# Latest papers in fluid mechanics

### Solving the population balance equation for non-inertial particles dynamics using probability density function and neural networks: Application to a sooting flame

Numerical modeling of non-inertial particles dynamics is usually addressed by solving a population balance equation (PBE). In addition to space and time, a discretization is required also in the particle-size space, covering a large range of variation controlled by strongly nonlinear phenomena. A novel approach is presented in which a hybrid stochastic/fixed-sectional method solving the PBE is used to train a combination of an artificial neural network (ANN) with a convolutional neural network (CNN) and recurrent long short-term memory artificial neural layers. The hybrid stochastic/fixed-sectional method decomposes the problem into the total number density and the probability density function of sizes, allowing for an accurate treatment of surface growth/loss. After solving for the transport of species and temperature, the input of the ANN is composed of the thermochemical parameters controlling the particle physics and of the increment in time. The input of the CNN is the shape of the particle size distribution (PSD) discretized in sections of size. From these inputs, in a flow simulation, the ANN–CNN returns the PSD shape for the subsequent time step or a source term for the Eulerian transport of the particle size density. The method is evaluated in a canonical laminar premixed sooting flame of the literature, and for a given level of accuracy (i.e., a given discretization of the size space), a significant computing cost reduction is achieved (six times faster compared to a sectional method with ten sections and 30 times faster for 100 sections).

### Inertial focusing of elliptical particles and formation of self-organizing trains in a channel flow

The inertial focusing of elliptical particles and the formation of self-organizing trains in a channel flow are studied by using the lattice Boltzmann method. The effects of particle aspect ratio (α), particle concentration (Φ), Reynolds number (Re), and blockage ratio (k) on self-organizing single-line and staggered particle trains are explored. The results show that a single-line particle train is dynamically formed mainly due to the inclination of height (IH) for the particles in the train. The elliptical particle with large α, Φ, Re, and small k facilitates self-organizing of the particle train with relatively stable spacing for a long travel distance. With increasing α, Φ, Re, and k, the value of IH increases and the interparticle spacing decreases. Four kinds of stability conditions for a self-organizing staggered particle train exist depending on Re, k, and α. The threshold Re to form the stable staggered particle train increases with increasing k and is insensitive to α. As Re increases, the spacing of the staggered particle train for the particles with low k and large α is more likely to fluctuate within a certain range. The staggered particle train can be dynamically formed when Re is larger than a critical value. This critical value of Re increases with increasing k and decreasing α. The interparticle spacing of the formed staggered particle train, which is insensitive to Φ, increases with increasing Re and α and decreasing k.

### Drop-on-demand assessment of microdrops of dilute ZnO–water nanofluids

Shrinking device dimensions demand a high level of control and manipulation of materials at microscale and nanoscale. Microfluidics has a diverse application spectrum including thermal management of chips, point-of-care diagnostics, and biomedical analysis, to name a few. Inkjet printing (IJP) is a manufacturing method used for micro-/nanofabrication and surface restructuring, and liquid inks are characterized based on their density, surface tension, and viscosity for their printability. Nanofluids as colloidal dispersions of nanoparticles hold potential in various heating, cooling, lubricating, and biomedical applications with the premise of nanoparticles’ size and concentration effects and interactions between nanoparticle–nanoparticle and nanoparticle–base fluid. In order to explore the microfluidic behavior of nanofluids, using micro-volumes of nanofluids and/or confining them in a micro-system is essential. With this motivation, we present a printability assessment on the potential of low concentration ZnO–water nanofluids by utilizing a combined theoretical and experimental approach. For 0.05 vol. %–0.4 vol. % of ZnO–water nanofluids, results showed that for a nozzle diameter of 25 μm, the samples do not exhibit the energy necessary for drop formation, while for 50 μm and 100 μm nozzle diameters, the samples behave as satellite droplets. Although satellite droplets were generally not desirable for IJP, the recently introduced satellite droplet printing concept may be applicable to the printing of aqueous nano-ZnO dispersions considered in this work.

### Collisional ferrohydrodynamics of magnetic fluid droplets on superhydrophobic surfaces

The study reports the aspects of post-impact hydrodynamics of ferrofluid droplets on superhydrophobic (SH) surfaces in the presence of a horizontal magnetic field. A wide gamut of dynamics was observed by varying the impact Weber number (We), the magnetic field strength (manifested through the magnetic Bond number (Bom), which is defined as the ratio of magnetic force to surface tension force), and the Hartmann number (Ha), defined as the ratio of magnetic force to the viscous force. For a fixed We ∼ 60, we observed that at moderately low Bom ∼300, droplet rebound off the SH surface is suppressed. The noted We is chosen to observe various impact outcomes and to reveal the consequent ferrohydrodynamic mechanisms. We also show that ferrohydrodynamic interactions lead to asymmetric spreading due to variation in magnitude of the Lorentz force, and the droplet spreads preferentially in a direction orthogonal to the magnetic field lines. We show analytically that during the retraction regime, the kinetic energy of the droplet is distributed unequally in the transverse (orthogonal to the external horizontal magnetic field) and longitudinal (along the direction of the magnetic field) directions. This ultimately leads to the suppression of droplet rebound. We studied the role of Bom at fixed We ∼ 60 and observed that the liquid lamella becomes unstable at the onset of retraction phase, through nucleation of holes, their proliferation and rupture after reaching a critical thickness only on SH surfaces, but is absent on hydrophilic surfaces. We propose an analytical model to predict the onset of instability at a critical Bom. The model shows that the critical Bom is a function of the impact We, and the critical Bom decreases with increasing We. We illustrate a phase map encompassing all the post-impact ferrohydrodynamic phenomena on SH surfaces for a wide range of We and Bom.

### Hydrodynamic analysis of nanofluid’s convective heat transfer in channels with extended surfaces

The effects of nanoparticles (NPs) on heat transfer in extended surface channels have been analyzed using a two-component (TC) model. The results show that unlike the single-component model, the TC model leads to more accurate predictions of the system’s heat transfer performance as a result of the direct influence of the NPs’ distribution on the hydrodynamics. It is found that the average Nusselt number varies non-monotonically with the block’s heights, and the trend is explained by the interplay between heat transfer mechanisms and the hydrodynamics. A similar non-monotonic trend observed in the case of the friction factor has been explained by the variations of the concentration- and temperature-dependent viscosity of the nanofluids. A guideline for an optimum design based on the combination of the variation of average Nusselt number and friction factor with respect to the geometrical parameters has also been presented.

### Collective locomotion of two uncoordinated undulatory self-propelled foils

Fish schooling with stable configurations is intriguing. How individuals benefit from hydrodynamic interactions is still an open question. Here, fish are modeled as undulatory self-propelled foils, which is more realistic. The collective locomotion of two foils in a tandem configuration with different amplitude ratios Ar and frequency ratios Fr is considered. Depending on Ar and Fr, the two foils without lateral or yaw motion may spontaneously form stable configurations, separate, or collide with each other. The phase diagram of the locomotion modes in the (Fr, Ar) plane is obtained, which is significantly different from that in Newbolt et al. [“Flow interactions between uncoordinated flapping swimmers give rise to group cohesion,” Proc. Natl. Acad. Sci. U. S. A. 116, 2419 (2019)]. For stable configurations, the gap spacing may be almost constant [stable position (SP) mode] or change dynamically and periodically [stable cycle (SC) mode]. In our diagram, the fast SP mode is found. Besides, the border between the separation and SP/SC modes is more realistic. In the fast SP cases, analyses of hydrodynamic force show the phenomenon of inverted drafting, in which the leader achieves hydrodynamic advantages. For the SC mode, the cruising speed increases piecewise linearly with FrAr. When Ar < 1, the linear slope is identical to that of the isolated leader, and the follower-control mechanism is revealed. Our result sheds some light on fish schooling and predating.

### Coherent vortex in a spatially restricted two-dimensional turbulent flow in absence of bottom friction

We investigate the coherent vortex produced by two-dimensional turbulence excited in a finite box. We establish analytically the mean velocity profile of the vortex for the case where the bottom friction is negligible and express its characteristics via the parameters of pumping. Our theoretical predictions are verified and confirmed by direct numerical simulations in the framework of two-dimensional weakly compressible hydrodynamics with zero boundary conditions.

### Appearance of the instability from the inhomogeneity in two-phase flow with gas-producing reaction in fractured porous medium

A homogenized model of incompressible two-phase flow accompanied by a gas-producing reaction in a double porosity medium with a chemically active skeleton is derived. The equations of the homogenized model contain non-local in time source terms corresponding to the contribution of the gas-producing chemical reaction in the matrix blocks. The time non-locality, which manifests itself as the appearance of a time delay between the change in reactant concentrations and the reaction rate, is shown to stimulate the instability of the one-dimensional two-phase flow initiated by injection of the acid solution into the double porosity medium with chemically active matrix blocks. The instability results in the development of the self-oscillating mode of the reaction wave propagation.

### Fully consistent Eulerian Monte Carlo fields method for solving probability density function transport equations in turbulence modeling

The probability density function (PDF) transport equation method is a sophisticated model for the closure of turbulent mixing and turbulent reactive flows. An efficient solution approach for solving the PDF transport equation has been vital for the method to be widely used in applications. The Eulerian Monte Carlo fields (EMCF) method has been developed to solve the PDF transport equation efficiently for decades. A recent work by Wang et al. [Phys. Fluids 30, 115106 (2018)] revealed a serious issue of the EMCF method for not being fully consistent with the PDF transport equation for which the method is designed to solve. This work advances the state of the art by introducing fully consistent EMCF methods for solving the PDF transport equation. The fully consistent EMCF formulations are derived for two different PDF equation forms. The consistency of the EMCF formulations is mathematically confirmed by examining the derived moment transport equations from the EMCF formulations and from the PDF transport equation. The method of manufactured solutions is employed to further verify the consistency and convergence of the different EMCF formulations numerically. The newly introduced EMCF formulations bring the EMCF method to full consistency with the PDF transport equations for the first time.

### Droplet evaporation residue indicating SARS-COV-2 survivability on surfaces

We conducted a systematic investigation of droplet evaporation on different surfaces. We found that droplets formed even with distilled water do not disappear with evaporation but instead shrink to a residue of a few micrometers lasting over 24 h. The residue formation process differs across surfaces and humidity levels. Specifically, under 40% relative humidity, 80% of droplets form residues on plastic and uncoated and coated glass, while less than 20% form on stainless steel and none on copper. The formation of residues and their variability are explained by modeling the evaporation process considering the presence of nonvolatile solutes on substrates and substrate thermal conductivity. Such variability is consistent with the survivability of SARS-CoV-2 measured on these surfaces. We hypothesize that these long-lasting microscale residues can potentially insulate the virus against environmental changes, allowing them to survive and remain infectious for extended durations.

### Droplet evaporation residue indicating SARS-COV-2 survivability on surfaces

We conducted a systematic investigation of droplet evaporation on different surfaces. We found that droplets formed even with distilled water do not disappear with evaporation but instead shrink to a residue of a few micrometers lasting over 24 h. The residue formation process differs across surfaces and humidity levels. Specifically, under 40% relative humidity, 80% of droplets form residues on plastic and uncoated and coated glass, while less than 20% form on stainless steel and none on copper. The formation of residues and their variability are explained by modeling the evaporation process considering the presence of nonvolatile solutes on substrates and substrate thermal conductivity. Such variability is consistent with the survivability of SARS-CoV-2 measured on these surfaces. We hypothesize that these long-lasting microscale residues can potentially insulate the virus against environmental changes, allowing them to survive and remain infectious for extended durations.

### Meshfree method based on discrete gas-kinetic scheme to simulate incompressible/compressible flows

A meshfree method based on the discrete gas-kinetic scheme (DGKS) (called the meshfree-DGKS) for simulation of incompressible/compressible flows is proposed in this work. In this approach, the governing equations are discretized using the meshfree method based on the least squares-based finite difference approach. To simulate compressible problems with discontinuities, the virtual mid-points between adjacent nodes, which are regarded as Riemann discontinuities, are established. Then, the concept of numerical flux is introduced, which enables computing both compressible and incompressible problems. The fluxes at the mid-points are calculated using the DGKS based on the discrete particle velocity model. The corresponding particle velocity components and distribution functions are integrated based on moment relations to obtain the flux. The meshfree-DGKS maintains the advantages of the meshless method as it is implemented at arbitrarily distributed nodes. This breaks through the limitations of the grid topology and is suitable to handle complex geometries. More importantly, the fluxes at the mid-point are reconstructed with the DGKS using the local solution of the Boltzmann equation, which can describe its physical properties well, thus easily and stably capturing the shock wave. In addition, the DGKS can simultaneously calculate inviscid and viscous fluxes when simulating viscous flow problems, which gives an improved algorithm consistency. Several representative examples, such as shock tube problems, implosion problem, couette flow, lid-driven cavity flow, flow in a channel with a backward-facing step, supersonic flow around a ramp segment, and flow around staggered NACA0012 biplane configuration, are simulated to validate the proposed meshfree-DGKS.

### Aerodynamic performance of profiled endwalls with upstream slot purge flow in a linear turbine cascade having pressure side separation

In aeroengines, purge flow directly fed from the compressor (which bypasses the combustor) is introduced through the disk space between blade rows to prevent the hot ingress. Higher quantity of purge gas fed through the wheel space can provide additional thermal protection to the passage endwall and blade surfaces. However, the interaction of purge flow with the mainstream flow leads to higher secondary losses. Secondary losses inside a turbine blade passage can be reduced effectively by endwall contouring. This paper presents computational investigation on the influence of non-axisymmetric endwall contouring over endwall secondary flow modification in the presence of purge flow with the pressure side bubble (PSB). The experimental analysis was conducted for the base case without purge and base case with purge (BCP) configurations having flat endwalls. The total pressure loss coefficient and exit yaw angle deviation were measured with the help of a five-hole pressure probe. Static pressure distribution over the blade midspan was obtained by 16 channel Scanivalve. Aerodynamic performances of three different profiled endwalls are numerically analyzed and are compared against the BCP configuration. The effects of different contoured endwall geometries on endwall static pressure distribution and secondary kinetic energy were also discussed. Analysis shows that in the first contoured endwall configuration (EC1), the formation of stagnation zones at a contour valley close to the suction surface causes the exit total pressure loss coefficient to increase. The shifting of the contour valley near to the pressure surface (EC2 configuration) has resulted in local acceleration of the diverted pressure side leg of the horseshoe vortex over the hump toward the end of the passage. In the third configuration (EC3 configuration), reduced valley depth and optimum hump height have effectively redistributed the endwall pitchwise pressure gradient. The increased static pressure coefficient at the endwall near to the pressure surface has eliminated the PSB formation. In addition, computational results of unsteady Reynolds averaged Navier–Stokes simulations are obtained for analyzing transient behavior of PSB, with more emphasis on its migration on the pressure surface and transport across the blade passage. The additional work done by the mainstream fluid to transport the low momentum PSB fluid has caused higher aerodynamic penalty at the blade exit region. In this viewpoint, the implementation of contoured endwalls has shown beneficial effects by eliminating the PSB and related secondary vortices. At 27% of axial chord downstream of the blade trailing edge, a 4.1% reduction in the total pressure loss coefficient was achieved with endwall contouring.

### Comparing flow characteristics of viscoelastic liquids in long and short capillaries (entrance effects)

This study is devoted to the analysis of the physical meaning of the difference in the results of the viscosity measurements obtained by using the two-capillary method—carrying on the same flow rate through two circular parallel capillaries of different lengths but the same diameter. Furthermore, there is the other approach for using short capillaries for determination of a value, known as “elongational viscosity,” based on the concept of domination extension at the convergent flow along the inlet of a capillary. The theory of this method dates back to the publication of Cogswell. However, it was shown that this approach is inadequate for polymeric liquids that demonstrate time-dependence effects. In this study, we present experimental data for melts of two polyethylenes of different molecular weights and their mixtures. The data of capillary viscometry are accompanied by measuring their viscoelastic properties. The obtained data showed that the two-capillary method cannot give a chance to estimate “elongational viscosity” for elastic liquids. The difference between the losses in flow through long and short capillaries is treated in the terms of “end correction,” which is determined by the Weissenberg number. Hence, elasticity is the main physical parameter responsible for additional losses in the flow through a short capillary.

### Viscoelastic flow instabilities in static mixers: Onset and effect on the mixing efficiency

Elastic instabilities are identified as flow instabilities occurring in the presence of low inertial effects, induced by the combination of strong elastic forces with nonlinearities of the flow. In continuous flow laminar mixing applications, the onset of these instabilities is likely to occur in the window of applied flow rates; therefore, it is important to understand the effects of their onset on the process efficiency. In this work, we investigated experimentally the onset of elastic instabilities in two tubular static mixers with different geometric features, i.e., the Kenics helical mixer and the SMB-R mixer, the latter characterized by a double X-shaped bar geometry. We obtained concentration maps at various mixer lengths by means of planar laser induced fluorescence techniques. To deduce a generalized effect of the fluid elasticity on the mixing patterns, we tested three fluids with different rheological behavior—a Boger fluid and two shear-thinning fluids. For all cases, we observed deviations from the Newtonian benchmark as soon as the Deborah number exceeded unity, even though different transitions occurred as the mean flow rate increased. The effect of the instability on the mixing patterns strongly depended on the different kinematics induced by the two geometries: for the helical mixer, the typical lamellar structure is not recovered and the two liquid streams remain unmixed, while for the SMB-R, the concentration maps are strongly unsteady, showing temporally and spatially chaotic fluctuations of the mass fraction. In both cases, the instabilities worsen the mixing efficiency compared to the Newtonian case.

### Dynamics of a rodlike deformable particle passing through a constriction

We use the finite element method to investigate the flow-induced translocation of a rodlike deformable particle through a narrow constriction in a microchannel from a dynamical perspective. Our results demonstrate that the deformable particle exhibits two translocation modes, one with folded deformation and one with unfolded deformation, depending mainly on the initial deflection angle. When the initial deflection angle is small, the deformable particle undergoes folded deformation, which changes to unfolded deformation as the angle increases. Depending on its initial location with respect to the axis of the microchannel, the deformable particle exhibits swinging motion or one of two types of tumbling motion: tumbling I (90° < θ < 180°, where θ is the rotation angle) and tumbling II (θ > 180°). Swinging motion occurs when the initial position is close to the axis, and this is converted to tumbling I and tumbling II motions when the initial position moves away from the axis. Our results provide a description of the deformation and motion of a rodlike deformable particle during its passage through a constriction, which can be useful for understanding the role of deformable particles in physiological processes, for cell separation, and for the application of deformable particles in drug delivery.

### Quantifying the destructuring of a thixotropic colloidal suspension using falling ball viscometry

The settling dynamics of falling spheres inside a Laponite suspension is studied. Laponite is a colloidal synthetic clay that shows physical aging in aqueous suspensions due to the spontaneous evolution of inter-particle electrostatic interactions. In our experiments, millimeter-sized steel balls are dropped in aqueous Laponite suspensions of different ages (i.e., time elapsed since sample preparation). The motion of the falling balls is captured using a high-speed camera, and the velocities of their centroids are estimated from the images. Interestingly, we observe that balls of larger diameters fail to achieve terminal velocity over the entire duration of the experiment. We propose a mathematical model that accounts for rapid structural changes (expected to be induced by the falling ball) in Laponite suspensions whose aging time scales are much slower than the time of fall of the ball. For a range of ball sizes and Laponite suspension ages, our model correctly predicts the time dependence of the ball velocity. Furthermore, fits to our model allow us to estimate the rates of destructuring of the thixotropic suspensions due to the passage of the falling ball.

### Effects of aspect ratio on rolling and twisting foils

Author(s): Andhini N. Zurman-Nasution, Bharathram Ganapathisubramani, and Gabriel D. Weymouth

Flow features and forces of three-dimensional flapping foils are presented with aspect-ratio (AR) variations. The similitude of rolling and twisting kinematics is maintained across different ARs. An AR correction analogous to Prandtl finite-wing theory is developed, enabling future use of strip theory in the analysis and design of finite flapping foils.

[Phys. Rev. Fluids 6, 013101] Published Fri Jan 15, 2021

### Adhesion, cavitation, and fibrillation during the debonding process of pressure sensitive adhesives

Author(s): S. Varchanis, A. Kordalis, Y. Dimakopoulos, and J. Tsamopoulos

Transient 3-dimensional simulations of viscoelastic fluidlike adhesives access the physics behind the debonding process of pressure sensitive adhesives (PSA). A novel finite element method is developed to simulate viscoelastic flows with multiple free surfaces and contact lines. The rheological properties of the adhesive are correlated with its tackiness and stickiness, targeting the development of materials with improved adhesive energy.

[Phys. Rev. Fluids 6, 013301] Published Fri Jan 15, 2021

### Diapycnal mixing efficiency in lock-exchange gravity currents

Author(s): Partho Mukherjee and Sridhar Balasubramanian

An experimental measure of irreversible mixing efficiency, Rif*, for a lock-exchange gravity current is presented. The results show that mixing efficiency increases and is sustained in the presence of strong stratification provided the flow energy is high. Following this, various turbulence regimes are mapped based on turbulent Froude number and turbulent Reynolds number. Additionally, Rif* is shown to have a strong dependence on turbulence regime and molecular effects induced by Prandtl number.

[Phys. Rev. Fluids 6, 013801] Published Fri Jan 15, 2021