The effect of viscosity and thermal conduction on the acoustics in a shear layer above an impedance wall is investigated numerically and asymptotically by solving the compressible linearised Navier-Stokes equations. It is found that viscothermal effects can be as important as shear, and therefore including shear while neglecting viscothermal effects by solving the linearised Euler equations is questionable. In particular, the damping rate of upstream propagating waves is found to be dramatically under-predicted by the LEE in certain instances. The effects of viscosity on stability are also found to be important. Short wavelength disturbances are stabilised by viscosity, greatly altering the characteristic wavelength and maximum growth rate of instability. For the parameters typical of aeroacoustic simulations considered here, the Reynolds number below which the flow stabilizes ranges from $10^5$ to $10^7$. By assuming a thin but nonzero-thickness boundary layer, asymptotic analysis leads to a system of boundary layer governing equations for the acoustics. This system may be solved numerically to produce an effective impedance boundary condition, applicable at the wall of a uniform inviscid flow, that accounts for both the shear and viscosity within the boundary layer. An alternative asymptotic analysis in the high frequency limit yields a different set of equations with analytic solutions. The acoustic mode shapes and axial wavenumbers from both asymptotic analyses compare well with numerical solutions of the full LNSE. A closed-form effective impedance boundary condition is derived from the high-frequency asymptotics, suitable for application in frequency-domain numerical simulations. Finally, surface waves are considered, and it is shown that a viscous flow over an impedance lining supports a greater number of surface wave modes than an inviscid flow.
Topology optimization offers optimal material layouts, enabling automation in the design of devices. Given the recent advances in computer technology and additive manufacturing, topology optimization is increasingly being used to design complex porous structures, for example, microfluidic devices. For the flow of fluids in such miniature-sized porous structures, viscous shearing stress will be significant. But the Darcy model -- the most popular mathematical model describing the flow of a single-phase incompressible fluid in rigid porous media -- neglects the internal friction arising from vis…
In this work, we extend the analyses devoted to Newtonian viscous fluids previously reported by Ribe [Physical Review E 68, 036305 (2003)], by investigating shear thickening (dilatant) and shear thinning (pseudoplastic) effects on the development of folding instabilities in non-Newtonian viscous sheets of which viscosity is given by a power-law constitutive equation. Such instabilities are trigged by compression stresses acting on viscous sheets that leave a channel at a very small initial velocity, fall, and then hit a solid surface or a fluid substrate. Our study is conducted through a mixed…
The input of the hydraulically induced shear traction into the elasticity equation is estimated for the entire fracture surface. It is established that, except for negligibly small vicinities of a pointed source and of the fluid front, the relative input of the viscous shear does not exceed 10 - 4 of the input of the conventionally accounted term. The estimation is true for Newtonian, as well as non-Newtonian thinning fluids. This implies that there is no need to account for viscous shear not only in the form of conventional near-front asymptotics and fracture conditions, but also in factors e…
Interfacial standing wave-patterns disentangle dilatational and shear surface viscous effects
Imperial College London（帝国理工学院）；Abdullah Al Salem University（阿卜杜拉·萨勒姆大学）；Université Paris Saclay（巴黎萨克雷大学）；Hongik University（弘益大学）；University of Cambridge（剑桥大学）
We experimentally and theoretically study impacts into dense cornstarch and water suspensions. We vary impact speed as well as intruder size, shape, and mass, and we characterize the resulting dynamics using high-speed video and an onboard accelerometer. We numerically solve previously proposed models, most notably the added-mass model as well as a class of {viscous-like} models. In the {viscous-like models}, the intruder dynamics are dominated by {large, viscous-like forces} at the boundary of the jammed front {where large shear rates and accompanying large viscosities are present.} We find t…
When a drop of fluid hits a small solid target of comparable size, it expands radially until reaching a maximum diameter and subsequently recedes. In this work, we show that the expansion process of liquid sheets is controlled by a combination of shear (on the target) and biaxial extensional (in the air) deformations. We propose an approach toward a rational description of the phenomenon for Newtonian and viscoelastic fluids by evaluating the viscous dissipation due to shear and extensional deformations, yielding a prediction of the maximum expansion factor of the sheet as a function of the re…
The traditional approach in the study of hydrodynamic stability of stratified fluids includes the stick boundary conditions between layers. However, this rule may be violated in polymer systems and as a consequence various instabilities may arise. The main objective of this paper is to analyze theoretically the influence of slip boundary conditions on the hydrodynamic stability of the interface between two immiscible viscous layers subjected to simple shear flow. It is found that the growth rate of long-wave disturbances is fairly sensitive to the slip at the interface between layers as well a…
A viscous instability in shearing laminar axisymmetric hydrodynamic flows around a gravitating center is described. In the linearized hydrodynamic equations written in the Boussinesq approximation with microscopic molecular transport coefficients, the instability arises when the viscous dissipation is taken into account in the energy equation. Using the local WKB approximation, we derive a third-order algebraic dispersion equation with two modes representing the modified Rayleigh modes R+ and R-, and the third X-mode. We show that in thin accretion flows the viscosity destabilizes one of the R…
The motion of a spheroid in a viscous incompressible fluid after a sudden small impulse in the direction of the symmetry axis is studied on the basis of the linearized Navier-Stokes equations. The time-dependence of the spheroid velocity follows by a Fourier transform from the frequency- dependence of the impedance involving friction coefficient, body mass, and added fluid mass. A term proportional to the square root of frequency in the asymptotic high frequency expansion of the impedance, with a Basset coefficient, describes the initial decay in time from the initial value determined by the e…
With flow, the acoustical effect of a lined wall cannot be described by a single quantity like the wall impedance. At least two quantities are required. In addition to the impedance, the unsteady tangential force exerted by the wall on the flow has to be taken into account. This force is due either to viscous effects or to the unsteady transfer of axial momentum from the flow into the lined wall. The paper describes a Stress--Impedance model where the two variables used are the impedance and the friction factor that links the pressure to a tangential stress at the wall. The use of a wall stres…
The shear induced collective diffusivity in an emulsion of viscous drops, specifically as a function of viscosity ratio, was numerically computed. An initially randomly packed layer of viscous drops spreading due to drop-drop interactions in an imposed shear has been simulated. The shear induced collective diffusivity coefficient was computed using a self-similar solution of the drop concentration profile. We also obtained the collective diffusivity computing the dynamic structure factor from the simulated drop positions--an analysis typically applied only to homogeneous systems. The two quant…
The use of tank-treading as a means of propulsion for microswimmers in viscous shear flows is taken into exam. We discuss the possibility that a vesicle be able to control the drift in an external shear flow, by varying locally the bending rigidity of its own membrane. By analytical calculation in the quasi-spherical limit, the stationary shape and the orientation of the tank-treading vesicle in the external flow, are determined, working to lowest order in the membrane inhomogeneity. The membrane inhomogeneity acts in the shape evolution equation as an additional force term, that can be used t…
We use matched asymptotics to derive analytical formulae for the acoustic impedance of a subwavelength orifice consisting of a cylindrical perforation in a rigid plate. In the inviscid case, an end correction to the length of the orifice due to Rayleigh is shown to constitute an exponentially accurate approximation in the limit where the aspect ratio of the orifice is large; in the opposite limit, we derive an algebraically accurate correction, depending upon the logarithm of the aspect ratio, to the impedance of a circular aperture in a zero-thickness screen. Viscous effects are considered in…
A self-consistent, thermodynamic approach is employed to derive the wave energy of a magnetohydrodynamic system within the harmonic approximation and to obtain the familiar dispersion relation from the resulting equation of motion. The evolution of the system due to an external perturbation is studied by a linear response formalism, that also gives the energy absorbed by the magnetohydrodynamic system from the external field. The calculated wave energy reveals the presence of positive and negative energy waves, that coalesce together to give rise to Kelvin - Helmholtz instability of the system…
The shear misfit model for the highly viscous flow is based upon a theoretical prediction for its terminal stage in terms of irreversible Eshelby relaxations in the five-dimensional shear space. The model is shown to predict a small delta-function (Debye peak) in the dielectric spectrum, in agreement with experimental evidence. It is extended from shear fluctuations to density fluctuations, a new relation between adiabatic and isothermal compressibility jumps at the glass transition is derived, and the model is brought into a form which requires only three temperature-dependent parameters for …
Background: The acoustic damping in gas turbines and aero-engines relies to a great extent on acoustic liners that consists of a cavity and a perforated face sheet. The prediction of the impedance of the liners by direct numerical simulations is nowadays not feasible due to the hundreds to thousands repetitions of tiny holes. We introduce a procedure to numerically obtain the Rayleigh conductivity for acoustic liners for viscous gases at rest, and with it define the acoustic impedance of the perforated sheet. Results: The proposed method decouples the effects that are dominant on different sca…
In a wide range of applications, microfluidic channels are implemented in soft substrates. In such configurations, where fluidic inertia and compressibility are negligible, the propagation of fluids in channels is governed by a balance between fluid viscosity and elasticity of the surrounding solid. The viscous-elastic interactions between elastic substrates and non-Newtonian fluids are particularly of interest due to the dependence of viscosity on the state of the system. In this work, we study the fluid-structure interaction dynamics between an incompressible non-Newtonian fluid and a slende…
Inhomogeneities in the velocity field of a moving fluid are dampened by the inherent viscous behaviour of the system. Both bulk and shear effects, related to the divergence and the curl of the velocity field, are relevant. On molecular time scales, beyond the Navier-Stokes description, memory plays an important role. We demonstrate here on the basis of molecular and overdamped Brownian dynamics many-body simulations that analogous viscous effects act on the acceleration field. This acceleration viscous behaviour is associated with the divergence and the curl of the acceleration field and it ca…
The Holmboe wave instability is one of the classic examples of a stratified shear instability, usually explained as the result of a resonance between a gravity wave and a vorticity wave. Historically, it has been studied by linear stability analyses at infinite Reynolds number, $Re$, and by direct numerical simulations at relatively low $Re$ in the regions known to be unstable from the inviscid linear stability results. In this paper, we perform linear stability analyses of the classical `Hazel model' of a stratified shear layer (where the background velocity and density distributions are assu…
Profound temperature peaks are observed in regions of high velocity shear in a 2D dusty plasma experiment with laser-driven flow. These are attributed to viscous heating, which occurs due to collisional scattering in a shear flow. Using measurements of viscosity, thermal conductivity, and spatial profiles of flow velocity and temperature, we determine three dimensionless numbers: Brinkman Br = 0.5, Prandtl Pr = 0.09, and Eckert Ec = 5.7. The large value of Br indicates significant viscous heating that is consistent with the observed temperature peaks.
The initial-boundary value problem of the vorticity equation has been solved numerically by an iterative method. A variety of initial vorticity distributions is specified. All of them can be described by simple mathematical functions: there are a vorticity source-sink pair, circular shears out of a localised monopole, a cat's-eye topology and a few flows originating from single or multiple vortices. Our computational results show diverse flow phenomena, such as roll-ups of shear layers, vortex merging or impingement, as well as birth of spiral structures.
Using absorbance measurements through a Couette cell containing an emulsion of buoyant droplets, volume fraction profiles are measured at various shear rates. These viscous resuspension experiments allow a direct determination of the normal stress in the vorticity direction in connection with the suspension balance model that has been developed for suspensions of solid particles. The results unambigously show that the normal viscosity responsible for the shear-induced migration of the droplets is independant on the capillary number, implying that particle deformation does not play a great role…
The rheological response under simple shear of an active suspension of Escherichia coli is determined in a large range of shear rates and concentrations. The effective viscosity and the time scales characterizing the bacterial organization under shear are obtained. In the dilute regime, we bring evidences for a low shear Newtonian plateau characterized by a shear viscosity decreasing with concentration. In the semi-dilute regime, for particularly active bacteria, the suspension display a "super-fluid" like transition where the viscous resistance to shear vanishes, thus showing that macroscopic…
We derive and study a new hyperbolic two-phase model of a porous deformable medium saturated by a viscous fluid. The governing equations of the model are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems and by generalizing the unified hyperbolic model of continuum fluid and solid mechanics. Similarly to the unified model, the presented model takes into account the viscosity of the saturating fluid through a hyperbolic reformulation. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturated flu…
From the Navier-Stokes-Korteweg (NSK) equations, the exact relations between the fundamental surface physical quantities for two-phase viscous flow with diffuse interface are derived, including density gradient, shear stress, vorticity, pressure, enstrophy flux and surface curvature. These theoretical results provide a solid foundation of the boundary/interfacial vorticity dynamics and a new tool for analysis of complex interfacial phenomena in two-phase viscous flows. To demonstrate the application of the developed results, simulation of a droplet impacting and spreading on a solid wall is co…
We study the effect of droplet coalescence on turbulent wall-bounded flows, by means of direct numerical simulations. In particular, the volume-of-fluid and front-tracking methods are used to simulate turbulent channel flows containing coalescing and non-coalescing droplets, respectively. We find that coalescing droplets have a negligible effect on the drag, whereas the non-coalescing ones steadily increase drag as the volume fraction of the dispersed phase increases: indeed, at 10\% volume fraction, the non-coalescing droplets show a 30\% increase in drag, whereas the coalescing droplets show…
A shear flow of particles in a laser-driven two-dimensional (2D) dusty plasma are observed in a further study of viscous heating and thermal conduction. Video imaging and particle tracking yields particle velocity data, which we convert into continuum data, presented as three spatial profiles: mean particle velocity (i.e., flow velocity), mean-square particle velocity, and mean-square fluctuations of particle velocity. These profiles and their derivatives allow a spatially-resolved determination of each term in the energy and momentum continuity equations, which we use for two purposes. First,…
In the present paper, we examine the viscous flow evolution in a square cavity. Coupled with the stream function, the initial-boundary value problem of the vorticity is numerically solved by a method of iteration. The only boundary condition is the wall velocity which in turn defines the Dirichlet value for the stream function. We assert that the corner singularity in the cavity flow is in fact a theoretical artefact. By adopting suitably regulated lid velocity, excellent comparison with experiments is found because of the converged palinstrophy field. Our calculations demonstrate that asymmet…
We use X-ray imaging to study viscous resuspension. In a Taylor-Couette geometry, we shear an initially settled layer of spherical glass particles immersed in a Newtonian fluid and measure the local volume fraction profiles. In this configuration, the steady-state profiles are simply related to the normal viscosity defined in the framework of the Suspension Balance Model (SBM). These experiments allow us to examine this fundamental quantity over a wide range of volume fractions, in particular in the semi-dilute regime where experimental data are sorely lacking. Our measurements unambiguously s…
The aim of this Letter is to characterize the flow regimes of suspensions of finite-size rigid particles in a viscous fluid at finite inertia. We explore the system behavior as function of the particle volume fraction and the Reynolds number (the ratio of flow and particle inertia to viscous forces). Unlike single phase flows where a clear distinction exists between the laminar and the turbulent states, three different regimes can be identified in the presence of a particulate phase, with smooth transitions between them. At low volume fractions, the flow becomes turbulent when increasing the R…
We present a detailed comparison of the rheological behaviour of sheared sediment beds in a pressure-driven, straight channel configuration based on data that was generated by means of fully coupled, grain-resolved direct numerical simulations and experimental measurements reviously published by Aussillous {\it et al.} (J. Fluid Mech., vol. 736, 2013, pp. 594-615). The highly-resolved simulation data allows to compute the stress balance of the suspension in the streamwise and vertical directions and the stress exchange between the fluid and particle phase, which is information needed to infer …
The morphological dynamics, instabilities and transitions of elastic filaments in viscous flows underlie a wealth of biophysical processes from flagellar propulsion to intracellular streaming, and are also key to deciphering the rheological behavior of many complex fluids and soft materials. Here, we combine experiments and computational modeling to elucidate the dynamical regimes and morphological transitions of elastic Brownian filaments in a simple shear flow. Actin filaments are employed as an experimental model system and their conformations are investigated through fluorescence microscop…
A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier, J. Fluid Mech., vol. 699, 2012, pp. 174-197). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr-Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a h…
We discuss a pure hyperbolic alternative to the Navier-Stokes equations, which are of parabolic type. As a result of the substitution of the concept of the viscosity coefficient by a microphysics-based temporal characteristic, particle settled life (PSL) time, it becomes possible to formulate a model for viscous fluids in a form of first order hyperbolic partial differential equations. Moreover, the concept of PSL time allows the use of the same model for flows of viscous fluids (Newtonian or non-Newtonian) as well as irreversible deformation of solids. In the theory presented, a continuum is …
We investigate freely expanding liquid sheets made of either simple Newtonian fluids or solutions of high molecular water-soluble polymer chains. A sheet is produced by the impact of a drop on a quartz plate covered with a thin layer of liquid nitrogen that suppresses shear viscous dissipation thanks to an inverse Leidenfrost effect. The sheet expands radially until reaching a maximum diameter and subsequently recedes. Experiments indicate the presence of two expansion regimes: the capillary regime, where the maximum expansion is controlled by surface tension forces and does not depend on the …
An exhaustive description of the dynamics under shear flow of a large number of red blood cells in dilute regime is proposed, which highlights and takes into account the dispersion in cell properties within a given blood sample. Physiological suspending fluid viscosity is considered, a configuration surprisingly seldom considered in experimental studies, as well as a more viscous fluid that is a reference in the literature. Stable and unstable flipping motions well described by Jeffery orbits or modified Jeffery orbits are identified, as well as transitions to and from tank-treading motion in …
