We investigate the spectral energy dynamics of turbulent flows with variable viscosity using direct numerical simulation of homogeneous isotropic turbulence of generalised Newtonian fluids described by the Carreau constitutive model, covering both shear-thinning and shear-thickening regimes. The spectral evolution equations for the variable viscosity Navier-Stokes system show that the viscous term becomes nonlinear and gives rise to a convolution product in spectral space, formally analogous to that of the convective term. Unlike the constant viscosity case, where it acts as a purely local dissipation mechanism, the variable viscosity term carries both conservative (transfer) and non-conservative (dissipation) contributions entangled in the convolution product. We present novel computations of the viscous \mtm coupling $\hat{V}(\k, \kP)$, which does not satisfy a detailed conservation property analogous to that of the convective term. The viscous coupling maps reveal two distinct spectral regions: a sign-definite non-conservative region near $\kP \approx \bm{0}$, and a transfer-like dipole near $\kP \approx \k$ in shear-thickening fluids. The dipole satisfies the approximate antisymmetry $\hat{V}(\k, \kP) \approx -\hat{V}(\kP, \k)$, which is the defining signature of a conservative energy transfer. This demonstrates that energy transfer across scales, a role traditionally attributed exclusively to the convective nonlinearity, can arise from any nonlinear term in the momentum equation. The viscous energy transfer participates in the forward cascade alongside the convective transfer, eventually taking over the latter in the dissipation range. Its presence is connected to the emergence of power-law spectral decay replacing the classical exponential cutoff in shear-thickening fluids.
In this article we challenge the claim that the previously proposed variational method to obtain flow solutions for generalized Newtonian fluids in circular tubes and plane slits is exact only for power law fluids. We also defend the theoretical foundation and formalism of the method which is based on minimizing the total stress through the application of the Euler-Lagrange principle.
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…
We model the flow of a bi-viscous non-Newtonian fluid in a porous medium by a square lattice where the links obey a piece-wise linear constitutive equation. We find numerically that the flow regime where the network transitions from all links behaving according to the first linear part of the constitutive equation to all links behaving according to the second linear part of the constitutive equation, is characterized by a critical point. We measure two critical exponents associated with this critical point, one of the being the correlation length exponent. We find that both critical exponents …
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 …
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…
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 …
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…
A flow vessel with an elastic wall can deform significantly due to viscous fluid flow within it, even at vanishing Reynolds number (no fluid inertia). Deformation leads to an enhancement of throughput due to the change in cross-sectional area. The latter gives rise to a non-constant pressure gradient in the flow-wise direction and, hence, to a nonlinear flow rate--pressure drop relation (unlike the Hagen--Poiseuille law for a rigid tube). Many biofluids are non-Newtonian, and are well approximated by generalized Newtonian (say, power-law) rheological models. Consequently, we analyze the proble…
This paper presents a new theory of turbulence in time-independent non-Newtonian fluids. The wall layer is modelled in terms of unsteady exchange of viscous momentum between the wall and the main stream, following the classic visualisation of inrush-sweep-ejection/burst. The thickness of the wall layer is found to be the same for Newtonian and purely viscous non-Newtonian fluids, when normalised with the instantaneous wall parameters at the onset of bursting. The results indicate that the mechanisms of turbulence in Newtonian and time-independent fluids are identical when structural similarity…
Murray's theory of constrained minimum-power branchings is critically reviewed in a generalised framework for a range of cases: channels with arbitrary cross-section shape, laminar flows of Newtonian and non-Newtonian fluids, and low and high Reynolds-number turbulent flows of Newtonian fluids. The theory states that the sum of hydraulic and metabolic power is minimised if and only if all channels satisfy the same relation between flow rate and effective radius. This relation leads to a generalised form of Murray's law. It is shown that, satisfying Murray's law is a necessary requirement for p…
Linear stability of horizontal and inclined stratified channel flows of Newtonian/non-Newtonian shear-thinning fluids is investigated with respect to all wavelength perturbations. The Carreau model has been chosen for the modeling of the rheology of a shear-thinning fluid, owing to its capability to describe properly the constant viscosity limits (Newtonian behavior) at low and high shear rates. The results are presented in the form of stability boundaries on flow pattern maps (with the phases' superficial velocities as coordinates) for several practically important gas-liquid and liquid-liqui…
In the above paper an analysis has been carried out to obtain results in the Couette flow of a Newtonian fluid with viscous dissipation and temperature dependent viscosity. The fluid viscosity is an exponential function of temperature. Exact analytical solutions are obtained for the calculation of velocity and temperature. However, there are some fundamental errors in this paper which are presented herein.
In this paper, we propose a new unified first order hyperbolic model of Newtonian continuum mechanics coupled with electro-dynamics. The model is able to describe the behavior of moving elasto-plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro-magnetic fields. It is actually a very peculiar feature of the proposed PDE system that viscous fluids are treated just as a special case of elasto-plastic solids. This is achieved by introducing a strain relaxation mechanism in the evolution equations of the distortion matrix. The model also contains a hyperbolic…
Blood flow, dam or ship construction and numerous other problems in biomedical and general engineering involve incompressible flows interacting with elastic structures. Such interactions heavily influence the deformation and stress states which, in turn, affect the engineering design process. Therefore, any reliable model of such physical processes must consider the coupling of fluids and solids. However, complexity increases for non-Newtonian fluid models, as used, e.g., for blood or polymer flows. In these fluids, subtle differences in the local shear rate can have a drastic impact on the fl…
In this video, Ray-tracing data visualization technique was used to obtain realistic and detailed flow motions during droplet collision. The differences of collision outcome between Newtonian and non-Newtonian were compared. Various types of droplet collision were presented, including bouncing, coalescence, and stretching separation. Because of the reducing of equivalent viscosity caused by shear stress, the gas film between shear-thinning droplet is thinner than Newtonian liquid. Since thinner gas film promotes coalescence, shear thinning liquid has smaller area of bouncing regime in the diag…
We investigate the two dimensional incompressible Navier-Stokes(NS) and the continuity equations in Cartesian coordinates and Eulerian description for non-Newtonian fluids. As a non-Newtonian viscosity we consider the Ladyzenskaya model with a non-linear velocity dependent stress tensor. The key idea is the multi-dimensional generalization of the well-known self-similar Ansatz, which was already used for non-compressible and compressible viscous flow studies. The geometrical interpretations of the trial function is also discussed. We compare our recent results to the former Newtonian ones.
The problem of buckling and coiling of jets of viscous, Newtonian liquids has received a substantial level of attention over the past two decades, both from experimental and theoretical points of view. Nevertheless, many industrial fluids and consumer products are non-Newtonian, and their rheological properties affect their flow behavior. The present work aims at studying the dynamics of cylindrical jets of a viscoelastic, shear-thinning solution of cetylpyridinium salt (CPyCl). In concentrated solutions, CPyCl surfactant molecules have been shown to assemble in long wormlike micellar structur…
Correlations of friction factors are presented for the general case of purely viscous non-Newtonian fluids without requiring a priori the adoption of a rheological model. They are based on an empirical estimate of the shift in the wall layer edge and the Kolmogorov point. The predictions of friction factors have the same level of accuracy as those of the Dodge-Metzner correlation but the visualisation is more compatible with measured velocity profiles. The general correlations obtained can be used to easily retrieve correlations for specific rheological models.
We investigate through Direct Numerical Simulations (DNS) the statistical properties of turbulent flows in the inertial subrange for non-Newtonian power-law fluids. The structural invariance found for the vortex size distribution is achieved through a self-organized mechanism at the microscopic scale of the turbulent motion that adjusts, according to the rheological properties of the fluid, the ratio between the viscous dissipations inside and outside the vortices. Moreover, the deviations from the K41 theory of the structure functions' exponents reveal that the anomalous scaling exhibits a sy…
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…
A stationary viscous jet falling from an oriented nozzle onto a moving surface is studied, both theoretically and experimentally. We distinguish three flow regimes and classify them by the convexity of the jet shape (concave, vertical and convex). The fluid is modeled as a Newtonian fluid, and the model for the flow includes viscous effects, inertia and gravity. By studying the characteristics of the conservation of momentum for a dynamic jet, the boundary conditions for each flow regime are derived, and the flow regimes are characterized in terms of the process and material parameters. The mo…
Non-Newtonian fluid flow might be driven by spatially nonlocal velocity, the dynamics of which can be described by promising fractional derivative models. This short communication proposes a space FrActional-order Constitutive Equation (FACE) that links viscous shear stress with the velocity gradient, and then interprets physical properties of non-Newtonian fluids for steady pipe flow. Results show that the generalized FACE model contains previous non-Newtonian fluid flow models as end-members by simply adjusting the order of the fractional index, and a preliminary test shows that the FACE mod…
The results studying various laminar flow regimes in diverging and converging plain channels (diffuser and confusor) with a small opening angle of channels (diverging and converging angles) are presented. The results are obtained for a viscous incompressible fluid by numerical simulation based on solving the Navier-Stokes equations. The paper presents the results concerning the change in the nature of flows from stationary - symmetric to stationary - asymmetric and to non-stationary in the diffuser and confusor in dependence on the Reynolds number. The ranges of existence of these flow regimes…
We analyze the stationary flow of a jet of Newtonian fluid that is drawn by gravity onto a moving surface. The situation is modeled by a third-order ODE on a domain of unknown length and with an additional integral condition; by solving part of the equation explicitly we can reformulate the problem as a first-order ODE, again with an integral constraint. We show that there are two flow regimes, and characterize the associated regions in the three-dimensional parameter space in terms of an easily calculable quantity. In a qualitative sense the results from the model are found to correspond with…
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…
Implicit integration of the viscous term can significantly improve performance in computational fluid dynamics for highly viscous fluids such as lava. We show improvements over our previous proposal for semi-implicit viscous integration in Smoothed Particle Hydrodynamics, extending it to support a wider range of boundary models. Due to the resulting loss of matrix symmetry, a key advancement is a more robust version of the biconjugate gradient stabilized method to solve the linear systems, that is also better suited for parallelization in both shared-memory and distributed-memory systems. The …
Exact solutions for laminar stratified flows of Newtonian/non-Newtonian shear-thinning fluids in horizontal and inclined channels are presented. An iterative algorithm is proposed to compute the laminar solution for the general case of a Carreau non-Newtonian fluid. The exact solution is used to study the effect of the rheology of the shear-thinning liquid on two-phase flow characteristics considering both gas/liquid and liquid/liquid systems. Concurrent and counter-current inclined systems are investigated, including the mapping of multiple solution boundaries. Aspects relevant to practical a…
Fluid-suspended microorganisms have evolved different swimming and feeding strategies in order to cope with an environment dominated by viscous effects. For instance ciliated organisms rely on the collective motion of flexible appendices to move and feed. By performing a non-reciprocal motion, flexible filaments can produce a net propulsive force, or pump fluid, in the absence of inertia. Inspired by such fundamental concept, we propose a strategy to produce macroscopic pumping and mixing in creeping flow. We measure experimentally the net motion of a Newtonian viscous fluid induced by the rec…
Numerous natural processes are contingent on microorganisms' ability to swim through fluids with non-Newtonian rheology. Here, we use the model organism Caenorhabditis elegans and tracking methods to experimentally investigate the dynamics of undulatory swimming in shear-thinning fluids. Theory and simulation have proposed that the cost of swimming, or mechanical power, should be lower in a shear-thinning fluid compared to a Newtonian fluid of the same zero-shear viscosity. We aim to provide an experimental investigation into the cost of swimming in a shear-thinning fluid from (i) an estimate …
In this chapter, we analyze the steady-state microscale fluid--structure interaction (FSI) between a generalized Newtonian fluid and a hyperelastic tube. Physiological flows, especially in hemodynamics, serve as primary examples of such FSI phenomena. The small scale of the physical system renders the flow field, under the power-law rheological model, amenable to a closed-form solution using the lubrication approximation. On the other hand, negligible shear stresses on the walls of a long vessel allow the structure to be treated as a pressure vessel. The constitutive equation for the microtube…
The composite torsional ultrasonic oscillator, a versatile experimental system, can be used to investigate slip of Newtonian fluid at a smooth surface. A rigorous analysis of slip-dependent damping for the oscillator is presented. Initially, the phenomenon of finite surface slip and the slip length are considered for a half-space of Newtonian fluid in contact with a smooth, oscillating solid surface. Definitions are revisited and clarified in light of inconsistencies in the literature. We point out that, in general oscillating flows, Navier's slip length b is a complex number. An intuitive vel…
Starch solutions, which are strongly non-Newtonian, show a surface instability, when subjected to a load. A droplet of the fluid is sandwiched between two glass plates and a weight varying from 1 to 5 kgs. is placed on the top plate. The area of contact between the fluid and plate increases in an oscillatory manner, unlike Newtonian fluids in a similar situation. The periphery moreover, develops a viscous fingering like instability, which is not expected under compression. We attempt to model the non-Newtonian nature of the fluid through a visco-elastic model incorporating generalized calculus…
Newtonian fluid invasion into a Yield-Stress Fluid-Saturated Medium
Université Paris-Saclay, CNRS, FAST（巴黎萨克雷大学，法国国家科学研究中心，FAST）；PoreLab, Department of Physics, Norwegian University of Science and Technology（孔隙实验室，挪威科技大学物理系）；Université Paris-Saclay, CNRS, LPTMS（巴黎萨克雷大学，法国国家科学研究中心，LPTMS）；PoreLab, Department of Physics, University of Oslo（孔隙实验室，奥斯陆大学物理系）
We investigate the enhancement of both mixing and heat transfer in a two-rod mixer for highly-viscous non-Newtonian fluids. The mixer is composed of two vertical, circular rods in a cylindrical tank. Chaotic flows are obtained by imposing the temporal modulations of the rotational velocities of the walls. We study the effects of different stirring protocols, which lead to non-chaotic and chaotic flows, on the efficiency of both mixing and heat transfer for three different rheological fluid behaviors: shear-thinning, Newtonian and shear-thickening. For this purpose, we use statistical indicator…
Conventionally, a microscopic particle that performs a reciprocal stroke cannot move through its environment. This is because at small scales, the response of simple Newtonian fluids is purely viscous and flows are time-reversible. We show that by contrast, fluid elasticity enables propulsion by reciprocal forcing that is otherwise impossible. We present experiments on rigid objects actuated reciprocally in viscous fluids, demonstrating for the first time a purely elastic propulsion set by the object's shape and boundary conditions. We describe two different artificial "swimmers" that experime…
We explore the behaviour of barotropic and irrotational fluids with a small viscosity under the effect of first-order acoustic perturbations. We discuss, following the extant literature, the difficulties in gleaning an acoustic geometry in the presence of viscosity. In order to obviate various technical encumbrances, when viscosity is present, for an extraction of a possible acoustic geometry, we adopted a method of double perturbations, whereby dynamical quantities such as the velocity field and potential undergo a perturbation both in viscosity and in an external acoustic stimulus. The resul…
The Saffman-Taylor viscous fingering problem is investigated for the displacement of a non-Newtonian fluid by a Newtonian one in a radial Hele-Shaw cell. We execute a mode-coupling approach to the problem and examine the morphology of the fluid-fluid interface in the weak shear limit. A differential equation describing the early nonlinear evolution of the interface modes is derived in detail. Owing to vorticity arising from our modified Darcy's law, we introduce a vector potential for the velocity in contrast to the conventional scalar potential. Our analytical results address how mode-couplin…
Being a major limiting factor for the efficiency of various technologies, such as Enhanced Oil Recovery, the viscous fingering (or Saffman--Taylor) instability has been extensively studied, especially for simple Newtonian fluids. Here, we experimentally and theoretically demonstrate a vital control of inhibiting the viscous fingering instability for complex (yield-stress) fluids to generate a complete sweep with a flat interface. Using a rectangular tapered cell, we first experimentally show the feasibility of controlling the primary fingering instability of a complex yield-stress fluid when p…
Viscous fingering (VF) is an interfacial instability that occurs in a narrow confinement or porous medium when a less-viscous fluid pushes a more viscous one, producing finger-like patterns. Controlling the VF instability is essential to enhance the efficiency of various technological applications. However, the control of VF instability has been challenging and so far focused on simple Newtonian fluids of constant viscosity. Here, we extend to complex yield-stress fluids and examine the controlling feasibility by carrying out a linear stability analysis using a radial cell with a converging ga…
When a droplet impacts a fabric mesh at a sufficiently high impact velocity, it not only spreads over the fabric but also penetrate its pores. To determine the influence of this liquid penetration of the fabric on droplet spreading on thin fabric meshes, we measured the droplet spreading ratio on fabric with and without an underlying substrate using a high-speed camera. For fabrics without a substrate, the droplet spreading ratio is reduced as the fabric penetration by the liquid reduces the droplet volume spreading on top of the fabric. Using entropic lattice Boltzmann simulations, we find th…
In the minutes immediately preceeding the rupture of a soap bubble, distinctive and repeatable patterns can be observed. These quasi-stable transient structures are associated with the instabilities of the complex Marangoni flows on the curved thin film in the presence of a surfactant solution. Here, we report a generalised Cahn-Hilliard-Swift-Hohenberg model derived using asymptotic theory which describes the quasi-elastic wrinkling pattern formation and the consequent coarsening dynamics in a curved surfactant-laden thin film. By testing the theory against experiments on soap bubbles, we fin…
Deformable bodies in a 3-dimensional viscous flow: Vorticity-Stream vector formulation
Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS, Inserm, Université PSL（跨学科生物研究中心，法兰西学院，法国国家科学研究中心，法国国家健康与医学研究院，巴黎文理研究大学）；Facultat de Física, Universitat de Barcelona（巴塞罗那大学物理学院）；Instituto de Investigación Tecnológica (IIT) and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Pontificia Comillas（技术研究所和复杂系统跨学科小组，科米利亚斯宗座大学）；Institute of Nanoscience and Nanotechnology (IN2UB)（纳米科学与技术研究所）
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…
Viscous heating can play an important role in the dynamics of fluids with strongly temperature-dependent viscosities because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease of the viscosity and a strong stratification in the viscosity profile. The problem of viscous heating in fluids was investigated and reviewed by Costa & Macedonio (2003) because of its important implications in the study of magma flows. Because of the strong coupling between viscosity and te…
Many cells exploit the bending or rotation of flagellar filaments in order to self-propel in viscous fluids. While appropriate theoretical modelling is available to capture flagella locomotion in simple, Newtonian fluids, formidable computations are required to address theoretically their locomotion in complex, nonlinear fluids, e.g.~mucus. Based on experimental measurements for the motion of rigid rods in non-Newtonian fluids and on the classical Carreau fluid model, we propose empirical extensions of the classical Newtonian resistive-force theory to model the waving of slender filaments in n…
