Normal Viscous Stresses Research Articles

Viscous stress distribution over a wavy gas–liquid interface

Viscous stress contributes to momentum transfer between two phases, which plays an important role in both industrial applications and environmental processes. Near a wavy interface, the flow is modulated and produces a spatially non-uniform normal and tangential viscous stress. This study presents measurements of these stresses at a liquid–gas interface populated with two-dimensional millimeter scale waves performed with multiphase particle image velocimetry. Large datasets enable conditional phase-averaging of the data based on wave steepness, which increases the precision of the results and allows statistical analysis. For the first time at this scale, the spatial distribution of normal and tangential viscous stress is obtained for a large range of wave steepness (ak = 0–1, with a the amplitude and k the wavenumber). As the steepness increases, the mean shear stress over a wavelength decreases in magnitude, while the normal viscous stress increases. These trends are linear for ak < 0.6, and correlations are proposed. At ak > 0.7, flow separation is observed in the gas phase near the troughs and drastically alters the viscous stress distribution.

The effect of a uniform through-surface flow on a cylinder and sphere

The effect of a uniform through-surface flow (velocity$U_{b}$) on a rigid and stationary cylinder and sphere (radius$a$) fixed in a free stream (velocity$U_{\infty }$) is analysed analytically and numerically. The flow is characterised by a dimensionless blow velocity${\it\Lambda}\,(=U_{b}/U_{\infty })$and Reynolds number$Re\,(=2aU_{\infty }/{\it\nu}$, where${\it\nu}$is the kinematic viscosity). High resolution numerical calculations are compared against theoretical predictions over the range$-3\leqslant {\it\Lambda}\leqslant 3$and$Re=1,10,100$for planar flow past a cylinder and axisymmetric flow past a sphere. For$-{\it\Lambda}\gg 1$, the flow is viscously dominated in a thin boundary layer of thickness${\it\nu}/|U_{b}|$adjacent to the rigid surface which develops in a time${\it\nu}/U_{b}^{2}$; the surface vorticity scales as$Re|{\it\Lambda}|U_{\infty }/a$for a cylinder and sphere. A boundary layer analysis is developed to analyse the unsteady viscous forces. Numerical results show that the surface pressure and vorticity distribution within the boundary layer agrees with a steady state analysis. The flow downstream of the body is irrotational so the wake volume flux,$Q_{w}$, is zero and the drag force is$F_{D}=-{\it\rho}U_{\infty }Q_{b}$, where${\it\rho}$is the density of the fluid and$Q_{b}$is the normal flux through the body surface. The drag coefficient is therefore$-2{\rm\pi}{\it\Lambda}$or$-8{\it\Lambda}$for a cylinder or sphere, respectively. A dissipation argument is applied to analyse the drag force; the rate of working of the drag force is balanced by viscous dissipation, flux of stagnation pressure and rate of work by viscous stresses due to sucking. At large$Re|{\it\Lambda}|$, the drag force is largely determined by viscous dissipation for a cylinder, with a weak contribution by the normal viscous stresses, while for a sphere, only$3/4$of the drag force is determined by viscous dissipation with the remaining$1/4$due to the flux of stagnation pressure through the sphere surface. When${\it\Lambda}\gg 1$, the boundary layer thickness initially grows linearly with time as vorticity is blown away from the rigid surface. The vorticity in the boundary layer is weakly dependent on viscous effects and scales as$U_{\infty }/a{\it\Lambda}$or$U_{\infty }/a{\it\Lambda}^{3/2}$for a cylinder and sphere, respectively. For large blow velocity, the vorticity is swept into two well-separated shear layers and the maximum vorticity decreases due to diffusion. The drag force is related to the vorticity distribution on the body surface and an approximate expression can be derived by considering the first term of a Fourier expansion in the surface vorticity. It is found that the drag coefficient$C_{D}$for a cylinder (corrected for flow boundedness) is weakly dependent on${\it\Lambda}$while for a sphere,$C_{D}$decreases with${\it\Lambda}$.

Влияние ПАВ на генерацию среднего течения около искривленной поверхности раздела

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 surfactants are 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 surfactants on the surface tension, the dominant mechanism for the generation is the Schlichting mechanisms under vibrations. Received 25.10.2016; accepted 29.11.2016

Cell mechanics in biomedical cavitation.

Studies on the deformation behaviours of cellular entities, such as coated microbubbles and liposomes subject to a cavitation flow, become increasingly important for the advancement of ultrasonic imaging and drug delivery. Numerical simulations for bubble dynamics of ultrasound contrast agents based on the boundary integral method are presented in this work. The effects of the encapsulating shell are estimated by adapting Hoff's model used for thin-shell contrast agents. The viscosity effects are estimated by including the normal viscous stress in the boundary condition. In parallel, mechanical models of cell membranes and liposomes as well as state-of-the-art techniques for quantitative measurement of viscoelasticity for a single cell or coated microbubbles are reviewed. The future developments regarding modelling and measurement of the material properties of the cellular entities for cutting-edge biomedical applications are also discussed.

Kinetic energy transport in Rayleigh–Bénard convection

The kinetic energy balance in Rayleigh–Bénard convection is investigated by means of direct numerical simulations for the Prandtl number range$0.01\leqslant \mathit{Pr}\leqslant 150$and for fixed Rayleigh number$\mathit{Ra}=5\times 10^{6}$. The kinetic energy balance is divided into a dissipation, a production and a flux term. We discuss the profiles of all the terms and find that the different contributions to the energy balance can be spatially separated into regions where kinetic energy is produced and where kinetic energy is dissipated. By analysing the Prandtl number dependence of the kinetic energy balance, we show that the height dependence of the mean viscous dissipation is closely related to the flux of kinetic energy. We show that the flux of kinetic energy can be divided into four additive contributions, each representing a different elementary physical process (advection, buoyancy, normal viscous stresses and viscous shear stresses). The behaviour of these individual flux contributions is found to be surprisingly rich and exhibits a pronounced Prandtl number dependence. Different flux contributions dominate the kinetic energy transport at different depths, such that a comprehensive discussion requires a decomposition of the domain into a considerable number of sublayers. On a less detailed level, our results reveal that advective kinetic energy fluxes play a key role in balancing the near-wall dissipation at low Prandtl number, whereas normal viscous stresses are particularly important at high Prandtl number. Finally, our work reveals that classical velocity boundary layers are deeply connected to the kinetic energy transport, but fail to correctly represent regions of enhanced viscous dissipation.

Adhesion force in fluids: effects of fingering, wetting, and viscous normal stresses.

Probe-tack measurements evaluate the adhesion strength of viscous fluids confined between parallel plates. This is done by recording the adhesion force that is required to lift the upper plate, while the lower plate is kept at rest. During the lifting process, it is known that the interface separating the confined fluids is deformed, causing the emergence of intricate interfacial fingering structures. Existing meticulous experiments and intensive numerical simulations indicate that fingering formation affects the lifting force, causing a decrease in intensity. In this work, we propose an analytical model that computes the lifting adhesion force by taking into account not only the effect of interfacial fingering, but also the action of wetting and viscous normal stresses. The role played by the system's spatial confinement is also considered. We show that the incorporation of all these physical ingredients is necessary to provide a better agreement between theoretical predictions and experiments.

The Boundary Layer Equations and a Dimensional Split Method for Navier-Stokes Equations in Exterior Domain of a Spheroid and Ellipsoid

In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.

Study on Capillary Instability With Heat and Mass Transfer Through Porous Media: Effect of Irrotational Viscous Pressure

In this paper, we investigate the effect of irrotational, viscous pressure on capillary instability of the interface between two viscous, incompressible, and thermally conducting fluids in a fully saturated porous medium when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface and when there is mass and heat transfer across the interface. The analysis extends our earlier work in which the capillary instability of two viscous and thermally conducting fluids in a fully saturated porous medium was studied assuming that the motion and pressure are irrotational and the viscosity enters through the jump in the viscous normal stress in the normal stress balance at the interface. 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 taking viscous contributions to the irrotational pressure. We use the Darcy's model, and a quadratic dispersion relation is obtained. It is observed that heat and mass transfer has a stabilizing effect on the stability of the system and this effect enhances in the presence of irrotational viscous pressure.

Nonlinear Rayleigh–Taylor instability of cylindrical flow with mass transfer through porous media

The work discusses nonlinear Rayleigh–Taylor instability of the interface between two viscous, incompressible and thermally conducting fluids in a fully saturated porous medium, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. We use viscous potential flow theory in which the flow is assumed to be irrotational and viscosity enters through normal viscous stresses at the interface. The perturbation analysis, in the light of the multiple expansions in both space and time, leads to imposing the well-known Ginzburg–Landau equation. The various stability conditions are discussed both analytically and numerically. The results are displayed in many plots showing the stability criteria in various parameter planes. It is observed that heat and mass transfer has stabilizing effect on the stability of the considered system while medium porosity destabilizes the interface. The flow through porous media is more stable than the pure flow.

Marangoni induced force on a drop in a Hele Shaw cell

We analyse the force balance on a cylindrical drop in a Hele-Shaw cell, subjected to a Marangoni flow caused by a surface tension gradient. Depth-averaged Stokes equations, called Brinkman equations, are introduced and a general closed form solution is obtained. The validity of the averaging procedure is ascertained by considering a linear surface tension gradient acting on a cylindrical flattened drop. The Marangoni-driven flow field and resulting force predicted by the Brinkman model are seen to match well a full three-dimensional direct numerical simulation. A closed form expression of the force acting on the drop is obtained, calculated from contributions due to the normal viscous stress, tangential viscous stress, and pressure fields, integrated on the drop perimeter. This expression is used to predict the force balance when a stationary droplet is submitted to both a carrier flow and a Marangoni flow. We show that previous results in the literature had underestimated by a factor two the Marangoni-induced force.

Nonlinear analysis of capillary instability with mass transfer through porous media

In this paper, we investigate the nonlinear capillary instability of the interface between two viscous, incompressible and thermally conducting fluids in a fully saturated porous medium, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. We use viscous potential flow theory in which the flow is assumed to be irrotational and viscosity enters through normal viscous stresses at the interface. The perturbation analysis, in the light of the multiple expansions, leads to imposing a first-order nonlinear partial differential equation. The various stability conditions are discussed both analytically and numerically. The results are displayed in many plots showing the stability criteria in various parameter planes. It is observed that the heat and mass transfer and porous medium both stabilize the interface while porosity supports the growth of disturbance waves.

Evaporative capillary instability for flow in porous media under the influence of axial electric field

We study the linear analysis of electrohydrodynamic capillary instability of the interface between two viscous, incompressible and electrically conducting fluids in a fully saturated porous medium, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface and, when there is mass and heat transfer across the interface. The fluids are subjected to a constant electric field in the axial direction. Here, we use an irrotational theory in which the motion and pressure are irrotational and the viscosity enters through the jump in the viscous normal stress in the normal stress balance at the interface. A quadratic dispersion relation that accounts for the growth of axisymmetric waves is obtained and stability criterion is given in terms of a critical value of wave number as well as electric field. It is observed that heat transfer has stabilizing effect on the stability of the considered system while medium porosity destabilizes the interface. The axial electric field has dual effect on the stability analysis.

Weakly nonlinear analysis on radially growing interface with the effect of viscous normal stress

Two-phase Hele-Shaw problem is studied as a model for a two-dimensional radially growing interface. Instead of the Young–Laplace equation, which has been employed in almost all previous studies concerning Hele-Shaw problem, we apply the normal stress balance and derive the new mode coupling equation for perturbation of the interface. In the present research, weakly nonlinear analysis is carried out and time evolution of the interfaces is numerically calculated. These numerical results suggest that our model reflects more exactly the nonlinear features of a radially growing interface, rather than the previous ones based on the Young–Laplace equation.

Three-dimensional magnetohydrodynamic Kelvin–Helmholtz instability of cylindrical flow with permeable boundaries

We study the linear magnetohydrodynamic Kelvin–Helmholtz instability of the interface between two viscous, incompressible, and electrically conducting fluids. The phases are enclosed between two coaxial cylindrical porous layers with the interface through which mass and heat transfer takes place. The fluids are subjected to a constant magnetic field parallel to the streaming direction, and the suction/injection velocities for the fluids at the permeable boundaries are also taken into account. Here, we use an irrotational theory in which the motion and pressure are irrotational, and the viscosity enters through the jump in the viscous normal stress in the normal stress balance at the interface. We consider both asymmetric and axisymmetric disturbances in our analysis. A quadratic dispersion relation is deduced and stability criterion is given in terms of a critical value of relative velocity, as well as, magnetic field. It has been observed that in the case of permeable boundaries, heat and mass transfer phenomena play a dual in the stability analysis. The flow through porous medium is more stable than the pure flow.

Determining the number of fingers in the lifting Hele-Shaw problem

The lifting Hele-Shaw cell flow is a variation of the celebrated radial viscous fingering problem for which the upper cell plate is lifted uniformly at a specified rate. This procedure causes the formation of intricate interfacial patterns. Most theoretical studies determine the total number of emerging fingers by maximizing the linear growth rate, but this generates discrepancies between theory and experiments. In this work, we tackle the number of fingers selection problem in the lifting Hele-Shaw cell by employing the recently proposed maximum-amplitude criterion [Dias and Miranda, Phys. Rev. E 88, 013016 (2013)]. Our linear stability analysis accounts for the action of capillary, viscous normal stresses, and wetting effects, as well as the cell confinement. The comparison of our results with very precise laboratory measurements for the total number of fingers shows a significantly improved agreement between theoretical predictions and experimental data.

Wavelength selection in Hele-Shaw flows: A maximum-amplitude criterion

As in most interfacial flow problems, the standard theoretical procedure to establish wavelength selection in the viscous fingering instability is to maximize the linear growth rate. However, there are important discrepancies between previous theoretical predictions and existing experimental data. In this work we perform a linear stability analysis of the radial Hele-Shaw flow system that takes into account the combined action of viscous normal stresses and wetting effects. Most importantly, we introduce an alternative selection criterion for which the selected wavelength is determined by the maximum of the interfacial perturbation amplitude. The effectiveness of such a criterion is substantiated by the significantly improved agreement between theory and experiments.

A full analytical solution for the force-driven compressible Poiseuille gas flow based on a nonlinear coupled constitutive relation

The compressible Poiseuille gas flow driven by a uniform force is analytically investigated using a phenomenological nonlinear coupled constitutive relation model. A new fully analytical solution in compact tangent (or hyperbolic tangent in the case of diatomic gases) functional form explains the origin behind the central temperature minimum and a heat transfer from the cold region to the hot region. The solution is not only proven to satisfy the conservation laws exactly but also well-defined for all physical conditions (the Knudsen number and a force-related dimensionless parameter). It is also shown that the non-Fourier law associated with the coupling of force and viscous shear stress in the constitutive relation is responsible for the existence of the central temperature minimum, while a kinematic constraint on viscous shear and normal stresses identified in the velocity shear flow is the main source of the nonuniform pressure distribution. In addition, the convex pressure profile with a maximum at the center is theoretically predicted for diatomic gases. Finally, the existence of the Knudsen minimum in the mass flow rate is demonstrated by developing an exact analytical formula for the average temperature of the bulk flow.

Effects of normal viscous stresses on radial viscous fingering

We revisit the radial viscous fingering problem in a Hele-Shaw cell, and consider the action of viscous stresses originated from velocity gradients normal to the fluid-fluid interface. The evolution of the interface during linear and weakly nonlinear stages is described analytically through a mode-coupling approach. We find that the introduction of normal stresses influences the stability and the ultimate morphology of the emerging patterns. Although at early stages normal stresses tend to stabilize the interface, they act to favor the development of tip-splitting phenomena at the weakly nonlinear regime. We have also verified that finger competition events are only significantly affected by normal stresses for circumstances involving the development of a large number of interfacial fingers.

A phenomenological and extended continuum approach for modelling non-equilibrium flows

This paper presents a new technique that combines Grad’s 13-moment equations (G13) with a phenomenological approach to rarefied gas flows. This combination and the proposed solution technique capture some important non-equilibrium phenomena that appear in the early continuum-transition flow regime. In contrast to the fully coupled 13-moment equation set, a significant advantage of the present solution technique is that it does not require extra boundary conditions explicitly; Grad’s equations for viscous stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations of transport. The relative computational cost of this novel technique is low in comparison to other methods, such as fully coupled solutions involving many moments or discrete methods. In this study, the proposed numerical procedure is tested on a planar Couette flow case, and the results are compared to predictions obtained from the direct simulation Monte Carlo method. This test case highlights the presence of normal viscous stresses and tangential heat fluxes that arise from non-equilibrium phenomena, which cannot be captured by the Navier–Stokes–Fourier constitutive equations or phenomenological modifications.

Purely irrotational theories for the viscous effects on the oscillations of drops and bubbles

In this paper, we apply two purely irrotational theories of the motion of a viscous fluid, namely, viscous potential flow (VPF) and the dissipation method to the problem of the decay of waves on the surface of a sphere. We treat the problem of the decay of small disturbances on a viscous drop surrounded by gas of negligible density and viscosity and a bubble immersed in a viscous liquid. The instantaneous velocity field in the viscous liquid is assumed to be irrotational. In VPF, viscosity enters the problem through the viscous normal stress at the free surface. In the dissipation method, viscosity appears in the dissipation integral included in the mechanical energy equation. Comparisons of the eigenvalues from VPF and the dissipation approximation with those from the exact solution of the linearized governing equations are presented. The results show that the viscous irrotational theories exhibit most of the features of the wave dynamics described by the exact solution. In particular, VPF and DM give rise to a viscous correction for the frequency that determines the crossover from oscillatory to monotonically decaying waves. Good to reasonable quantitative agreement with the exact solution is also shown for certain ranges of modes and dimensionless viscosity: For large viscosity and short waves, VPF is a very good approximation to the exact solution. For ‘small’ viscosity and long waves, the dissipation method furnishes the best approximation.