A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a `center mode' with phase speed close to the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re = ρU_{max} H/η$, the elasticity number $E = λη/(H^2ρ)$, and the ratio of solvent to solution viscosity $η_s/η$; here, $λ$ is the polymer relaxation time, $H$ is the channel half-width, and $ρ$ is the fluid density. For experimentally relevant values (e.g., $E \sim 0.1$ and $β\sim 0.9$), the predicted critical Reynolds number, $Re_c$, for the center-mode instability is around $200$, with the associated eigenmodes being spread out across the channel. In the asymptotic limit of $E(1 -β) \ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $Re_c \propto (E(1-β))^{-\frac{3}{2}}$ and the critical wavenumber $k_c \propto (E(1-β))^{-\frac{1}{2}}$. The unstable eigenmode in this limit is confined in a thin layer near the channel centerline. The above features are largely analogous to the center-mode instability in viscoelastic pipe flow (Garg et al., Phys. Rev. Lett., 121, 024502 (2018)), and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of suffciently elastic dilute polymer solutions.
Particles migrate in the transverse direction of the flow due to the existence of normal stress anisotropy in weakly viscoelastic liquids. We test the ability of theoretical predictions to predict the transverse velocity migration of particles in a confined Poiseuille flow according to the viscoelastic constitutive parameters of dilute polymers solutions. Firstly, we carefully characterize the viscoelastic properties of two families of dilute polymer solutions at various concentrations using shear rheometry and capillary breakup experiments. Secondly, we develop a specific 3D particle tracking…
For Gagen-Poiseuille flow, we show that exponential instability (to extremely small, axially symmetric disturbances represented by Galerkin's approximation) is possible only if there exists bi-periodic variability of the disturbances along the pipe axis when the threshold Reynolds number depends on the ratio of two longitudinal periods. Absolute minimum (for) is obtained that corresponds to the observed conditions of transition from the laminar resistance law to the turbulent one and Tollmien-Schlichting waves exciting in the boundary layer.
We show possibility of the Plane Couette (PC) flow instability for Reynolds number Re>Reth=140. This new result of the linear hydrodynamic stability theory is obtained on the base of refusal from the traditionally used assumption on longitudinal periodicity of the disturbances along the direction of the fluid flow. We found that earlier existing understanding on the linear stability of this flow for any arbitrary large Reynolds number is directly related with an assumption on the separation of the variables of the spatial variability for the disturbance field and their periodicity in linear th…
The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour-liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide (CO$_2$) at pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number ($Re=ρ_{w}^{*}u_{r}^{*}h^{*}/μ_{w}^{*}$), the product of Prandtl ($Pr=μ_{w}^{*}C_{pw}^{*}/κ_{w}^{*}$) and …
Here we report some interesting new features of the spatio-temporal instability of the incompressible plane Couette-Poiseuille flow (CPF). First of all, this flow represents the first instance of a "non-inflectional" absolute instability, within constant-viscosity formulation, which is triggered when one of the plates moves opposite to the bulk motion. More strikingly, with further increase in the negative plate motion, the absolute instability ($\textrm{AI}$) transitions to an "upstream" convective instability ($\textrm{CI}^-$), wherein an unstable wave packet moves opposite to the direction …
In this paper I analyze the onset of Rayleigh-Taylor instability between two linear viscoelastic fluids assuming that the perturbations at the interface are small. In the first half, the paper analyzes a stratified viscoelastic fluid in which I prove that the perturbations rise or fall exponentially without oscillating. The second half of the paper examines the effect of electric and magnetic fields on viscoelastic fluids. I show that it is possible to choose electric or magnetic field gradient such that the effective acceleration due to gravity is zero. If a heavy Newtonian fluid rests on top…
The instability of a streak and its nonlinear evolution are investigated by direct numerical simulation (DNS) for plane Poiseuille flow at Re=3000. It is suggested that there exists a traveling-wave solution (TWS). The TWS is localized around one of the two walls and notably resemble to the coherent structures observed in experiments and DNS so far. The phase space structure around this TWS is similar to a saddle point. Since the stable manifold of this TWS is extended close to the quasi two dimensional (Q2D) energy axis, the approaching process toward the TWS along the stable manifold is appr…
In this work, we report numerical results on the flow instability and bifurcation of a viscoelastic fluid in the upstream region of a confined cylinder in a narrow channel. Two-dimensional direct numerical simulations based on the FENE-P model (the finite-extensible nonlinear elastic model with the Peterlin closure) are conducted with numerical stabilization techniques. Our results show that the macroscopic viscoelastic constitutive relation can capture the viscoelastic upstream instability reported in previous experiments for low-Reynolds-number flows. The numerical simulations reveal that th…
Interior stagnation point flows of viscoelastic liquids arise in a wide variety of applications including extensional viscometry, polymer processing and microfluidics. Experimentally, these flows have long been known to exhibit instabilities, but the mechanisms underlying them have not previously been elucidated. We computationally demonstrate the existence of a supercritical oscillatory instability of low-Reynolds number viscoelastic flow in a two-dimensional cross-slot geometry. The fluctuations are closely associated with the "birefringent strand" of highly stretched polymer chains associat…
We show that viscoelastic plane Poiseuille flow becomes linearly unstable in the absence of inertia, in the limit of high elasticities, for ultra-dilute polymer solutions. While inertialess elastic instabilities have been predicted for curvilinear shear flows, this is the first ever report of a purely elastic linear instability in a rectilinear shear flow. The novel instability continues upto a Reynolds number ($Re$) of $O(1000)$, corresponding to the recently identified elasto-inertial turbulent state believed to underlie the maximum-drag-reduced regime. Thus, for highly elastic ultra-dilute …
We report experimental results on elastic instability in a viscoelastic channel shear flow due to only a natural non-smoothed inlet and small holes along the channel for pressure measurements. We show that non-normal mode instability results in elastic waves and chaotic flow self-organized into periodically cycled stream-wise streaks synchronized by elastic wave frequency. The chaotic flow persists above the transition with increasing $Wi$ further into elastic turbulence and drag reduction regimes. Thus, we resolve the recent puzzle whether strong prearranged perturbations are necessary to get…
A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number, the elasticity number, and the ratio of solvent to solution viscosity. The unstable mode has a phase speed close to the base-state maximum over the entire unstable region in the relevant parameter space, implying that the unstable mode belongs to a class of viscoelastic center modes. Unlike t…
A critical-layer instability of structural acoustic wall modes in a two-flexible-plate channel with Poiseuille flow: closed-form low-Mach corrections and parity-protected absorption
Birla Institute of Technology and Science Pilani（比拉理工学院皮拉尼校区）
It is well known that the Poiseuille flow of a visco-elastic polymer fluid between plates or through a tube is linearly stable in the zero Reynolds number limit, although the stability is weak for large Weissenberg numbers. In this paper we argue that recent experimental and theoretical work on the instability of visco-elastic fluids in Taylor-Couette cells and numerical work on channel flows suggest a scenario in which Poiseuille flow of visco-elastic polymer fluids exhibits a nonlinear "subcritical" instability due to normal stress effects, with a threshold which decreases for increasing Wei…
Recently, detailed experiments on visco-elastic channel flow have provided convincing evidence for a nonlinear instability scenario which we had argued for based on calculations for visco-elastic Couette flow. Motivated by these experiments we extend the previous calculations to the case of visco-elastic Poiseuille flow, using the Oldroyd-B constitutive model. Our results confirm that the subcritical instability scenario is similar for both types of flow, and that the nonlinear transition occurs for Weissenberg numbers somewhat larger than one. We provide detailed results for the convergence o…
The recently-discovered centre-mode instability of rectilinear viscoelastic shear flow (Garg et al. Phy. Rev. Lett. 121, 024502, 2018) has offered an explanation for the origin of elasto-inertial turbulence (EIT) which occurs at lower Weissenberg ($Wi$) numbers. In support of this, we show using weakly nonlinear analysis that the subcriticality found in Page et al. (Phys. Rev. Lett. 125, 154501, 2020) is generic across the neutral curve with the instability only becoming supercritical at low Reynolds ($Re$) numbers and high $Wi$. We demonstrate that the instability can be viewed as purely elas…
We review three different types of viscoelastic surface instabilities: The Rayleigh -- Plateau, the Saffman -- Taylor and the Faraday instability. These instabilities are classical examples of hydrodynamic surface instabilities. The addition of a small amount of polymers to pure water can alter its flow behavior drastically and the type of instability may change not only quantitatively but also qualitatively. We will show that some of the observed new phenomena can be explained by the use of simple rheological models that contain most of the underlying physical mechanisms leading to the instab…
We study the Marangoni instability in a thin polymeric liquid film heated from the free surface. Polymeric solutions are usually the binary mixture of a Newtonian solvent with polymeric solute and exhibit viscoelastic behaviour. In the presence of a temperature gradient, stratification of these polymeric solutes can take place via the Soret effect, which may give rise to the solutocapillary effect at the free surface. Considering the Soret effect and incorporating the effects of gravity, here we analyse the stability characteristics of this polymeric thin film bounded between its deformable fr…
The viscoelastic fluids are usually the blends of a polymeric solute and a Newtonian solvent. In the presence of a temperature gradient, stratification of these solutes can take place via the Soret effect. Here, we investigate the classical Marangoni instability problem for a thin viscoelastic film considering this binary aspect of the fluid. The film, bounded above by a deformable free surface, is subjected to heating from below by a solid substrate. Linear stability analysis performed numerically for perturbations of finite wavelength (short-wave perturbations) reveals that both monotonic an…
When two fluids of different electrical conductivities are transported side by side in a microfluidic device under the influence of an electric field, an electrokinetic instability (EKI) is often generated after some critical values of the applied electric field strength and conductivity ratio. Many prior experimental and numerical studies show that this phenomenon results in a chaotic flow field inside a microdevice, thereby facilitating the mixing of two fluids if they are Newtonian in behaviour. However, the present numerical study shows that this chaotic convection arising due to the elect…
The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Po…
In this fluid dynamics video prepared for the APS-DFD Gallery of Fluid Motion we study the Kelvin-Helmholtz instability for both Newtonian and viscoelastic jets. The nonlinear dynamics of the jet motion are slowed down by orders of magnitude using a synchronized strobe effect coupled with precise timing control of perturbation frequencies. Our results show that at high wave-numbers the imposed perturbations initially grow linearly with time and the jet axis remains straight while the Kelvin-Helmholtz wave amplitude grows and rolls up into bags that encapsulate the central jet within themselves…
The effects of polymer additives on Rayleigh--Taylor (RT) instability of immiscible fluids is investigated using the Oldroyd-B viscoelastic model. Analytic results obtained exploiting the phase-field approach show that in polymer solution the growth rate of the instability speeds up with elasticity (but remains slower than in the pure solvent case). Numerical simulations of the viscoelastic binary fluid model confirm this picture.
The study of nonmodal amplification of distributed body forces in channel flows of viscoelastic fluids has provided useful insights into the mechanisms that may govern the initial stages of transition to elastic turbulence. However, distributed body forces are not easy to implement in experiments and so there is a need to examine amplification of localized body forces. In this work, we use the linearized governing equations to examine such amplification in Poiseuille flow of FENE-CR fluids. We first identify the wall-normal location at which the impulsive excitations experience the largest amp…
I shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertia-less flows with curved streamlines. Then I describe a theory of elastic turbulence and prediction of elastic waves at Re<<1 and Wi>>1, which speed depends on the elastic stress similarly to the Alfven waves in magneto-hydrodynamics and in in a contrast to all other fluid flows with wave speed depending on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvatu…
We revisit the problem of the two-dimensional spatiotemporal linear stability of viscoelastic slender jets obeying linear Phan-Thien-Tanner (PTT) stress constitutive equation, and investigate the role of finite stresses on the elasto-capilliary stability of the Beads-on-a-String (BOAS) structure (structures which include the formation of very thin filament between drops) and identify the regions of topological transition of the advancing jet interface, in the limit of low to moderate Ohnesorge number ($Oh$) and high values of Weissenberg number ($We$). The Briggs idea of analytic continuation …
Newtonian pipe flow is known to be linearly stable at all Reynolds numbers. We report, for the first time, a linear instability of pressure driven pipe flow of a viscoelastic fluid, obeying the Oldroyd-B constitutive equation commonly used to model dilute polymer solutions. The instability is shown to exist at Reynolds numbers significantly lower than those at which transition to turbulence is typically observed for Newtonian pipe flow. Our results qualitatively explain experimental observations of transition to turbulence in pipe flow of dilute polymer solutions at flow rates where Newtonian …
The interaction of flexible structures with viscoelastic flows can result in very rich dynamics. In this paper, we present the results of the interactions between the flow of a viscoelastic polymer solution and a cantilevered beam in a confined microfluidic geometry. Cantilevered beams with varying length and flexibility were studied. With increasing flow rate and Weissenberg number, the flow transitioned from a fore-aft symmetric flow to a stable detached vortex upstream of the beam, to a time-dependent unstable vortex shedding. The shedding of the unstable vortex upstream of the beam imposed…
We investigate the collective behavior of magnetic swimmers, which are suspended in a Poiseuille flow and placed under an external magnetic field, using analytical techniques and Brownian dynamics simulations. We find that the interplay between intrinsic activity, external alignment, and magnetic dipole-dipole interactions leads to longitudinal structure formation. Our work sheds light on a recent experimental observation of a clustering instability in this system.
New condition Re>Re_th_min=124 of linear (exponential) instability of the Hagen-Poisseuille (HP) with respect to extremely small by magnitude axially-symmetric disturbances of the tangential component of the velocity field is obtained. For this, disturbances must necessarily have quasi-periodic longitude variability (not representable as a Fourier series or integral) along the pipe axis that complies with experimental data and differs from the usually considered idealized case of pure periodic disturbances for which HP flow is stable for arbitrary large Reynolds numbers Re. Obtained minimal th…
We present a detailed study of the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow. The base flow is characterised by the cross flow Reynolds number, $R_{inj}$ and the dimensionless wall velocity, $k$. Squire's transformation may be applied to the linear stability equations and we therefore consider 2D (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, $k\in[0,1]$, two ranges of $R_{inj}$ exist where unconditional stability is observed. In the lower range of $R_{inj}$, for modest $k$ we have a stabilisation of long wavelen…
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…
When a dilute polymer solution experiences capillary thinning, it forms an almost uniformly cylindrical thread, which we study experimentally. In the last stages of thinning, when polymers have become fully stretched, the filament becomes prone to instabilities, of which we describe two: A novel "breathing" instability, originating from the edge of the filament, and a sinusoidal instability in the interior, which ultimately gives rise to a "blistering" pattern of beads on the filament. We describe the linear instability with a spatial resolution of 80 nm in the disturbance amplitude. For suffi…
The convection velocity of localized wave packet in plane-Poiseuille flow is found to be determined by a solitary wave at the centerline of a downstream vortex dipole in its mean field after deducting the basic flow. The fluctuation component following the vortex dipole oscillates with a global frequency selected by the upstream marginal absolute instability, and propagates obeying the local dispersion relation of the mean flow. By applying localized initial disturbances, a nonzero wave-packet density is achieved at the threshold state, suggesting a first order transition.
Viscoelastic flows occur widely, and numerical simulations of them are important for a range of industrial applications. Simulations of viscoelastic flows are more challenging than their Newtonian counterparts due to the presence of exponential gradients in polymeric stress fields, which can lead to catastrophic instabilities if not carefully handled. A key development to overcome this issue is the log-conformation formulation, which has been applied to a range of numerical methods, but not previously applied to Smoothed Particle Hydrodynamics (SPH). Here we present a 2D incompressible SPH alg…
The rimming flow of a viscoelastic thin film inside a rotating horizontal cylinder is studied theoretically. Attention is given to the onset of non-Newtonian free-surface instability in creeping flow. This non-inertial instability has been observed in experiments, but current theoretical models of Newtonian fluids can neither describe its origin nor explain its onset. This study examines two models of non Newtonian fluids to see if the experimentally observed instability can be predicted analytically. The non-Newtonian viscosity and elastic properties of the fluid are described by the Generali…
For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat-plate boundary layer is an important issue. In the present work we investigate the stability of boundary layer in Poiseuille flow. We normalize pressure and time by inertial and viscous effects. The disturbances are taken to be periodic in the spanwise direction and time. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances. Then, we derive modified Orr-Sommerfeld equations that can be applie…
We report quantitative evidence of mixing-layer elastic instability in a viscoelastic fluid flow between two widely spaced obstacles hindering a channel flow at $Re\ll1$ and $Wi\gg1$. Two mixing layers with nonuniform shear velocity profiles are formed in the region between the obstacles. The mixing-layer instability arises in the vicinity of an inflection point on the shear velocity profile with a steep variation in the elastic stress. The instability results in an intermittent appearance of small vortices in the mixing layers and an amplification of spatio-temporal averaged vorticity in the …
In this paper, we discuss whether the instability of viscoelastic flow around a circular cylinder is subcritical or supercritical by numerical simulation. The Oldroyd-B model is selected to describe the viscoelastic constitutive relationship. The Log-conformation reformulation is employed to stabilize numerical simulation. The parameter ranges investigated are the Reynolds numbers ($Re$) spanning from 5 to 100 and the Weissenberg number ($Wi$) spanning from 0 to 10, with a fixed viscosity ratio of $β= 0.9$. Simulations are performed under two paths, i.e., 1) increasing $Re$ (or $Wi$) and 2) de…
The Oldroyd-B model has been used extensively to predict a host of instabilities in shearing flows of viscoelastic fluids, often realized experimentally using polymer solutions. The present review, written on the occasion of the birth centenary of James Oldroyd, provides an overview of instabilities found across major classes of shearing flows. These comprise (i) the canonical rectilinear shearing flows including plane Couette, plane and pipe Poiseuille flows; (ii) viscometric shearing flows with curved streamlines such as those in the Taylor-Couette, cone-and-plate and parallel-plate geometri…
