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A new analysis is made on the viscous dissipation within a microchannel parallel plate gap induced by an oscillating lower plate and a stationary upper plate, assuming a one-dimensional, incompress...

Hot accretion flows contain collisionless plasmas that are believed to be capable of accelerating particles to very high energies, as a result of turbulence generated by the magnetorotational instability (MRI). We conduct unstratified shearing-box simulations of the MRI turbulence in ideal magnetohydrodynamic (MHD), and inject energetic (relativistic) test particles in simulation snapshots to conduct a detailed investigation on particle diffusion and stochastic acceleration. We consider different amount of net vertical magnetic flux to achieve different disk magnetizations levels at saturated states, with sufficiently high resolution to resolve the gyro-radii ($R_g$) of most particles. Particles with large $R_g$ ($\gtrsim0.03$ disk scale height $H$) show spatial diffusion coefficients of $\sim30$ and $\sim5$ times Bohm values in the azimuthal and poloidal directions, respectively. We further measure particle momentum diffusion coefficient $D(p)$ by applying the Fokker-Planck equation to particle momentum evolution. For these particles, contribution from turbulent fluctuations scales as $D(p)\propto p$, and shear acceleration takes over when $R_g\gtrsim0.1H$, characterized by $D(p)\propto p^3$. For particles with smaller $R_g$ ($\lesssim0.03H$), their spatial diffusion coefficients roughly scale as $\sim p^{-1}$, and show evidence of $D(p)\propto p^2$ scaling in momentum diffusion but with large uncertainties. We find that multiple effects contribute to stochastic acceleration/deceleration, and the process is also likely affected by intermittency in the MRI turbulence. We also discuss the potential of accelerating PeV cosmic-rays in hot accretion flows around supermassive black holes., Comment: 22 pages, 18 figures, accepted for publication in MNRAS

An expression for the momentum diffusion coefficient for an ultrarelativistic electron beam moving in the spatially periodic magnetic field of an undulator has been obtained taking into account the initial energy spread of electrons. The dependences of the diffusion coefficient both on the distance traveled by electrons in the undulator and on the initial energy spread of electrons have been analyzed. It has been shown that the longitudinal momentum spread of electrons increases in the process of motion of the beam in the undulator. It has been found that the diffusion coefficient in the kinetic stage is independent of the distance traveled by the beam in the undulator. The possibility development of X-ray free electron lasers is discussed.

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

PurposeThe purpose of this study is to investigate the unsteady stagnation-point flow and heat transfer of fractional Maxwell fluid towards a time power-law-dependent stretching plate. Based on the characteristics of pressure in the boundary layer, the momentum equation with the fractional Maxwell model is firstly formulated to analyze unsteady stagnation-point flow. Furthermore, generalized Fourier’s law is considered in the energy equation and boundary condition of convective heat transfer.Design/methodology/approachThe nonlinear fractional differential equations are solved by the newly developed finite difference scheme combined with L1-algorithm, whose convergence is verified by constructing a numerical example.FindingsSome interesting results can be revealed. The larger fractional derivative parameter of velocity promotes the flow, while the smaller fractional derivative parameter of temperature accelerates the heat transfer. The temperature boundary layer is thicker than the velocity boundary layer, and the velocity enlarges as the stagnation parameter raises. This is because when Prandtl number < 1, the capacity of heat diffusion is greater than that of momentum diffusion. It is to be observed that all the temperature profiles first enhance a little and then reduce rapidly, which indicates the thermal retardation of Maxwell fluid.Originality/valueThe unsteady stagnation-point flow model of Maxwell fluid is extended from integral derivative to fractional derivative, which has more flexibility to describe viscoelastic fluid’s complex dynamic process and provide a theoretical basis for industrial processing.

Direct numerical simulations for a round jet controlled by half delta-wing tabs are performed to elucidate the transition of turbulent diffusion near the jet exit. The aim also involves clarifying the function of the tabs in round jet control. The jet Reynolds number is set as 10,000. The number of the tab is set to 3 and 6. The results reveal that the installation of the tabs generally decreases the turbulent diffusion coefficient except immediately after the jet exit of $$x/d

We simulate the acceleration of atoms by the counter-propagating frequency-modulated (FM) waves. We show that the standard deviation of the velocity of atoms has a non-monotonic time dependence, and explain this phenomenon. The momentum diffusion coefficient of atoms in the field of FM waves is estimated from the analogy with the behaviour of atoms in the field of counter-propagating $$\pi $$ -pulses. If the parameters of the atom–field interaction are optimal, the momentum diffusion coefficient linearly depends on light intensity. We obtain our results with the use of the Monte Carlo wave-function method and compare them with the results obtained by the solution of the optical Bloch equations. We compare our results with the behaviour of atoms in the field of the counter-propagating bichromatic waves.

Diffusion of momentum gives rise to viscosity. This article presents a solution in the explicit form of the equation of the momentum diffusion for a viscous fluid flowing around a plate taking into account deceleration. Three characteristic regions of a viscous flow have been described: the mantle, the body of the boundary layer, the viscous sublayer. In the mantle, the effect of viscosity is significant at a considerable distance from the plate. The momentum diffusion is focused in the body of the boundary layer. The diffusion force that produces the momentum of force giving rise to eddies is localized in the viscous sublayer. At the beginning of the plate, a moment of force twists the liquid along the flow, creating eddies that roll along the plate. For this reason, they are pressed against the surface of the plate. But at some distance from the beginning of the plate, the moment of force changes its orientation to the opposite and twists the vortices in the opposite direction, causing the vortices to roll along the plate against the flow. This causes the liquid to detach from the surface of the plate. This is the beginning of turbulence. The diameter of the vortex produced in the viscous sublayer is small being of the order of the thickness of the viscous sublayer. The vortex possesses a large angular velocity. Due to the momentum diffusion and the effect of the eddies combined in passing along the plate, its diameter increases up to the size of the thickness of the boundary layer and even more, whereas its angular velocity decreases down to the values really observed. The value of the critical Reynolds number of the transition from the laminar flow to the turbulent one has been found, and it agrees with the experimental data. The value of the shear stress produced by the viscous fluid on the plate surface has also been obtained. The way of measurement of the friction coefficient characterizing the effect of the plate on the flow has been proposed. It has been shown that the boundary condition of adhesion to the surface of a body flown around, that is applied in the estimation of viscous flows, contradicts the real processes of the flow.

The spatial variation in the properties of an arrested salt wedge have been investigated, both analytically and in the laboratory. In the laboratory particle image velocimetry and laser induced fluorescence were used to obtain flow velocities and the height of the density interface. An analytical solution for the profile of interface height, in the absence of interfacial instabilities, has been developed from two-layer internal hydraulic theory. The evolution of the velocity profile is predicted using a momentum diffusion equation following a Lagrangian frame of reference along the interface of the salt wedge. The centre of the shear layer is predicted to lie above the density interface, with this offset decreasing in the downstream direction. Our theoretical predictions are in good agreement with our laboratory measurements. Keywords: Arrested salt wedge, Internal hydraulics, Shear layer, Asymmetry

In this paper, we discuss a model of a nonlinear interaction between a λ-type four-level atom interacting with a cavity field. The analytical solution of the system under particular conditions has been obtained. Numerical treatments have been discussed and new features of the entanglement degree are obtained using entropy and Mandel Q-parameter. Moreover, we evaluate a few of their non-classical properties such as momentum increment, momentum diffusion, mean photon number and normal squeezing. The Kerr medium impact has been discussed through different phenomena. It is shown that Kerr-like medium parameter can be used as a controller parameter of the system dynamics.

The flow conditions in and around a suspended canopy, resembling those formed by aquaculture structures such as rafts cages and longlines, were modeled using an augmented version of the hydrodynamic model Environmental Fluid Dynamics Code. The model was calibrated using vertical profiles of horizontal velocities, Reynolds stresses, and turbulent kinetic energies obtained from prior laboratory flume experiments. The parameter estimation code, PEST, was used to optimize various model parameters including horizontal momentum diffusivity, vertical eddy viscosity, turbulence closure constants, and, most importantly, depth-dependent drag coefficients. An increasing average drag coefficient was observed with decreasing canopy blockage ratio, and an empirical relationship for the vertical variation of drag coefficient was developed that may be appropriate for use in full-scale models of aquaculture systems. Overall, the calibrated canopy-turbulence parameters and drag-coefficient empiricisms may yield improved predictions of alterations to hydrodynamic and nutrient-transport conditions due to various aquaculture structures. Such predictions will help develop methods to minimize environmental impacts and to increase production from aquaculture farms.

The simple shear response of soft polymers under large deformation (>50%) and strain rates spanning 10−3 – 103 s−1 is characterized by developing quasi-static and split-Hopkinson pressure bar based single-pulse dynamic simple shear experiments rooted in continuum mechanics fundamentals. Cross-linked polydimethylsiloxane (PDMS) is chosen as a model material. By examining the evolution of stress, strain and strain rate, the latter two parameters measured using two-dimensional digital image correlation (DIC), it is demonstrated that dynamic simple shear deformation consists of four distinct stages: momentum diffusion, inertia effect, steady-state material response, and strain rate decay. By isolating the unsteady and steady-state deformation stages, inertia-free material response is captured under a uniform strain rate. It is shown that the shear response of PDMS is nearly linear with a weakly rate-sensitive shear modulus in the investigated strain rate range. Further, by analyzing the DIC strain-field and comparing the kinematic experimental results with those predicted by classical continuum mechanics, it is demonstrated that the proposed experiments not only achieve a nearly theoretical simple shear state that is uniform across the specimen, but also allow for post-test validation of individual experiments based on these criteria.

To reveal the void dominated fracture mechanism, cavitation instabilities in metallic glasses are studied through analytical and numerical approaches, with particular attention on surface energy and thermal effects. The critical pressure for unbounded growth of voids is determined, which increases apparently as surface energy is taken into account. A dimensionless number Iγ which is a ratio of the energy required to form new void surface and the energy dissipated by plastic deformation is proposed. It is found that the surface energy significantly impedes void growth at the early stage of void growth when Iγ is large. Besides, to address the thermal effects, another dimensionless number Ith, which reflects the competition of momentum diffusion and thermal diffusion, is presented. Results of numerical simulations show that the thermal effects first promote and finally impede the void growth at the late stage of void growth when Ith ≤ 1. Further study on combined influence of inertia, surface energy and thermal effects reveals the factors which are dominant as the process develops.

Statistical analysis of pressure fluctuations in spouted beds has been used as a well-established diagnostic tool to determine bed and flow characteristics because of its smooth operation. However, in many recent and conventional applications of spouted beds such as drying, coal gasification, catalytic conversion, biomass treatment, and chemical vapor deposition, direct estimation of the heat transfer rate from the solid bed to the gas or vice versa has proven to be difficult. A variance and spectral analysis of pressure fluctuation is extended here to characterize the heat transfer phenomena in a spouted bed. In the present study, zirconia and alumina were used as the bed materials, and argon and nitrogen were used as the spouting gases. Experiments were conducted at various heating rates for different superficial gas velocities for a range of temperatures up to 300 °C. Significant changes in the gas density and viscosity with different extents of heat transfer were observed to affect the momentum diffusivity and gas–particle interaction, which in turn led to local pressure fluctuations, causing the bed to behave differently. In the present work, a novel approach is proposed to establish a link between local pressure fluctuation and the extent of heat transfer in the bed. This method shows potential for correlation of the statistics of pressure fluctuation with the thermal properties of individual solids and gases. Thus, the technique can be extended to many industrial applications for the indirect estimation of the extent of heat transfer and prediction of unknown thermal properties of products in solids or gases.

We study numerically the evolution of the velocity distribution of atoms under the action of the bichromatic force. The comparison of the time dependencies of the distribution width and the average acceleration of atoms reveal the correlation of these quantities. We show that the estimation of the momentum diffusion coefficient on the basis of the analogy between the interaction of atoms with the counter-propagating bichromatic waves and the interaction of atoms with the counter-propagating sequences of the $\pi$-pulses roughly corresponds to the results of numerical calculations. To separate the influence of the momentum diffusion on the evolution of atomic momentum distribution from the influence of the time-dependent Doppler shift, we study the motion of a ``heavy'' atom, for which the velocity change during the interaction of an atom with the field can be neglected. Provided that the parameters of the atom-field interaction are optimal, we show that the momentum diffusion coefficient is proportional to the intensity of the laser radiation. We used the Monte Carlo wave-function method for the numerical simulation of the atomic motion., Comment: 32 pages, 13 figures

Recently, a direct signature of chaos in many body system has been realized from the energy density retarded Green's function using the phenomenon of `pole skipping'. Moreover, special locations in the complex frequency and momentum plane are found, known as the pole skipping points such that the retarded Green's function can not be defined uniquely there. In this paper, we compute the correction/shift to the pole skipping points due to a spatial anisotropy in a holographic system by performing near horizon analysis of EOMs involving different bulk field perturbations, namely the scalar, the axion and the metric field. For vector and scalar modes of metric perturbations we construct the gauge invariant variable in order to obtain the master equation. Two separate cases for every bulk field EOMs is considered with the fluctuation propagating parallel and perpendicular to the direction of anisotropy. We compute the dispersion relation for momentum diffusion along the transverse direction in the shear channel and show that it passes through the first three successive pole skipping points. The pole skipping phenomenon in the sound channel is found to occur in the upper half plane such that the parameters Lyapunov exponent $\lambda_{L}$ and the butterfly velocity $v_{B}$ are explicitly obtained thus establishing the connection with many body chaos., Comment: 41 pages, 3 figures, Conclusion modified, Accepted for publication in JHEP

We have studied the effect of strong magnetic field on the viscous properties of hot QCD matter at finite chemical potential by calculating the shear ($\eta$) and bulk ($\zeta$) viscosities. The viscosities are calculated using kinetic theory in the relaxation time approximation. The interactions are incorporated through the quasiparticle masses of partons at strong magnetic field and finite chemical potential. From this study, one can understand the effects of strong magnetic field and chemical potential on the sound attenuation through the Prandtl number (Pl), on the nature of the flow by the Reynolds number (Rl), and on the relative behavior between shear viscosity and bulk viscosity through the ratio $\zeta/\eta$. We have found that, $\eta$ and $\zeta$ get increased in a strong magnetic field and the additional presence of chemical potential further enhances their magnitudes. With the increase of temperature, $\eta$ increases in a strong magnetic field as well as in the absence of magnetic field, whereas $\zeta$ decreases with the temperature, contrary to its increase in the absence of magnetic field. We have observed that, the Prandtl number gets increased in the presence of strong magnetic field and chemical potential as compared to the isotropic one, but it always remains larger than 1, thus the momentum diffusion largely affects the sound attenuation in the medium. The Reynolds number gets lowered than 1 in a strong magnetic field and it becomes further decreased in an additional presence of chemical potential, so the kinematic viscosity dominates over the characteristic length scale of the system. Finally, $\zeta/\eta$ becomes larger than 1, contrary to its value in the absence of magnetic field and chemical potential where it is less than 1, so the bulk viscosity prevails over the shear viscosity for the hot and dense QCD matter in the presence of a strong magnetic field., Comment: 39 pages, 7 figures

We apply the gradient flow on a color-electric two-point function that encodes the heavy quark momentum diffusion coefficient. The simulations are done on fine isotropic lattices in the quenched approximation at $1.5\,T_c$. The continuum extrapolation is performed at fixed flow time followed by a second extrapolation to zero flow time. Perturbative calculations of this correlation function under Wilson flow are used to enhance the extrapolations of the non-perturbative lattice correlator. The final estimate for the continuum correlator at zero flow time largely agrees with one obtained from a previous study using the multi-level algorithm. We perform a spectral reconstruction based on perturbative model fits to estimate the heavy quark momentum diffusion coefficient. The approach we present here yields high-precision data for the correlator and is also applicable for actions with dynamical fermions., Published version, 14 pages, 15 figures

Hydrogels exhibit a fluid-like viscous response under high shear strain rates with significant rate- and microstructure-dependent rheological properties. In this study, the transient shear response of poly(ethylene glycol) diacrylate (PEGDA) hydrogels is characterized by application of the power-law fluid model to incompressible Navier–Stokes equation of start-up planar Couette flow. To impart the necessary boundary conditions for model calibration, a split-Hopkinson pressure bar based single-pulse dynamic simple shear experiment is developed, in which unsteady momentum diffusion between two shear plates is measured using two-dimensional digital image correlation (DIC). The measured shear profiles and their characteristic self-similarity under these boundary conditions are utilized to calculate power-law exponent and transient-state viscosity via finite difference simulations.

Sufficient confinement can completely suppress the Rayleigh-Taylor instability between two density-inverted miscible fluids., The prevention of hydrodynamic instabilities can lead to important insights for understanding the instabilities’ underlying dynamics. The Rayleigh-Taylor instability that arises when a dense fluid sinks into and displaces a lighter one is particularly difficult to arrest. By preparing a density inversion between two miscible fluids inside the thin gap separating two flat plates, we create a clean initial stationary interface. Under these conditions, we find that the instability is suppressed below a critical plate spacing. With increasing spacing, the system transitions from the limit of stability where mass diffusion dominates over buoyant forces, through a regime where the gap sets the wavelength of the instability, to the unconfined regime governed by the competition between buoyancy and momentum diffusion. Our study, including experiment, simulation, and linear stability analysis, characterizes all three regimes of confinement and opens new routes for controlling mixing processes.

We consider the out-of-equilibrium transport in $T\bar{T}$-deformed (1+1)-dimension conformal field theories (CFTs). The theories admit two disparate approaches, integrability and holography, which we make full use of in order to compute the transport quantities, such as the the exact non-equilibrium steady state currents. We find perfect agreements between the results obtained from these two methods, which serve as the first checks of the $T\bar{T}$-deformed holographic correspondence from the dynamical standpoint. It turns out that integrability also allows us to compute the momentum diffusion, which is given by a universal formula. We also remark on an intriguing connection between the $T\bar{T}$-deformed CFTs and reversible cellular automata., v1: 6 pages, 1 figure. v2: typos corrected, v3: Fig. 1 corrected, published version

The drag and diffusion coefficients are studied within the framework of Fokker-Planck dynamics for the case of a charm quark propagating in an expanding quark-gluon plasma. The space-time evolution of the nuclear matter created in the relativistic heavy-ion collision is modeled using music, a 3+1D relativistic viscous hydrodynamic approach. The effect of viscous corrections to the heavy quark transport coefficients is explored by considering scattering processes with thermal quarks and gluons in the medium. It is observed that the momentum diffusion of the heavy quarks is sensitive to the shear and bulk viscosity to entropy ratios. The collisional energy loss of the charm quark in the viscous quark-gluon plasma is analyzed.

In a recent paper published in Environmental Earth Sciences, Jain et al. (Environ Earth Sci 77:253, 2018) proposed an interesting study about the $$\beta$$-factor (the inverse of the turbulent Schmidt number, i.e. the ratio of momentum diffusivity to mass diffusivity in a turbulent flow). They proposed an equation for the depth-averaged $$\beta$$-factor which was used in the Rouse equation to calculate concentration profiles of suspended sediments in open-channel turbulent flows. Despite the interest, the study shows a weakness related to some inconsistencies and contradictions in the method. The main result should be improved. In this note, the weakness in this study will be pointed out. Some used equations are in contradiction with the initial assumptions. The Rouse equation is a solution of the steady-state one-dimensional convection–diffusion equation with a parabolic eddy viscosity which is based on a logarithmic velocity profile. In the same study, Jain et al. (2018) used two different mixing length equations, parabolic and linear relations. The parabolic mixing length and a constant mixing velocity are in contradiction with the used exponential-type eddy viscosity profile. The used equation for coefficient $$\gamma$$ is based on the finite mixing length model and the related assumptions of a linear eddy viscosity and a logarithmic velocity distribution (Nielsen and Teakle in Phys Fluids 16(7):2342–2348, 2004) which are in contradiction with the used exponential-type eddy viscosity profile and velocity distribution. An improved approach and new equations for the depth-averaged $$\beta$$-factor are proposed. The first is based on the equation for the coefficient $$A$$ proposed by Jain et al. (2018) and a second based on a linear function for $$A$$ which gives an equation for $$\beta$$ similar to that of van Rijn (J Hydraul Eng ASCE 1104(11):1613–1641, 1984).

We analyze the interplay between fluid inertia and thixotropy during the transient start-up dynamics of a fluidity model which captures the rheological behavior of aging soft (glassy) materials, in a rectilinear shear flow upon application of a step shear rate. We observe that when the steady-state flow curve is nonmonotonic, the system shows transient and/or apparent steady-state shear banding in close qualitative agreement with experimental observations. Due to competition between inhomogeneous aging and rejuvenation caused by fluid inertia during start up, we show that there is an apparent steady-state banding at large times (compared to the relevant momentum diffusion timescale) even for shear rates such that the steady state flow curve allows for a homogeneous flow. Thus, for aging soft glassy materials, the shear rate for achieving homogeneous flow is not necessarily given by the steady-state flow curve. We also observe that the transient and apparent steady-state shear banding behavior is not correlated to the negative slope of the stress-strain dependence during the transient. This work also emphasizes that in order to arrive at a realistic description of shear banding behavior in aging (thixotropic) soft materials, inertia may not always be neglected during the start-up dynamics.We analyze the interplay between fluid inertia and thixotropy during the transient start-up dynamics of a fluidity model which captures the rheological behavior of aging soft (glassy) materials, in a rectilinear shear flow upon application of a step shear rate. We observe that when the steady-state flow curve is nonmonotonic, the system shows transient and/or apparent steady-state shear banding in close qualitative agreement with experimental observations. Due to competition between inhomogeneous aging and rejuvenation caused by fluid inertia during start up, we show that there is an apparent steady-state banding at large times (compared to the relevant momentum diffusion timescale) even for shear rates such that the steady state flow curve allows for a homogeneous flow. Thus, for aging soft glassy materials, the shear rate for achieving homogeneous flow is not necessarily given by the steady-state flow curve. We also observe that the transient and apparent steady-state shear banding behavior is not corre...

We present unstratified 3D MHD simulations of an accretion disk with a boundary layer (BL) that have a duration $\sim 1000$ orbital periods at the inner radius of the accretion disk. We find the surprising result that angular momentum piles up in the boundary layer, which results in a rapidly rotating belt of accreted material at the surface of the star. The angular momentum stored in this belt increases monotonically in time, which implies that angular momentum transport mechanisms in the BL are inefficient and do not couple the accretion disk to the star. This is in spite of the fact that magnetic fields are advected into the BL from the disk and supersonic shear instabilities in the BL excite acoustic waves. In our simulations, these waves only carry a small fraction ($\sim 10 \%$) of the angular momentum required for steady state accretion. Using analytical theory and 2D viscous simulations in the $R-\phi$ plane, we derive an analytical criterion for belt formation to occur in the BL in terms of the ratio of the viscosity in the accretion disk to the viscosity in the BL. Our MHD simulations have a dimensionless viscosity ($\alpha$) in the BL that is at least a factor of $\sim 100$ smaller than that in the disk. We discuss the implications of these results for BL dynamics and emission., Comment: 29 pages, 11 figures, published in MNRAS

A two-dimensional study of mass transfer from a circular composite particle settling in a vertical channel is conducted with the lattice Boltzmann method. The particle is composed of two materials, one insoluble while the other soluble in the ambient fluid. In the problem, mass transfer, particle motion and fluid flow are closely coupled, where the concentration at the particle surface and particle properties vary with mass transfer. It is observed that mass transfer follows a Schmidt number independent scaling law of [ t / ( t ν Sc ) ] - 1.5 ( t is the physical time, t ν is the momentum diffusion time scale and Sc is the Schmidt number), which is quite different from the pure diffusion of a stationary particle, ∼ ( t / t α ) - 0.5 ( t α is the mass diffusion time scale). An analysis of the concentration and flow pattern around the particle suggests that the scaling law for a settling particle is related to both diffusion distance and convection distance, while it is only relevant to the diffusion distance in the case of a stationary particle. For a settling particle, mass transfer is enhanced by two mechanisms due to convection, i.e., the concentration around the particle surface transported to the downstream of the particle by the fluid flow and the interface of mass transfer stretched with the particle motion, which are absent in the case of a stationary particle. Thus, the rate of mass transfer in the case of a settling particle is much higher than that for a stationary particle.

The combustion of a freely falling dodecane droplet has been studied experimentally in a droptower-like facility under ambient conditions. A unique ignition mechanism is used by igniting the droplet in pendant mode and releasing it to fall freely. This unveils a different type of droplet wake flame behavior which is explored in this study. Initially, the droplet flame transitions from fully enveloped to a wake flame configuration due to forward extinction. The wake flame has similar characteristics as a laminar lifted triple-flame. As the droplet accelerates, the flame stand-off increases continuously. The change in wake flame topology and intensity occurs in two different regimes corresponding to different droplet diameters. A new non-dimensional parameter has been derived to account for the local balance between buoyancy and momentum diffusion that alters the fuel availability. To explain the flame topological evolutions and transitions for different droplet diameters and Reynolds numbers, a theoretical formulation has been proposed based on the momentum diffusion from surrounding due to relative motion. Further, at very high Reynolds number, flame stretching or shedding regime occurs, causing momentary spikes in flame intensity due to the interaction with asymmetric vortex shedding induced by the Bernard–Von Karman instability. Interestingly, the flame shedding height follows the buoyant flickering scaling, even for the momentum-dominant droplet wake flame. Additionally, the circulation build-up mechanisms are shown to be responsible for the flame shedding events for droplet wake flame at high Reynolds number.

Theoretical hydrodynamics may lead one into serious delusions. This article is focused on three of them. First, using flowing around a sphere as an example it is shown that the known potential solutions of the flow-around problems are not unique and there exist nonpotential solutions. A nonpotential solution has been obtained for flowing around a sphere. A general solution of the problem of flowing around an arbitrary surface has been obtained in the quadrature form. To single out a physically realisable solution among a great number of others, it is necessary to add supplementary conditions to the known boundary ones, in particular, to find a solution with the minimum total energy. The hypothesis explaining the reason for stalled flows by viscosity is erroneous. When considering a flow-around problem one should use stalled and broken solutions of the continuity equation along with the continuous ones. If the minimum total energy is achieved by the continuous solution, it is a continuous flow that will be implemented. If it is achieved by the broken solution, a stalled flow will be realised. Second, the hydrodynamics of a flow is considered exclusively at each point of it. Differential equations are used to describe the flows that are written for a randomly small volume of a flow, i.e., for a point. The integral characteristics of a flow and its inertial properties are neglected in the consideration, which results in the misunderstanding of the mechanism of the formation of a vortex. The reason for the formation of vortices is related to viscosity, which is a mistake. The formation of vortices is the result of the inhomogeneity of the acceleration field and the inertial properties of a flow. Third, the fictitious values of viscous stresses are used in hydrodynamics. As a matter of fact, viscosity is the momentum diffusion and it should be described by the diffusion equation included into the Euler system of equations for a viscous fluid. The momentum diffusion leads to the necessity of including the volume momentum sources produced by diffusion into the continuity equation and excluding the viscosity forces from the equation of motion. The problem of a viscous fluid flowing around a thin plate has been solved analytically, the velocity profiles satisfying the experiment have been obtained. The superfluidity of helium is not its property. It is the interaction of helium with a streamlined surface that is responsible for the mechanism of superfluidity. At low temperatures when the quantum properties are most pronounced the momentum transfer from the helium atoms to the streamlined wall becomes impossible, since the value of the energy transferred in the collision of a helium atom with that of the wall is smaller than the permitted quantum of energy. This mechanism takes place in the case of a flow in capillaries. Under a hydrodynamic flow-around superfluidity does not manifest due to the occurrence of stalled flows. The hypothesis of the disappearance of the viscous stresses at low temperatures is erroneous. The viscous stresses cannot disappear since they do not exist in nature. The theory of representing superfluidity as a phase transition accompanied by the formation of the combined viscous and nonviscous phases is a mistake.

The Schmidt number, defined as the ratio of scalar to momentum diffusivity, varies by multiple orders of magnitude in real-world flows, with large differences in scalar diffusivity between temperature, solute, and sediment driven flows. This is especially crucial in gravity currents, where the flow dynamics may be driven by differences in temperature, solute, or sediment, and yet the effect of Schmidt number on the structure and dynamics of gravity currents is poorly understood. Existing numerical work has typically assumed a Schmidt number near unity, despite the impact of Schmidt number on the development of fine-scale flow structure. The few numerical investigations considering high Schmidt number gravity currents have relied heavily on two-dimensional simulations when discussing Schmidt number effects, leaving the effect of high Schmidt number on three-dimensional flow features unknown. In this paper, three-dimensional direct numerical simulations of constant-influx solute-based gravity currents with Reynolds numbers 100≤Re≤3000 and Schmidt number 1 are presented, with the effect of Schmidt number considered in cases with (Re,Sc)=(100,10), (100,100), and (500, 10). These data are used to establish the effect of Schmidt number on different properties of gravity currents, such as density distribution and interface stability. It is shown that increasing Schmidt number from 1 leads to substantial structural changes not seen with increased Reynolds number in the range considered here. Recommendations are made regarding lower Schmidt number assumptions, usually made to reduce computational cost.