# Latest papers in fluid mechanics

### Origin of hydrodynamic instability from noise: From laboratory flow to accretion disk

Author(s): Subham Ghosh and Banibrata Mukhopadhyay

The problem of the origin of turbulence, and hence, transport of angular momentum, in accretion flows (particularly cold flows) as well as laboratory flows like plane Couette flow, prevails due to their stability under linear perturbation. We attempt to resolve this long-standing issue with linear analysis by considering an extra stochastic force with a nonzero mean (m). We show that these flows become unstable, and the corresponding maximum growth rate (Im(β)max) increases, if the mean is increased, with other parameters fixed. Since accretion flow has a central sink a fluid parcel must take less time to become nonlinear than to cross the local analysis region, consistent with our analysis.

[Phys. Rev. Fluids 6, 013903] Published Tue Jan 19, 2021

### Pattern selection in oscillatory longwave Marangoni convection with nonlinear temperature dependence of surface tension

Author(s): Alexander B. Mikishev and Alexander A. Nepomnyashchy

Pattern selection in oscillatory long-wave Marangoni convection in a heated thin layer of liquid with weak heat flux from the free surface is investigated. The research is performed under the assumption that the surface tension of the liquid nonlinearly depends on the temperature. The pattern selection is analyzed for square, rhombic, and hexagonal planforms.

[Phys. Rev. Fluids 6, 014002] Published Tue Jan 19, 2021

### Triple-deck analysis of the steady flow over a rotating disk with surface roughness

Author(s): Claudio Chicchiero, Antonio Segalini, and Simone Camarri

A theory based on a triple-deck approach is developed to rapidly assess the velocity field over a rotating disk with surface roughness. The theory results are validated with numerical simulations and suggest new ways to incorporate roughness in flow calculations.

[Phys. Rev. Fluids 6, 014103] Published Tue Jan 19, 2021

### Application of the Gram–Schmidt factorization of the deformation gradient to a cone and plate rheometer

In this paper, we study the cone and plate rheometer using the Gram–Schmidt factorization of the deformation gradient. This new solution has several advantages over the traditional approach. It is shown that with the use of these kinematics, one can avoid the need for using a convected, curvilinear, coordinate system, which often leads to cumbersome calculations. Here, the use of a convected coordinate system has been replaced with a certain orthonormal coordinate system that arises from the Gram–Schmidt factorization of the deformation gradient. Moreover, by using this solution procedure, it is possible to obtain the normal stress differences and shear stress explicitly. Therefore, this solution procedure opens up a possibility for characterizing material properties by using only a cone and plate rheometer.

### Instability of a tangential discontinuity surface in a three-dimensional compressible medium

Compressible flows appear in many natural and technological processes, for instance, the flow of natural gases in a pipe system. Thus, a detailed study of the stability of tangential velocity discontinuity in compressible media is relevant and necessary. The first early investigation in two-dimensional (2D) media was given more than 70 years ago. In this article, we continue investigating the stability in three-dimensional (3D) media. The idealized statement of this problem in an infinite spatial space was studied by Syrovatskii in 1954. However, the omission of the absolute sign of cos θ with θ being the angle between vectors of velocity and wave number in a certain inequality produced the inaccurate conclusion that the flow is always unstable for entire values of the Mach number M. First, we revisit this case to arrive at the correct conclusion, namely that the discontinuity surface is stabilized for a large Mach number with a given value of the angle θ. Next, we introduce a real finite spatial system such that it is bounded by solid walls along the flow direction. We show that the discontinuity surface is stable if and only if the dispersion relation equation has only real roots, with a large value of the Mach number; otherwise, the surface is always unstable. In particular, we show that a smaller critical value of the Mach number is required to make the flow in a narrow channel stable.

### Classification of the major nonlinear regimes of oscillations, oscillation properties, and mechanisms of wave energy dissipation in the nonlinear oscillations of coated and uncoated bubbles

Acoustic waves are dissipated when they pass through bubbly media. Dissipation by bubbles takes place through thermal damping (Td), radiation damping (Rd), and damping due to the friction of the liquid (Ld) and friction of the coating (Cd). Knowledge of the contributions of Td, Rd, Ld, and Cd during nonlinear bubble oscillations will help in optimizing bubble and ultrasound exposure parameters for the relevant applications by maximizing a desirable outcome or oscillation pattern. In this work, we investigate the mechanisms of dissipation in bubble oscillations and their contribution to the total damping (Wtotal) in various nonlinear regimes. By using a bifurcation analysis, we have classified nonlinear dynamics of bubbles that are sonicated with their third superharmonic (SuH) and second SuH resonance frequency (fr), pressure dependent resonance frequency (PDfr), fr, subharmonic (SH) resonance (fsh = 2fr), pressure dependent SH resonance (PDfsh), and 1/3 order SH resonance, which are important exposure ranges for various applications. The corresponding Td, Rd, Ld, Cd, Wtotal, scattering to dissipation ratio, maximum wall velocity, and maximum backscattered pressure from non-destructive oscillations of bubbles were calculated and analyzed using the bifurcation diagrams. Universal ultrasound exposure parameter ranges are revealed in which a particular non-destructive bubble related phenomenon (e.g., wall velocity) is enhanced. The enhanced bubble activity is then linked to relevant ultrasound applications. This paper represents the first comprehensive analysis of the nonlinear oscillations regimes, the corresponding damping mechanisms, and the bubble related phenomena.

### How to deform an egg yolk? On the study of soft matter deformation in a liquid environment

In this paper, we report a novel experimental study to examine the response of a soft capsule bathed in a liquid environment to sudden external impacts. Taking an egg yolk as an example, we found that the soft matter is not sensitive to translational impacts but is very sensitive to rotational, especially decelerating-rotational, impacts, during which the centrifugal force and the shape of the membrane together play a critical role in causing the deformation of the soft object. This finding, as the first study of its kind, reveals the fundamental physics behind the motion and deformation of a membrane-bound soft object, e.g., egg yolk, cells, and soft brain matter, in response to external impacts.

### Studying the flow dynamics and heat transfer of stranded conductor cables using large eddy simulations

Stranded cables are widely used in applications where their heat transfer and fluid dynamics are important, but they have not been extensively studied. This paper investigates, using large eddy simulations with the dynamic Smagorinsky sub-grid scale model, a helically wound stranded conductor cable in comparison to a circular cylinder at a Reynolds number of 1000 and Prandtl number of 0.7. The cylinder and the cable were normal to the flow. The triply decomposed heat transport equations were derived, and proper orthogonal decomposition was applied to the fluctuating vorticity and temperature fields to determine the total, coherent, and incoherent terms in the heat transport equations. The results showed that the stranded cable, relative to circular cylinder, has (i) three-dimensional mean flow and heat transfer, especially within and around recirculation region, (ii) 9% higher drag and 8% higher base pressure magnitude, (iii) near-stagnant flow in the gaps between the strands, which results in a significant variation in the local Nusselt number, (iv) ∼15% lower span-wise averaged local Nusselt number in the attached boundary layer, suggesting that surface modifications should be addressed to enhance heat transfer, (v) ∼36° variation in the separation angle along the span, (vi) 12% higher turbulent kinetic energy and 39% higher spanwise normal Reynolds stresses, (vii) insignificant difference in shedding frequency, suggesting similar flow induced vibrations to the cylinder, (viii) asymmetry in the flow and heat fields around the x axis, (ix) significantly different coherent temperature fields and dynamics, and (x) in general, high heat energy transport close to the cable rear side.

### Studying the flow dynamics and heat transfer of stranded conductor cables using large eddy simulations

Stranded cables are widely used in applications where their heat transfer and fluid dynamics are important, but they have not been extensively studied. This paper investigates, using large eddy simulations with the dynamic Smagorinsky sub-grid scale model, a helically wound stranded conductor cable in comparison to a circular cylinder at a Reynolds number of 1000 and Prandtl number of 0.7. The cylinder and the cable were normal to the flow. The triply decomposed heat transport equations were derived, and proper orthogonal decomposition was applied to the fluctuating vorticity and temperature fields to determine the total, coherent, and incoherent terms in the heat transport equations. The results showed that the stranded cable, relative to circular cylinder, has (i) three-dimensional mean flow and heat transfer, especially within and around recirculation region, (ii) 9% higher drag and 8% higher base pressure magnitude, (iii) near-stagnant flow in the gaps between the strands, which results in a significant variation in the local Nusselt number, (iv) ∼15% lower span-wise averaged local Nusselt number in the attached boundary layer, suggesting that surface modifications should be addressed to enhance heat transfer, (v) ∼36° variation in the separation angle along the span, (vi) 12% higher turbulent kinetic energy and 39% higher spanwise normal Reynolds stresses, (vii) insignificant difference in shedding frequency, suggesting similar flow induced vibrations to the cylinder, (viii) asymmetry in the flow and heat fields around the x axis, (ix) significantly different coherent temperature fields and dynamics, and (x) in general, high heat energy transport close to the cable rear side.

### A modal wave-packet model for the multi-mode Richtmyer–Meshkov instability

A model for multimode perturbations subject to the Richtmyer–Meshkov (RM) instability is presented and compared with simulations and experiments for conditions relevant to inertial confinement fusion. The model utilizes the single mode response to the RM impulse whereby its amplitude h(k, t) first grows with an initial velocity V0 ∝ kh(k, 0) that eventually decays in time as 1/kV0t. Both the growth and saturation stages are subject to nonlinearities since they depend explicitly on the initial amplitude. However, rather than using the individual mode amplitude h(k, t), nonlinearity is taken to occur when the root-mean-square amplitude hrms(k, t) of a wave-packet within wavenumbers k ± δk becomes comparable to 1/k. This is done because nearby sidebands can act in unison for an auto-correlation distance 1/δk beyond nonlinearity as observed in the beam-plasma instability. Thus, the nonlinear saturation amplitude for each mode is reduced from the usual 1/k by a phase space factor that depends on the physical dimensionality, as in the Haan model for the Rayleigh–Taylor instability. In addition, for RM, the average value of khrms for the initial spectrum is used to calculate a nonlinear factor FNL that reduces V0, as observed for single modes. For broadband perturbations, the model describes self-similar growth ∝tθ as successively longer wavelength modes reach saturation. The growing and saturated modes must be discerned because only the former promote θ and are enhanced by reshock and spherical convergence. All of these flows are described here by the model in good agreement with simulations and experiments.

### Impacts of fuel nonequidiffusivity on premixed flame propagation in channels with open ends

The present study scrutinizes premixed flame dynamics in micro-channels, thereby shedding light on advanced miniature micro-combustion technologies. While equidiffusive burning (when the Lewis number Le = 1) is a conventional approach adopted in numerous theoretical studies, real premixed flames are typically non-equidiffusive (Le ≠ 1), which leads to intriguing effects, such as diffusional-thermal instability. An equidiffusive computational study [V. Akkerman et al., Combust. Flame 145, 675–687 (2006)] reported regular oscillations of premixed flames spreading in channels having nonslip walls and open extremes. Here, this investigation is extended to non-equidiffusive combustion in order to systematically study the impact of the Lewis number on the flame in this geometry. The analysis is performed by means of computational simulations of the reacting flow equations with fully-compressible hydrodynamics and one-step Arrhenius chemical kinetics in channels with adiabatic and isothermal walls. In the adiabatic channels, which are the main case of study, it is found that the flames oscillate at low Lewis numbers, with the oscillation frequency decreasing with Le, while for the Le > 1 flames, a tendency to steady flame propagation is observed. The oscillation parameters also depend on the thermal expansion ratio and the channel width, although the impacts are rather quantitative than qualitative. The analysis is subsequently extended to the isothermal channels. It is shown that the role of heat losses to the walls is important and may potentially dominate over that of the Lewis number. At the same time, the impact of Le on burning in the isothermal channels is qualitatively weaker than that in the adiabatic channels.

### 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.