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We report experimental microfluidic measurements and theoretical modeling of elastoviscoplastic materials under steady, planar elongation. Employing a theory that allows the solid state to deform, we predict the yielding and flow dynamics of such complex materials in pure extensional flows. We find a significant deviation of the ratio of the elongational to the shear yield stress from the standard value predicted by ideal viscoplastic theory, which is attributed to the normal stresses that develop in the solid state prior to yielding. Our results show that the yield strain of the material governs the transition dynamics from the solid state to the liquid state. Finally, given the difficulties of quantifying the stress field in such materials under elongational flow conditions, we identify a simple scaling law that enables the determination of the elongational yield stress from experimentally measured velocity fields.

Viscoplasticity is characterized by a yield stress, below which the materials will not deform and above which they will deform and flow according to different constitutive relations. Viscoplastic models include the Bingham plastic, the Herschel-Bulkley model and the Casson model. All of these ideal models are discontinuous. Analytical solutions exist for such models in simple flows. For general flow fields, it is necessary to develop numerical techniques to track down yielded/unyielded regions. This can be avoided by introducing into the models a regularization parameter, which facilitates the solution process and produces virtually the same results as the ideal models by the right choice of its value. This work reviews several benchmark problems of viscoplastic flows, such as entry and exit flows from dies, flows around a sphere and a bubble and squeeze flows. Examples are also given for typical processing flows of viscoplastic materials, where the extent and shape of the yielded/unyielded regions are clearly shown. The above-mentioned viscoplastic models leave undetermined the stress and elastic deformation in the solid region. Moreover, deviations have been reported between predictions with these models and experiments for flows around particles using Carbopol, one of the very often used and heretofore widely accepted as a simple “viscoplastic” fluid. These have been partially remedied in very recent studies using the elastoviscoplastic models proposed by Saramito.

The translational velocities of two spherical gas bubbles oscillating in water, which is irradiated by a high-intensity acoustic wave field, are calculated. The two bubbles are assumed to be located far enough apart so that shape oscillations can be neglected. Viscous effects are included owing to the small size of the bubbles. An asymptotic solution is obtained that accounts for the viscous drag on each bubble, for large ${\\it Re}$ based on the radial part of the motion, in a form similar to the leading-order prediction by Levich (1962), $C_{D} = 48/{\\it Re}_{T}$; ${\\it Re}_{T} \\to \\infty$ based on the translational velocity. In this context the translational velocity of each bubble, which is a direct measure of the secondary Bjerknes force between the two bubbles, is evaluated asymptotically and calculated numerically for sound intensities as large as the Blake threshold. Two cases are examined. First, two bubbles of unequal size with radii on the order of $100\\,\\umu$m are subjected to a sound wave with amplitude $P_{A} < 1.0$ bar and forcing frequency $\\omega_{f} = 0.51\\omega_{10}$, so that the second harmonic falls within the range defined by the eigenfrequencies of the two bubbles, $\\omega_{10} < 2\\omega_{f} < \\omega_{20}$. It is shown that their translational velocity changes sign, becoming repulsive as $P_{A}$ increases from 0.05 to 0.1 bar due to the growing second harmonic, $2\\omega_{f}$, of the forcing frequency. However, as the amplitude of sound further increases, $P_{A} \\approx 0.5$ bar, the two bubbles attract each other due to the growth of even higher harmonics that fall outside the range defined by the eigenfrequencies of the two bubbles. Second, the case of much smaller bubbles is examined, radii on the order of $10\\,\\umu$m, driven well below resonance, $\\omega_{f}/2\\pi = 20$ kHz, at very large sound intensities, $P_{A} \\approx 1$ bar. Numerical simulations show that the forces between the two bubbles tend to be attractive, except for a narrow region of bubble size corresponding to a nonlinear resonance related to the Blake threshold. As the distance between them decreases, the region of repulsion is shifted, indicating sign inversion of their mutual force. Extensive numerical simulations indicate the formation of bubble pairs with constant average inter-bubble distance, consisting of bubbles with equilibrium radii determined by the primary and secondary resonance frequencies for small and moderate sound amplitudes or by the Blake threshold for large sound amplitudes. It is conjectured that in experiments where ‘acoustic streamers’ are observed, which are filamentary structures consisting of bubbles that are aligned and move rapidly in a cavitating fluid at nearly constant distances from each other, bubbles with size determined by the Blake threshold are predominant because those with size determined by linear resonance are larger and therefore become unstable due to shape oscillations.

This work focuses on the advanced modeling of the thixotropic nature of blood, coupled with an elasto-visco-plastic formulation by invoking a consistent and validated model for TEVP materials. The proposed model has been verified for the adequate description of the rheological behavior of suspensions, introducing a scalar variable that describes dynamically the level of internal microstructure of rouleaux at any instance, capturing accurately the aggregation and disaggregation mechanisms of the RBCs. Also, a non-linear fitting is adopted for the definition of the model’s parameters on limited available experimental data of steady and transient rheometric flows of blood samples. We present the predictability of the new model in various steady and transient rheometric flows, including startup shear, rectangular shear steps, shear cessation, triangular shear steps and LAOS tests. Our model provides predictions for the elasto-thixotropic mechanism in startup shear flows, demonstrating a non-monotonic relationship of the thixotropic index on the shear-rate. The intermittent shear step test reveals the dynamics of the structural reconstruction, which in turn is associated with the aggregation process. Moreover, our model offers robust predictions for less examined tests such as uniaxial elongation, in which normal stress was found to have considerable contribution. Apart from the integrated modeling of blood rheological complexity, our implementation is adequate for multi-dimensional simulations due to its tensorial formalism accomplished with a single time scale for the thixotropic effects, resulting in a low computational cost compared to other TEVP models.

The present work focuses on the in-silico investigation of the steady-state blood flow in straight microtubes, incorporating advanced constitutive modeling for human blood and blood plasma. The blood constitutive model accounts for the interplay between thixotropy and elasto-visco-plasticity via a scalar variable that describes the level of the local blood structure at any instance. The constitutive model is enhanced by the non-Newtonian modeling of the plasma phase, which features bulk viscoelasticity. Incorporating microcirculation phenomena such as the cell-free layer (CFL) formation or the Fåhraeus and the Fåhraeus-Lindqvist effects is an indispensable part of the blood flow investigation. The coupling between them and the momentum balance is achieved through correlations based on experimental observations. Notably, we propose a new simplified form for the dependence of the apparent viscosity on the hematocrit that predicts the CFL thickness correctly. Our investigation focuses on the impact of the microtube diameter and the pressure-gradient on velocity profiles, normal and shear viscoelastic stresses, and thixotropic properties. We demonstrate the microstructural configuration of blood in steady-state conditions, revealing that blood is highly aggregated in narrow tubes, promoting a flat velocity profile. Additionally, the proper accounting of the CFL thickness shows that for narrow microtubes, the reduction of discharged hematocrit is significant, which in some cases is up to 70%. At high pressure-gradients, the plasmatic proteins in both regions are extended in the flow direction, developing large axial normal stresses, which are more significant in the core region. We also provide normal stress predictions at both the blood/plasma interface (INS) and the tube wall (WNS), which are difficult to measure experimentally. Both decrease with the tube radius; however, they exhibit significant differences in magnitude and type of variation. INS varies linearly from 4.5 to 2 Pa, while WNS exhibits an exponential decrease taking values from 50 mPa to zero.

The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water film that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, r˙, compared to the free-stream velocity, U∞, and (c) small film thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid film like x3/4; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the film. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the triple-deck regime. Two distinct families of eigenvalues are identified, one corresponding to the well-known Tollmien–Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air–water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We−1) increases, indicating the increasing effect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We−1 ≈ We−1c, beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr−1), indicating the increasing effect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr−1 ≈ Fr−1c, beyond which the family of interfacial waves becomes convectively unstable. As We−1 and Fr−1 are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.

We examine the displacement by pressurized air of a liquid, which only partially occupies straight or complex tubes, according to the Gas‐Assisted Injection Molding (GAIM) process. The process involves the formation and continuous elongation of a gaseous finger, which sets in motion the liquid, which, in turn, forms a second moving interface with gas downstream, the advancing front, and is simultaneously deposited on the tube walls. The motion of the advancing front is simulated using a Navier‐type slip condition. A complete parametric analysis is performed in order to examine the effects of the initial amount of liquid, inertia, liquid compressibility, and the slip coefficient. Simulations under creeping flow conditions and relatively large initial amounts of liquid show that the thickness of the deposited film on the inner tube wall is uniform for the most part, except near the tube entrance and where the still moving portion of the liquid is nearly depleted. Increasing inertia causes flattening of the liquid front, non‐uniform film distribution along the wall and eventually a tip‐splitting instability. Liquid compressibility influences the phenomenon only slightly. The difference between the two interfacial velocities increases as the no‐slip condition is approached and eventually leads to their collision. Finally, coating of an expanding tube of finite length and either closed or open downstream is examined for various amounts of liquid initially placed in it. POLYM. ENG. SCI., 46:47–68, 2006. © 2005 Society of Plastics Engineers

Nonlinear dynamics of the concentric, two-phase flow of two immiscible fluids in a circular tube of variable cross-section is studied for parameter values where the steady core–annular flow (CAF) is linearly unstable. The simulations are based on a pseudo-spectral numerical method. They are carried out assuming axial symmetry, that the total flow rate remains constant and that all dependent variables are periodic in the axial direction, which includes the minimum necessary number of repeated units so that the obtained solution is independent of this number. The time integration originates with the numerically computed steady CAF or the steady CAF seeded with either the most unstable mode or random small disturbances. Only a limited number of the most interesting cases are presented. For the most part, the values of the majority of the dimensionless parameters are such that oil flows in the centre of the tube driven by an applied pressure gradient against gravity, whereas water is flowing in the annulus. It is shown that, whereas the steady (unstable) solution may indicate that the heavier water flows countercurrently with respect to the oil, the time periodic (observable) solution may indicate the same, albeit at a much smaller core flow rate or that concurrent flow occurs. This is due to the water being trapped between the large-amplitude interfacial waves that are generated and being convected by the oil. It is also shown that increasing the inverse Weber number increases the wave amplitude to the point that the flow of the core fluid may become discontinuous with a mechanism that depends on the viscosity ratio between the two fluids. Increasing the amplitude of the sinusoidal variation of the tube leads to a combination of travelling and standing waves, which interact to produce a time periodic solution with a long period associated with the time it takes the travelling wave to travel through the computational domain and a second much shorter period that is related to their interaction time. Qualitative agreement has been obtained upon comparing our numerical simulations with limited experimental reports, even though the experimental conditions were not identical to those in our model.

We consider the buoyancy-driven rise and interaction between two gravity-aligned bubbles of wide radii ratio and constant volume in an elasto-visco-plastic (EVP) material, extending our previous work on equal bubbles [Kordalis et al., Phys. Rev. Fluids 8, 083301, (2023)]. Primarily we consider a 0.1% aqueous Carbopol solution and model it with the Saramito-Herschel-Bulkley model. Initially, we investigate the dynamics for a specific initial separation distance in a wide range of bubble radii and we determine the conditions leading to three distinct patterns: bubble approach, bubble separation and establishment of a constant distance between them. Specifically, when the leading bubble (LB) is smaller than the trailing bubble (TB), the bubbles approach each other due to the smaller buoyancy of the LB. Strong attraction also occurs when the ratio of buoyant over viscous force of both bubbles is considerable. On the other hand, when the size of the TB is such that this ratio is moderate or small, the pattern is dictated by the size of the LB: A significantly larger LB compared to the TB causes separation of the pair. On the contrary, an only slightly larger LB may result in the bubbles rising with the same terminal velocity establishing a constant distance between them, the magnitude of which is mainly determined by the elastic response of the surrounding medium. The coupling of a negative wake behind the LB with a slight modification of the stresses exerted at its rear pole generates this dynamic equilibrium. The same equilibrium may be achieved by other specific pairs of bubble sizes for different initial distances of the pair, if a critical initial distance is exceeded. Below this critical value, the bubbles approach each other. Finally, we construct maps of the three patterns with TB radius versus bubble radii ratio for different initial separation distances and material properties.