Viscoelastic Shear Flow Research Articles

Inertial enhancement of the polymer diffusive instability

Beneitezet al.(Phys. Rev. Fluids, vol. 8, 2023, L101901) have recently discovered a new linear ‘polymer diffusive instability’ (PDI) in inertialess rectilinear viscoelastic shear flow using the finitely extensible nonlinear elastic constitutive model of Peterlin (FENE-P) when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette and plane Poiseuille flows under varying Weissenberg number${W}$, polymer stress diffusivity$\varepsilon$, solvent-to-total viscosity ratio$\beta$and Reynolds number${Re}$, considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with${Re}$. For instance, as$Re$increases with$\beta$fixed, the instability emerges at progressively lower values of$W$and$\varepsilon$than in the inertialess limit, and the associated growth rates increase linearly with$Re$when all other parameters are fixed. For finite$Re$, it is also demonstrated that the Schmidt number$Sc=1/(\varepsilon Re)$collapses curves of neutral stability obtained across various$Re$and$\varepsilon$. The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabilizer, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.

A combined electro-optical deformability micro-cytometer.

We have developed a deformability cytometer that simultaneously measures the optical and electrical shape change of single cells in a viscoelastic shear flow. The optical deformability of single cells is measured using a low-cost CMOS camera illuminated with a high-power LED triggered from an electrical impedance signal created by a passing cell. Simultaneously the electrical deformability of the cell is determined using electrode arrays that measure shape changes along different axes. This is achieved by correlating the optical and electrical signals captured without complicated synchronisation. The system was characterised by measuring the deformability of HL-60 cells treated with cytochalasin D, latrunculin B and glutaraldehyde. Results demonstrate excellent correlation between the optical and electrical methods.

Orientational dynamics and rheology of active suspensions in weakly viscoelastic flows

Microswimmer suspensions in Newtonian fluids exhibit unusual macroscale properties, such as a superfluidic behavior, which can be harnessed to perform work at microscopic scales. Since most biological fluids are non-Newtonian, here we study the rheology of a microswimmer suspension in a weakly viscoelastic shear flow. At the individual level, we find that the viscoelastic stresses generated by activity substantially modify the Jeffery orbits well-known from Newtonian fluids. The orientational dynamics depends on the swimmer type; especially pushers can resist flow-induced rotation and align at an angle with the flow. To analyze its impact on bulk rheology, we study a dilute microswimmer suspension in the presence of random tumbling and rotational diffusion. Strikingly, swimmer activity and its elastic response in polymeric fluids alter the orientational distribution and substantially amplify the swimmer-induced viscosity. This suggests that pusher suspensions reach the superfluidic regime at lower volume fractions compared to a Newtonian fluid with identical viscosity.

Effect of a Change in the CaCl2/Pectin Mass Ratio on the Particle Size, Rheology and Physical Stability of Lemon Essential Oil/W Emulgels

A three-step (rotor-stator-microfluidization-rotor stator) protocol was used to prepare 15% lemon essential oil in water emulgels using a mixture of Tween 80 and Span 20 surfactants as low molecular mass emulsifiers and 0.4% low-methoxyl citrus peel pectin as a gelling agent. Ca2+ was used as a gel-promoting agent. Different CaCl2/pectin mass ratio values from 0.3 to 0.7 were used. Emulgels showed a microstructure consisting of oil droplets embedded in a sheared gel matrix, as demonstrated by bright field optical microscopy. Laser diffraction tests showed multimodal particle size distributions due to the coexistence of oil droplets and gel-like particles. Multiple light scattering tests revealed that the physical stability of emulgels was longer as the CaCl2/pectin mass ratio decreased and that different destabilization mechanisms took place. Thus, incipient syneresis became more important with increasing CaCl2 concentration, but a parallel creaming mechanism was detected for CaCl2/pectin mass ratio values above 0.5. Dynamic viscoelastic and steady shear flow properties of the emulgels with the lowest and highest CaCl2/pectin mass ratio values were compared as a function of aging time. The lowest ratio yielded an emulgel with enhanced connectivity among fluid units as indicated by its wider linear viscoelastic region, higher storage modulus, loss modulus and viscosity values, and more shear thinning properties than those of the emulgel formulated with the highest CaCl2/pectin mass ratio. The evolution of the dynamic viscoelastic properties with aging time was consistent with the information provided by monitoring scans of backscattering as a function of sample height.

Rotational dynamics of a neutrally buoyant prolate spheroid in viscoelastic shear flows at finite Reynolds numbers

Non-spherical particles exhibit peculiar behaviour in non-Newtonian flows. In this paper, we numerically investigate the dynamics of a neutrally buoyant prolate spheroid immersed in viscoelastic shear flows at finite Reynolds numbers by means of the immersed boundary method. Our results show that the period of particle rotation changes monotonically with the solvent viscosity ratio but non-monotonically with the mobility factor. Furthermore, we find five rotation modes of the spheroid under the effects of fluid inertia and fluid rheology in the present flow configuration. With weak fluid inertia, the particle rotation rate is remarkably reduced by fluid elasticity, which also induces asymmetric rotational behaviour. While the particle tends to tumble in the shear plane with weak fluid elasticity and moderate fluid inertia. However, as the fluid elasticity increases, the particle rotates with a newly observed mode, named the asymmetric-kayaking mode, which is classified by two additional critical elastic numbers that differ from the earlier studies on Stokesian viscoelastic shear flows. The present findings imply the importance of fluid inertia and fluid elasticity on the particle dynamics and could be potentially used to control the particle orientations in viscoelastic fluid flows.

Master curves for FENE-P fluids in steady shear flow

The FENE-P (Finitely-Extensible Nonlinear Elastic) dumbbell constitutive equation is widely used in simulations and stability analyses of free and wall-bounded viscoelastic shear flows due to its relative simplicity and accuracy in predicting macroscopic properties of dilute polymer solutions. The model contains three independent material parameters, which expressed in dimensionless form correspond to a Weissenberg number ( Wi ), i.e. , the ratio of the dumbbell relaxation time scale to a characteristic flow time scale, a finite extensibility parameter ( L ), corresponding to the ratio of the fully extended dumbbell length to the root mean square end-to-end separation of the polymer chain under equilibrium conditions, and a solvent viscosity ratio, commonly denoted β . An exact solution for the rheological predictions of the FENE-P model in steady simple shear flow is available (Sureshkumar et al., 1997), but the resulting nonlinear and nested set of equations do not readily reveal the key shear-thinning physics that dominates at high Wi as a result of the finite extensibility of the polymer chain. In this note we review a simple way of evaluating the steady material functions characterizing the nonlinear evolution of the polymeric contributions to the shear stress and first normal stress difference as the shear rate increases, provide asymptotic expansions as a function of Wi , and show that it is in fact possible to construct universal master curves for these two material functions as well as the corresponding stress ratio. Steady shear flow experiments on three highly elastic dilute polymer solutions of different finite extensibilities also follow the identified master curves. The governing dimensionless parameter for these master curves is Wi / L and it is only in strong shear flows exceeding Wi / L ≳ 1 that the effects of finite extensibility of the polymer chains dominate the evolution of polymeric stresses in the flow field. We suggest that reporting the magnitude of Wi / L when performing stability analyses or simulating shear-dominated flows with the FENE-P model will help clarify finite extensibility effects. • Asymptotic solutions for FENE-P model in steady shear flow are calculated. • Master curves exist for material functions of FENE-P fluids in steady shear flow. • Governing dimensionless parameter is (Weissenberg number)/(finite extensibility). • Reporting this dimensionless parameter clarifies the magnitude of nonlinear effects.

Self-sustained oscillations of active viscoelastic matter

Models of active nematics in biological systems normally require complexity arising from the hydrodynamics involved at the microscopic level as well as the viscoelastic nature of the system. Here we show that a minimal, space-independent, model based on the temporal alignment of active and polymeric particles provides an avenue to predict and study their coupled dynamics within the framework of dynamical systems. In particular, we examine, using analytical and numerical methods, how such a simple model can display self-sustained oscillations in an activity-driven viscoelastic shear flow.

Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers

The recently discovered centre-mode instability of rectilinear viscoelastic shear flow (Garget al.,Phys. Rev. Lett., vol. 121, 2018, 024502) has offered an explanation for the origin of elasto-inertial turbulence that occurs at lower Weissenberg numbers ($Wi$). In support of this, we show using weakly nonlinear analysis that the subcriticality found in Pageet al.(Phys. Rev. Lett., vol. 125, 2020, 154501) is generic across the neutral curve with the instability becoming supercritical only at low Reynolds numbers ($Re$) and high$Wi$. We demonstrate that the instability can be viewed as purely elastic in origin, even for$Re=O(10^3)$, rather than ‘elasto-inertial’, as the underlying shear doesnotfeed the kinetic energy of the instability. It is also found that the introduction of a realistic maximum polymer extension length,$L_{max}$, in the FENE-P model moves the neutral curve closer to the inertialess$Re=0$limit at a fixed ratio of solvent-to-solution viscosities,$\beta$. At$Re=0$and in the dilute limit ($\beta \rightarrow 1$) with$L_{max} =O(100)$, the linear instability can be brought down to more physically relevant$Wi\gtrsim 110$at$\beta =0.98$, compared with the threshold$Wi=O(10^3)$at$\beta =0.994$reported recently by Khalidet al.(Phys. Rev. Lett., vol. 127, 2021, 134502) for an Oldroyd-B fluid. Again, the instability is subcritical, implying that inertialess rectilinear viscoelastic shear flow is nonlinearly unstable – i.e. unstable to finite-amplitude disturbances – for even lower$Wi$.

Spatiotemporal linear stability of viscoelastic free shear flows: Nonaffine response regime

We provide a detailed comparison of the two-dimensional, temporal, and spatiotemporal linearized analyses of the viscoelastic free shear flows (inhomogeneous flows with mean velocity gradients that develop in the absence of boundaries) in the limit of low to moderate Reynolds number and elasticity number obeying four different types of stress–strain constitutive equations: Oldroyd-B, upper convected Maxwell, Johnson–Segalman (JS), and linear Phan-Thien–Tanner (PTT). The resulting fourth-order Orr–Sommerfeld equation is transformed into a set of six auxiliary equations that are numerically integrated via the compound matrix method. The temporal stability analysis suggests (a) elastic stabilization at higher values of elasticity number {shown previously in the dilute regime [Sircar and Bansal, “Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime,” Phys. Fluids 31, 084104 (2019)]} and (b) a nonmonotonic instability pattern at low to intermediate values of elasticity number for the JS as well as the PTT model. To comprehend the effect of elasticity, Reynolds number, and viscosity on the temporal stability curves of the PTT model, we consider a fourth parameter, the centerline shear rate, ζc. The “JS behavior” is recovered below a critical value of ζc, and above this critical value, the PTT base stresses (relative to the JS model) are attenuated thereby explaining the stabilizing influence of elasticity. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability and absolute and convective instabilities, as well as evanescent modes, and the results are compared with previously conducted experiments for Newtonian as well as viscoelastic flows past a cylinder. The phase diagrams reveal the two familiar regions of inertial turbulence modified by elasticity and elastic turbulence as well as (a recently substantiated) region of elastoinertial turbulence and the unfamiliar temporally stable region for intermediate values of Reynolds and elasticity number.

Comment on “Migration of an electrophoretic particle in a weakly inertial or viscoelastic shear flow”

A recent article [Khair and Kabarowski Phys. Rev. Fluids 5, 033702 (2020)] has studied the cross-streamline migration of electrophoretic particles in unbounded shear flows with weak inertia or viscoelasticity. That work compares their results with those reported in two of our previous studies [Choudhary et al. J. Fluid Mech. 874, 856 (2019); J. Fluid Mech. 898, A20 (2020)] and reports a disagreement in the derived analytical expressions. In this comment, we resolve this discrepancy. For viscoelastic flows, we show that Khair and Kabarowski have not accounted for a leading order surface integral of polymeric stress in their calculation of first-order viscoelastic lift. When this integral is included, the resulting migration velocity matches exactly with that reported in our work [J. Fluid Mech. 898, A20 (2020)]. This qualitatively changes the migration direction that is reported by Khair and Kabarowski for viscoelastic flows. For inertial flows, we clarify that Khair and Kabarowski find the coefficient of lift to be $1.75\ensuremath{\pi}$ compared to $2.35\ensuremath{\pi}$ in our previous work [J. Fluid Mech. 874, 856 (2019)]. We show that this difference occurs because Khair and Kabarowski accurately include the effect of a rapidly decaying $\ensuremath{\sim}O(1/{r}^{4})$ velocity field (a correction to the stresslet field $\ensuremath{\sim}1/{r}^{2}$), which was neglected in our work.

Fractional oscillator noise and its applications

It is shown that a fractional oscillator (FO) noise, which is a generalization of the ordinary overdamped linear oscillator driven by the white noise may be ‘applied to diverse systems; its stationary correlation function presentspower-law-like function, exponential-like function, exponential function, and oscillatory decays. The model may be employed to describe the fluctuation of the distance between a fluorescein–tyrosine pair within a single protein complex and the internal dynamics of a lysozyme molecule in solution. It also has the possibility of describing a Brownian particle in an oscillatory viscoelastic shear flow.

Transport and tumbling of polymers in viscoelastic shear flow.

Polymers in shear flow are ubiquitous and we study their motion in a viscoelastic fluid under shear. Employing Hookean dumbbells as representative, we find that the center-of-mass motion follows: 〈x_{c}^{2}(t)〉∼γ[over ̇]^{2}t^{α+2}, generalizing the earlier result: 〈x_{c}^{2}(t)〉∼γ[over ̇]^{2}t^{3}(α=1). Here 0<α<1 is the coefficient defining the power-law decay of noise correlations in the viscoelastic media. Motion of the relative coordinate, on the other hand, is quite intriguing in that 〈x_{r}^{2}(t)〉∼t^{β} with β=2(1-α), for small α. This implies nonexistence of the steady state, making it inappropriate for addressing tumbling dynamics. We remedy this pathology by introducing a nonlinear spring with FENE-LJ interaction and study tumbling dynamics of the dumbbell. We find that the tumbling frequency exhibits a nonmonotonic behavior as a function of shear rate for various degrees of subdiffusion. We also find that this result is robust against variations in the extension of the spring. We briefly discuss the case of polymers.

Lift and drag force on a spherical particle in a viscoelastic shear flow

We present a comprehensive 3D numerical study of particles with imposed velocities relative to the local bulk flow (termed “slip velocities”) in a viscoelastic shear flow. We consider the force on a spherical particle sedimenting, a spherical bubble rising, and a spherical neutral squirmer swimming in an imposed viscoelastic shear flow. We demonstrate that any particle moving with a slip velocity in the flow or gradient direction of the shear flow experience a lateral lift force. We calculate and compare the magnitude and direction of the lift force in all situations. At small Deborah (De) and Weissenberg (Wi) numbers, our results show good agreement with an existing perturbation theory for rigid particles (Einarsson and Mehlig, 2017 [1]) and new perturbation theories for drops and for squirmers respectively. Our simulations extend these results to higher De and Wi regimes. Through our simulations, we uncover the physical mechanism of the lateral force on all particles. For rigid particles, we find the lift force arises from an imbalance in polymer stress on either side of the particle, which in turn is due to the imbalance of polymer stretch surrounding the particle. If this lift force is not balanced by an external force, a lateral drift velocity arises. We further consider the implication of this lateral drift for rigid particles hydrodynamically forced in a viscoelastic Poiseuille flow, where particles can migrate either toward the channel center plane or toward the wall, depending on whether the direction of the applied force is in the direction aligned or opposite to the direction of the Poiseuille flow, respectively. We study both the migration of a single particle as well as a suspension of particles in channel flow. Even with the addition of hydrodynamic interactions, we show that particles forced in the direction of the Poiseuille flow migrate towards the channel center.

Migration of an electrophoretic particle in a weakly inertial or viscoelastic shear flow

Migration of an electrophoretic particle in a weakly inertial or viscoelastic shear flow

Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime

We report the temporal and spatiotemporal stability analyses of antisymmetric, free shear, viscoelastic flows obeying the Oldroyd-B constitutive equation in the limit of low to moderate Reynolds number (Re) and Weissenberg number (We). The resulting fourth-order Orr-Sommerfeld equation is reduced to a set of six auxiliary equations that are numerically integrated starting from the rescaled far-field conditions, i.e., via Compound Matrix Method. The temporal stability analysis indicates that with increasing We, (a) the entire range of the most unstable mode is shifted toward longer waves (i.e., the entire region of temporal instability is gradually concentrated near zero wavenumber), (b) the vorticity structure contours are dilated, and (c) the residual Reynolds stresses are diminished. All these analogous observations previously reported in the inertial limit [J. Azaiez and G. M. Homsy, “Linear stability of free shear flow of viscoelastic liquids,” J. Fluid Mech. 268, 37–69 (1994).] suggest a viscoelastic destabilization mechanism operating at low and moderate Re. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilities, as well as evanescent modes. The main result is that the free shear flow of dilute polymeric liquids is either (absolutely/convectively) unstable for all Re or the transition to instability occurs at comparatively low Re, a finding attributed to the fact that viscoelasticity aggravates instabilities via shear-induced anisotropy and the slow relaxation effects.

Viscoelastic planar elongational flow past an infinitely long cylinder

Following our previous work [K. D. Housiadas and R. I. Tanner, “Viscoelastic shear flow past an infinitely long and freely rotating cylinder,” Phys. Fluids 30, 073101 (2018)], we study analytically the effect of steady planar elongational flow past an infinitely long circular cylinder using asymptotic methods. The ambient fluid is assumed viscoelastic and modelled with the Upper Convected Maxwell, Oldroyd-B, exponential Phan-Thien and Tanner, Giesekus, and Finite Extensibility Nonlinear Elastic model with the Peterlin approximation constitutive equations under isothermal and creeping flow conditions. The solution for all the dependent variables is expanded as an asymptotic power series with the small parameter being the Weissenberg number, Wi, which is defined as the product of the single relaxation time of the fluid times the constant rate of elongation. The resulting sequence of equations is solved analytically up to fourth order in the Weissenberg number. The solution derived here is the first analytical result in the literature for the planar elongational flow of viscoelastic fluids past a cylinder. It reveals the effect of viscoelasticity and all the relevant rheological parameters on the flow variables and the extensional viscosity of the fluid.

“Phase diagram” for viscoelastic Poiseuille flow over a wavy surface

We experimentally examine the Poiseuille flow of viscoelastic fluids over wavy surfaces. Five precision microfabricated flow channels are utilized, each of depth 2d = 400 μm, spanwise width w = 10d and with a sinusoidal undulation of amplitude A = d/20 on one of the spanwise walls. The undulation wavelength λ is varied between each of the channels, providing dimensionless channel depths α in the range 0.2π ≤ α = 2πd/λ ≤ 3.2π. Nine viscoelastic polymer solutions are formulated, spanning more than four orders in elasticity number El and are tested in the wavy channels over a wide range of Reynolds and Weissenberg numbers. Flow velocimetry is used to observe and measure the resulting flow patterns. Perturbations to the Poiseuille base flow caused by the wavy surfaces are quantified by the depth of their penetration P into the flow domain. Consistent with theoretical predictions made for wavy plane-Couette flow [J. Page and T. A. Zaki, “Viscoelastic shear flow over a wavy surface,” J. Fluid Mech. 801, 392–429 (2016)], we observe three distinct flow regimes (“shallow elastic,” “deep elastic” and “transcritical”) that can be assembled into a “phase diagram” spanned by two dimensionless parameters: α and the depth of the theoretically predicted critical layer Σ∼El. Our results provide the first experimental verification of this phase diagram and thus constitute strong evidence for the existence of the predicted critical layer. In the inertio-elastic transcritical regime, a surprising amplification of the perturbation occurs at the critical layer, strongly influencing P. These effects are of likely importance in widespread inertio-elastic flows in pipes and channels, such as in polymer turbulent drag reduction.

Viscoelastic shear flow past an infinitely long and freely rotating cylinder

We study analytically the effect of steady shear flow on an infinitely long and freely rotating circular cylinder using asymptotic methods. The ambient fluid is assumed viscoelastic and modeled with the Oldroyd-B constitutive equation under isothermal and creeping flow conditions. The solution for all the dependent variables is expanded as an asymptotic power series with the small parameter being the Weissenberg number, Wi, which is defined as the product of the single relaxation time of the fluid times the externally imposed shear rate. The resulting sequence of equations is solved analytically up to the eighth order in the Weissenberg number, leading to a series solution, the accuracy and validity of which is shown to be linked to the loss of positive definiteness of the conformation tensor. The solution derived here is the first analytical result in the literature for viscoelastic fluids past a freely rotating cylinder. It reveals the slowdown of the rotation of the cylinder with respect to the Newtonian case, as previously found for the same type of flow past a rigid sphere. It is also seen that the first correction to the velocity profile is of second order in Wi, while that for the pressure is of first order. This contrasts with the solution for the viscoelastic flow past a sphere for which the correction for both the velocity and pressure profiles are of first order in Wi. The analytical solution also shows that the region of closed streamlines around the cylinder is enlarged compared to the Newtonian case. Last, it is seen that the up-and-down and fore-and-aft symmetries of the streamlines and the vorticity contours, observed for Newtonian fluids, break in the viscoelastic case.

Growth of viscoelastic wings and the reduction of particle mobility in a viscoelastic shear flow

The sedimentation of a rigid, spherical particle in a cross sheared elastic fluid is studied. By increasing the shear rate (shear Weissenberg number, Wi), the growth of viscoelastic wake structures, resembling wings, is observed, which are linked to a dramatic decrease in the particle mobility.

On the use of a diffuse-interface model for the simulation of rigid particles in two-phase Newtonian and viscoelastic fluids

For the simulation of particles in two-phase flows, the diffuse-interface model is frequently employed to describe the fluid-fluid interface. The diffuse-interface model can naturally handle moving contact lines and topological changes, but the thickness of the interface is in general chosen larger than the physical one. We systematically investigated the effect of both the interface thickness and the diffusion of the fluids on the motion of rigid particles in two-phase flows, using a diffuse-interface model for the fluid-fluid interface. A sharp-interface model is considered for the fluid-fluid interface as well, which is expected to behave as the limiting case of the interface thickness going to zero.The first case that was investigated is a spherical particle in a closed cylindrical container filled with two Newtonian fluids, where the particle is moved toward the fluid-fluid interface by a force. The second case that was investigated is that of the migration of a rigid particle in a two-phase viscoelastic shear flow. The migration of the particle is due a contrast in the viscoelastic properties of the fluids, a phenomenon that was not reported before in the literature. For both cases, the results for the diffuse-interface model converge to the sharp-interface model when the interface thickness is decreased. However, it is shown that both the interface thickness and the diffusion of the fluids play crucial roles in the resulting dynamics of a particle interacting with a diffuse fluid-fluid interface.