of interesting properties: it is thermodynamically consistent, it is non-singular and vanishes at equilibrium and it predicts non-zero granular bulk viscosity and shear-rate-dependent normal viscous stresses. Another feature of the theory is that the resulting model incorporates a rate equation for the evolution of the volume fraction of the granular phase. As a result, the velocity fields of both the granular material and the carrier fluid are divergent even for constant-density flows. Further, in this article we present the incompressible limit of our model which is derived via low-Mach-number asymptotics. The reduced equations for the important special case of constant-density flows are also presented and discussed. Finally, we apply the proposed model to two test cases, namely, steady shear flow of a homogeneous suspension and fully developed pressure-driven channel flow, and compare its predictions with available experimental and numerical results.