Fractional calculus has become a powerful tool to describe viscoelastic mechanical behaviors. The key to its practical application lies in determining the number and combination form of fractional operators. In this paper, a systematic comparative study on the application effects of different forms of single and double fractional models has been conducted. To this end, various concentrations of polyacrylamide (PAAm) solutions were prepared as representatives, and the storage modulus and loss modulus were obtained through frequency sweep experiments. Subsequently, the single and double fractional models in Maxwell and Kelvin-Voigt types were used separately to describe the experimental data, and the goodness of fit was employed for the evaluation of the results. The results indicated that the double fractional models can provide more accurate results than the single fractional models, especially the double fractional Maxwell model, which demonstrated significant advantages in both the storage and loss moduli. Therefore, compared with the other three models, the double fractional Maxwell model is the optimal modeling tool for the dynamic mechanical analysis of viscoelastic fluids.

ABSTRACT Viscoelastic fluids, exhibiting both elastic and viscous properties, play a fundamental role in various industrial and biological applications. Accurate modeling of their rheological behavior requires constitutive equations that capture the complex interplay between these properties. The present study focuses on the analysis of incompressible, isothermal, two‐dimensional, planar, laminar, submerged jet flow of viscoelastic fluids. A computational methodology is adopted to determine the polymer stress‐tensor distribution for different viscoelastic models, including Oldroyd‐B, UCM, Giesekus, Phan‐Thien‐Tanner (PTT), and finitely extensible nonlinear elastic (FENE). These models are chosen to represent a diverse range of viscoelastic behaviors. The Navier–Stokes equations, coupled with the appropriate constitutive model, are solved numerically. The proposed method allows one to access the distribution of the polymer stress‐tensor components with very low computational cost. Results demonstrate the accuracy of the computational method for various models and their parameter values. The findings provide valuable insights into the fundamental behavior of viscoelastic jets and can serve as a foundation for subsequent linear and nonlinear stability investigations.

A microfluidic approach to probing the first normal stress difference from single-point pressure measurements in transient shear flows is presented. Using an original experimental design, we examine the near-zero-mean pulsatile flow of polymeric solutions in a straight microchannel at low Reynolds and Womersley numbers. An important aspect of this work is that the enhanced fluid elastic stresses can be efficiently determined via the pressure shift measured from pressure-controlled pulsatile shear experiments. We find a scaling law that collapses pressure-shift data from viscoelastic fluids of different molecular weights onto a single master curve that can then be used to predict this phenomenology. Taken together, these results could help shed light on our understanding of the nonlinear normal stress responses in time-dependent flows.

Abstract This study investigates the linear stability of mixed convection
flow of an Oldroyd-B viscoelastic fluid in a vertical porous channel
subjected to differential heating. To understand how thermal properties
influence stability, two representative Prandtl numbers, Pr = 0.7
and Pr = 7 are considered. The governing equations for basic flow and
generalised eigenvalue problem, based on the viscoelastic form of the
volume averaged Navier-Stokes equations, are solved using a spectral
collocation approach. The analysis shows that variations in Reynolds
number strongly impact the critical Grashof number (Gr′), the propagation
speed of disturbances, and the overall instability mechanisms.
The results further highlight that flow stability is governed by a delicate
balance between fluid elasticity, buoyancy-driven forces, and the
permeability of the porous medium. The fluids with lower Prandtl
number display a stronger sensitivity to elasticity effects and further,
the fluid elasticity has significant impact on the disturbance convection
patterns inside the channel. The nature of instability is commonly
found to be thermal-bouyant for the range of parameters considered
in this study.

The motility of bacteria in viscoelastic fluids plays a crucial role in diverse biological processes such as infection and biofilm formation. In such environments, the rotation of long helical flagella is often employed as a primary means of bacterial propulsion. In this paper, we present a general version of the immersed boundary method incorporating the finitely extensible nonlinear elastic (FENE) constitutive equations for viscoelastic fluids. We first verify that the method solves correctly the rotational dynamics of a helical flagellum in a FENE fluid by conducting a convergence study. We then investigate the hydrodynamic interactions of single and multiple flagella in Newtonian and FENE viscoelastic fluids. For a single flagellum at the same solvent viscosity, the fluid flux generated by the flagellum is higher in FENE fluid than in Newtonian fluid for nearly identical rotation rates, implying enhanced bacterial swimming speeds in a viscoelastic fluid. The dependence of rotational dynamics on applied motor torque, relaxation time, and polymer viscosity is systematically analyzed. For multiple flagella, the time required for bundle formation is shown to increase with hook rigidity, initial separation, and the number of flagella. In the case of two flagella rotating counterclockwise under different applied torques, bundling occurs only when the torque difference is within a limited range, producing a bundle whose axis is tilted from the flagellar axes. We further examine conditions leading to bundling failure, providing insights into the mechanical and hydrodynamic factors governing flagellar coordination.

This study investigates the impact of elasticity and plasticity on two-dimensional flow past a circular cylinder at Reynolds number Re = 100. Ten direct numerical simulations were performed using the Saramito-Herschel–Bulkley model to represent viscoelastic and elastoviscoplastic (EVP) fluids. The flow evolves from a periodic von Kármán vortex street to chaotic-like regimes. Proper Orthogonal Decomposition (POD) and Higher Order Dynamic Mode Decomposition (HODMD) are applied to extract dominant flow structures and their temporal dynamics. For viscoelastic fluids, increasing the Weissenberg number Wi elongates the recirculation bubble and shifts it downstream, resulting in more intricate but still periodic behavior. In EVP fluids, seven cases explore variations in Bingham number Bn, solvent viscosity ratio beta _s, and power law index n, aiming to qualitatively assess their influence rather than determine critical thresholds. Results indicate that stronger plastic effects, especially with n ge 1, lead to increased flow complexity. Three dynamic regimes are identified: (i) periodic; (ii) transitional, with elongated recirculation and disrupted periodicity; and (iii) fully complex, with breakdown of recirculation. Overall, the study highlights the interplay between inertia, elasticity, and yield stress in non-Newtonian flows past obstacles and identifies key parameters driving the transition from periodic to complex regimes.

Microorganisms such as bacteria and spermatozoa often inhabit confined viscoelastic environments. These organisms exhibit self-organization/collective dynamics in such complex surroundings. Here, we report a simulation study of active particle suspensions in viscoelastic fluids under confinement-representing an experimental scenario where motile organisms suspended in aqueous viscoelastic fluid are surrounded by an oil medium. We employed dissipative particle dynamics, a particle-based mesoscopic approach, to model the system with minimalistic ingredients and qualitatively reproduced some of the experimental observations by Liu et al. Nature, 2021, 590 (7844), 80-84. The collective dynamics within the suspended drop, mediated by the viscoelastic nature of the medium, manifest into two steady state configurations, namely a unidirectional vortex or an oscillatory vortex. We present a phase diagram for the drop's steady state configuration as a function of system parameters, such as strength and packing fraction of active agents, polymer concentration etc.

Viscoelastic fluids, such as polymer solutions, surfactant mixtures, colloidal suspensions, emulsions, and biological fluids like blood, are frequently transported in microfluidic systems using external electric fields. In such flows, two distinct types of instabilities can emerge, namely, electro-elastic instabilities (EEI), arising from the interaction between elastic stresses and streamline curvature, and electrokinetic instabilities (EKI), triggered by electrical conductivity gradients once the external electric field exceeds a critical value. Both instabilities can promote fluid mixing by inducing chaotic flow structures; however, their roles are not always complementary. Recent experimental and numerical studies have shown that increasing fluid viscoelasticity can suppress EKI, leading to reduced mixing efficiency in a microfluidic T-junction. However, this study demonstrates that while viscoelasticity initially hinders mixing by damping EKI, a further increase in the Weissenberg number (a measure of fluid elasticity) leads to the onset of EEI, which in turn again increases mixing. Therefore, a non-monotonic relationship between mixing efficiency and Weissenberg number is found in the present study. Furthermore, although both EEI and EKI promote mixing, they differ significantly in their coherent flow structures and regions of origin within the microdevice. To elucidate these differences, we employ the data-driven dynamic mode decomposition (DMD) technique to characterise the underlying instability modes and their influence on the mixing dynamics. Overall, this study provides fundamental insights into how viscoelasticity modulates flow instabilities in electrokinetically driven microflows and offers strategies to optimise mixing by tuning fluid properties and operating conditions.

Friction factor for pulsatile flow of viscoelastic fluids in circular tubes and concentric annuli using a dynamic slip model at the walls

A central-moment lattice Boltzmann study of bubble interaction and coalescence in shear-thinning viscoelastic fluid

Dynamics of cavitation/Air bubbles in seawater-based viscoelastic fluids

Global well-posedness of the compressible electrically conducting viscoelastic fluids subject to the Coulomb force in the half space

A fluctuating lattice Boltzmann method for viscoelastic fluid flows

To meet the underwater drag reduction needs in engineering applications, one can draw on the drag reduction strategies of marine animals such as sharks, utilizing their surface groove structures and the secretion of viscoelastic drag-reducing mucus to achieve drag reduction effects. However, in the field of engineering research, efficient and accurate simulation methods for viscoelastic fluid flow are relatively scarce. There is an urgent need to develop efficient numerical simulation methods for viscoelastic fluids, and to analyse the drag reduction characteristics of the coupling between surface grooves and drag-reducing agents. Through large eddy simulation and re-development of the ANSYS Fluent platform, efficient and accurate simulation of viscoelastic fluid flow as well as wall infiltration method has been successfully realized. Quantitative analysis of drag reduction characteristics of surface structures and viscoelastic fluid additives within the Reynolds number range of 4×104 to 4×105 has been conducted. Moreover, the synergistic effect of micro-grooves and drag reducer infiltration on drag reduction has been deeply explored. It is found that the maximum drag reduction by the groove surface solely is below 20%, whilst the maximum drag reduction by viscoelastic drag-reducing agent infiltration on the smooth flat plate is up to 28.9%. Notably, when combining the groove structure with drag-reducing agent infiltration, the maximum drag reduction reaches up to approximately 70%. The mechanism for the drag reducing effect is analysed from the aspects of vortex structure evolution on the wall, velocity profile and drag-reducing agent diffusion. This establishes an efficient and accurate simulation method for viscoelastic fluid flow suitable for engineering research, providing an important reference and foundation for underwater drag reduction studies.

This study investigates the collective dynamics of the binary pusher-puller mixtures in Giesekus viscoelastic fluids using the fictitious domain method, with a focus on the synergistic effects of the mixture composition and fluid elasticity on clustering behaviours and energy transfer mechanisms. Key findings reveal that an increasing of the puller proportion induces a phase transition from dynamic clustering to stable clusters, ultimately leading to phase separation. Pushers disrupt cluster stability through intensified local rotational disturbances. Cluster formation universally enhances particle motility and turbulent kinetic energy, whereas fluid elasticity systematically suppresses these effects. Energy spectrum analysis reveals distinct signatures of active turbulence: a constant slope of 1 at low wavenumbers, indicating an inverse energy cascade, and a steep decay slope of − 4 at high wavenumbers, reflecting the typical energy dissipation mechanism in active turbulence. These findings establish a multiscale framework that links microswimmers interactions to macroscopic rheological responses in complex biological fluids.

In this paper, we seek to determine the critical Reynolds number of a viscoelastic fluid flowing in a cylindrical pipe with a horizontal axis. The problem obtained is a generalized eigenvalue problem . A Gauss-Lobatto-Tchebyshev method was adopted to discretize this equation and the QZ algorithm combined with the Newton-Raphson method was used to determine this critical value of the Reynolds number. It is obtained by searching for two successive and very close values ​​for which correspond two eigenvalues ​​whose maximum real parts are respectively negative and positive. In other words, the critical value is the smallest value of the Reynolds number for which instability occurs. The code for performing this calculation was written in FORTRAN. The flow is stable if all the real parts of the eigenvalues obtained are negative and unstable if only one of these values is positive.

Predicting Oil Droplet Mobilization in Viscoelastic Fluid

An Analytical Study of Mass Transport in Viscoelastic Fluid Under the Influence of Gravity Modulation

Abstract Stochastic resetting (SR), in which a system intermittently returns to a fixed location, is a powerful strategy for optimizing search processes. While extensively studied in memoryless (Markovian) systems, its behavior in complex media with memory remains largely unexplored. Here, we experimentally investigate SR in a viscoelastic fluid by tracking a colloidal particle subjected to intermittent resets. In this non-Markovian environment, the fluid’s memory gives rise to elastic restoring forces that oppose the reset, pulling the particle back toward its prior position and hindering efficient exploration. We show that these memory effects can be actively controlled: by holding the particle at the trap center for a sufficient time, the fluid relaxes, erasing its memory and allowing the system to re-equilibrate. When introducing a fixed target site, we find that this memory control enables a significant reduction in the mean passage time (MPT), with optimal search performance emerging at intermediate resetting frequencies. In this regime, memory enhances performance through a "bunching" effect, in which the particle rapidly revisits the target due to temporal correlations in its trajectory. These results highlight the dual role of memory in resetting dynamics—as both a hindrance and a resource—and suggest new strategies for optimizing search in non-Markovian systems, with potential applications in soft matter, biological transport, and stochastic control.

The dispersion of solutes has been extensively studied due to its important applications in microfluidic devices for mixing, separation and other related processes. Solute dispersion in fluids can be analysed over multiple time scales; however, Taylor dispersion specifically addresses long-term behaviour, which is primarily influenced by advective dispersion. This study investigates Taylor–Aris dispersion in a viscoelastic fluid flowing through axisymmetric channels of arbitrary shape. The fluid’s rheology is described using the simplified Phan-Thien–Tanner (sPTT) model. Although the channel walls are axisymmetric, they can adopt any geometry, provided they maintain small axial slopes. Drawing inspiration from the work of Chang & Santiago (2023 J. Fluid Mech. vol. 976, p. A30) on Newtonian fluids, we have developed a governing equation for solute dynamics that accounts for the combined effects of fluid viscoelasticity, molecular diffusivity and channel geometry. This equation is expressed using key dimensionless parameters: the Weissenberg number, the Péclet number and a shape-dependent dimensionless function. Solving this model allows us to analyse the temporal evolution of the solute distribution, including its mean and variance. Our analysis shows that viscoelasticity significantly decreases the effective solute diffusivity compared with that observed in a Newtonian fluid. Additionally, we have identified a specific combination of parameters that results in zero or negative transient growth of the variance. This finding is illustrated in a phase diagram and provides a means for transient control over dispersion. We validated our results against Brownian dynamics simulations and previous literature, highlighting potential applications for the design and optimisation of microfluidic devices.