The rheological characterisation of complex fluids is mostly performed under simple shear flow in rotational rheometers. Their modern commercial versions are extremely sensitive instruments which are able to provide very accurate measurements of low values of different material functions, such as viscosity, viscoelastic moduli, etc., under certain ideal flow conditions. Nevertheless, they fail in providing reliable data when characterising the response of complex fluids at short time scales, due to artefacts induced by either instrument or fluid inertia. This is crucial in the analysis of the rheological properties of new formulations of shear thickening fluids specifically developed for protective applications, in which the performance is extremely linked to the time dependent structural changes provoked by the sudden impact loads. Thus, the necessity of providing a reliable experimental tool able to impose large, rapid, transient shear deformation for their rheological characterisation in conditions similar to those of the applications becomes evident. This numerical study aims at assessing the potential use of the penetroviscometer for the measurement of the transient shear viscosity of complex fluids at short time scales beyond the current limits of commercial rotational rheometers.
We analyze transient dynamics during shear start-up in viscoelastic flows between two parallel plates, with a specific focus on the signatures for the onset of transient shear banding using the Johnson-Segalman, non-stretching Rolie-Poly and Giesekus models. We explore the dynamics of shear start-up in monotonic regions of the constitutive curves using two different methodologies: (i) the oft-used `frozen-time' linear stability (eigenvalue) analysis, wherein we examine whether infinitesimal perturbations imposed on instantaneous stress components (treated as quasi steady states) exhibit expone…
Two approaches to the problem of transition to turbulence of shear flows are popular in the literature. The first is the linear one of transient growth which focuses on the likely form of the most 'dangerous' (lowest energy) turbulence-triggering disturbances. The second is the nonlinear calculation of the laminar-turbulent boundary which instead focuses on their typical amplitudes. We look to bridge the gap between these two perspectives by considering the fully nonlinear transient growth problem to estimate both the form and amplitude of the most dangerous disturbance. We thereby discover a …
Spatially inhomogeneous shear flow occurs in entangled polymer solutions, both as steady state shear banding and transiently after a large step strain or during start up to a steady uniform shear rate. Steady state shear banding is a hallmark of models with a non-monotonic constitutive relation between total shear stress and applied shear rate, but transient banding is sometimes seen in fluids that do not shear band at steady state. We model this behavior using the diffusive Rolie-Poly model in a Newtonian solvent, whose constitutive behavior can be monotonic or non-monotonic depending on the …
We compute the rheological properties of inelastic hard spheres in steady shear flow for general shear rates and densities. Starting from the microscopic dynamics we generalise the Integration Through Transients (\textsc{itt}) formalism to a fluid of dissipative, randomly driven granular particles. The stress relaxation function is computed approximately within a mode-coupling theory---based on the physical picture, that relaxation of shear is dominated by slow structural relaxation, as the glass transition is approached. The transient build-up of stress in steady shear is thus traced back to …
We investigate the effects of micellar entanglement number on the kinetics of shear banding flow formation in a Taylor-Couette flow. Three sets of wormlike micellar solutions, each set with a similar fluid elasticity and zero-shear-rate viscosity, but with varying entanglement densities, are studied under start-up of steady shear. Our experiments indicate that in the set with the low fluid elasticity, the transient shear banding flow is characterized by the formation of a transient flow reversal in a range of entanglement densities. Outside of this range, the transient flow reversal is not obs…
A linear non-modal mechanism for transient amplification of perturbation energy is known to trigger sub-critical transition to turbulence in many shear flows. Feedback control strategies for minimizing this transient energy growth can be formulated as convex optimization problems based on linear matrix inequalities. Unfortunately, solving the requisite linear matrix inequality problem can be computationally prohibitive within the context of high-dimensional fluid flows. In this work, we investigate the utility of control-oriented reduced-order models to facilitate the design of feedback flow c…
In linearly stable shear flows at moderate Re, turbulence spontaneously decays despite the existence of a codimension-one manifold, termed the edge of chaos, which separates decaying perturbations from those triggering turbulence. We statistically analyse the decay in plane Couette flow, quantify the breaking of self-sustaining feedback loops and demonstrate the existence of a whole continuum of possible decay paths. Drawing parallels with low-dimensional models and monitoring the location of the edge relative to decaying trajectories we provide evidence, that the edge of chaos separates state…
Dense emulsions, colloidal gels, microgels, and foams all display a solid-like behavior at rest characterized by a yield stress, above which the material flows like a liquid. Such a fluidization transition often consists of long-lasting transient flows that involve shear-banded velocity profiles. The characteristic time for full fluidization, $τ_\text{f}$, has been reported to decay as a power-law of the shear rate $\dot γ$ and of the shear stress $σ$ with respective exponents $α$ and $β$. Strikingly, the ratio of these exponents was empirically observed to coincide with the exponent of the He…
We present a reformulation of the 'reactive rod model' (RRM) of Dutta and Graham [Dutta, Sarit and Graham, Michael D., JNNFM 251 (2018)], a constitutive model for describing the behavior of dilute wormlike micelle solutions. The RRM treats wormlike micelle solutions as dilute suspensions of rigid Brownian rods undergoing reversible scission and growth in flow. Evolution equations for micelle orientation and stress contribution are coupled to a kinetic reaction equation for a collective micelle length, producing dynamic variations in the length and rotational diffusivity of the rods. This model…
This study focuses on a parametric study of the laminar fast transient flow of non-Newtonian fluids through helical pipes. Classical simulations of fluid hammer do not deal with the pipeline helicity and non-Newtonian characteristics of the fluid, while the present work addresses those features. To this end, the power-law model is employed to accommodate the non-Newtonian behavior of the fluid. Effects of the pipe wall elasticity and compressibility of the working fluid are taken into account through a modified bulk modulus elasticity of the fluid. The results of the three-dimensional numerica…
We report a non-perturbative study of the effects of shear flows on turbulence reduction in a decaying turbulence in two dimensions. By considering different initial power spectra and shear flows (zonal flows, combined zonal flows and streamers), we demonstrate how shear flows rapidly generate small scales, leading to a fast damping of turbulence amplitude. In particular, a double exponential decrease in turbulence amplitude is shown to occur due to an exponential increase in wavenumber. The scaling of the effective dissipation time scale $τ_{e}$, previously taken to be a hybrid time scale $τ_…
We obtain a general solution for the water waves resulting from a general, time-dependent surface pressure distribution, in the presence of a shear current of uniform vorticity beneath the surface, in three dimensions. Linearized governing equations and boundary conditions including the effects of gravity, a distributed external pressure disturbance, and constant finite depth, are solved analytically, and particular attention is paid to classic initial value problems: an initial pressure impulse and a steady pressure distribution which appears suddenly. In the present paper, good agreement wit…
Coriolis force effects on shear flows are important in geophysical and astrophysical contexts. We here report a study on the linear stability and the transient energy growth of the plane Couette flow with system rotation perpendicular to the shear direction. External rotation causes linear instability. At small rotation rates, the onset of linear instability scales inversely with the rotation rate and the optimal transient growth in the linearly stable region is slightly enhanced, ~Re^2. The corresponding optimal initial perturbations are characterized by roll structures inclined in the stream…
This paper reports experiments on the shear transient response of concentrated non-Brownian suspensions. The shear viscosity of the suspensions is measured using a wide-gap Couette rheometer equipped with a Particle Image Velocimetry (PIV) device that allows measuring the velocity field. The suspensions made of PMMA particles (31$μ$m in diameter) suspended in a Newtonian index- and density-matched liquid are transparent enough to allow an accurate measurement of the local velocity for particle concentrations as high as 50%. In the wide-gap Couette cell, the shear induced particle migration is …
We investigate the weakly nonlinear dynamics of transient gravity waves at infinite depth under the influence of a shear current varying linearly with depth. An analytical solution is permitted via integration of the Euler equations. Although similar problems were investigated in the 1960's and 70's for special cases of resonance, this is to our knowledge the first general wave interaction (mode coupling) solution derived to second order with a shear current present. Wave interactions are integrable in a spectral convolution to yield the second order dynamics of initial value problems. To seco…
Linear stability and the non-modal transient energy growth in compressible plane Couette flow are investigated for two prototype mean flows: (a) the {\it uniform shear} flow with constant viscosity, and (b) the {\it non-uniform shear} flow with {\it stratified} viscosity. Both mean flows are linearly unstable for a range of supersonic Mach numbers ($M$). For a given $M$, the critical Reynolds number ($Re$) is significantly smaller for the uniform shear flow than its non-uniform shear counterpart. An analysis of perturbation energy reveals that the instability is primarily caused by an excess t…
We investigate the flow evolution of a linear and a branched wormlike micellar solution with matched rheology in a Taylor-Couette (TC) cell using a combination of particle-tracking velocimetry, birefringence, and turbidity measurements. Both solutions exhibit a stress plateau within a range of shear rates. Under startup of a steady shear rate flow within the stress plateau, both linear and branched samples exhibit strong transient shear thinning flow profiles. However, while the flow of the linear solution evolves to a banded structure at longer times, the flow of the branched solution transit…
The present thesis deals with the non-modal linear analysis of 3D perturbations in wall flows. In the first part,a solution to the Orr-Sommerfeld and Squire IVP, in the form of orthogonal functions expansion, is researched. The Galerkin method is successfully implemented to numerically compute approximate solutions for bounded flows. The Chandrasekhar functions revealed to ensure a fifth order of accuracy. The focus of the subsequent analysis is on the transient behavior of the perturbation frequency and phase velocity. The results confirm recent observations about a jump in the temporal evolu…
We propose a general strategy for determining the minimal finite amplitude isturbance to trigger transition to turbulence in shear flows. This involves constructing a variational problem that searches over all disturbances of fixed initial amplitude, which respect the boundary conditions, incompressibility and the Navier--Stokes equations, to maximise a chosen functional over an asymptotically long time period. The functional must be selected such that it identifies turbulent velocity fields by taking significantly enhanced values compared to those for laminar fields. We illustrate this approa…
A highly sheared dense aqueous suspension of granular cornstarch particles displays rich nonlinear rheology. We had previously demonstrated the growth and onset of interfacial instabilities when shear-thinning cornstarch suspensions were displaced by a Newtonian fluid, and had suggested methods to maximise displacement efficiency [Palak, R. Sathayanath, S. K. Kalpathy and R. Bandyopadhyay, Colloids Surf. A Physicochem. Eng. Asp., 629 (2021) 127405]. In the present work, we explore the miscible displacement of a dense aqueous cornstarch suspension in its discontinuous shear-thickening regime in…
We study the waves and wave-making forces acting on ships travelling on currents which vary as a function of depth. Our concern is realism; we consider a real current profile from the Columbia River, and model ships with dimensions and Froude numbers typical of three classes of vessels operating in these waters. To this end we employ the most general theory of waves from free-surface sources on shear current to date, which we derive and present here. Expressions are derived for ship waves which satisfy an arbitrary dispersion relation and are generated by a wave source acting on the free surfa…
Our goal is to gain new insights into the physics of wave overreflection phenomenon in MHD nonuniform/shear flows changing the existing trend/approach of the phenomenon study. The performed analysis allows to separate from each other different physical processes, grasp their interplay and, by this way, construct the basic physics of the overreflection in incompressible MHD flows with linear shear of mean velocity, ${\bf U}_0=(Sy,0,0)$, that contain two different types of Alfv${\rm \acute{e}}$n waves. These waves are reduced to pseudo- and shear shear-Alfv${\rm \acute{e}}$n waves when wavenumbe…
Recent studies have brought into question the view that at sufficiently high Reynolds number turbulence is an asymptotic state. We present the first direct observation of the decay of turbulent states in Taylor-Couette flow with lifetimes spanning five orders of magnitude. We also show that there is a regime where Taylor-Couette flow shares many of the decay characteristics observed in other shear flows, including Poisson statistics and the coexistence of laminar and turbulent patches. Our data suggest that characteristic decay times increase super-exponentially with increasing Reynolds number…
Non-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical and analytical computations of linear transient growth covering all linearly stable regimes of Taylor--Couette flow. Our numerical experiments reveal comparable energy amplifications in the different regimes. For high shear Reynolds numbers Re the optimal transient energy growth always follows a 2/3-scaling with Re, which allows for large amplifications even…
In order to understand whether, and to what extent, spectral representation can effectively highlight the nonlinear interaction among different scales, it is necessary to consider the state that precedes the onset of instabilities and turbulence in flows. In this condition, a system is still stable, but is however subject to a swarming of arbitrary 3D small perturbations. These can arrive any instant, and then undergo a transient evolution which is ruled out by the initial-value problem associated to the Navier-Stokes linearized formulation. The set of 3D small perturbations constitutes a syst…
Turbulent state of spectrally stable shear flows may be developed and sustained according to the bypass scenario of transition. If it works in non-magnetised boundless and homogeneous quasi-Keplerian flow, transiently growing shearing vortices should supply turbulence with energy. Employing the large shearing box approximation, as well as a set of global disc models, we study the optimal growth of the shearing vortices in such a flow in the whole range of azimuthal length-scales, $λ_y$, as compared to the flow scale-height, $H$. It is shown that with the account of the viscosity the highest po…
The flow driven by a rapidly expanding and collapsing cavitation bubble in a narrow cylindrical gap is studied with the volume of fluid method. The simulations reveal a developing plug flow during the early expansion followed by flow reversal at later stages. An adverse pressure gradient leads to boundary layer separation and flow reversal, causing large shear stress near the boundaries. Analytical solution to a planar pulsating flow shows qualitative agreement with the CFD results. The shear stress close to boundaries has implications to deformable objects located near the bubble: experiments…
We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the non-monotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are refor…
Sommerfeld paradox (turbulence paradox) roughly says that mathematically the Couette linear shear flow is linearly stable for all values of the Reynolds number, but experimentally transition from the linear shear to turbulence occurs under perturbations of any size when the Reynolds number is large enough. In [Li, Lin 2011], we offered a resolution of this paradox. The aim of this paper is to provide a numerical implementation of the resolution. The main idea of the resolution is that even though the linear shear is linearly stable, slow orbits (also called quasi-steady states) in arbitrarily …
The classical Squire transformation is extended to the entire eigenfunction structure of both Orr-Sommerfeld and Squire modes. For arbitrary Reynolds numbers Re, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number Re2D. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal g…
We investigated the transient relaxation of a Discontinuous Shear Thickening suspension of cornstarch in water. We performed 2 types of relaxation experiments starting from a steady shear in a parallel plate rheometer, followed by either stopping the top plate rotation and measuring the transient torque relaxation, or removing the torque on the plate and measuring the transient tool rotation. We found that at low weight fraction $ϕ_{eff}<58.8\pm0.4\%$, the suspensions exhibited a relaxation behavior consistent with a generalized Newtonian fluid. However, for larger weight fraction $58.8\% < ϕ_…
We present a reduced model for the transition to turbulence in shear flow that is simple enough to admit a thorough numerical investigation while allowing spatio-temporal dynamics that are substantially more complex than those allowed in previous modal truncations. Our model allows a comparison of the dynamics resulting from initial perturbations that are localised in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localised initial conditions the subcritical transition to a `turbulent' state (i) takes place more abruptly, with a boundary between laminar…
We conduct laboratory experiments on the time evolution of an ice layer cooled from below and subjected to a turbulent shear flow of warm water from above. Our study is motivated by observations of warm water intrusion into the ocean cavity under Antarctic ice shelves, accelerating the melting of their basal surfaces. The strength of the applied turbulent shear flow in our experiments is represented in terms of its Reynolds number $\textit{Re}$, which is varied over the range $2.0\times10^3 \le \textit{Re} \le 1.0\times10^4$. Depending on the water temperature, partial transient melting of the…
Linear transient growth analysis is commonly used to suggest the structure of disturbances which are particularly efficient in triggering transition to turbulence in shear flows. We demonstrate that the addition of nonlinearity to the analysis can substantially change the prediction made in pipe flow from simple 2 dimensional streamwise rolls to a spanwise and cross-stream localised 3 dimensional state. This new nonlinear optimal is demonstrably more efficient in triggering turbulence than the linear optimal indicating that there are better ways to design perturbations to achieve transition.
A subcritical transition to turbulence in magnetized Keplerian shear flows is investigated using a statistical approach. Three-dimensional numerical simulations of the shearing box equations with zero net magnetic flux are employed to determine the transition from decaying to sustained turbulence as a function of the magnetic Reynolds number Rm. The results reveal no clear transition to sustained turbulence as the average lifetime of the transients grows as an exponential function of Rm, in accordance with a type-II supertransient law.
Global modal/non-modal analysis of high-enthalpy Martian entry vehicles
Graduate Aerospace Laboratories, California Institute of Technology（加州理工学院高等航空实验室）；Department of Aeronautics and Astronautics, Massachusetts Institute of Technology（麻省理工学院航空航天系）
Recent step strain experiments in well-entangled polymeric liquids demonstrated a bulk fracture-like phenomenon. We have studied this instability using a modern version of the Doi-Edwards theory for entangled polymers, and we find close quantitative agreement with the experiments. The phenomenon occurs because the viscoelastic liquid is sheared into a rubbery state that possesses an elastic constitutive instability (Marrucci and Grizzuti, 1983). The fracture is a transient manifestation of this instability, which relies on the amplification of spatially inhomogeneous fluctuations. This mechani…
Continuum modeling of Soft Glassy Materials under shear
Università di Roma “Tor Vergata”（罗马托尔维尔加塔大学）；ENSL, CNRS, Laboratoire de physique（里昂高等师范学校，法国国家科学研究中心，物理实验室）；Univ. de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière（里昂大学，克莱贝尔·伯纳德里昂第一大学，法国国家科学研究中心，光物质研究所）；Institut Universitaire de France (IUF)（法兰西学院）；Eindhoven University of Technology（埃因霍温理工大学）；CNR-IAC（意大利国家研究委员会应用化学研究所）
Numerous studies in the past 40 years have established that turbulent flow fields are populated by transient coherent structures that represent patches of fluids moving cohesively for significant distances before they are worn out by momentum exchange with the surrounding fluid. Two particular well-documented structures are the hairpin vortices that move longitudinally above the wall and ejections inclined with respect to the wall that bring the fluid from the transient viscous layers underneath these vortices into the outer region of the boundary layer. It is proposed that the Karman universa…
Structured on the paradigmatic Navier-Stokes flow model, we study a stochastically forced Taylor-Couette system in the narrow gap limit, in order to analyze the simultaneous impact of a non-conserved (Gaussian) force and a nonlinear perturbation, in determining linear stability. Our analysis identifies key parametric windows within which the model remarkably retains its linear stability, even against jointly varying stochastic forcing and nonlinear fluctuations. We identify this feature as a {\it latent universality}, that we then utilize to answer the elusive question as to how a recent groun…
The choice and placement of sensors and actuators is an essential factor determining the performance that can be realized using feedback control. This determination is especially important, but difficult, in the context of controlling transitional flows. The highly non-normal nature of the linearized Navier-Stokes equations makes the flow sensitive to small perturbations, with potentially drastic performance consequences on closed-loop flow control performance. Full-information controllers, such as the linear quadratic regulator (LQR), have demonstrated some success in reducing transient energ…
The question of optimal spanwise-periodic modification for the stabilisation of spanwise-invariant flows is addressed. A 2nd-order sensitivity analysis is conducted for the linear temporal stability of parallel flows U0 subject to small-amplitude spanwise-periodic modification e*U1, e<<1. Spanwise-periodic modifications have a quadratic effect on stability, i.e. the 1st-order eigenvalue variation is zero. A 2nd-order sensitivity operator is computed from a 1D calculation, allowing one to predict how eigenvalues are affected by any U1, without actually solving for modified eigenvalues/eigenmode…
Droplets confined in a microfluidic channel often exhibit intriguing shapes, primarily attributable to complex hydrodynamic interactions over small scales. We show that effect of varied substrate wettability conditions may further complicate these interactions remarkably, and often non-trivially. Our studies reveal that the combined influence of substrate wettability and fluidic confinement eventually culminates towards influencing the droplet transients, distortion of the local shear flow field, as well as drop stabilization against breakup or detachment, allowing one to develop different reg…
In the finite element analysis with fast decoupled time integration scheme for viscoelastic fluid (the Leonov model) flow, we investigate strong nonlinear behavior in 2D creeping contraction flow. The algorithm is applicable in the whole range of the retardation parameter. In the analysis of steady solutions, there exists upper convergence limit of available numerical solutions in this contraction flow that is free from the frustrating mesh dependence when we incorporate the tensor-logarithmic formulation [Fattal and Kupferman, JNNFM, 2004]. With adjustment of a nonlinear parameter, 2 kinds of…
We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\| U(y) - y\|_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon \ll ν^{1/2}$, where $ν$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-close in $H^1$ to $(e^{t ν\partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \gg ν^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient growth of gra…
When a microorganism begins swimming from rest in a Newtonian fluid such as water, it rapidly attains its steady-state swimming speed since changes in the velocity field spread quickly when the Reynolds number is small. However, swimming microorganisms are commonly found or studied in complex fluids. Because these fluids have long relaxation times, the time to attain the steady- state swimming speed can also be long. In this article we study the swimming startup problem in the simplest liquid crystalline fluid: a two-dimensional hexatic liquid crystal film. We study the dependence of startup t…
